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The Fiber-Optic Gyroscope
Second Edition
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The Fiber-Optic Gyroscope
Second Edition
Hervé C. Lefèvre
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ISBN-13: 978-1-60807-695-6
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10 9 8 7 6 5 4 3 2 1
To Sophie, Charlotte, and Elliot
“Eppur si muove!”—Galileo Galilei
Chambre Mentale #79
Photograph by Marc Le Mené, 2000
Contents
Preface to the Second Edition
CHAPTER 1
Introduction
CHAPTER 2
Principle of the Fiber-Optic Gyroscope
2.1 Sagnac Effect
2.1.1 A History of Optics from Aether to Relativity
2.1.2 Sagnac Effect in a Vacuum
2.1.3 Sagnac Effect in a Medium
2.2 Active and Passive Ring Resonators
2.2.1 Ring-Laser Gyroscope (RLG)
2.2.2 Resonant Fiber-Optic Gyroscope (R-FOG)
2.3 Passive Fiber-Ring Interferometer
2.3.1 Principle of the Interferometric Fiber-Optic Gyroscope (I-FOG)
2.3.2 Theoretical Sensitivity of the I-FOG
2.3.3 Noise, Drift, and Scale Factor
2.3.4 Evaluation of Noise and Drift by Allan Variance (or Allan Deviation)
2.3.5 Bandwidth
References
CHAPTER 3
Reciprocity of a Fiber Ring Interferometer
3.1 Principle of Reciprocity
3.1.1 Single-Mode Reciprocity of Wave Propagation
3.1.2 Reciprocal Behavior of a Beam Splitter
3.2 Minimum Configuration of a Ring Fiber Interferometer
3.2.1 Reciprocal Configuration
3.2.2 Reciprocal Biasing Modulation-Demodulation
3.2.3 Proper (or Eigen) Frequency
3.3 Reciprocity with All-Guided Schemes
3.3.1 Evanescent-Field Coupler (or X-Coupler or Four-Port Coupler)
3.3.2 Y-Junction
3.3.3 All-Fiber Approach
3.3.4 Hybrid Architectures with Integrated Optics: Y-Coupler Configuration
3.4 Problem of Polarization Reciprocity
3.4.1 Rejection Requirement with Ordinary Single-Mode Fiber
3.4.2 Use of Polarization-Maintaining (PM) Fiber
3.4.3 Use of Depolarizer
3.4.4 Use of an Unpolarized Source
References
CHAPTER 4
Backreflection and Backscattering
4.1 Problem of Backreflection
4.1.1 Reduction of Backreflection with Slant Interfaces
4.1.2 Influence of Source Coherence
4.2 Problem of Backscattering
4.2.1 Coherent Backscattering
4.2.2 Use of a Broadband Source
4.2.3 Evaluation of the Residual Rayleigh Backscattering Noise
References
CHAPTER 5
Analysis of Polarization Nonreciprocities with Broadband Source and High-Birefringence
Polarization-Maintaining Fiber
5.1 Depolarization Effect in High-Birefringence Polarization-Maintaining Fibers
5.2 Analysis of Polarization Nonreciprocities in a Fiber Gyroscope Using an All-PolarizationMaintaining Waveguide Configuration
5.2.1 Intensity-Type Effects
5.2.2 Comment About Length of Depolarization Ld Versus Length of Polarization Correlation
Lpc
5.2.3 Amplitude-Type Effects
5.3 Use of a Depolarizer
5.4 Testing with Optical Coherence Domain Polarimetry (OCDP)
5.4.1 OCDP Based on Path-Matched White-Light Interferometry
5.4.2 OCDP Using Optical Spectrum Analysis
References
CHAPTER 6
Time Transience-Related Nonreciprocal Effects
6.1 Effect of Temperature Transience: The Shupe Effect
6.2 Symmetrical Windings
6.3 Stress-Induced T-Dot Effect
6.4 Basics of Heat Diffusion and Temporal Signature of the Shupe and T-Dot Effects
6.5 Effect of Acoustic Noise and Vibration
References
CHAPTER 7
Truly Nonreciprocal Effects
7.1 Magneto-Optic Faraday Effect
7.2 Transverse Magneto-Optic Effect
7.3 Nonlinear Kerr Effect
References
CHAPTER 8
Scale Factor Linearity and Accuracy
8.1 Problem of Scale Factor Linearity and Accuracy
8.2 Closed-Loop Operation Methods to Linearize the Scale Factor
8.2.1 Use of a Frequency Shift
8.2.2 Use of an Analog Phase Ramp (or Serrodyne Modulation)
8.2.3 Use of a Digital Phase Ramp
8.2.4 All-Digital Closed-Loop Processing Method
8.2.5 Control of the Gain of the Modulation Chain with Four-State Modulation
8.2.6 Potential Spurious Lock-In (or Deadband) Effect
8.3 Scale Factor Accuracy
8.3.1 Problem of Scale Factor Accuracy
8.3.2 Wavelength Dependence of an Interferometer Response with a Broadband Source
8.3.3 Effect of Phase Modulation
8.3.4 Wavelength Control Schemes
8.3.5 Mean Wavelength Change with a Parasitic Interferometer or Polarimeter
References
CHAPTER 9
Recapitulation of the Optimal Operating Conditions and Technologies of the I-FOG
9.1 Optimal Operating Conditions
9.2 Broadband Source
9.2.1 Superluminescent Diode
9.2.2 Rare-Earth Doped Fiber ASE Sources
9.2.3 Excess RIN Compensation Techniques
9.3 Sensing Coil
9.4 The Heart of the Interferometer
9.5 Detector and Processing Electronics
References
CHAPTER 10
Alternative Approaches for the I-FOG
10.1 Alternative Optical Configurations
10.2 Alternative Signal Processing Schemes
10.2.1 Open-Loop Scheme with Use of Multiple Harmonics
10.2.2 Second Harmonic Feedback
10.2.3 Gated Phase Modulation Feedback
10.2.4 Heterodyne and Pseudo-Heterodyne Schemes
10.2.5 Beat Detection with Phase Ramp Feedback
10.2.6 Dual-Phase Ramp Feedback
10.3 Extended Dynamic Range with Multiple Wavelength Source
References
CHAPTER 11
Resonant Fiber-Optic Gyroscope (R-FOG)
11.1 Principle of Operation of an All-Fiber Ring Cavity
11.2 Signal Processing Method
11.3 Reciprocity of a Ring Fiber Cavity
11.3.1 Introduction
11.3.2 Basic Reciprocity Within the Ring Resonator
11.3.3 Excitation and Detection of Resonances in a Ring Resonator
11.4 Other Parasitic Effects in the R-FOG
Acknowledgments
References
CHAPTER 12
Conclusions
12.1 The State of Development and Expectations in 1993
12.2 The Present State of the Art, Two Decades Later
12.2.1 FOG Versus RLG
12.2.2 FOG Manufacturers
12.3 Trends for the Future and Concluding Remarks
References
APPENDIX A
Fundamentals of Optics for the Fiber Gyroscope
A.1 Basic Parameters of an Optical Wave: Wavelength, Frequency, and Power
A.2 Spontaneous Emission, Stimulated Emission, and Related Noises
A.2.1 Fundamental Photon Noise
A.2.2 Spontaneous Emission and Excess Relative Intensity Noise (Excess RIN)
A.2.3 Resonant Stimulated Emission in a Laser Source
A.2.4 Amplified Spontaneous Emission (ASE)
A.3 Propagation Equation in a Vacuum
A.4 State of Polarization of an Optical Wave
A.5 Propagation in a Dielectric Medium
A.5.1 Index of Refraction
A.5.2 Chromatic Dispersion, Group Velocity, and Group Velocity Dispersion
A.5.3 E and B, or E and H?
A.6 Dielectric Interface
A.6.1 Refraction, Partial Reflection, and Total Internal Reflection
A.6.2 Dielectric Waveguidance
A.7 Geometrical Optics
A.7.1 Rays and Phase Wavefronts
A.7.2 Plane Mirror and Beam Splitter
A.7.3 Lenses
A.8 Interferences
A.8.1 Principle of Two-Wave Interferometry
A.8.2 Most Common Two-Wave Interferometers: Michelson and Mach-Zehnder
Interferometers, Young Double-Slit
A.8.3 Channeled Spectral Response of a Two-Wave Interferometer
A.9 Multiple-Wave Interferences
A.9.1 Fabry-Perot Interferometer
A.9.2 Ring Resonant Cavity
A.9.3 Multilayer Dielectric Mirror and Bragg Reflector
A.9.4 Bulk-Optic Diffraction Grating
A.10 Diffraction
A.10.1 Fresnel Diffraction and Fraunhofer Diffraction
A.10.2 Knife-Edge Fresnel Diffraction
A.11 Gaussian Beam
A.12 Coherence
A.12.1 Basics of Coherence
A.12.2 Mathematical Derivation of Temporal Coherence
A.12.3 The Concept of a Wave Train
A.12.4 The Case of an Asymmetrical Spectrum
A.12.5 The Case of Propagation in a Dispersive Medium
A.13 Birefringence
A.13.1 Birefringence Index Difference
A.13.2 Change of Polarization with Birefringence
A.13.3 Interference with Birefringence
A.14 Optical Spectrum Analysis
Reference
Selected Bibliography
APPENDIX B
Fundamentals of Fiber Optics for the Fiber Gyroscope
B.1 Main Characteristics of a Single-Mode Optical Fiber
B.1.1 Attenuation of a Silica Fiber
B.1.2 Gaussian Profile of the Fundamental Mode
B.1.3 Beat Length and h Parameter of a PM Fiber
B.1.4 Protective Coating
B.1.5 Temperature Dependence of Propagation in a PM Fiber
B.2 Discrete Modal Guidance in a Step-Index Fiber
B.3 Guidance in a Single-Mode (SM) Fiber
B.3.1 Amplitude Distribution of the Fundamental LP01 Mode
B.3.2 Equivalent Index neq and Phase Velocity vϕ of the Fundamental LP01 Mode
B.3.3 Group Index ng of the Fundamental LP01 Mode
B.3.4 Case of a Parabolic Index Profile
B.3.5 Modes of a Few-Mode Fiber
B.4 Coupling in a Single-Mode Fiber and Its Loss Mechanisms
B.4.1 Free-Space Coupling
B.4.2 Misalignment Coupling Losses
B.4.3 Mode-Diameter Mismatch Loss of LP01 Mode
B.4.4 Mode Size Mismatch Loss of LP11 and LP21 Modes
B.5 Birefringence in a Single-Mode Fiber
B.5.1 Shape-Induced Linear Birefringence
B.5.2 Stress-Induced Linear and Circular Birefringence
B.5.3 Combination of Linear and Circular Birefringence Effects
B.6 Polarization-Maintaining (PM) Fibers
B.6.1 Principle of Conservation of Polarization
B.6.2 Residual Polarization Crossed-Coupling
B.6.3 Depolarization of Crossed-Coupling with a Broadband Source
B.6.4 Polarization Mode Dispersion (PMD)
B.6.5 Polarizing (PZ) Fiber
B.7 All-Fiber Components
B.7.1 Evanescent-Field Coupler and Wavelength Multiplexer
B.7.2 Piezoelectric Phase Modulator
B.7.3 Polarization Controller
B.7.4 Lyot Depolarizer
B.7.5 Fiber Bragg Grating (FBG)
B.8 Pigtailed Bulk-Optic Components
B.8.1 General Principle
B.8.2 Optical Isolator
B.8.3 Optical Circulator
B.9 Rare-Earth-Doped Amplifying Fiber
B.10 Microstructured Optical Fiber (MOF)
B.11 Nonlinear Effects in Optical Fibers
Selected Bibliography
APPENDIX C
Fundamentals of Integrated Optics for the Fibergyroscope
C.1 Principle and Basic Functions of LiNbO3 Integrated Optics
C.1.1 Channel Waveguide
C.1.2 Coupling Between an Optical Fiber and an Integrated-Optic Waveguide
C.1.3 Fundamental Mode Profile and Equivalence with an LP11 Fiber Mode
C.1.4 Mismatch Coupling Attenuation Between a Fiber and a Waveguide
C.1.5 Low-Driving-Voltage Phase Modulator
C.1.6 Beam Splitting
C.1.7 Polarization Rejection and Birefringence-Induced Depolarization
C.2 Ti-Indiffused LiNbO3 Integrated Optics
C.2.1 Ti-Indiffused Channel Waveguide
C.2.2 Phase Modulation and Metallic-Overlay Polarizer with Ti-Indiffused Waveguide
C.3 Proton-Exchanged LiNbO3 Integrated Optics
C.3.1 Single-Polarization Propagation
C.3.2 Phase Modulation in Proton-Exchanged Waveguide
C.3.3 Theoretical Polarization Rejection of a Proton-Exchanged LiNbO3 Circuit
C.3.4 Practical Polarization Rejection of Proton-Exchanged LiNbO3 Circuit
C.3.5 Improved Polarization Rejection with Absorbing Grooves
C.3.6 Spurious Intensity Modulation
Selected Bibliography
APPENDIX D
Electromagnetic Theory of the Relativistic Sagnac Effect
D.1 Special Relativity and Electromagnetism
D.2 Electromagnetism in a Rotating Frame
D.3 Case of a Rotating Toroidal Dielectric Waveguide
Selected Bibliography
APPENDIX E
Basics of Inertial Navigation
E.1 Introduction
E.2 Inertial Sensors
E.2.1 Accelerometers (Acceleration Sensors)
E.2.2 Gyroscopes (Rotation Rate Sensors)
E.2.3 Classification of the Inertial Sensor Performance
E.3 Navigation Computation
E.3.1 A Bit of Geodesy
E.3.2 Reference Frames
E.3.3 Orientation, Velocity, and Position Computation
E.3.4 Altitude Computation
E.4 Attitude and Heading Initialization
E.4.1 Attitude Initialization
E.4.2 Heading Initialization with Gyrocompassing
E.5 Velocity and Position Initialization
E.6 Orders of Magnitude to Remember
Selected Bibliography
List of Abbreviations
List of Symbols
About the Author
Preface to the First Edition
Fifteen years of research and development have established the potential of the fiber-optic
gyroscope, which is now considered a privileged technology for future applications of inertial
guidance and control. Its “solid-state” configuration brings crucial advantages over previous
approaches using spinning wheels or gas ring lasers.
Interest in the fiber-optic gyroscope is growing rapidly in many companies around the world.
Development, production, and system engineers are now getting involved, in addition to the
scientific and technological communities that have conducted the research. Therefore, this a good
time to present a detailed description of the analysis thas has been carried out to achieve a
practical device. Despite the relative simplicity of the final scheme, the fiber-optic gyro is a
sophisticated instrument with many subtle error sources which must be understood and
controlled. The subject requires a multidisciplinary approach involving physics guided optics,
opto-electronic technology, signal processing theory, and electronic design. The variety of topics
involved is a good example of a thorough system analysis, and the study of the fiber gyro would
be a very formative theoretical and experimental program for graduate students in fiber optics
and opto-electronics.
To help the reader, I have included detailed appendixes that provide information on optics,
single-mode fiber optics, and integrated optics necessary for understanding the fiber gyro, and
the vocabulary for communicating with opto-electronic component designers. For the newcomer
to the field, this material will help him or her avoid having to go through general text books to
find specific basics. However, based on my own experience preparing these basics, the
appendixes could also be a useful review for those already involved in the subject area. I have
also tried (except in Appendix 4) to avoid as much cumbersome mathematical calculations and
formulae as possible. The many figures act as visual aids to simplify the explanations and help the
reader grasp the important ideas dictating the design rules.
Because this is a single-author book, the analysis may be slightly influenced by my personal
views. However, I have chosen to share with the reader the results of my fifteen years of
experience, and I have clearly indicated my preferences instead of giving a strictly impartial
description, which would have resulted in a dreariness this subject does not deserve.
This book is based on research experience that has been shared with scientists who, like me,
have been fascinated by the technical serendipity of the device. I owe a very special tribute to H.J.
Arditty of Photonetics for our continuous fruitful collaboration. My postdoctoral scholarship at
Stanford University was an essential experience, marked by the profound influence of Professor
H.J. Shaw. I would also like to acknowledge the crucial contributions of M. Papuchon and G.
Pircher of Thomson-CSF, R.A. Bergh, now with Fibernetics, and Ph. Martin and Ph. Graindorge
of Photonetics. It is important to recall that this research has been carried out with very open
exchanges within the international scientific community, which may be one reason for its success.
Finally, the efficiency of C. Hervé was essential in the preparation of the manuscript.
Hervé C. Lefèvre
Paris, France
March 1992
Preface to the Second Edition
Two decades after the first edition of this book in the early 1990s, it seemed timely to prepare a
revised version. If the basic design rules of the fiber-optic gyroscope (often abbreviated FOG)
have remained unchanged, the technology has matured, and the expectations presented in the first
edition have been largely exceeded. Navigation-grade bias stability (0.01°/hour) was seen as the
goal, when today strategic grade looks accessible with a bias better than 0.0001°/hour. The
technical serendipity of the fibergyro, which was outlined early on, is clearly confirmed.
This revised edition is enlarged by 50%, and about 300 figures ease the understanding of this
multidisciplinary system.
I would like to recall the very special tribute I owe to Hervé J. Arditty, the president of iXCore
(the parent company of iXBlue), for our continuous fruitful collaboration and the profound
influence of the late Professor H. John Shaw during my postdoctoral scholarship at Stanford
University in the early 1980s.
I would like also to outline the crucial contribution of Thierry Gaiffe, who led the development
of iXSea, the origin of iXBlue, during the 2000s.
This revised edition is the result of the scientific interaction with my R&D colleagues of the
Inertial Division and the Photonic Division of iXBlue: Benoît Cadier, Cédric Moluçon, Eric
Ducloux, Frédéric Guattari, Henri Porte, Jean-Jacques Bonnefois, Jérôme Hauden, Joachin
Honthaas, Maxime Rattier, Pascal Simonpiétri, Pham Van Doug, Robert Blondeau, Sébastien
Ferrand, Thierry Robin, and Thomas Villedieu.
Finally, I wish to thank Glen Sanders, of Honeywell, who revised Chapter 11, and Yves Paturel,
of iXBlue, who prepared the new Appendix E. The efficiency of Chantal Allano was also essential
for the preparation of the revised manuscript.
Hervé C. Lefèvre
Paris, France
September 2014
CHAPTER 1
Introduction
The laws of mechanics show that an observer kept locked up inside a black box in uniform linear
translation has no way to know his or her movement. However, it is possible to detect an
acceleration or a rotation. Precise measurements may be performed with mechanical
accelerometers and gyroscopes. This is the basis of inertial guidance and navigation. Knowing
the initial orientation and position of the vehicle, the (mathematic) integration of the acceleration
and rotation rate measurements yields the attitude and the trajectory of the vehicle. Such inertial
techniques are completely autonomous and need no external reference: they do not suffer from
any shadow effect nor jamming. For nearly a century, they have been a key technology in
aeronautics, naval, terrestrial, and space systems for civilian and military applications.
In 1913, Sagnac [1, 2] demonstrated experimentally that it is also possible to detect rotation with
respect to inertial space with an optical system that has no moving part. He used a ring
interferometer and showed that rotation induces a phase difference between the two
counterpropagating paths. However, the original setup was far from a practical rotation rate
sensor, because of its very limited sensitivity. In 1925, Michelson and Gale [3] were able to
measure Earth rotation with a gigantic ring interferometer of almost 2 km in perimeter to
increase the sensitivity, but the Sagnac effect has remained a rarely observed physic curiosity for
many decades, because it was not possible to get usable performance from a reasonably compact
device.
This possibility of getting a gyroscope without moving parts to replace the spinning wheel
mechanical gyro was indeed very attractive, and in 1962 Rosenthal proposed to enhance the
sensitivity with a ring laser cavity [4] where the counterpropagating waves recirculate many times
along the closed resonant path instead of once in the original Sagnac interferometer, and this was
first demonstrated by Macek and Davis [5] in 1963. Entering the market in the early 1980s [6, 7],
ring laser gyro (RLG) technology has reached its full maturity and is nowadays the dominant
gyroscope technology for inertial guidance and navigation.
However, because of the huge technological effort devoted to the development of low-loss
optical fibers and solid-state semiconductor light sources and detectors for telecommunication
applications, it has become possible to use a multiturn optical-fiber coil instead of a ring laser to
enhance the Sagnac effect by multiple recirculation. Proposed early in 1967 by Pircher and
Hepner [8] and demonstrated experimentally by Vali and Shorthill in 1976 [9], the fiber-optic
gyroscope (FOG) has since attracted a lot interest because it provides unique advantages due to its
solid-state configuration.
The first decade of research and development generated 770 publications devoted to the subject
[10], and the most significant contributions were compiled in a single SPIE “Milestone” volume
in 1989 [11], which is very convenient when working in this field. The proceedings of the four
“anniversary” conferences [12–15] and two special conference sessions [16, 17] specifically
dedicated to the subject are also a good indication of the progress of the technology over nearly
40 years.
A critical step was passed in the early 1990s when several companies started industrial
production [14]. At that stage, it seemed useful to present a thorough analysis of the results of the
R&D phase with the first edition of this book in 1993, which emphasized the concepts that had
emerged as the preferred solutions. The main applications were foreseen in the medium accuracy
range (bias drift from 0.1°/h to 10°/h), but there were also expectations to become a significant
contender in the high-accuracy navigation-grade domain (bias drift below 0.01°/h) despite the fact
that the RLG was, and still is, a tough competitor.
Today, two decades later, one may say that expectations have been fulfilled and even exceeded.
Nearly 500,000 FOG axes have been produced, taking an estimated 40% share of the tacticalgrade market (1°/h to 10°/h), versus 30% for the RLG; about 30% of the intermediate-grade
market (0.01°/h to 1°/h), versus 50% for the RLG; and entering, with about a 20% share, the highaccuracy navigation-grade market (0.001°/h to 0.01°/h) where the RLG is still clearly dominant
with 65%. However, the FOG and the RLG have only a small share (less than 10%) of the
industrial-grade market (10°/h to 100°/h) where they face the rapidly developing
microelectromechanical systems (MEMS) technology that offers high compactness and very low
cost.
Over the years, the specifications of the fiber-gyro have continuously improved, and they even
get to strategic-grade performance (<0.001°/h) that is barely reachable for RLG technology. To
the question of whether the fiber-optic gyroscope is actually better than the ring-laser gyroscope
[18], the answer is yes. The dream of a strap-down navigation system getting to a performance of
a nautical mile per month with an FOG bias stability of 10-5°/h becomes accessible [19].
These progresses have been made possible by an important engineering effort over the years,
but also because the fiber-gyro has continued to take advantage of the progress of optical-fiber
communications. As one example, the erbium-doped fiber ASE source which is a quasi-ideal
source for the FOG, is derived from the erbium-doped fiber amplifier (EDFA) that
revolutionized optical-fiber communications in the 1990s [20] and is produced in huge quantities.
Pump laser diode, isolators, and even fusion splicers would not have been developed just for the
FOG.
This revised edition does incorporate the new developments over these last two decades, but
also tries to further simplify the presentation. For example, Appendix A on the fundamentals of
optics does not start at once with Maxwell’s equations anymore, but with a section presenting the
basic parameters of an optical wave: wavelength, frequency, and power. Appendix B on the
fundamentals of fiber-optics begins now with a section summarizing the main characteristics of a
single-mode fiber and not with overlap integrals or the concept of eigenvectors in a multimode
fiber.
Chapter 2 describes the general principle of optical gyros (RLG, R-FOG, and I-FOG) that are
all based on a recirculating Sagnac effect and have similar theoretical performance. Some
historical background of this Sagnac effect has been added and shows the important contribution
of von Laue to clarify what should be called the Sagnac-Laue effect. It also explained the FresnelFizeau drag effect that limits the bias performance of the laser gyro. It is why the fiber gyro has
the potential to surpass it by one to two orders of magnitude. Finally, a new section describes
Allan variance, which is used to test bias noise and drift of inertial sensors.
Chapter 3 analyzes the fundamental concept of single-mode reciprocity and how it is
implemented in a fiber gyro. The importance of the reciprocal biasing modulation-demodulation
scheme at the proper (or eigen) frequency of the sensing coil acting as a delay-line filter is
outlined, as well as the difficult requirement on the rejection of the polarizer.
Chapter 4 deals with the first noise sources that were encountered, backreflection and
backscattering, and shows the importance of a broadband source that eliminates these parasitic
effects because of its low temporal coherence.
Chapter 5 details the difficult question of reciprocity for the polarization modes, and the way it
is solved is by combining conservation of polarization with PM (polarization maintaining) fiber
and depolarization brought by the birefringence of components (PM fiber and integrated-optic
circuit) and the low temporal coherence of broadband sources. It also describes optical coherence
domain polarimetry (OCDP), which was used very early to understand this problem of
polarization nonreciprocity and can be performed easily today with the new tools of optical
spectrum analysis developed for the telecom industry.
Chapter 6 presents the parasitic effect a fiber gyro may face with temperature transience: the
famous Shupe effect often used as a critic of FOG technology by its competitors. It is outlined that
rather than a strict Shupe effect that is actually small, temperature-transience error is related to
temperature-induced stresses in the coil. However, it can be reduced and modeled to get to
strategic-grade performance (better than 0.001°/h), even if it is clearly the last challenge to solve
to obtain an ultimate performance that no other technology could surpass.
Chapter 7 analyzes the magneto-optic Faraday effect and the nonlinear Kerr effect, which yield
true nonreciprocities similar to the Sagnac effect. A section has been added about an additional
transverse magneto-optic effect about 10 times smaller than the Faraday effect, which has a
longitudinal dependence.
Chapter 8 is concerned with the crucial problem of scale factor linearity and accuracy, which is
preferably solved with closed-loop processing using digital phase-ramp feedback. A new section
describes a four-state modulation yielding an efficient control of the gain of the feedback
modulation chain. The delicate problem of mean wavelength control is also emphasized.
Chapter 9 recapitulates the optimal operating conditions and technologies for the I-FOG and I
did not resist adding photographs of early breadboards and prototypes to remember where we are
coming from. A new section treats compensation techniques of the excess relative intensity noise
(RIN) inherent in the broadband light source used to reduce bias noise and drift.
Chapter 10 briefly discusses the alternative approaches that have been proposed.
Chapter 11 details the advancement of R&D of the competing resonant approach, the R-FOG,
where a passive resonant ring cavity is used instead of the two-wave interferometer of the I-FOG.
This chapter was revised by my long-term colleague and friend, Dr. Glen Sanders from
Honeywell, who is one of the world-class experts in this domain.
Finally, Chapter 12 concludes with the present state of the art and future trends for the FOG,
motivating at the least.
Appendixes A, B, and C detail the fundamentals of optics, fiber optics, and integrated optics that
are important to understand the design rules of the fiber gyro. They have been enlarged by almost
100% in this revised edition, based on my own experience about what was missing while training
new young scientists starting to work on the subject. They include also information about
components or functions that were not mature, not to say unknown, two decades ago: rare-earthdoped amplifying fiber, fiber Bragg grating, isolator, circulator, and microstructured fiber.
Appendix D on the electromagnetic theory of the relativistic Sagnac effect has not changed
from the earlier edition and confirms that the Sagnac effect is a pure temporal delay independent
of matter, which is actually very important.
Appendix E, prepared by Dr. Yves Paturel, my colleague at iXBlue, presents the basics of
inertial guidance and navigation techniques, to understand how a gyro is used in practical
applications: attitude and heading reference systems (AHRS), gyrocompassing, and inertial
navigation systems (INS).
References
[1] Sagnac, G., “L’éther lumineux démontré par l’effet du vent relatif d’éther dans un interféromètre en rotation uniforme,” Comptes
rendus de l’Académie des Sciences, Vol. 95, 1913, pp. 708–710.
[2] Sagnac, G., “Sur la preuve de la réalité de l’éther lumineux par l’expérience de l’interférographe tournant,” Comptes rendus de
l’Académie des Sciences, Vol. 95, 1913, pp. 1410–1413.
[3] Michelson, A. A., and H. G. Gale, “The effect of the Earth’s rotation on the velocity of light” Journal of Astrophysics, Vol. 61,
1925, p. 401.
[4] Rosenthal, A. H., “Regenerative Circulatory Multiple Beam Interferometry for the Study of Light Propagation Effects,”
J.O.S.A., Vol. 52, 1962, pp. 1143–1148.
[5] Macek, W. M., and D. T. M. Davis, “Rotation-Rate Sensing with Travelling-Wave Ring Lasers,” Applied Physics Letters, Vol.
2, 1963, pp. 67–68.
[6] Ezekiel, S., and G. E. Knausenberger, (eds.), “Laser Inertial Rotation Sensors,” SPIE Proceedings, Vol. 157, 1978.
[7] Chow, W. W., et al., “The Ring Laser Gyro,” Review of Modern Physics, Vol. 57, 1985, p. 61.
[8] Pircher, G., and G. Hepner, “Perfectionnements aux dispositifs du type gyromètre interféromètrique à laser,” French patent
1.563.720, 1967.
[9] Vali, V., and R. W. Shorthill, “Fiber Ring Interferometer,” Applied Optics, Vol. 15, 1976, pp. 1099–1100.
[10] Smith, R. B., “Fiber-Optic Gyroscopes 1991: A Bibliography of Published Literature,” SPIE Proceedings, Vol. 1585, 1991, pp.
464–503.
[11] Smith, R. B., (ed.), “Selected Papers on Fiber-Optic Gyroscopes,” SPIE Milestone Series, Vol. MS8, 1989.
[12] Ezekiel, S., and H. J. Arditty, (eds.), “Fiber-Optic Rotation Sensors and Related Technologies,” Proceedings of the First
International Conference, Springer Series in Optical Sciences, Vol. 32, 1981.
[13] Udd, E., (ed.), “Fiber Optic Gyros: 10th Anniversary Conference,” SPIE Proceedings, Vol. 719, 1986.
[14] Ezekiel, S., and E. Udd, (eds.), “Fiber Optic Gyros: 15th Anniversary Conference,” SPIE Proceedings, Vol. 1585, 1991.
[15] Udd, E., H. C. Lefèvre, and K. Hotate, eds., “Fiber Optic Gyros: 20th Anniversary Conference,” SPIE Proceedings, Vol. 2837,
1996.
[16] Sanders, G., chair, “Fiber Optic Gyros: 30th Anniversary Symposium,” Proceedings of OFS 18 Conference, Sessions MA-MC
and MD, Cancun, Mexico, 2006.
[17] Sanders, G., chair, “Fiber Optic Gyros: 35th Anniversary Workshop,” SPIE Proceedings OFS 22 Conference, Beijing, China,
Vol. 8421, 2012.
[18] Lefèvre, H. C., ‘The Fiber-Optic Gyroscope: Actually Better Than the Ring-Laser Gyroscope?” Fiber Optic Gyros 35th
Anniversary Workshop, OFS 22 Conference, SPIE Proceedings, Vol. 8421, Paper 842104, 2012.
[19] Paturel, Y., et al., “One Nautical Mile per Month FOG-Based Strapdown Inertial Navigation System: A Dream Already within
Reach ?” ISSN 2013 Conference, Saint Petersburg, Gyroscopy and Navigation, Vol. 5, No. 1, 2014, pp. 1–8.
[20] Desurvire, E., Erbium-Doped Fiber Amplifier, New York: Wiley-Interscience, 2002.
Introduction
Introduction
CHAPTER 2
Principle of the Fiber-Optic Gyroscope
2.1 Sagnac Effect
2.1.1 A History of Optics from Aether to Relativity
If Huygens proposed in the seventeenth century a wave theory of light, Newton imposed his views
of a corpuscular theory in the early eighteenth century. It is only in the early nineteenth century
that Young’s double-slit experiment reopened the wave theory, knowing that it was not easily
admitted; you did not contradict Newton. It required the exceptional quality of the theoretical and
experimental work of Fresnel to convince the physicist community.
However, for the minds of that time, a wave needed some kind of propagation medium, as for
acoustic waves. It was called “luminiferous aether,” and light was seen as propagating at a
constant velocity c with respect to this fixed aether.
Even when Maxwell showed in 1864 the electromagnetic nature of light wave, Aether was not
questioned. It required the famous Michelson and Morley experiment in 1887 to have a clear
demonstration that the concept of aether should be revised, and this yielded, in 1905, the special
theory of relativity, when, based on earlier theoretical works of Lorentz, Poincaré, Planck, and
Minkowsky, Einstein abandoned the concept of aether and stated that light is propagating at the
same velocity c in any inertial frame of reference in linear translation, despite its own velocity.
This revolutionary conceptual leap was very difficult to admit, and Sagnac’s experiment, the
base of present optical gyroscopes, was actually performed to demonstrate that aether did exist as
clearly stated in the title of his publication, “The Luminiferous Aether Demonstrated by the Effect
of the Relative Wind of Aether in an Interferometer in Uniform Rotation” [1, 2].
As it will be detailed later, Sagnac’s experiment, which takes place in a vacuum (actually in air,
but it can be considered as in a vacuum), can be either explained by the relativity theory or by the
classical aether theory, and does not allow one to demonstrate which theory is right or wrong. It
was explained clearly by von Laue in 1911 [3], two years before Sagnac’s publication, and Sagnac
effect should perhaps be renamed as the Sagnac-Laue effect.
Another important point of the aether theory was the hypothesis made by Fresnel in 1818 of the
drag of aether by matter that was demonstrated experimentally by Fizeau in 1851 [4]. The velocity
v of a wave that propagates in a medium of index n that moves at a speed vm is not c/n anymore
but:
(2.1)
This Fresnel-Fizeau drag effect was explained by von Laue in 1907 [5] as resulting from the
law of addition of speeds of special relativity:
(2.2)
where v1 is the speed of a mobile in a frame moving at v2 with respect to the rest frame, and v is
the speed of this mobile in this rest frame. One sees that (2.2) yields (2.1) to the first order,
considering v1 = c/n, v2 = vm, and vm << c. The Fresnel-Fizeau drag effect is actually a
relativistic effect.
We shall see later that the Sagnac effect does not depend on the index of refraction of the
propagation medium as it was stated by von Laue [6] as early as in 1920, and as it is clearly
experienced in a fiber gyroscope.
If the Sagnac effect in a vacuum can be explained by the aether theory, the Sagnac effect in a
medium is related to Fresnel-Fizeau drag effect and is then also a relativistic effect.
2.1.2 Sagnac Effect in a Vacuum
The fiber-optic gyroscope (FOG) is based on the Sagnac effect, which produces a phase
difference ∆ΦR proportional to the rotation rate Ω in a ring interferometer [1, 2]. Sagnac’s
original setup was composed of a collimated source and a beam-splitting plate to separate the
input beam into two waves that propagate in opposite directions along a closed path defined by
mirrors (Figure 2.1).
Figure 2.1 The original Sagnac setup [1, 2] of a ring interferometer to demonstrate sensitivity to rotation rate (S stands for
surface, which means “area” in French).
A pattern of straight interference fringes was obtained with a slight misalignment of one mirror
as explained in Section A.8.2, and a lateral shift of the fringe pattern was observed as the whole
system was rotated. This fringe shift corresponds to an additional phase difference ∆ΦR between
the two counterpropagating waves, depending on the area A enclosed by the path and the angular
frequency ω:
(2.3)
Following von Laue (Figure 2.2) [3], this can be explained by considering a regular polygonal
path M0M1 … MN–1M0. At rest, both opposite paths are equal, but, in rotation around the center,
the co-rotating path is increased to M0M1′K MN–1′MN′ and the counter-rotating path is decreased
to M0M1″K MN–1″MN″ (Figure 2.3). As a matter of fact, for an observer in the inertial rest frame
of reference, the points Mi move on a circle of radius R, and light propagates along polygon
sides Mi′Mi+1′ or Mi″Mi+1″ instead of MiMi+1.
Figure 2.2 Explanation of the Sagnac effect in the von Laue publication of 1911 [3].
Figure 2.3 Path change in a ring interferometer with a regular polygonal path: (a) at rest; (b) co-rotating path; and (c) counterrotating path.
In particular, the first side of the co-rotating polygonal path becomes M0M1′ (Figure 2.4).
Using 2θ to represent the angle M0OM1, δθ to represent the angle M0OM1′, LM to represent the
length M0M1, and δLM to represent the path length increase M0M1′ − M0M1, there are:
(2.4)
Figure 2.4 Geometrical analysis of the Sagnac effect along one side of a polygonal path.
This angle δθ is, to the first order, the angle of rotation during the propagation between M0
and M1:
(2.5)
and since LM = 2Rsinθ, and the area of the triangle M0OM1 is At = (Rsinθ)(Rcosθ), this yields:
(2.6)
The phenomenon is observed in the rest frame, where light propagates always at velocity c;
therefore, the path increase δLM corresponds to an increase δt+ of the propagation time:
(2.7)
There is this same increase for each side of the polygon and the opposite variation δt– = −δt+
in the counter-rotating direction. The difference ∆tR of propagation time between the two
opposite closed paths in a vacuum is then:
(2.8)
where ∑At is the sum of all the triangular areas (i.e., the whole enclosed area A). Measured in an
interferometer, this time difference yields the phase difference:
(2.9)
where ω is the angular frequency of the wave. It can be shown that this result is very general and
can be extended to any axis of rotation and to any closed path, even if they are not contained in a
plane, using the scalar product A ⋅ Ω:
(2.10)
(2.11)
where Ω is the rotation rate vector and A is the equivalent area vector of the closed path defined
in terms of the line integral:
(2.12)
where r is the radial coordinate vector. The Sagnac effect appears as the flux of the rotation
vector Ω through the closed path of the equivalent area vector A.
To get a simple understanding of the Sagnac effect, it is possible to consider the case of an
“ideal” circular path [7, 8], which would be the limit of a polygonal path with an infinite number
of sides. At rest, light entering the system is divided into two counterpropagating waves which
return in phase after having traveled at the same velocity c along the same path in opposite
directions [Figure 2.5(a)].
Figure 2.5 Sagnac effect in a vacuum considering an “ideal” circular path: (a) system at rest; and (b) system rotating.
Now, when the interferometer is rotating, an observer at rest in the inertial frame of reference
sees the light entering the interferometer at point M [Figure 2.5(b)] and still traveling with the
same vacuum velocity c in opposite directions; however, during the transit time tv through the
loop, the beam splitter has moved to M′, and our observer sees that the co-rotating wave has had
to propagate over a path longer than one turn, while the counter-rotating one has had to propagate
over less than one turn. This path difference 2∆lv can be measured by interferometric means.
This explanation is straightforward, but we must not forget the fundamental point: this is
observed in the inertial frame but still observed in the rotating frame, because both events
(returns of the co-rotating and counter-rotating waves onto the beamsplitter) take place at the
same point, and the principle of causality can be applied: if two events take place at the same point
in space, their difference of time of occurrence is conserved (to the first order in v/c) in any
frame of reference. It is actually interesting to compare the Sagnac effect with the well-known
problem of relativistic kinematics, which explains that simultaneity of events is not an absolute
notion.
Let us consider a system composed of a source S placed at equal distance from two mirrors M1
and M2 [Figure 2.6(a)]. Light is emitted by the source in opposite directions, and, after reflection,
both waves return to the source at the same time. Now, if the system moves laterally [Figure
2.6(b)], an observer in the “laboratory” frame will observe the light hitting first the mirror, M1,
moving toward the incoming wave, and then the other mirror, M2. The delay between both events
is essentially the same as the Sagnac delay, replacing the circumference of the circular path by the
distance between the source and the mirrors, and the tangential speed due to rotation by the
translation speed. However, in the case of translation, both events take place at two different
points, and the principle of causality cannot be applied. An observer in the co-moving system
frame has to wait for both returns of the light to the source that will, to the observer, occur at the
same time. Then this observer can only deduce that, in his or her moving frame of reference,
light hit both mirrors at the same time. Note that the source is also moving for the observer in the
laboratory frame, and he or she sees that the light returns from both ways at the same time. This is
consistent with what was previously said, because the returns on the source are two events taking
place at the same point, and if they are simultaneous, this is observed in any frame of reference.
Figure 2.6 Problem of synchronization in relativistic kinematics: (a) system at rest; and (b) system in uniform linear translation.
Note that the Sagnac effect can alternatively be interpreted as a double Doppler effect on the
beamsplitter. Instead of the temporal approach we detailed above, this can be analyzed spatially,
considering the system “frozen” at a given instant (Figure 2.7). The observer in the laboratory
frame measures one wave transmitted twice and keeping the same wavelength, while the opposite
wave is reflected twice on the moving splitter, resulting in a Doppler shift ∆λD of wavelength
along the circular path and a second opposite shift at the output, resulting in a return to the
original wavelength. The temporal approach considers two waves that propagate at the same
velocity along different paths; this spatial approach considers two waves along the same path but
with different wavelengths. These two explanations are equivalent, but care must be taken not to
use both at the same time.
Figure 2.7 Sagnac effect interpreted as a double Doppler effect: (a) system at rest; and (b) system rotating.
2.1.3 Sagnac Effect in a Medium
Now if light propagates in a medium, as in the case of the fiber-optic gyroscope, it can be
demonstrated that the Sagnac phase difference remains unchanged [6–9]. Considering again a
circular path for simplicity, at rest, both waves propagate at the same velocity v = c/n, where n is
the index of the medium, and return on the beam splitter after the same time tm = 2πR/v = 2πnR/c
= n ⋅ tv. When the interferometer is rotating, the beam splitter moves along a length ∆lm = RΩtm
during the propagation time tm (Figure 2.8).
Figure 2.8 Sagnac effect in a medium: (a) case of a vacuum; and (b) case of a co-rotating medium.
This length ∆lm is n times longer than ∆lv, but, in this case, the light velocity is no longer the
same in both directions. Indeed, this experiment is observed in the inertial frame of reference and
a Fresnel-Fizeau drag occurs because of the motion of the co-rotating medium. It depends on the
relative directions of light propagation and medium motion. In the fixed laboratory frame, the
velocities vcr of the co-rotating wave and vccr of the counter-rotating wave are, respectively:
(2.13)
where αF is the Fizeau drag coefficient and RΩ is the tangential speed of the medium. The
difference of propagation time becomes:
∆tRm = ∆tRn2 (1 − αF) (2.14)
where ∆tR would be the value in a vacuum. Since αF = 1 − n–2, the Fresnel-Fizeau drag
compensates for the effect of the index n, and there is:
∆tm = ∆tv
This perfect compensation is not fortuitous as it may look in this analysis. A rigorous approach
[7, 10, 11] (detailed in Appendix D) has to consider the laws of electromagnetism in a rotating
frame and solve the propagation equation in this frame. Such an analysis shows without any
ambiguity that the Sagnac effect is a pure temporal delay that does not depend on the medium or
on wave guidance if an optical fiber is used.
Note that it is necessary to be careful because the value of the Fresnel-Fizeau drag coefficient is
often given as:
(2.15)
As a matter of fact, in the original Fizeau experiment [4], the light velocity v is first measured
with the medium at rest, and:
(2.16)
where ω0 is the light angular frequency in the laboratory frame. When the medium is moving in
the laboratory frame at a speed vm, the light velocity becomes:
(2.17)
There is a dispersion term
because the frequency “seen” by the medium in its proper
frame of reference is not ω0 anymore, but a frequency ωp, which is shifted because of a Doppler
effect:
(2.18)
However, the pure Fresnel-Fizeau effect is actually:
(2.19)
where ωp is the light frequency “seen” by the medium. In the case of the Sagnac interferometer,
the light frequency has the same ωp value for both counterpropagating waves in the rotating
frame, where the medium is at rest and one has to use [8]:
(2.20)
2.2 Active and Passive Ring Resonators
2.2.1 Ring-Laser Gyroscope (RLG)
The Sagnac effect is very small; the original experiment required a high rotation rate to
demonstrate the phenomenon. Assuming an enclosed area as large as 1 m2, a rate as high as 2π
rad/s, and a wavelength of 1 μm, the phase difference is only 0.5 rad, that is, about one-tenth of a
fringe. The effect has to be greatly enhanced to make a practical rotation sensor with good
sensitivity and compactness.
In the early 1960s, it was proposed to use a ring laser cavity to increase the Sagnac effect,
because the light recirculates many times around the resonant cavity [12, 13]. This technology
reached maturity in the 1980s [14, 15], and the ring-laser gyroscope (or RLG or laser gyro) has
brought major improvements of performance and reliability to inertial navigation systems. It is
the dominant technology today, especially in aeronautics.
In an ordinary laser, the emission wavelength is an integral submultiple of the Fabry-Perot
cavity round-trip length (see Section A.9.1). It is possible to make a ring cavity working on this
principle of optical resonance (see Section A.9.2) (Figure 2.9). The cavity has mirrors with a
quasi-total reflectivity and one output mirror with a small transmissivity. The two
counterpropagating beams are emitted through the output mirror. At rest, the emitted frequencies
(or wavelengths) are equal, since the cavity length is the same in both directions. When it is
rotated, there is a small difference of cavity lengths because of the Sagnac effect, which yields a
frequency difference between both output beams:
(2.21)
where A is the area enclosed by the ring cavity, P is the perimeter, and λ is the wavelength at rest.
Figure 2.9 Ring laser cavity.
This frequency difference is measured by combining the two output beams to get interferences.
Since the beams have different frequencies, their phase difference varies as:
∆ϕ = 2π∆fRt (2.22)
and the interference intensity I is modulated at the beat frequency ∆fR:
I = I1[1 + cos(2π∆fRt)] (2.23)
The counting of the beats gives the angle of rotation, since ∆fR is proportional to the rotation
rate Ω. The angle value corresponding to one modulation period is called the angular increment
θinc, with:
(2.24)
Most high-performance laser gyros have a triangular cavity with a perimeter of about 20 to 30
cm (Honeywell and Sagem). There are also square cavities and in particular an interesting
“octohedral” design integrating three orthogonal square cavities with only six mirrors (Kearfott
and Thales). They operate at a wavelength of 633 nm with a helium-neon (HeNe) amplifying
medium. We have:
θinc ≈ 10–5 rad ≈ 2 arcsec
A rotation of 1°/h (i.e., 1 arcsec/s) gives a beat frequency of 0.5 Hz.
The effect can be understood very simply by considering an “ideal” circular cavity. Both
counterpropagating beams create a standing wave with a space of λ/2 between nodes (Figure
2.10). When the gyro is rotating, the standing wave remains at rest in inertial space, but the
detector rotates and gives one count each time it is passing a length of λ/2. Therefore, the angular
increment θinc is simply:
(2.25)
where R is the radius of the cavity, which is consistent with the general formula
θinc = λP/4A,
since in this case P = 2πR and A = πR2.
Figure 2.10 Simple case of an ideal circular cavity.
The well-known problem of the laser-gyro is the phenomenon of mode lock-in between the
counterpropagating beams. These are oscillators with a very high resonance frequency (in the
range of 500 THz) and a very small frequency difference. As with any resonance effect
(mechanical and electrical), if there is some weak coupling between both oscillators, they become
locked together and oscillate at the same frequency, creating a dead zone at low rotation rate. The
main source of coupling is the backscattering of the mirrors. An intense technological effort was
required to improve the quality of the reflective coating using multidielectric layer technology
(Section A.9.3). However, even with very low scattering mirrors, there is still a dead zone
(typically hundreds of degrees per hour) much wider than the potential sensitivity of the device,
but this is solved with a very efficient technique of mechanical dither that vibrates the gyro at a
rate outside of the dead zone. Present dithered laser gyros have very good performance (bias
stability better than 10–2°/h and scale factor accuracy of 1 ppm over a dynamic range of
±400°/s); but it remains a complex technology, with a limited lifetime because of the wearing-out
of the discharge electrodes that excite the amplifying HeNe plasma, and with some reliability
problems because of leakage of helium.
In addition, the He-Ne laser-gyro faces long-term drift of its bias: the electrical discharge
creates an ionic flow, and because of the Fresnel-Fizeau drag effect that we just saw [4], this
matter flow yields a velocity difference between counterpropagating waves as outlined by
Aronowitz [16]. It is only on the order of 10–15 in terms of relative velocity value, but it creates a
spurious effect equivalent to about a rate of 1°/h. It is counterbalanced by using a common
cathode and two symmetrical anodes (Figure 2.11), but this balancing cannot be perfect and there
is a residual bias instability on the order of few thousandths of a degree per hour.
Figure 2.11 Symmetrical electrical discharges to balance Fresnel-Fizeau drag effect due to ionic flow.
One could think: Why not use a solid-state laser to avoid this drag effect of the He-Ne plasma?
After all, since the early 1960s when the He-Ne laser-gyro was invented, numerous kinds of
lasers have been developed, but there is a key problem in laser behavior: mode competition. In
principle, a CW ring laser should not work because both directions have the same lasing
conditions and they “compete,” that is, it is unstable.
He-Ne ring lasers work because of a very subtle effect: with the flow, the moving amplifying
ions see different frequencies for both opposite directions because of Doppler effect, and the use
of 20Ne and 22Ne isotopes with shifted-frequency gain curves allows one to get two
“superimposed” lasers: one isotope amplifying one direction and the other one amplifying the
opposite one, which avoids mode competition. This is magic, but within the limit of the bias drift
induced by the Fresnel-Fizeau drag.
2.2.2 Resonant Fiber-Optic Gyroscope (R-FOG)
To avoid the problems of lock-in and Fresnel-Fizeau drag, the use of a passive ring cavity instead
of an active system was proposed [7], an external source being fed in both directions into the
cavity. As already seen with the laser-gyro, such a cavity is very similar to a Fabry-Perot
interferometer (Figure 2.12), with resonance frequencies or wavelengths that are transmitted
when the cavity length is equal to a multiple number of wavelengths. When the cavity is rotated,
the Sagnac effect yields a difference ∆fR between the resonance frequencies of the opposite
directions. This difference has the same value as the one in the laser gyro:
(2.26)
The width of the resonance is given by the finesse F and the free spectral range (FSR), just as
for Fabry-Perot cavities (see Section A.9.1). The rotation sensitivity is amplified by half the
finesse, as it corresponds approximately to the number of recirculations in the ring cavity.
Figure 2.12 Similarity between a Fabry-Perot cavity and a ring cavity (R is the reflectivity of the mirrors): (a) Fabry-Perot
cavity; and (b) counterpropagating resonant paths in a ring cavity.
This passive approach in a bulk form does not bring significant advantages over the active
device. However, the use of a single-mode fiber permits increasing the sensitivity further with a
multiturn coil [18]. The frequency difference is still:
(2.27)
where D is the diameter of the coil and n is the index of refraction. This frequency difference
does not depend on the number of fiber turns, but increasing the number of turns (N) increases the
length of the cavity and then reduces its free spectral range, which decreases the absolute width of
the resonance for a given finesse F. Compared to the original Sagnac interferometer, the potential
improvement is the product of half the finesse F by the number N of turns.
An analysis of this resonant fiber-optic gyroscope, often abbreviated R-FOG, will be provided
in Chapter 11. This approach faces a very difficult problem: to fully exploit the system, a very
narrow source spectrum is required, and the related large coherence length induces various
sources of noise that degrade the performance.
2.3 Passive Fiber-Ring Interferometer
2.3.1 Principle of the Interferometric Fiber-Optic Gyroscope (I-FOG)
As the Sagnac effect is proportional to the flux of the rotation rate vector Ω, it can be enhanced
with a multiturn path, just as the flux of a B field is enhanced in a multiturn inductance coil. With a
low-loss single-mode fiber, the enhancement can be made so large that it does not require the use
of a resonant cavity, and a two-wave ring interferometer with a multiturn fiber coil may provide
adequate sensitivity, as proposed very early by Pircher and Hepner [19], and demonstrated
experimentally by Vali and Shorthill in 1976 [20] (Figure 2.13).
Figure 2.13 Two-wave ring interferometer: (a) bulk form with an enclosed area A; and (b) enhanced sensitivity with a multiturn
fiber coil that has an enclosed area N ⋅ A.
The Sagnac phase difference is still ∆ϕ R = 4ωAΩ/c2, but now the actual area  is N-fold the area
of a single loop. It is often expressed as:
(2.28)
where λ is the wavelength in a vacuum, D is the coil diameter, L = NπD is the fiber coil length, A
= πD2/4 is the area of a single fiber loop, and N is the number of loops.
A constant rate yields a constant phase difference. The response is (co)sinusoidal, just as for
any two-wave interferometer (see Section A.8.1), with an interference power given by:
(2.29)
There is an unambiguous range of phase measurement of ±π rad around zero which
corresponds to an unambiguous operating range of ±Ωπ for the rotation rate (Figure 2.14):
Figure 2.14 Response of an interferometric fiber-optic gyroscope.
(2.30)
Let us give some orders of magnitude. A very high-sensitivity fiber-gyro could have a coil
length L up to 10 km and a coil diameter D up to 30 cm (i.e., an area of N ⋅ A = 750 m2). With a
wavelength of 1,550 nm, this yields:
Ωπ = 0.0775 rad/s = 4.4 deg/s
A phase difference of a micro-radian being a good order of magnitude of sensitivity, this
corresponds to a rate of Ωμ of:
A high-sensitivity fiber-gyro uses typically L = 1 km and D = 10 cm (i.e., an area N ⋅ A = 25
m2). At 1,550 nm, it yields:
Ωπ = 130 deg/s and Ωμ = 0.15 deg/h
For medium grade applications, a fiber-gyro would have a short coil length of 100 m and a
small coil diameter of 3 cm (i.e., an area N ⋅ A = 0.75 m2) and it is usually operated at 850 nm. It
yields:
Ωπ = 2400 deg/s and Ωμ = 2.8 deg/h
This geometrical flexibility is a very important advantage of the FOG technology, because, as
we shall see, the same basic components and the same assembly techniques may be used for
various devices without a complete redesign. By simply scaling up or down the effective area of
the fiber coil, the actual sensitivity may be shifted and tuned to the need of the application.
In practice, the I-FOG works over a few fringes about zero path difference, and thus does not
require the use of a very narrow spectrum source, contrary to the case of the resonant fiber
gyroscope (R-FOG). As we shall see, this is a fundamental advantage, since many parasitic effects
are greatly reduced with the low temporal coherence obtained with a broad-spectrum source.
Furthermore, a fiber ring interferometer behaves like a vacuum interferometer despite
propagation in a medium. The Sagnac phase difference ∆ϕ R may be expressed with an equivalent
geometrical path length difference ∆LR without any dispersion effect that could be encountered
with a broad spectrum:
(2.31)
The power P(∆LR) of the interfering wave is:
(2.32)
where C(∆LR) is the coherence function of the source (this problem of temporal coherence in
interferometers is detailed in Section A.12).
2.3.2 Theoretical Sensitivity of the I-FOG
As in any passive optical system, the theoretical sensitivity of the I-FOG is limited by the photon
shot noise. Assuming a phase bias of π/2 rad to operate the interferometer at the inflection point
of the cosine response for maximum sensitivity, the detected power is:
(2.33)
As detailed in Section A.2.1, an optical beam may be regarded as a stream of photons, which
behaves statistically like any ensemble of uncorrelated discrete particles. Any flow Ṅ = dN/dt
yields a random counting with a standard deviation
of Ṅ following:
(2.34)
where ∆fd is the counting bandwidth (i.e., the inverse of the duration of the counting). For photons
with an energy h ⋅ f = hc/λ (where h = 6.63 × 10–34 J⋅s is the Planck constant), the optical power P
and its standard deviation σ Ph follow:
(2.35)
This can also be expressed in power spectral density (PSD with Hz–1 as the unit) of relative
noise:
The theoretical relative photon noise is 10–14/Hz (i.e., −140 dB/Hz) in PSD value, or
in σ value, for P = 25 μW at 1,550 nm or P = 46 μW at 850 nm.
Now the flow of photons Ṅp is converted by the detector into a primary current of electrons
Ṅe, which is also shot-noise limited. The detector quantum efficiency η = Ṅe /Ṅp must be as close
as possible to 1 to limit the degradation of the theoretical photon noise. A perfect detector would
have a responsivity of 0.68 A/W at 850 nm, and 1.2 A/W at 1,550 nm. Practical semiconductor
PIN diodes have a responsivity of 0.55 A/W at 850 nm and 1 A/W at 1,550 nm, which yields a
slight increase by a factor
of the actual detected photon noise. A σ
value of
is obtained in practice for 50 μW at 850 nm and 30 μW at 1,550 nm. On a
π/2 bias, the signal slope is unity with respect to the phase difference, and such a relative photon
noise of
yields a noise of the measured phase difference of
.
Present technology gives a typical returning bias power of 1 to 100 μW, as seen in Table 2.1.
Comparing this result with the Ωμ values, which range, as we have seen, between 0.01°/h and
1°/h, it can be seen that the fiber-gyro technology has a very good theoretical sensitivity, which
has motivated an important R&D effort around the world over the past 35 years.
Note that the wavelength has a little influence on the theoretical sensitivity (if, of course, it is
within a transparency window of the fiber). For the same coil dimension and the same returning
power, the Sagnac phase difference is inversely proportional to the wavelength and the signal-tonoise ratio is proportional to the square root of the wavelength, because, as the wavelength
increases, the photon energy decreases, increasing their number for a given power. Any of the
usual transparency windows (i.e., 850, 1,060, 1,300, and 1,550 nm) of silica fibers have been used.
Today, I-FOGs use 850 nm for medium-grade performance with a coil length of few hundreds of
meters and 1,550 nm for high-grade with a coil of 1 km or more.
Note that for a given fiber attenuation per unit length α (in decibels per kilometer), it is
possible to define an optimal fiber length Lop. For a given diameter, the Sagnac phase difference
increases proportionally to the length L, but the power decreases as 10–αL/10, which accordingly
reduces the signal-to-noise ratio of phase detection to (10–αL/10)1/2 = 10–αL/20.
Table 2.1 Photon Noise as a Function of Power
Anticipated Bias
Photon Noise Equivalent Phase Difference at 850 nm Photon Noise Equivalent Phase Difference at 1,550 nm
Power
1 μW
0.7
0. 6
10 μW
0.22
0.18
100 μW
0.07
0.06
The optimal length is defined by:
(2.36)
where f(L) is a function defined by:
f(L) = L ⋅ 10–αL/20 (2.37)
which yields:
where Lop is in kilometers and α is in decibels per kilometer (see Table 2.2).
Table 2.2 Optimal Length as a Function of Wavelength
850 nm 1,300 nm 1,550 nm
α
3 dB/km 1 dB/km 0.5 dB/km
Lop 3 km
8 km
17 km
This optimal length is much longer than what is used in practice, because this would
significantly reduce the unambiguous dynamic range ±Ωπ, and the coil volume would become
too large to fit within reasonable overall dimensions. With coil lengths ranging between 100m to
a few kilometers, the I-FOG has adequate performance for most applications, but, as we can see,
it still remains possible to improve sensitivity if there are relaxed size and dynamic range
constraints for some specific applications.
Note also that operation around the π/2 radian bias point yields the highest sensitivity, and the
theoretical photon noise has been calculated for this value. However, the optimal performance is
obtained for the best signal-to-noise ratio, which is not a priori to this π/2 bias.
Let us consider a perfectly contrasted interferometer. The response is a raised cosine. The
sensitivity at a given phase bias ϕ b is proportional to the slope (i.e., the derivative sinϕ b of the
raised cosine (1 + cosϕ b)), but the theoretical photon noise is proportional to the square root of
the bias power (i.e.,
). Thus, the theoretical signal-to-noise ratio,
which is proportional to the ratio between the sensitivity and the noise, follows (Figure 2.15):
(2.38)
Therefore, the theoretical signal-to-noise ratio is optimal for ϕ b = π rad (i.e., on a black
fringe) and not for ϕ b = π/2 rad, where it is
times lower [21].
In practice, it is not desirable to operate very close to the black fringe, because of the thermal
noise of the detector, but this shows that the bias point may be chosen between 3π/4 and typically
7π/8 with a slight improvement of the theoretical signal-to-noise ratio.
To conclude, theory shows that there is the same theoretical noise limit for all the optical
gyros, RLG, R-FOG, and I-FOG when they have the same equivalent enclosed area, that is, the
area of the cavity time the Q of this cavity (typically 104) for the RLG, the area of a single fiber
loop time the number of loops and the finesse for the R-FOG, and the area of a single fiber loop
time the number of loops (about 103 to more than 104 depending on performance range) for the
I-FOG.
However, the mechanical dithering of RLGs degrades their theoretical noise by at least one
order of magnitude [16], and, as seen in Chapter 11, R-FOG is sensitive to coherence related
noises. On the contrary, the I-FOG can get very close to its theoretical photon noise, which makes
possible noise level tenfold to a hundredfold lower than the ones of RLG or R-FOG.
Figure 2.15 Optimal signal-to-noise ratio as a function of the phase bias ϕ b with (a) the actual optical power; (b) the photon
noise; (c) the sensitivity; and (d) the signal-to-noise ratio (the vertical coordinates are normalized).
2.3.3 Noise, Drift, and Scale Factor
The output signal at rest (called also the signal bias, but it should not be confused the π/2 phase
bias seen in the previous section) of a fiber gyro is a random function that is the sum of a white
noise (with the theoretical limit of the photon shot noise) and a slowly varying function to take
into account the long-term drift of the mean value. The white noise can be expressed in terms of
the standard deviation σ Ω(f) of equivalent rotation rate per square root of bandwidth of detection
(i.e., degrees per hour per square root of hertz or (°/h)/
Equivalent noise power spectral
density could be used instead, by taking the square of the standard deviation (i.e., (°/h)2/Hz).
However, the habit in the world of inertial navigation is to use the angular random walk (ARW)
performance in °/
which has the same dimension as the standard deviation, but with l (°/h)/
is equal to 1/60°/
Angular random walk must not be confused with the drift, which evaluates the boundaries of the
long-term variations of the mean value of the output signal (Figure 2.16). Bias drift is usually
expressed in degrees per hour. In the fiber gyro, the noise limit is the detection noise, which
depends mainly on the amount of returning optical power; but the drift, which can be theoretically
null, corresponds to a residual lack of reciprocity, which we will analyze.
Noise and drift are different requirements which depend on applications. Low bias noise is
important for fast response stabilization and control, but, for navigation, low bias drift is a more
fundamental parameter. The rotation rate signal is mathematically integrated to get the change in
angular orientation, and this process of integration produces an averaging of the white noise
which renders the effect of the drift predominant in the long term.
Figure 2.16 Example of bias variation with noise and long-term drift.
Another very important characteristic of a gyro is the scale factor. Compared to other sensors,
a gyroscope needs a much better accuracy over a much wider dynamic range: the important
measurement is the integrated rotation angle, and any past error degrades future information. It is
important to have low noise and low drift to measure a very low rate, but it is also important to
have an accurate measurement of high rates (i.e., an accurate scale factor). The required
performance depends on the kind of trajectory, and it is precisely defined by complex system
analysis and modeling, but most applications are usually classified into five grades (see Section
E.2.3), as seen in Table 2.3.
Table 2.3 Performance of the Various Gyro Grades
Angular Random
Walk (White Noise)
Bias Drift
(σ Value)
Rate grade
>0.5° /
10° /h to 1,000° /h 0.1% to 1%
Tactical grade
0.5° to 0.05° /
1° /h to 10° /h
100 to 1,000 ppm
0.01° /h to 1° /h
10 to 100 ppm
Intermediate grade0.05° to 0.005° /
Scale Factor
Accuracy (σ Value)
Inertial grade
<0.005° /
<0.01° /h
5 ppm
Strategic grade
<0.0003° /
<0.001° /h
1 ppm
As stated in the first edition of this book 20 years ago, fiber-gyro technology was first seen as
particularly suitable for tactical-grade applications. Inertial-grade performance was seen as
possible, but difficult because of the competition with the laser gyro, which had reached a very
strong position in this market segment.
Today, the situation has greatly evolved, and the fiber gyro did get a very strong position in
tactical-grade applications, but also in intermediate and navigation grades, especially for marine
and space applications. It is also viewed as able to reach the ultimate high end of the strategic
grade (bias stability as low as 10-5°/h) [22], while the performance of the laser gyro remains
limited to the inertial grade (bias stability of 10−3°/h).
2.3.4 Evaluation of Noise and Drift by Allan Variance (or Allan Deviation)
The basic limit of a measurement instrument is its white noise: the term “white” is used, because
as white light has a uniform light spectral density, the power spectral density (PSD) of this white
noise is also uniform. As we saw, the PSDΩ of a gyro is expressed in terms of (°/h)2/Hz, and is
also used the square root of this value
the root-mean-square (rms) value, also called the
σ value, expressed in term of
However a practical device experiences other sources of random variation. Allan variance is
used to calculate these variations [23]. It was first proposed in the 1960s to evaluate the long-term
stability of atomic clocks [24].
The random measurement signal of a gyro is Ω(t), and can be averaged over a time τ, yielding
a series of averaged values
The Allan variance, AVAR (τ), is defined as half the mean
value of the square of the difference between two successive averaged values
Is also used its square root, called the Allan deviation, ADEV(τ):
Note that Allan deviation is very often called Allan variance, even if, strictly speaking, it is its
square root.
The Allan deviation of a white noise is, in logarithmic scale, a straight line with a −1/2 slope as
a function of the averaging time τ. The power spectral density of the measured rotation rate Ω is:
there is:
AVAR(τ) = PSDΩ/τ
and
because the samples Ωk+1 and Ωk are not correlated:
< Ωk ⋅ Ωk+1 > = 0
and then:
⟨(Ωk+1 − Ωk2)⟩ = ⟨Ωk2+1 + Ωk + 2Ωk Ωk+1⟩
⟨(Ωk+1 − Ωk)2⟩ = 2⟨(Ωk − ⟨Ωk⟩)2⟩
⟨(Ωk+1 − Ωk)2⟩ = 2PSDΩ/τ
The numerical value of the Allan deviation expressed in degrees per hour for τ = 1 hour is the
numerical value of the corresponding ARW expressed in °/
The white noise in (°/h)/
is
the value of the Allan deviation for τ = 1 second (Figure 2.17).
Now, when there is not a pure white noise, the Allan deviation does not follow this −1/2 slope
decay (Figure 2.18).
The actual curve (Figure 2.19) is decomposed with:
The −1/2 slope of the theoretical white noise (ARW), with a unit in °/
A flattening, called simply bias drift, but also flicker or 1/f noise, with a unit in degrees per hour
and the time corresponding to the minimum of Allan deviation called the correlation time;
A +1/2 slope increase, called a rate random walk (RRW), with a unit in °/h3/2;
A +1 slope increase, called a rate ramp, with a unit in °/h2.
Figure 2.17 Example of Allan deviation of a white noise of 0.06 (°/h)/
corresponding to an ARW of 10−3°/
.
There is also a quantization error with a +1 slope at short averaging time. Note that sometimes
the term quantization “noise” is used, but actually quantization is not random and then “error” is
more appropriate.
Figure 2.18 Example of Allan deviations: (a) case of a white noise signal (100 seconds of integration time) and its −1/2 slope
Allan deviation (ARW = 0.005°/
) (b) case of a long-term drift with a flattening and an increase of Allan deviation on the long
term. The bias drift is 0.022°/h and the correlation time is 200 seconds.
The fact that the numerical value of ARW in °/
is the numerical value of the corresponding
Allan deviation in degrees per hour for τ = 1 hour can be generalized to all the modes of Allan
deviation: quantization error in degrees, bias drift in degrees per hour, RRW in °/h3/2, and rate
ramp in °/h2 (Figure 2.20).
Figure 2.19 Modes of Allan deviation.
Figure 2.20 Example of numerical values of the various modes of Allan deviation, looking at their corresponding deviation
values for τ = 1 hour.
As we shall see, FOG technology allows one to get pure white noise (ARW) without
quantization error, bias drift, RRW, or rate ramp over weeks of integration time.
2.3.5 Bandwidth
The minimum response time of the interferometric fiber gyroscope is the transit time through the
fiber coil (i.e., 1 μs for a length of about 200m). This yields a very high theoretical bandwidth of
several hundreds of kilohertz. As will be seen later, signal processing techniques have to be used,
which reduces the bandwidth, but in practice frequency ranges of several kilohertz are reached,
which is a very significant improvement over previous technologies.
It is important to note that the rate signal is averaged over the transit time. Therefore, signal
sampling with this periodicity does not yield any loss of information for the integrated angle of
rotation [25], and subsequent averaging does yield the exact averaged rate. Some proposed signal
processing techniques use signal gating and sampling with a longer periodicity (see Section
10.2), and it is important to be aware that may yield error if the frequency band of the rate signal
is too high, as the variation of the rate during the gating is not taken into account.
As pointed out [26], the interferometric fiber gyro is usually viewed as a rate gyro; that is, the
basic measurement is a rotation rate signal. However, considered only over the transit time
through the fiber coil, the rate is averaged and the fiber gyro may be viewed in this case as a rate
integrating gyro; that is, the basic measurement is an angle of rotation, as, mathematically, an
average is equivalent to an integration.
References
[1] Sagnac, G., “L’éther lumineux démontré par l’effet du vent relatif d’éther dans un interféromètre en rotation uniforme,” Comptes
rendus de l’Académie des Sciences, Vol. 95, 1913, pp. 708–710.
[2] Sagnac, G., “Sur la preuve de la réalité de l’éther lumineux par l’expérience de l’interférographe tournant,” Comptes rendus de
l’Académie des Sciences, Vol. 95, 1913, pp. 1410–1413.
[3] von Laue, M., “Über einen Versuch zur Optik der bewegten Körper,” Münchener Sitzungsberichte, 1911, pp. 405–411.
[4] Fizeau, H., “Sur les hypothèses relatives à l’éther lumineux, et sur une expérience qui parait démontrer que le mouvement des
corps change la vitesse avec laquelle la lumière se propage dans leur intérieur,” Comptes rendus de l’Académie des Sciences,
Vol. 33, 1851, pp. 349–355.
[5] von Laue, M., “Die Mitführung des Lichtes durch bewegte Körper nach dem Relativitätsprinzip,” Annalen der Physik, Vol. 328,
No. 10, 1907, pp. 989–990.
[6] von Laue, M., “Zum Verzuch von F. Harress,” Annalen der Physik, 1920, pp. 448–463.
[7] Post, E. J., “Sagnac Effect,” Review of Modern Physics, Vol. 39, 1967, pp. 475–494.
[8] Arditty, H. J., and H. C. Lefèvre, “Sagnac Effect in Fiber Gyroscopes,” Optics Letters, Vol. 6, 1981, pp. 401–403.
[9] Leeb, W. R., G. Schiffner, and E. Scheiterer, “Optical Fiber Gyroscopes: Sagnac or Fizeau Effect,” Applied Optics, Vol. 18,
1979, pp. 1293–1295.
[10] Post, E. J., “Interferometric Path-Length Changes Due to Motion,” J.O.S.A., Vol. 62, 1972, pp. 234–239.
[11] Arditty, H. J., and H. C. Lefèvre, “Theoretical Basis of Sagnac Effect in Fiber Gyroscopes,” Fiber-Optic Rotation Sensors,
Springer Series in Optical Sciences, Vol. 32, 1982, pp. 44–51.
[12] Rosenthal, A. H., “Regerative Circulatory Multiple Beam Interferometry for the Study of Light Propagation Effects,” J.O.S.A.,
Vol. 52, 1962, pp. 1143–1148.
[13] Macek, W. M., and D. T. M. Davis, “Rotation Rate Sensing with Travelling-Wave Ring Lasers,” Applied Physics Letters, Vol.
2, 1963, pp. 67–68.
[14] Ezekiel, S., and G. E. Knausenberger, (eds.), “Laser Inertial Rotation Sensors,” SPIE Proceedings, Vol. 157, 1978.
[15] Chow, W. W., et al., “The Ring Laser Gyro,” Review of Modern Physics, Vol. 57, 1985, p. 61.
[16] Aronowitz, F., “Fundamentals of the Ring Laser Gyro,” Optical Gyros and Their Applications, RTO AGARDograph, Vol.
339, 1999, pp. 3-1 to 3-45.
[17] Ezekiel, S., and S. R. Balsamo, “Passive Ring Resonator Laser Gyroscope,” Applied Physics Letters, Vol. 30, 1977, pp. 478–
480.
[18] Meyer, R. E., et al., “Passive Fiber-Optic Ring Resonator for Rotation Sensing,” Optics Letters, Vol. 8, 1983, pp. 644–646.
[19] Pircher, G., and G. Hepner, “Perfectionnements aux dispositifs du type gyromètre interférométrique à laser,” French patent
1.563.720, 1967.
[20] Vali, V., and R. W. Shorthill, “Fiber Ring Interferometer,” Applied Optics, Vol. 15, 1976, pp. 1099–1100.
[21] Lefèvre, H. C., et al., “Integrated Optics: A Practical Solution for the Fiber-Optic Gyroscope,” SPIE Proceedings, Vol. 719,
1986, pp. 101–112.
[22] Lefèvre, H. C., “The Fiber-Optic Gyroscope: Challenges to Become the Ultimate Rotation-Sensing Technology,” OFT, Vol.
19, 2013, pp. 828–832.
[23] “Format Guide and Test Procedure for Single-Axis Interferometric Fiber Optic Gyros,” IEEE Standard Specification 952,
1997.
[24] Allan, D., “Statistics of Atomic Frequency Standards,” Proceedings of IEEE, Vol. 54, 1966, pp. 221–230.
[25] Lefèvre, H. C., et al., “Double Closed-Loop Hybrid Fiber Gyroscope Using Digital Phase Ramp,” Proceedings of OFS 3/’85,
OSA/IEEE, San Diego, postdeadline paper 7, 1985.
[26] Fidric, B., et al., “A Rate Integrating Fiber Optic Gyro: From the Theoretical Concept to System Mechanization,” SPIE
Proceedings, Vol. 1585, 1991, pp. 437–448.
CHAPTER 3
Reciprocity of a Fiber Ring Interferometer
3.1 Principle of Reciprocity
3.1.1 Single-Mode Reciprocity of Wave Propagation
As we have seen, the theoretical noise of a fiber ring interferometer is, in practice, on the order
of few tenths of
. Through integration, phase differences of 10–8 to 10–9 rad should
be measurable while the absolute phase accumulated by the wave along 100m to 10 km of fiber is
109 to 1011 rad. The sensitivity limit is 18 to 20 orders of magnitude below the actual
propagation path when the temperature dependence is already as high as 10–5/°C, as seen in
Section B.1.5. Such a performance could look unrealistic, but the fundamental principle of
reciprocity of light propagation in a linear medium is the key to the solution of this problem.
As a matter of fact, in a linear medium, the propagation equation of the wave amplitude A is
(see Section A.3):
(3.1)
Looking for harmonic solutions, A(x, y, z, t) = As(x, y, z)eiωt, where As depends only on the
spatial coordinates and ω is the angular frequency, the propagation equation is reduced to:
(3.2)
Therefore, any solution A(x, y, z, t) = As(x, y, z)eiωt has a perfectly reciprocal solution A′(x, y,
z, t) = As(x, y, z)e–iωt, as the reduced propagation equation depends on the square ω2 = (−ω)2.
Physically, this mathematical change of sign for ω corresponds to a propagation in the opposite
direction with exactly the same propagation delay and the same attenuation of the phase front. In
free space, difficult alignments are required to excite both reciprocal opposite solutions, and it is
never perfect (except when phase-conjuguate mirrors are used [1]). However, if the system is
single-mode (i.e., “single-solution”), alignments are only needed to optimize the throughput
power, but once it is coupled, both opposite waves are automatically reciprocal. The single-mode
waveguide filters the exact part of both opposite input waves corresponding to the same unique
propagating solution.
The Sagnac effect is a very small first-order effect (in DΩ/c), which should be buried in the
changes of the zero order, the absolute phase accumulated in the propagation, but single-mode
reciprocity provides perfect common-mode rejection and will permit the nulling out of this zero
order and its variations almost perfectly, yielding a sensitive measurement of the rotation-induced
nonreciprocal phase difference.
Figure 3.1 Reciprocity of a bulk-optic ring interferometer.
An RLG works also because of reciprocity: a resonant laser cavity has also modes like a
waveguide, and an RLG is a single-longitudinal/single-transverse mode laser.
3.1.2 Reciprocal Behavior of a Beam Splitter
Reciprocity of counterpropagation is fundamental, but it is also important to understand the
behavior of a beam splitter. Considering a simple ring interferometer in bulk optics (Figure 3.1),
the two waves returning through the common input-output port are in phase because they have
propagated along the same path, but also because they have both experienced one reflection and
one transmission on the beam splitter.
Assuming a 50-50 (or 3 dB) separation, the intensity of these returning waves is Iin/4, if Iin is
the input intensity. Because they interfere in phase (i.e., ∆ϕ = 0), the intensity Iint of the total
interfering wave is equal to the input intensity Iin:
(3.3)
and in the absence of a nonreciprocal effect, a ring interferometer with a 50-50 beam splitting
behaves like a mirror. Therefore, because of conservation of energy, the intensity at the second
output port (the free port) of the ring interferometer must be zero. As stated early on by Sagnac in
his original publications [2, 3], there is always a black fringe at the free port, independent of the
exact nature of the 3-dB splitter. This implies that the waves have a π-radian phase difference at
this port. As their propagation paths are identical, this difference is due to a basic phase shift
between the reflected and transmitted waves on the beam splitter.
This basic phase difference may be viewed directly, considering a symmetrical system with two
input waves which arrive in phase on an ideal symmetrical splitter with zero thickness (Figure
3.2). Because of symmetry, the power of both interference waves must be equal, which implies
that the split waves that interfere are in phase quadrature at both output ports. Because there is no
difference of path length, this phase shift must be a basic π/2 difference between the reflected and
the transmitted waves. As this π/2 difference is experienced twice at the free port of the ring
interferometer, a π difference is yielded at the output.
It is interesting to compare this effect with the Huygens’ principle, which states that the
propagation of a wave may be analyzed with virtual sources that re-emit spherical waves from the
phase front. It is well known that to restore the exact phase of the pursuant wave front, a π/2 phase
shift must be added to the virtually re-emitted spherical waves. With an actual reflection from a
splitter, matter is re-emitting real waves without this −π/2 phase shift; therefore, reflection has a
phase quadrature with respect to transmission.
Figure 3.2 Intrinsic phase shift on a beam splitter.
It is interesting to note that Fizeau used actually a ring interferometer in his experiment of 1851
[4] to demonstrate the drag effect of matter. Reading his publication, one sees that he clearly
understood the importance of reciprocity.
Finally, we just saw that a ring interferometer with a 50-50 splitter is a perfect mirror, but it can
make a partial reflector with a non-50-50 splitter. For example, a 15-85 ratio yields about 50%
reflection in the common port and 50% in the free port. This can be used to extract light from a
laser cavity as described for example [5] (Figure 3.3).
Figure 3.3 Tunable laser using a ring interferometer to extract light from the middle of a laser cavity with a unique output port,
according to [5].
3.2 Minimum Configuration of a Ring Fiber Interferometer
3.2.1 Reciprocal Configuration
The first experimental demonstration of a fiber gyro [6] showed that a simple fiber ring
interferometer is not perfectly reciprocal. Complementary interference fringes were observed at
both ports of the interferometer, depending on the alignments of the fiber ends (Figure 3.4). At the
input, a parallel Gaussian laser beam is split and focused on both fiber ends which filter the
unique propagation mode, and, at the output, the beams are recombined to interfere. A small
change in the alignments does not strongly modify the input power coupling, but it has a
significant effect on the matching of both output phase fronts, modifying the fringe pattern and
consequently producing a high parasitic variation of the measured phase difference.
All these problems could have been a severe limitation to high performance, but can be solved
very simply with a reciprocal configuration proposed independently by Ulrich [7] and by Arditty
et al. [8]. It is sufficient to feed light into the interferometer through a truly single-mode
waveguide and look at the returning interference wave, which is filtered through this same
waveguide in the opposite direction (Figure 3.5). In this case, the alignments are needed solely to
optimize the throughput power (and its related signal-to-noise ratio), which requires difficult but
nevertheless reasonable mechanical tolerances (see Section B.4). It is now ensured that both
returning waves have propagated along exactly the same path in the opposite direction and that
they interfere perfectly in phase when the system is at rest. This simple modification has made
both opposite paths identical, zero rotation giving zero phase difference. This common inputoutput port is often called the reciprocal port, and the other free port is often called the
nonreciprocal port, or the fourth port.
Figure 3.4 Free-space interference pattern in a fiber-ring interferometer.
A reciprocal operation of the fiber gyro does not require a continuous single-mode
propagation, but merely a single-mode filter at the common input-output port. A reasonable
amount of power must remain in the primary mode to get some light through the output filter, but
any undesirable signals are perfectly eliminated. For example, assuming that 90% of the light
remains in the primary mode, there is only a slight decrease by a factor of
in the theoretical
signal-to-noise ratio, whereas the rejected 10% may carry a spurious signal equivalent to a phase
difference as high as 0.1 radian (i.e., at least six orders of magnitude above the theoretical
sensitivity).
Spatially, a short length (about 1m) of single-mode fiber can be considered as a perfect filter,
but there is also a need for polarization filtering [7–9], because a single-spatial-mode fiber is
actually a dual-polarization-mode fiber and the fiber birefringence may yield a spurious phase
difference (see Section B.5). However, a polarizer rejection is limited in practice, and we will
explain further how this important problem may be solved with a compromise among
polarization filtering, polarization conservation, and statistical depolarization. In addition to the
coil splitter, the complete reciprocal configuration needs a second splitter (a source splitter) to
tap off part of the interference wave returning through the filtered input-output port of the ring
interferometer. This gives an intrinsic 6-dB attenuation, 50% of the useful light being lost at the
input and at the output, but it is a very moderate drawback compared to the crucial improvement
brought about by reciprocity. Note that the returning power is maximum when both splitters are
50-50 (or 3 dB), but this value does not have to be very precise. For example, with a 60-40
splitter, the transmission of the returning power is 24% instead of 25% with 50-50.
Figure 3.5 Reciprocal configuration of a fiber-ring interferometer (a circulator may be used instead of the source splitter).
To avoid this 6-dB attenuation, one can use a circulator (see Section B.8.3). Such a component
was not available 20 years ago, but it is now quite common because of its use in
telecommunications.
Note that reciprocity is a much more powerful principle than this simple case of single-mode
reciprocity. Considering a multimode filter at the common input-output port, it is possible to
show that light entering in any mode Mi and leaving the interferometer through this same mode
Mi does not carry any spurious phase difference. This result is simply the generalization of the
single-mode case.
However, general linear network theory [10] shows that there is also a reciprocity phenomenon
on the crossed terms. Light entering in any mode Mi and leaving through another mode Mj
because of parasitic couplings is carrying a spurious phase difference signal ∆ϕ ij, but light
entering in Mj and leaving through Mi is carrying exactly the opposite phase difference:
∆ϕ ji = −∆ϕ ij
Therefore, if Mi and Mj have equal power at the input and equal attenuation in the system, both
detected signals cancel out the spurious term due to ∆ϕ ij.
This applies to spatial modes with multimode fibers [11], but also to polarization modes with an
unpolarized source [12] as analyzed by Pavlath and Shaw.
This point is a clear advantage of erbium ASE sources that are naturally unpolarized (see
Section B.9).
3.2.2 Reciprocal Biasing Modulation-Demodulation
The reciprocal configuration provides an interference signal of the Sagnac effect with perfect
contrast, as the phases and the amplitudes of both counterpropagating waves are perfectly equal at
rest. The optical power response is then a raised cosine function, P(∆ϕ R) = P0/2 [1 + cos∆ϕ R], of
the rotation-induced phase difference ∆ϕ R, which is maximum at zero. To get high sensitivity, this
signal must be biased about an operating point with a nonzero response slope:
(3.4)
where ϕ b is the phase bias. However, ϕ b must be as stable as the anticipated sensitivity, that is,
significantly better than 0.1 μrad. For example, the use of the nonreciprocal Faraday effect
(explained in Section 7.1), controlled with an electrical current, was proposed [13], but this would
require a control of the biasing current with an accuracy better than 1 ppm.
Figure 3.6 Generation of the biasing phase modulation using the delay through the fiber coil.
The problem of drift of the phase bias is completely overcome with the use of a reciprocal
phase modulator placed at one end of the coil that acts as a delay line (Figure 3.6) as proposed
very early by Martin and Winkler [14]. Because of reciprocity, both interfering waves carry
exactly the same phase modulation ϕ m(t), but shifted in time. The delay is equal to the difference
∆τg of group transit time (see Section A.5) between the long and short paths that connect the
modulator and the splitter. This yields a biasing modulation ∆ϕ m(t) of the phase difference:
∆ϕ m(t) = ϕ m(t) − ϕ m(t − ∆τg) (3.5)
and the interference signal becomes:
(3.6)
This technique may be implemented with a square-wave modulation ϕ m = ±(ϕ b/2), where the
half-period is equal to ∆τg. This corresponds to the proper or eigenfrequency fp of the coil, fp =
1/2∆τg, and the product, coil length × proper frequency, is about 100 kHz ⋅ km with silica fibers.
This yields a biasing modulation ∆ϕ m = ±ϕ b. At rest, both modulation states give the same signal
(Figure 3.7):
Figure 3.7 Square-wave biasing modulation.
P(0,−ϕ b) = P(0,ϕ b) = P0/2(1 + cosϕ b) (3.7)
but in rotation:
P(∆ϕ R, ϕ b) = P0/2[1 + cos(∆ϕ R + ϕ b)]
P(∆ϕ R, −ϕ b) = P0/2[1 + cos(∆ϕ R − ϕ b)] (3.8)
and the difference between both states becomes:
∆P(∆ϕ R, ϕ b) = P0/2[cos(∆ϕ R − ϕ b) − cos(∆ϕ R + ϕ b)] (3.9)
∆P(∆ϕ R, ϕ b) = P0sinϕ bsin∆ϕ R (3.10)
This biased signal ∆P is measured by demodulating the detector signal with a lock-in amplifier,
and the maximum sensitivity is obtained for ϕ b = π/2, where sinϕ b = 1.
This modulation-demodulation technique is now widely accepted as the optimal biasing
technique, because it yields a sine response (derivative of the unmodulated cosine response) with
a very stable bias (Figure 3.8). The power dependence P0 and the phase bias dependence sinϕ b are
multiplicative, and therefore have no influence on the bias stability. The use of a reciprocal phase
modulator is fundamental, as it yields a modulation of the phase difference perfectly centered
around zero. For any square-wave modulation
difference is always ∆ϕ m(t) = ±ϕ b = ±(ϕ m1 − ϕ m2).
the modulation of the phase
Figure 3.8 Unmodulated response and demodulated biased signal of a fiber-ring interferometer. The dotted curves show a
change of power.
This reciprocal biasing technique may be alternatively implemented with a (co)sine modulation
∆ϕ m(t) = ϕ b ⋅ cos(2πfmt) [7, 8], which does not require a phase modulator with a flat frequency
response. The detected signal may be decomposed in harmonic components of the modulation
frequency fm:
P(∆ϕ R) = P0/2[1 + cos(∆ϕ R + ϕ bcos(2πfmt))] (3.11)
P(∆ϕ R) = P0/2[1 + cos(∆ϕ R)cos(ϕ bcos2πfmt))
−sin(∆ϕ R)sin(ϕ bcos(2πfmt))] (3.12)
Using the Jn Bessel function expansion, this becomes:
P(∆ϕ R) = P0/2 + P0/2cos(∆ϕ R)[J0(ϕ b) + 2J2(ϕ b)cos(4πfmt) + …]
+ P0sin(∆ϕ R)[2J1(ϕ b)sin(2πfmt) + 2J3(ϕ b)sin(6πfmt) + …](3.13)
The even harmonics are still proportional to cos(∆ϕ R) as the unbiased response, but the odd
harmonics and particularly the fundamental frequency are proportional to sinϕ R. With a
synchronous demodulation, this yields a biased signal:
P1(∆ϕ R) = P0J1(ϕ b)sin(∆ϕ R) (3.14)
The maximum sensitivity is now obtained for ϕ b ≈ 1.8 rad (instead of π/2 ≈ 1.5 with square
wave) and J1(1.8) = 0.53. At rest, the detector signal is mainly composed of the second harmonic
component, but, in rotation, it appears to be an unbalanced modulation that contains a signal at the
fundamental frequency fm (Figure 3.9). (Note that we call first harmonic the fundamental
frequency and second harmonic the double of fundamental frequency.)
3.2.3 Proper (or Eigen) Frequency
For a sinusoidal phase modulation ϕ m = ϕ m0sin(2πfmt) applied on both counterpropagating
waves, the modulation of the phase difference is:
Figure 3.9 Sine-wave biasing modulation: (a) at rest and (b) in rotation.
∆ϕ m(t) = ϕ m0[sin(2πfmt) − sin(2πfm(t − ∆τg))] (3.15)
and after applying trigonometric identities:
(3.16)
The ring interferometer behaves like a perfect delay line filter with a sinusoidal transfer
function 2sin(πϕ m ⋅ ∆τg) that is maximum (in absolute value) at the proper (or eigen) frequency
fp [defined previously as fp = 1/(2∆τg)] and its odd harmonics, and that is null at dc and all the
even harmonics (Figure 3.10).
Figure 3.10 Normalized transfer function of the modulation of the phase difference.
From this result, it seems that the choice of the operating frequency is not critical, but spurious
nonlinearity or amplitude modulation in the phase modulation may degrade the quality of the bias
[15]. The use of an additional modulation to detect the maximum of a response curve is actually
very common, but an accuracy of a few tenths of percent of the width is usually sufficient. In a
fiber gyro, we are looking for less than 10–8, and potential parasitic effects must be evaluated
more carefully.
A basic point deserves to be repeated: because the interference signal is an autocorrelation
function, the response is perfectly symmetrical about zero, even with an asymmetrical spectrum
(see Appendix A).
Now, a first cause of spurious bias is the nonlinearity of the phase modulator or of the
electronic generator. A pure (co)sine modulation centered about zero yields only even harmonics
in the modulated interferometer response, and the demodulated response at the fundamental (and
also odd harmonic) frequency is perfectly zero. However, if there is a spurious even harmonic
content due to a nonlinear response of the modulation chain, additional odd frequency
components are generated in the response that are seen as a nonzero bias in the demodulation.
This can be easily seen in Figure 3.11, which shows that a second-harmonic content yields an
unbalanced modulation, as with an offset, due to rotation. In this worst case, the second harmonic
content must be less than 160 dB (electrical) to limit the bias below 10–8 rad. This effect also
depends on the respective phase of the main fundamental-frequency modulation and on the
spurious even harmonic.
A bias offset is not a priori detrimental to the gyro performance if it is stable, but in practice it
is susceptible to drift, particularly with changing environmental conditions. As will be seen
throughout this analysis, the best way to suppress the drift is, after all, to suppress the offset. In the
case of a nonlinear modulator response, a simple solution is to operate the system at the proper
(also called eigen) frequency [15] (or its odd harmonics), and any spurious even harmonics of the
modulation ϕ m(t) will be nulled out in the phase difference ∆ϕ m(t) because, as we saw, the
interferometer behaves as a delay line filter. Note that since this modulation propagates on a very
high-frequency carrier, the optical wave, there is no dispersion for these subcarrier harmonic
components, which all propagate at the same group velocity, and the notch filtering is perfectly
periodic.
Figure 3.11 Biasing modulation imbalance: (a) with a constant offset and (b) with an additional second-harmonic modulation.
(The solid curve in each graph is the sum of the two dashed curves.)
With a square-wave modulation, a nonlinear phase modulation response is not harmful, since
ϕ m takes only two values, ϕ m1 and ϕ m2, and the modulation of the phase difference ∆ϕ m = ±
(ϕ m1 − ϕ m2) is always balanced around zero. If the modulation frequency fm is not equal to the
proper frequency fp, the modulation of the phase difference takes four values (Figure 3.12):
Figure 3.12 Square-wave biasing modulation with (a) fm < fp and (b) fm > fp.
(3.17)
but it is still balanced about zero without any demodulated bias offset.
However, a spurious offset may arise with an asymmetry of the duty cycle of the square-wave
modulation [16]:
(3.18)
where Tm is the period of the modulation. At the proper frequency (where the period Tm = 2 ⋅
∆τg), the phase difference ∆ϕ m(t) takes four values (Figure 3.13):
Figure 3.13 Modulated phase difference with an asymmetrical duty cycle of the square wave.
(3.19)
When the gyro is at rest, the output signal is composed of spikes of equal width ∈ ⋅ ∆τg, but
when the modulation frequency fm is not equal to fp, one spike width is reduced while the other
one is enlarged (Figure 3.14), which yields a very strong parasitic signal at the modulation
frequency, and thus a spurious offset on the demodulated signal. This is actually a very sensitive
way to measure the proper frequency [16]. However, the duty cycle of a square wave cannot be
made perfectly equal to 50-50 (particularly because rise time and fall time are usually not equal in
electronics), and it yields a spurious bias which depends on the modulation frequency and goes
through zero at the proper frequency. As this proper frequency depends on temperature (about
10–5/°C) because of the thermal dependence of the index of refraction, and therefore of the transit
time, this spurious bias offset is not stable. To eliminate this defect, an electronic gate has to be
used to suppress the spikes that carry the spurious effect of an imperfect 50-50 duty cycle.
Figure 3.14 Effect of an asymmetrical square-wave modulation.
This problem makes the use of square-wave modulation more delicate than that of sine-wave
modulation, but, as will be seen later, closed-loop digital processing techniques that use squarewave biasing are so advantageous that this additional complexity is a reasonable drawback.
A last parasitic effect is due to residual intensity modulation in the phase modulator and may be
detected as a spurious signal at the fundamental frequency [15]. This modulation is also
reciprocal and is seen equally by both counterpropagating waves, but the interference signal is
derived from the difference of the wave phases, whereas the intensities are added. For intensity
modulation, the system also behaves as a delay line filter, but in this case it rejects the proper
frequency and its odd harmonics. For intensity modulation, the transfer function is 2cos(πfm ⋅
∆τg); therefore, operation at the proper frequency (or at an odd harmonic) also eliminates the
effect of residual intensity modulation. Note that this is strictly valid only at rest, where the
rotation-induced phase difference is zero. While rotating, intensity modulation may yield some
parasitic effects even at fp, which modifies the scale factor [17].
To summarize, the use of a truly single-mode filter at the common input-output port of the
interferometer renders a fiber ring interferometer sensitive only to nonreciprocal effects as the
Sagnac effect. Furthermore, a modulation-demodulation at the proper or eigenfrequency of the
coil (or one of its odd harmonics) provides a biased signal that does not degrade the original
perfection of the system. These two simple conditions, combined in the minimum configuration
[18], make high performance possible despite the various defects of the components.
As is discussed in Section A.5.2, phase modulation propagates at the group velocity vg as any
modulation and not at the phase velocity vϕ , with:
(3.20)
where ne is the equivalent index of the mode, which depends mostly on the material but also on
the guidance characteristics. When a broad spectrum is used, it is possible to consider that the
modulation propagates at the group velocity of the mean wavelength and that there is a temporal
spreading ∆τs of the modulation waveform, because vg(λ) is not constant over the spectrum width
∆λ when the second-order derivative d2ω/dk2 or d2ne/dλ2, called the propagation dispersion, is
not zero:
(3.21)
This value is often given with ∆τs/(∆λ ⋅ L) in ps ⋅ nm–1 ⋅ km–1. The silica material dispersion
is given in Table 3.1.
Table 3.1 Group Velocity Dispersion in Silica as a Function of Wavelength
λ (in nm)
(in ps ⋅ nm –1 ⋅ km)
850
1,060 1,300 1,550
−100 −30
≈0
+20
This effect is very important in telecommunication applications where modulation frequency
reaches the gigahertz range with propagation along kilometers of fiber, but it is negligible in the
fiber gyro case.
The modulation period is preferably the double of the group transit time ∆τg through the coil,
and because ∆τg the temporal spreading ∆τs is also proportional to the coil length, the ratio ∆τs/
∆τg is independent of the coil length. At 850 nm, where dispersion has the highest value, and with
a spectrum width as broad as 50 nm, this ratio is only 0.1%. In the case of square-wave
modulation, this induces finite rise and fall times, but this does not influence the performance of
the fiber gyro.
3.3 Reciprocity with All-Guided Schemes
3.3.1 Evanescent-Field Coupler (or X-Coupler or Four-Port Coupler)
To avoid the difficulties of the coupling stability of free-space waves into single-mode fibers, it is
desirable to use an all-guided scheme that improves ruggedness. This requires the duplication in a
guided form of the various functions required in the interferometer. In particular, an evanescent-
field coupler may replace the 3-dB splitter. The principle of such couplers, also called X-coupler
or four-port coupler, is usually explained with the coupling overlap of the evanescent tail of a
waveguide’s fundamental mode with a second adjacent waveguide (see Section B.7.1). These
couplers may be realized in an all-fiber form, but also on an integrated optic substrate (see
Appendix C.1.6).
Figure 3.15 Decomposition of the input light with the modes of a two-waveguide coupling structure.
However, to understand its reciprocity behavior, it is interesting to use an alternative
explanation proposed by Youngquist et al. [19]. Two parallel single-mode waveguides may be
regarded as a two-mode waveguide. When light is fed into one input port, it excites the
fundamental symmetrical mode of the coupler and the second-order antisymmetrical mode
(Figure 3.15). The lobes of the modes are in phase in the input lighted waveguide and π (or 180°)
out of phase in the other waveguide without light. As the two modes propagate with different
velocities, their phase difference varies linearly, and the amplitude in each waveguide can be
evaluated using a phasor diagram (Figure 3.16).
Figure 3.16 Propagation of the modes in a two-waveguide coupling structure with a phasor diagram for light propagating in
each waveguide.
At the half-coupling length, there is a 50-50 splitting: both modes are in phase quadrature, and
the modulus of the wave amplitude is the same in both waveguides. At the coupling length Lcp,
light has been completely transferred into the second waveguide. Because the wave amplitudes in
both waveguides are the two diagonals of the rhombus describing the vector sum of the mode
amplitudes, they are always perpendicular, which means that the coupled wave always has a phase
quadrature with respect to the transmitted wave.
The use of an evanescent-field coupler in a fiber-ring interferometer (Figure 3.17) seems very
advantageous, as there is no fringe pattern in free space and the free port could be exactly
complementary to the reciprocal port, with a stable π-radian phase difference, getting twice the
π/2 shift of the coupler.
Figure 3.17 All-fiber ring interferometer with a 3-dB coupler.
This could allow the use of the more complex minimum configuration to be avoided. However,
as will be shown later, polarization still has to be filtered at the same reciprocal input-output port,
and, furthermore, the phase shift of the coupled wave in the coupler is not perfectly equal to π/2
in practice. As a matter of fact, there is always a residual differential loss between the
symmetrical and antisymmetrical modes of the coupler regarded as a two-mode waveguide. The
ideal rhombus of the phasor diagram is transformed into an ordinary parallelogram with unequal
sides, where the diagonals are not perpendicular anymore (Figure 3.18) [19].
Figure 3.18 Phasor diagram of the transmitted and coupled waves of a coupler: (a) lossless coupler and (b) differential loss.
This yields a spurious phase difference at the free port of a ring interferometer. With a very
low-loss coupler, this effect is small, but it remains significant compared to the very small bias
change (less than 1 μrad, which corresponds to less than 10–6 in differential loss), which is
sought to get high performance. Therefore, even with an evanescent-field four-port coupler, a
reciprocal configuration has to be used to get a low bias drift.
3.3.2 Y-Junction
As we have seen, only three ports are actually useful in an all-guided ring interferometer, so
evanescent field couplers can be replaced by Y-junctions. In an all-fiber form, evanescent-field
four-port X-couplers are easier to fabricate than three-port Y-couplers; however, with integrated
optics (see Appendix C), which, as will be seen, is a crucial technology in implementing highperformance signal processing schemes, Y-junctions are preferred because of their simplicity
and stability.
The Y-junction, also called a branching waveguide, was proposed early on [20] as a very useful
integrated optic component. It is composed of a base single-mode waveguide connected to two
single-mode branch waveguides (Figure 3.19). This is fabricated very simply with a Y-mask, and
symmetry ensures 3-dB splitting for any wavelength, while evanescent-field couplers require
careful control of the diffusion process to get the adequate coupling ratio and are wavelength
dependent. The principle of direct operation is simple: light that propagates in the single-mode
base waveguide is split equally into the two symmetrical single-mode branch waveguides, which
diverge with a very small angle (typically 1°) to minimize the loss. The behavior of the reverse
operation [21, 22] is not as straightforward to explain, but as 50% of the base waveguide light is
coupled into each branch waveguide, reciprocity arguments show that the same percentage has to
be coupled from one branching waveguide into the base waveguide in the opposite direction.
Figure 3.19 Integrated optic Y-junction.
This can be better understood by considering an all-guided Mach-Zehnder interferometer
composed of two Y-junctions [Figure 3.20(a)]. Light is split at the first junction, and with two
equal optical paths the two waves are recombined in phase at the second junction. This is
equivalent to a bulk Mach-Zehnder interferometer [Figure 3.20(b)], in which the two waves
interfere in phase at one port and there is no light at the other port because of destructive
interference. Now, if there is an additional π-rad phase difference between the two paths, they are
recombined into a two-lobe second-order antisymmetrical mode that radiates into the substrate
because it cannot be guided into the single-mode base waveguide of the output junction (Figure
3.21). This is equivalent to what happens in a bulk Mach-Zehnder, in which the induced phase
difference switches the output light at the fourth port of the output splitter. A Y-junction is a fourport device, as is any 3-dB splitter, but in this case there are three guided ports and the fourth port
is an antisymmetrical wave radiated into the substrate.
Figure 3.20 Mach-Zehnder interferometer with equal paths: (a) with two Y-junctions and (b) in a bulk form.
Figure 3.21 Mach-Zehnder interferometer with π rad phase difference: (a) with two Y-junctions and (b) in a bulk form.
The reverse operation can be understood by regarding the pair of branch waveguides as a twomode waveguide, similar to the case of an evanescent-field coupler. Coupling light into one of the
branches can be interpreted as the superposition of the fundamental symmetrical mode and the
second-order antisymmetrical mode. Both modes are in phase in the lighted waveguide, while
they are π rad (or 180°) out of phase in the other one, which is not lighted (Figure 3.22). At the
junction, the symmetrical mode that carries 50% of the optical power can be coupled to the base
single-mode waveguide, while the antisymmetrical mode above the cutoff is radiated into the
substrate.
Figure 3.22 Reverse operation of a Y-junction: (a) principle and (b) photograph of the output face of the circuit with the guided
single mode and the radiated antisymmetrical wave.
This radiated antisymmetrical mode experiences a Lloyd mirror effect similar to what happens
for the nonguided TM mode of a proton-exchanged LiNbO3 circuit, which can be considered as
an antisymmetrical LP11 mode with a virtual lobe 180° out of phase (see Section C.3.3).
With this effect, it can be considered as a four-lobe LP21 mode (see Section B.3.5) with a
double antisymmetry, which is preserved while it diffracts. It yields two real lobes into the
substrate and two virtual lobes imaged 180° out of phase by the top surface of the circuit (Figure
3.23). The full width at the maximum 2w1′ = θD1 ⋅ L is at the output facet typically 300 to 400 μm
for a length L of the base branch equal to 10 mm.
Figure 3.23 Radiated fourth port of a Y-junction considered as a four-lobe LP 21 mode with two real lobes and two virtual
lobes.
Because, as we have seen, the gyro architecture requires the use of only three ports, the Yjunction is perfectly adequate. It naturally yields a reciprocal configuration because the base
waveguide is used as the common input-output port, and the nonreciprocal free port is lost in the
substrate (Figure 3.24). It is the optimal technological solution because it is much more stable and
much easier to fabricate than an evanescent-field coupler in integrated optics. Such couplers are
used for fast switching, but the simplicity of Y-junction makes it optimal for permanent 50-50
splitting.
Figure 3.24 Reciprocal configuration with a Y-junction and its single-mode fiber lead.
3.3.3 All-Fiber Approach
The all-fiber approach first appeared as the ideal technological choice for the fiber gyro because
of the very low loss of the components, which provides a very good signal-to-noise ratio, due to
the high returning power [15, 23]. The all-fiber architecture (Figure 3.25) makes use of a first
coil coupler to split and recombine the interfering wave and of a second source coupler to send
the signal coming back through the common input-output port onto a detector. The polarization is
filtered at this reciprocal port with an all-fiber polarizer, which was originally made with a
birefringent crystal facing a fiber polished laterally to extract one polarization by prism
outcoupling of the evanescent tail of the mode [24]. Present all-fiber systems now prefer a
polarizing (PZ) fiber [25], which works on the differential curvature loss that occurs under
special conditions between polarizations in stress-induced high-birefringence PM fibers (see
Section B.6.5). If the early brass-boards were using ordinary single-mode (SM) fiber with in-line
polarization controllers, in particular λ/4 and λ/2 loops [26] (see Section B.7.3), this constraint is
now avoided with the availability of polarization maintaining (PM) fibers (see Section B.6).
Figure 3.25 All-fiber reciprocal configuration according to [15] (from Stanford University).
The main limitation of this approach is the phase modulator. The only practical technique is to
wind a fiber around a piezoelectric tube (or disk) [27] (see Section B.7.2), which modifies the
fiber length by controlling the tube diameter with a driving voltage. This method is perfectly
adequate for the biasing modulation-demodulation, but obtaining an accurate scale factor requires
the use of more sophisticated signal processing techniques. Numerous publications have
described processing schemes that are compatible with piezoelectric modulators (see Section
10.2), but by far the highest scale factor performance is obtained with phase-ramp closed-loop
techniques (see Section 8.2), which require a broad modulation band, while piezoelectric
modulators experience narrow mechanical resonances.
An all-fiber approach yields very good bias performance and navigation grade ARW was
already obtained in the early 1980s at Stanford University [23], but the scale factor accuracy is
limited in practice to about 1,000 ppm.
3.3.4 Hybrid Architectures with Integrated Optics: Y-Coupler Configuration
Integrated optics, particularly on a lithium niobate (LiNbO3) substrate (see Appendix C), was
recognized early on by Papuchon and Puech [28] as a very promising technology for the fiberoptic gyroscope, because a single multifunction circuit could be used to implement all the
functions needed to make the device work, thus yielding a very simple, all-guided configuration
with a sensing fiber coil connected to an integrated-optic circuit. However, the decisive advantage
of integrated optics over the all-fiber approach is its phase modulator with a flat response over a
large bandwidth, which permits the use of efficient signal-processing techniques that yield high
performance over the whole potential dynamic range of the fiber gyro.
It is possible to use an all-fiber approach and limit the use of integrated optics to a phase
modulator on a straight waveguide, but this technology provides the useful advantage of
permitting the integration of several other functions onto a single circuit, which improves the
compactness and reduces the connections. Nevertheless, the pursuit of maximum integration is
perhaps a technological challenge, but it is not a goal in itself, and an optimal hybrid compromise
had to be found to take advantage of this possibility of integration without degrading the
performance.
As we have already seen, the splitter-combiner of an interferometer can be realized very simply
with a Y-junction, and the use of two junctions connected by their base branch [20] has been
proposed. However, this double-Y configuration (Figure 3.26) led to disappointing performances
because of the limited rejection of the common base waveguide that acts as the spatial filter to
ensure reciprocity [29]. The input light is split in the first source junction, and half the power
remains in the base branch to be split again in the second coil junction. The rest of the input
power forms an antisymmetrical mode that is radiated from the first junction, because it is above
cutoff, but it is partially recoupled in the second junction, which acts as a receiving antenna,
because it may again guide this antisymmetric diverging wave (Figure 3.26).
Figure 3.26 Double-Y configuration.
This recoupling is very small, but the parasitic waves added in the two output branches of the
coil junction are π (or 180°) out of phase because of the antisymmetry of the diverging wave. The
total wave amplitude in each branch is the sum of the main amplitude term Am and the additional
parasitic term As coming from the spurious recoupling. Considering a phasor diagram (Figure
3.27), it can be seen that the phase of total amplitude At or At′ in each branch is modified and that
it also yields a spurious phase difference ∆ϕ e between both branches while it should be zero with
a perfect rejection.
Figure 3.27 Limitation of single-mode rejection in a double-Y-junction.
The maximum phase error is proportional to the amplitude ratio:
∆ϕ e ≤ 2As/Am (3.22)
To limit this error below 10–8 rad, the rejection should be better than 166 dB. In practice, a
common base branch of 10 to 20 mm yields a rejection limited to about 60 to 70 dB, and
therefore a phase bias offset as high as several 10–4 rad [26]. As we have already seen, an offset
is not, strictly speaking, detrimental, but in practice it is susceptible to drift. In this case, the actual
phase error ∆ϕ e depends on the phase difference ∆ϕ ms between the main amplitude Am, which
has been continuously guided, and the spurious term As, which has propagated freely between
both junctions:
∆ϕ e = 2(As/Am)sin∆ϕ ms (3.23)
(It is maximum for ∆ϕ ms = π/2, where As is perpendicular to Am, and it is zero for ∆ϕ ms = 0
or π, where As is parallel to Am.) The phase difference ∆ϕ ms is related to the path difference
∆nms ⋅ Lbb, where the difference of relative index ∆nms may be estimated as half the index step of
the waveguide (i.e., a few 10–3). With 10 mm of common base-branch length Lbb, this path
difference is a few tens of micrometers (i.e., a few tens of wavelengths), and this changes with
temperature, which induces drift.
This double-Y configuration looked at first like a very tempting solution, but it is actually
showing that the search for maximal integration may encounter drastic performance limitations.
At this stage, one may choose to improve the technology by implementing filtering devices about
the central common base waveguide or by using evanescent-field couplers, which avoids the
radiating fourth port of Y-junctions; but this increases drastically the complexity of the process
and of its control.
The optimal compromise between integration and performance is obtained with the Y-coupler
configuration [30–32], where the coil splitter is an integrated optic Y-junction and the source
splitter is an all-fiber 3-dB coupler (Figure 3.28). The spatial filtering required for reciprocity is
simply obtained with the fiber lead connected on the base waveguide of the Y-junction, and the 3dB coupler is used to extract 50% of the returning power with low additional loss and send it onto
a detector. Phase modulators are fabricated on both branches of the Y-junction. It is possible to
use one modulator for the biasing modulation and the other one for the feedback modulation
required with closed-loop processing techniques, but it is preferable to connect both modulators
in a push-pull configuration. The voltage is applied between the central electrodes and both
external electrodes, which automatically drives the two modulators with opposite polarity (Figure
3.29).
This doubles the efficiency, but, above all, this eliminates the second-order nonlinearity of the
response of each single modulator, which is very useful to get a good scale-factor linearity with
closed-loop schemes.
Note that if a single-mode 3-dB coupler is commonly used for the tap function, it is not strictly
required. The light returning through the input fiber lead of the Y-junction has already interfered,
and the signal is carried by the intensity modulation. Therefore, the coupler may be replaced with
a 3-dB tap, which has a single-mode output port and may be easier to fabricate than a coupler
[31].
Figure 3.28 Y-coupler configuration with a 3-dB coupler as the source splitter.
This Y-coupler configuration has been widely accepted since the 1990s [32] as the optimal
technological compromise for high-performance fiber gyroscopes that have to use integrated
optic phase modulators to get a good scale factor with closed-loop processing schemes.
Figure 3.29 Push-pull connection of two integrated optic phase modulators: (a) x-cut and y-propagating LiNbO3 circuit and (b)
z-cut and y-propagating LiNbO3 circuit.
The preferred integrated optic technology is LiNbO3 (see Appendix C), which yields very
efficient phase modulation. The waveguides were first fabricated with titanium indiffusion, and a
polarizer was realized with a metallic overlay that absorbs the TM mode, but today proton
exchange [33] is the privileged alternative, as it yields single-polarization guidance that provides
a very high polarization rejection. This point is very important, as we shall see, for getting good
bias stability.
The optimal multifunction gyro circuit (Figure 3.30) has typical dimensions of 1 mm in
thickness, a few millimeters in width, and 15 to 20 mm in length at 850 nm and 30 to 40 mm in
length at 1,550 nm.
Figure 3.30 The multifunction integrated-optic circuit (MIOC) for the fiber-gyro; Y-junction with single-polarization protonexchange waveguides and a pair of push-pull electrodes. The substrate has a parallelogram shape to avoid backreflection.
It is composed of a Y-junction with a separation of 200 to 400 μm between the two branches to
connect the two fiber coil ends. Push-pull phase modulators are fabricated on the branches, and
polarization rejection is directly obtained with proton exchange. Finally, as will be shown later,
the gyro circuit has a parallelogram shape to avoid backreflection at the interfaces between the
circuit and the fibers [32].
3.4 Problem of Polarization Reciprocity
3.4.1 Rejection Requirement with Ordinary Single-Mode Fiber
A single-mode fiber has actually two polarization-modes. With an ordinary fiber these two modes
are almost degenerate, but there is a residual birefringence that modifies the state of polarization
as the light propagates. In a ring interferometer, at a given position, both counterpropagating
waves have a different state of polarization and therefore do not see exactly the same index of
refraction, because of birefringence, which yields a spurious phase difference at the output.
Applying the general linear network theory, it is possible to show that if a polarizer is placed at
the input and at the output of a fiber, the phase of the waves transmitted in opposite directions are
then perfectly equalized [10]. To take into account the birefringence of the splitter combiner of
the ring interferometer, a single polarizer has to be placed at the common reciprocal port [7–10]:
light is filtered at the input, and the two waves that are coming back through this same polarizer at
the output are perfectly in phase. Some polarization control is necessary, as in the worst case,
light may come back in the crossed state, yielding signal fading.
However, the rejection of a practical polarizer is not infinite, and there remains a residual
phase difference between both counterpropagating waves. It was shown by Kintner [34] that with
ordinary single-mode fibers the bias error is limited by the amplitude rejection ratio ∈ of the
polarizer and not by its intensity ratio ∈2: a maximum phase measurement error of 10–8 rad
would then require a rejection on the order of 160 dB and not 80 dB, as might at first be hoped.
As a matter of fact, the input amplitude A is filtered by the polarizer at the input, and its parallel
component A1 is transmitted while its crossed component A2 is attenuated (Figure 3.31). The
primary component A1 is split and propagates in opposite directions along the ring
interferometer and yields at the output two waves having, respectively, an amplitude component
A11 and A11′ in the transmitted state of the polarizer, and a component A12 and A12′ in the
crossed state:
The two primary components A11 and A11′, which are in the same state at the output as A1 at the
input, interfere in phase (in absence of rotation) and with the same modulus ⎪A11⎪ = ⎪A11′⎪
because of reciprocity.
The two secondary components A12 and A12′ that have been coupled in the crossed mode have
the same modulus (⎪A12⎪ = ⎪A12′⎪) but they have a spurious phase difference ∆ϕ 12 since the
waves have not followed exactly the same path in opposite direction.
Similarly, the input crossed component A2 yields two amplitude components A22 and A22′ in
the same crossed state and two components A21 and A21′, which correspond to light that has been
coupled back in the transmitted state:
The two components A22 and A22′ are also in phase because of reciprocity, but because they are
attenuated twice (at the input and at the output), these terms may be ignored.
The two secondary components A21 and A21′ also have the same modulus, but they also interfere
with a spurious phase difference ∆ϕ 21, which is equal to −∆ϕ 12 because of reciprocity of
crossed coupling (see Section 3.2.1).
Figure 3.31 Polarization coupling in a ring interferometer (for clarity the Aij′ amplitudes are not shown).
These two parasitic couples, (A12, A12′) and (A21, A21′), behave very differently. The first
couple, (A12, A12′), is cross-polarized with respect to the main couple, (A11, A11′), and their
intensities are simply added. Because the main interference signal is 2⎪A11⎪2sin∆ϕ R and the
spurious signal is 2∈2⎪A12⎪2sin(∆ϕ R + ∆ϕ 12), a phase error ∆ϕ e is yielded in the total signal:
⎪∆ϕ e⎪ < ∈2 ρcr (3.24)
where ρcr = ⎪A12⎪2/⎪A11⎪2 is the polarization intensity cross-coupling in the ring
interferometer. This error term is reduced by the intensity rejection ratio ∈2 of the polarizer.
However, the second couple, (A21, A21′), has the same state of polarization as the main couple,
(A11, A11′), and there is interference between (A11 + A21) and (A11′ + A21′) instead of between
A11 and A11′. A phasor diagram (Figure 3.32) shows that it yields a phase error ∆ϕ e′ bound by:
(3.25)
Note that its exact value depends on ∆ϕ 21, but also on the phase difference between the primary
components, (A11, A11′), and the spurious components, (A21, A21′).
It is also possible to consider that the main interference wave, ⎪A11 + A11′⎪2, serves as a local
oscillator for the coherent detection (see Section A.8.1) of the spurious signal, ⎪A21 + A21′⎪2
[34]. Using ρm = ⎪A2⎪2/⎪A1⎪2 to represent the intensity polarization ratio at the input, we have
and:
(3.26)
This second error term is reduced only by the amplitude rejection ratio ∈ instead of ∈2 in the
first case, which makes a big difference, as ∈ is much smaller than unity.
The first error term, ∆ϕ e, is often classified as an intensity-type error, while the second one,
∆ϕ e′, is classified as an amplitude-type error. Both effects may be further decreased with
polarization control to optimize the polarization alignment at the input and at the output, which
reduces ρin and ρr; however, this improvement is limited, as it is difficult to limit their value
below 10–2 in the long term.
Figure 3.32 Phasor diagram of the nonreciprocity induced by A21 and A21′.
3.4.2 Use of Polarization-Maintaining (PM) Fiber
In addition to the problem of lack of polarization rejection, the use of ordinary fiber requires
polarization control to avoid signal fading, which increases the complexity and the overall
dimensions of the system. The development of high-quality polarization-maintaining (PM) fibers
(see Section B.6) was an important step toward a compact and practical device. Singlepolarization or polarizing (PZ) fibers would be even better in theory, but they have to be used
with great care to get good rejection and avoid bending loss at the same time, which does not
make them very practical, except for making an all-fiber polarizer at the common input-output
port.
Nevertheless, the conservation of polarization in polarization-maintaining fibers used in coils
or with in-line components is typically in the 20-dB range, that is, the intensity ratio ρin and ρr
are still on the order of 10–2, which does not relax very much the requirement on polarization
rejection. To limit the phase error to 10–7 rad, this requires a polarizer rejection of about 100 dB,
which is still much too high to be readily achieved in practice. However, polarization-maintaining
fibers are birefringent in principle: the two states of polarization propagate with a slightly
different velocity, and when a broadband source is used, as seen in Section B.6.3 the spurious
crossed-state waves are losing their coherence with the main primary signal, which brings a
major reduction of the actual phase error signal, even with a medium rejection of the polarizer.
This applies mainly to the amplitude-type error, since it is coming from a coherent detection
process, but also to the intensity-type term through averaging effects.
As already stated, a broadband source is very advantageous in an interferometric fiber
gyroscope, and its influence on birefringence-induced phase error is crucial for good
performance. This very important point will be discussed in detail in Chapter 5.
3.4.3 Use of Depolarizer
If a polarization-maintaining fiber is obviously the optimal technical choice, its potential for very
significant cost reduction with mass production is not fully accepted by the entire community of
fiber gyro developers, and some teams still prefer to use an ordinary fiber. The problem of
signal fading is solved with the use of Lyot depolarizers [35, 36] (see Section B.7.4), which yield
a random state of polarization that is evenly distributed over all the possible states. When it is
filtered by a polarizer, half the power is always transmitted, independent of the effect of the
propagation along a fiber.
A first depolarizer has to be used at the input before the polarizer and a second one in the coil.
There is no more signal fading, but 50% of the power is lost through the polarizer at the input
and again at the output, which induces an additional loss of 6 dB. This degrades the signal-tonoise ratio that depends on the returning power, but there is another important drawback: this
does not take full advantage of the relaxation of the requirement for the polarizer rejection
brought by polarization-maintaining fiber. The comparison between both approaches will be
presented in Chapter 5.
3.4.4 Use of an Unpolarized Source
An unpolarized source has a completely random state of polarization. It is possible to consider it
the combination of two incoherent sources with orthogonal states of polarization. With a fiber
gyro we can use the same analysis as in Section 3.4.1, but now the two components A1 and A2 are
incoherent. At the output, there is no more coherent detection of the spurious term, (A21, A21′),
coming from A2 with (A11, A11′), coming from A1, which was serving as a local oscillator. This
(A21, A21′) term still yields its own intensity-type effect, as does the term, (A12, A12′); however,
we have seen that the two spurious phase differences are opposite, ∆ϕ 12 = −∆ϕ 21, and as their
power is balanced because each parasitic wave passes once through the polarizer [at the input for
(A21, A21′) and at the output for (A12, A12′)], the two effects complement each other, which
cancels out the total error signal.
In theory, this could even avoid the use of a polarizer as analyzed by Pavlath and Shaw [12], but
this requires that the light remains perfectly unpolarized, which implies that there is no
differential loss (PDL for polarization dependent loss) in the propagation or in the components.
As stated earlier, in the general multimode case, it is easier in practice to eliminate the crossed
polarization with a rejection of 10–x than to ensure that the differential loss is smaller than 10–x,
particularly in components.
In any case, the effect of an unpolarized source and that of the polarizer are cumulative, and this
applies to any scheme, particularly to the use of polarization-maintaining fiber coil. A real source
has a certain degree of polarization P defined with the respective intensities of its polarized
components. For example, a super-luminescent diode (SLD), which is a very popular broadband
source for the fiber gyro, emits partially unpolarized light, with the most powerful polarization
being parallel to the junction. It can be regarded as the sum of two incoherent sources with a
power P// for the parallel polarization and P⊥ for the perpendicular polarization. In a gyro, it
must be regarded as the sum of a perfectly unpolarized source of intensity 2P⊥ and of a perfectly
polarized source of intensity P// - P⊥ (Figure 3.33), the degree of polarization being:
(3.27)
The perfectly polarized component of power P// - P⊥ has to be analyzed according to the
previous explanation of Section 3.4.1, while the perfectly unpolarized component of intensity
2P⊥ does not yield any spurious birefringence-induced phase error. This shows that an
unpolarized source is very advantageous for reducing the phase error caused by a lack of
polarization rejection, although it has the drawback of an additional 3 dB of loss because half the
power is attenuated through the polarizer at the input.
Figure 3.33 Decomposition of a partially unpolarized source.
It also simplifies the system, because the fiber pigtail and the source coupler that send the light
to the polarizer of the ring interferometer do not have to preserve polarization, because
unpolarized light remains unpolarized when it propagates in ordinary single-mode fiber.
As we see in Sections A.2.4 and B.9, unpolarization is one of the key advantages of erbium ASE
source derived from erbium amplifiers (EDFAs) of telecoms.
References
[1] Graindorge, P., et al., “Forced Reciprocity Using Phase Conjugation,” Fiber-Optic Rotation Sensors, Springer Series in Optical
Sciences, Vol. 32, 1982, pp. 368–374.
[2] Sagnac, G., “L’éther lumineux démontré par l’effet du vent relatif d’éther dans un interféromètre en rotation uniforme,” Comptes
rendus de l’Académie des Sciences, Vol. 95, 1913, pp. 708–710.
[3] Sagnac, G., “Sur la preuve de la réalité de l’éther lumineux par l’expérience de l’interférographe tournant,” Comptes rendus de
l’Académie des Sciences, Vol. 95, 1913, pp. 1410–1413.
[4] Fizeau, H., “Sur les hypothèses relatives à l’éther lumineux, et sur une expérience qui parait démontrer que le mouvement des
corps change la vitesse avec laquelle la lumière se propage dans leur intérieur,” Comptes rendus de l’Académie des Sciences,
Vol. 33, 1851, pp. 349–355.
[5] Lefèvre, H., “Wavelength Tuneable Laser Source,” U.S. Patent # 6,519,269 B1, 2003.
[6] Vali, V., and R. W. Shorthill, “Fiber Ring Interferometer,” Applied Optics, Vol. 15, 1976, pp. 1099–1100.
[7] Ulrich, R., “Fiber-Optic Rotation Sensing with Low Drift,” Optics Letters, Vol. 5, 1980, pp. 173–175.
[8] Arditty, H., M. Papuchon, and C. Puech, “Ring Interferometer Device and Its Application to the Detection of Non-Reciprocal
Effects,” U.S. Patent # 4,480,915, Foreign Application Priority of December 1979, 1984.
[9] Ulrich, R., and M. Johnson, “Fiber-Ring Interferometer: Polarization Analysis,” Optics Letters, Vol. 4, 1979, pp. 152–154.
[10] Ulrich, R., “Polarization and Depolarization in the Fiber-Optic Gyroscope,” Fiber-Optic Rotation Sensors, Springer Series in
Optical Sciences, Vol. 32, 1982, pp. 52–77.
[11] Pavlath, G. A., and H. J. Shaw, “Multimode Fiber Gyroscope,” Fiber-Optic Rotation Sensors, Springer Series in Optical
Sciences, Vol. 32, 1982, pp. 364–367.
[12] Pavlath, G. A., and H. J. Shaw, “Birefringence and Polarization Effects in Fiber Gyroscopes,” Applied Optics, Vol. 21, 1982,
pp. 1752–1757.
[13] Davis, W. C., W. L. Pondrom, and D. E. Thompson, “Fiberoptic Gyro Using Magneto-Optic Phase-Nulling Feedback,” FiberOptic Rotation Sensors, Springer Series in Optical Sciences, Vol. 32, 1982, pp. 308–315.
[14] Martin, J. M., and J. T. Winkler, “Fiber-Optic Laser Gyro Signal Detection and Processing Technique,” SPIE Proceedings, Vol.
139, 1978, pp. 98–102.
[15] Bergh, R. A., H. C. Lefèvre, and H. J. Shaw, “All-Single-Mode Fiber-Optic Gyroscope with Long-Term Stability,” Optics
Letters, Vol. 6, 1981, pp. 502–504.
[16] Lefèvre, H. C., “Comments About Fiber-Optic Gyroscopes,” SPIE Proceedings, Vol. 838, 1987, pp. 86–97.
[17] Kiesel, E., “Impact of Modulation Induced Signal Instabilities on Fiber-Gyro Performances,” SPIE Proceedings, Vol. 838,
1987, pp. 129–139.
[18] Ezekiel, S., and H. J. Arditty, “Fiber-Optic Rotation Sensors,” Fiber-Optic Rotation Sensors, Springer Series in Optical
Sciences, Vol. 32, 1982, pp. 2–26.
[19] Youngquist, R. C., L. F. Stokes, and H. J. Shaw, “Effects of Normal Mode Loss in Dielectric Waveguide Directional Couplers
and Interferometers,” Journal of Quantum Electronics, Vol. QE 19, 1983, pp. 1888–1896.
[20] Yajima, H., “Dielectric Thin-Film Optical Branching Waveguide,” Applied Physics Letters, Vol. 22, 1973, pp. 647–649.
[21] Arditty, H. J., M. Papuchon, and C. Puech, “Reciprocity Properties of a Branching Waveguide,” Fiber-Optic Rotation Sensors,
Springer Series in Optical Sciences, Vol. 32, 1982, pp. 102–110.
[22] Izutzu, M., Y. Nakai, and T. Sueta, “Operating Mechanism of the Single-Mode Optical Waveguide Y Junction,” Optics Letters,
Vol. 7, 1982, pp. 136–138.
[23] Lefèvre, H. C., R. A. Bergh, and H. J. Shaw, “All-Fiber Gyroscope with Inertial-Navigation Short-Term Sensitivity,” Optics
Letters, Vol. 7, 1982, pp. 454–456.
[24] Bergh, R. A., H. C. Lefèvre, and H. J. Shaw, “Single-Mode Fiber-Optic Polarizer,” Optics Letters, Vol. 5, 1980, pp. 479–481.
[25] Varnham, M.P., D. N. Payne, and E. J. Tarbox, “Coiled Birefringent Fiber Polarizer,” Optics Letters, Vol. 9, 1984, pp. 306–
308.
[26] Lefèvre, H. C., “Single-Mode Fibre Fractional Wave Devices and Polarization Controllers,” Electronics Letters, Vol. 16,
1980, pp. 778–780.
[27] Davies, D. E. N., and S. A. Kingsley, “Method of Phase Modulating Signals in Optical Fibres: Application to Optical Telemetry
Systems,” Electronics Letters, Vol. 10, 1974, pp. 21–22.
[28] Papuchon, M., and C. Puech, “Integrated Optics: A Possible Solution for the Fiber Gyroscope,” SPIE Proceedings, Vol. 157,
1978, pp. 218–222.
[29] Arditty, H. J., et al., “Integrated-Optic Fiber Gyroscope: Progresses Towards a Tactical Application,” Proceedings of OFS
2’84, Stuttgard, VDE Verlag, 1984, pp. 321–325.
[30] Lefèvre, H. C., et al., “Progress in Optical Fiber Gyroscopes Using Integrated Optics,” AGARD-NATO Proceedings, Vol.
CCP-383, 1985, pp. 9A1–9A13.
[31] Lefèvre, H. C., et al., “Integrated Optics: A Practical Solution for the Fiber-Optic Gyroscope,” SPIE Proceedings, Vol. 719,
1986, pp. 101–112.
[32] “Fiber Optic Gyros: 15th Anniversary Conference,” SPIE Proceedings, Vol. 1585, 1991.
[33] Suchosky, P. G., T. K. Findakly, and F. L. Leonberger, “LiNbO3 Integrated Optical Components for Fiber-Optic Gyroscopes,”
SPIE Proceedings, Vol. 993, 1988, pp. 240–243.
[34] Kintner, E. C., “Polarization Control in Optical-Fiber Gyroscopes,” Optics Letters, Vol. 6, 1981, pp. 154–156.
[35] Fredricks, R. J., and R. Ulrich, “Phase-Error Bounds of Fibre Gyro with Imperfect Polarizer/Depolarizer,” Electronics Letters,
Vol. 20, 1984, pp. 330–332.
[36] Böhm, K., et al., “Low-Drift Fibre Gyro Using a Superluminescent Diode,” Electronics Letters, Vol. 17, 1981, pp. 352–353.
CHAPTER 4
Backreflection and Backscattering
4.1 Problem of Backreflection
4.1.1 Reduction of Backreflection with Slant Interfaces
Reciprocity applies only to transmitted waves, and in the early days of fiber gyro research, the
first problem encountered was the 4% Fresnel backreflection (see Section A.6.1) at the air-silica
interface of the fiber coil ends, which superimposes a parasitic Michelson interferometer. This
was solved by polishing the ends at a sufficient slant angle to avoid backward reflection and
aligning them according to refraction laws so as not to degrade the throughput coupling, as
analyzed by Arditty et al. [1] (Figure 4.1).
Figure 4.1 Suppression of backreflection at the fiber ends with a slant angle.
Antireflection coating cannot provide a low enough level of backreflected light. With an allfiber approach, this problem of index mismatch is avoided, but this method is still used at the
interface between silica fiber and the LiNbO3 integrated optic circuit [2] (Figure 4.2), which has
been found to be the optimal technological approach, as seen earlier.
Figure 4.2 Suppression of backreflection between a fiber and an integrated optic waveguide.
To evaluate the reduction of backreflection as a function of the slant angle, it is necessary to
consider that it is equivalent to the coupling between the real waveguide and its virtual mirror
image through the interface plane (Figure 4.3).
Figure 4.3 Estimation of the backreflection with an image fiber.
The throughput coupling loss Γ θ between two identical waveguides with an angular
misalignment θm is analyzed in Section B.4.2, assuming that the mode is pseudo-Gaussian.
It yields:
(4.1)
where θD′ is the full divergence angle at 1/e2 in power of the mode in an index-matched medium:
θD′ = θD/n
θD ≈ 1.5 NA (4.2)
and where θD is the divergence in a vacuum, n is the medium index, and NA is the fiber numerical
aperture. Because θm is twice the slant angle θs, the additional loss of backreflection may be
estimated as:
(4.3)
The almost standard slant angle of the integrated optic circuit is 10°, which implies that the slant
angle of the fiber is 15° (i.e., 0.26 rad), to align both waveguides according to refraction law:
nSiO2sin15° = nLiNbO3sin10° (4.4)
with nSiO2 ≈ 1.45 and nLiNbO3 ≈ 2.2. Single-mode fibers used in the fiber gyro have a typical
numerical aperture NA = 0.17; therefore, the attenuation Γ br of the backreflection is:
Γ br = Γ F + Γ θ (15°) (4.5)
With 4% between LiNbO3 and SiO2, the Fresnel backreflection Γ F is 14 dB, and computing Γ θ
it is found that Γ br (15°) = 165 dB.
However, the validity of the pseudo-Gaussian approximation of the mode shape is lost in the
tails, and the formula of Γ θ applies only for small angular misalignments. Furthermore, spurious
scattering at the interface increases the backreflection, but, in practice, the standard 10°/15°
combination at the integrated optic interface is perfectly adequate to reduce the backreflection
below 80 to 90 dB without decreasing the throughput coupling, compared to the case of
perpendicular interfaces.
4.1.2 Influence of Source Coherence
Assuming two backreflection points at the ends of the fiber coil (Figure 4.4), this actually yields
six waves at the output of the ring interferometer:
Figure 4.4 Explanation of phase error induced by two backreflections.
The two primary transmitted reciprocal counterpropagating waves that have exactly the same
amplitude A and A′ (same modulus and same phase);
Two waves backreflected at the inputs with different amplitudes A1 and A1′;
Two waves backreflected at the outputs with different amplitudes A2 and A2′.
If these waves are coherent, there is interference at the output between (A + A1′ + A2′) and (A′ +
A1 + A2) instead of only A and A′. Using once again a phasor diagram, it is shown that the phase
difference error may depend on the amplitude ratio between the primary waves and the
backreflected waves. Limiting this error below 10–8 rad would require an attenuation of at least
160 dB of the backreflected waves.
The path difference between the primary waves and these spurious backreflections is
considerable, because it corresponds to the whole optical length of the coil (i.e., hundreds of
meters to kilometers), and most sources are not coherent enough. In practice, there is a
superposition of three independent interferometers:
The main Sagnac interferometer with A and A′;
Two spurious Michelson interferometers with, respectively, (A1, A1′) and (A2, A2′)
The spurious Michelson signals are simply added in optical intensity to the Sagnac signal,
instead of amplitude, and an attenuation of 80 dB is then good enough to get 10–8 rad instead of
160 dB.
Furthermore, if a source of particularly short coherence length is used, it becomes possible to
reduce the contrast of these parasitic Michelson interferometers by unbalancing their arms. In
particular, it is advantageous to get the slant angle of the integrated optic interface with a
parallelogram substrate shape [3]. With a typical separation of 200 μm between the branch
waveguides of the Y-junction and a slant angle of 10°, the optical path length unbalance ∆Lunb of
the spurious Michelson interferometer is (Figure 4.5):
∆Lunb = 2nLiNbO3 200 μm tan 10° = 150 μm (4.6)
while the decoherence length Ldc of broadband sources like superluminescent diodes is on the
order of 30 to 50 µm.
Figure 4.5 Unbalancing of the parasitic Michelson interferometers with a parallelogram-shaped circuit.
An erbium ASE source at 1,550 nm has a longer decoherence length (typically 200 μm) and it
is then useful to increase the separation between the branch waveguides or the slant angle.
This analysis of backreflection is a good example of the way parasitic effects related to
coherence are suppressed in an interferometric fiber gyro, which involves three steps:
1. The spurious waves have to be reduced as much as possible with a reasonably simple
technological approach. The slant interface is a good solution in the case of backreflection.
2. The coherence between the primary waves and the spurious waves has to be destroyed to avoid
direct coherent detection that is very harmful.
3. Some further improvements may be obtained if the coherence between the spurious waves is also
destroyed.
4.2 Problem of Backscattering
4.2.1 Coherent Backscattering
Propagation in a material suffers various kinds of scattering processes. In a fiber, the light that is
scattered inside the numerical aperture of the fiber remains guided in the forward and backward
directions. In a fiber gyro, problems arise with Rayleigh backscattering, which preserves the
same optical frequency as the throughput wave and can be considered as a randomly distributed
backreflection. The backscattered power in a single-mode fiber is typically 10–7/m at 1,550 nm
yielding typically 40 dB of backscattering light for a 1-km coil. With a CW coherent source, this
may yield a spurious phase difference error as high as 10–2 rad because of the coherent detection
process, with the primary waves acting as a local oscillator as analyzed by Cutler et al. [4].
Furthermore, because the path differences are not stable, this phase error fluctuates inducing
noise. To reduce this effect, it was proposed that additional frequency modulation [5] or phase
modulation [6, 7] be used to displace this noise outside of the detection bandwidth.
4.2.2 Use of a Broadband Source
This reduction may be obtained much more efficiently with an adequate source. It was first
proposed to use a pulsed source: in that case, only the light backscattered from the middle of the
coil (within the pulse length Lp) will interfere with the primary pulses (Figure 4.6) [4]. However,
a broadband source of a short coherence length yields a similar reduction [6, 7], considering that
the pulses are replaced by wave trains. All these spurious signals arise from parasitic
interferometers with various path differences, and, as detailed in Section A.12.3, the interference
contrast is proportional to the autocorrelation function of the source, which is the Fourier
transform of the power (or intensity) spectrum. A pulse has an amplitude spectrum with all the
frequency components in phase, while an incoherent broadband source has an amplitude spectrum
with all the frequency components in a random phase, but the phase information is lost in the
autocorrelation process involved in the interferences and both pulsed and incoherent sources
reduce the contrast of the spurious interferometers because of their narrow autocorrelation
function.
Figure 4.6 Limitation of coherent backscattering with a pulsed source.
However, a broadband source is more advantageous, because its autocorrelation width, called
in optics its coherence length, is much narrower in practice (20 to 200 μm) than that of a pulse.
An equivalent pulse duration should be in the range of 0.1 to 1 picosecond. Furthermore, for the
same mean power, a pulse has a much higher peak power, which induces nonlinearities, as will be
seen later in Section 7.3.
4.2.3 Evaluation of the Residual Rayleigh Backscattering Noise
Rayleigh scattering is caused by dipolar antenna radiation of the material atomic bindings excited
by the incoming wave. It has a λ–4 dependence in wavelength (it is why the sky is blue), and there
is a cumulative effect only in fluids or amorphous glasses that have a random structure. Inversely,
with an ordered crystal matrix, the various scattered waves would interfere destructively except in
the forward direction.
In a fiber, part of the scattered light remains guided in the forward and backward directions.
The forward scattering is reciprocal and does not yield any spurious effect in the fiber gyro. On
the contrary, the backward scattering behaves like a randomly distributed backreflection. With a
Rayleigh scattering attenuation αR and considering a source with a coherence length Lc, the
coherent backscattered power Pcb coming from the middle of the coil is:
(4.7)
where Pm is the intensity of the main wave and S is the recapture factor, which may be grossly
approximated as the ratio of the acceptance solid angle of the fiber, about equal to NA2, to a full
solid angle of 4π steradian. In practice, S is about 10–3.
Taking practical numerical values αR = 2 dB/km (for λ = 850 nm) and Lc = 20 μm, giving:
which still yields an amplitude ratio:
(4.8)
Assuming two main counterpropagating amplitudes Am and Am′, there are interferences
between (Am + Acb′) and (Am′ + Acb) at the output. It seems at first that the phase error could be as
high as 6 × 10–6 rad, even with a coherence length as small as 20 μm. However, this is not taking
into account the benefit brought by the biasing modulation at the proper frequency [8] and the fact
that the phase difference between Am and Acb′ is correlated with the one between Am′ and Acb [9].
As a matter of fact, when the phase modulator is operated at the proper frequency fp of the coil,
a wave backscattered from the middle of the coil experiences the phase modulation twice, but with
a half-period delay which cancels out the total modulation (Figure 4.7). Furthermore, the main
interference wave resulting from Am + Am′ is modulated in intensity but is not modulated in
phase; therefore, the coherent detection process between (Am + Am′) as the local oscillator and
Acb and Acb′ remains at low frequencies and does not yield any spurious signal in the
demodulation at fp [8].
Figure 4.7 Effect of a phase modulator on the waves backscattered in the middle of the coil.
Some benefit also comes from the correlation between the phases of both backscattered waves.
Let us consider a scattering point: if the phase accumulated in the propagation by the main
primary waves is ϕ p, the phases ϕ cb and ϕ cb′ due to the propagation of the two waves
backscattered at a given point, are such as:
(4.9)
and therefore the phase differences due to propagation between the main and backscattered waves
are opposite:
(4.10)
However, the scattering process adds a π/2 phase lag on each spurious wave, and the actual
phase differences are [9]:
(4.11)
Using again a phasor diagram (Figure 4.8), it can be seen that if only the propagation is taken
into account, the phase errors carried in each direction are opposite, and therefore the error on
the phase difference would be doubled. However, with the additional π/2 phase lag, the phase
errors ϕ e and ϕ e′ are actually:
Figure 4.8 Phasor diagram of the main primary amplitudes Am and Am ′, and the amplitudes Acb and Acb′ backscattered in the
middle of the coil: (a) effect of propagation and (b) additional effect of the π/2 phase lag.
(4.12)
(4.13)
As sinϕ cb′ = sin(−π − ∆ϕ ch) = sin∆ϕ cb, both phase errors become equal when the moduli
⎪Acb′⎪ and ⎪Acb⎪ are equal. Because on the beamsplitter one backscattered wave is transmitted
twice while the other is reflected (or coupled) twice, these moduli are equal when the splitter is
precisely 50-50 (assuming that the loss is symmetrical in the coil), which eliminates the effect of
Rayleigh backscatter.
To summarize, the low coherence length of the broadband source preferably used in an
interferometric fiber gyro decreases considerably the Rayleigh backscattering noise, and a
further reduction is obtained with a biasing modulation at the proper frequency and with a precise
50-50 separation in the coil splitter. Other reduction techniques remain possible if a longer
coherence length is used [10], but they are difficult to reduce to practice.
In any case, the reduction of Rayleigh backscattering noise is reasonably demanding in terms
of short coherence length, compared to other effects, and a multimode laser diode is sufficient to
get low noise. This spurious effect does not require a coherence length as short as that of a
superluminescent diode or an erbium ASE source (see Section 9.2).
References
[1] Arditty, H. J., et al., “Re-Entrant Fiber-Optic Approach to Rotation Sensing,” SPIE Proceedings, Vol. 157, No. 2978, pp. 138–
148.
[2] Lefèvre, H. C., et al., “Progress in Optical Fiber Gyroscopes Using Integrated Optics,” AGARD-NATO Proceedings, Vol. CPP383, 1985, pp. 9A1–9A13.
[3] Lefèvre, H. C., “Comments About Fiber-Optic Gyroscopes,” SPIE Proceedings, Vol. 838, 1987, pp. 86–97.
[4] Cutler, C. C., S. A. Newton, and H. J. Shaw, “Limitation of Rotating Sensing by Scattering,” Optics Letters, Vol. 5, 1980, pp.
488–490.
[5] Davis, J. L., and S. Ezekiel, “Closed-Loop, Low-Noise Fiber-Optic Rotation Sensor,” Optics Letters, Vol. 6, 1981, pp. 505–
507.
[6] Böhm, K., et al., “Low-Noise Fiber-Optic Rotation Sensing,” Optics Letters, Vol. 6, 1981, pp. 64–66.
[7] Bergh, R. A., H. C. Lefèvre, and H. J. Shaw, “All-Single-Mode Fiber-Optic Gyroscope,” Optics Letters, Vol. 6, 1981, pp. 198–
200.
[8] Lefèvre, H. C., R. A. Bergh, and H. J. Shaw, “All-Fiber Gyroscope with Inertial-Navigation Short-Term Sensitivity,” Optics
Letters, Vol. 7, 1982, pp. 454–456.
[9] Takada, K., “Calculation of Rayleigh Backscattering Noise in Fiber-Optic Gyroscopes,” J.O.S.A., Vol. A2, 1985, pp. 872–877.
[10] Mackintosh, J. M., and B. Culshaw, “Analysis and Observation of Coupling Ratio Dependence of Rayleigh Backscattering
Noise in a Fiber-Optic Gyroscope,” Journal of Lightwave Technology, Vol. 7, 1989, pp. 1323–1328.
CHAPTER 5
Analysis of Polarization Nonreciprocities with
Broadband Source and High-Birefringence
Polarization-Maintaining Fiber
5.1 Depolarization Effect in High-Birefringence PolarizationMaintaining Fibers
The principle of polarization maintaining (PM) fibers (see Section B.6) is to create a strong
birefringence that splits the polarization-mode degeneracy of ordinary single-mode fiber. When
light is coupled in one principal (or eigen) polarization state, it will remain mostly in this input
state, as the induced birefringence is much higher than the spurious defects and yields phase
mismatch.
The behavior of polarization-maintaining fibers may be described with two orthogonal
principal polarization modes and random coupling points, which transfer a small amount of light
from one mode to the other. The mean rate of power transfer is called the h-parameter of the
fiber as analyzed by Kaminow [1], “h” standing for holding, with:
dP⟘/dL = h ⋅ P// (5.1)
where P// is the power of the primary input polarization mode and dP⟘ is the power transferred
in crossed mode over the elementary length dL. One calls the polarization extinction ratio (PER):
A typical value of h is 10–5 m–1 (a polarization conservation of 20 dB over 1 km).
The most popular polarization-maintaining fibers used in a fiber gyro are based on linear
birefringence induced by an additional stressing structure in the cladding. They conserve a linear
state of polarization when this state is coupled at the input along one principal axis of fiber
birefringence.
In a fiber gyro, this polarization conservation maintains most of the power in the primary
reciprocal wave, avoiding signal fading. Only a small part is transferred in the crossed
nonreciprocal wave, and, in addition, when a broadband source is used, it gets depolarized
because it propagates at a different velocity compared to the main mode, as outlined by Rashleigh
et al. [2]. The propagation of a broad-spectrum wave may be analyzed by considering that it is
composed of wave trains with a length equal to the decoherence length Ldc. As described in
Section A.12, the coherence function is a bell-shaped function without strict width boundary. The
definition of the coherence length varies depending on publications. In this book, we use the
classical Born and Wolf definition for the coherence length Lc [3], which is actually an rms half-
width or half-width at 1σ: this is the length that preserves a good interference contrast. We prefer
to call the decoherence length Ldc, the other usual definition based on the simple formula λ2/
∆λFWHM, where ∆λFWHM is the full width at the half-maximum of the source intensity
spectrum. Ldc is actually a half-width at about 4σ: this is the length that reduces the contrast down
to a few percentages. Note that some authors are using a third definition for a Gaussian coherence
function: the half-width at 1/e, which is actually a half-width at 2σ.
When there is a crossed-polarization coupling point, the main and crossed wave trains happen
to lose their overlap (there is a phase mismatch) because of their velocity difference (Figure 5.1),
which yields statistical decorrelation between both crossed polarizations, called depolarization.
Figure 5.1 Propagation of a wave train in high-birebringence fiber.
This is obtained after a propagation length, called the depolarization length Ld [2], and that is
defined with:
Ld/Ldc = 1/∆nb = Λ/λ (5.2)
where ∆nb is the birefringence index difference and Λ is the fiber birefringence beat length.
As explained in Section B.6.3, this simple formula applies only if the birefringence index
difference ∆nb is wavelength-independent, which is the case with stress-induced birefringence
fiber commonly used in the fiber gyro. In the most general case, dispersion has to be taken into
account and there is Ld/Ldc = 1/[∆nb + ω(d(∆nb)/dω)] [2], which can be simply expressed with
the difference of the group index ∆nbg instead of the phase index ∆nb:
Ld/Ldc = 1/∆nbg
For example, an SLD at 850 nm has a typical spectrum width ∆λFWHM of 20 nm, which gives a
decoherence length:
Ldc = 35 μm
and a typical ∆nb value is 5 × 10–4; therefore, the depolarization length is:
Ld = 7 cm
An erbium-doped-fiber ASE source at 1,550 nm has a spectral width of about 15 nm, yielding:
Ldc = 150 μm and Ld = 30 cm
However, this depolarization is not completely equivalent to a naturally unpolarized light.
Unpolarized light remains perfectly unpolarized independently of the birefringence of the
propagation medium (if the attenuation is not polarization-dependent): two orthogonal
polarizations are always statistically decorrelated. With depolarization of a broad spectrum with a
birefringent propagation, a good correlation may be restored by compensating for the delay. For
example, if, after a propagation in a birefringent medium, light propagates along the same length
of birefringent medium, but the principal axes have been rotated by 90°, both polarization modes
propagate half the way in the fast mode and half the way in the slow mode: they are depolarized in
the middle, but they get polarized again at the end, as the first delay is compensated for by the
second one.
The whole analysis of the polarization problem in the fiber gyro aims at avoiding such
compensation to take full advantage of this depolarization effect which drastically reduces
nonreciprocities.
5.2 Analysis of Polarization Nonreciprocities in a Fiber
Gyroscope Using an All-Polarization-Maintaining
Waveguide Configuration
5.2.1 Intensity-Type Effects
Let us first consider a ring interferometer using a polarization-maintaining fiber or waveguide
with a perfect input; that is, the state of polarization of the light source is maintained at the input in
the same mode, mode 1, and, in particular, is aligned on the transmission mode of the polarizer.
When there is one polarization coupling point M in the coil, the input wave train of the broadband
source yields at the output two primary wave trains that are still in mode 1 and that are perfectly
in phase because of reciprocity. There are also two coupled wave trains in mode 2 that have
propagated along different paths [Figure 5.2(a)]. This should induce a phase difference, but
because of their short coherence length, they do not overlap and these spurious interferences lose
their contrast as analyzed by Burns and Moeller [4] (except for a coupling in the middle of the
coil, which may be disregarded for the moment).
If there are two coupling points M and M′ placed at about the same distance from the splitter
[Figure 5.2(b)], the cross-coupled wave trains overlap and interfere [5, 6]. This is similar to the
problem of backreflection, in which one reflecting point is not harmful but two symmetrical
reflection points yield a spurious Michelson interferometer. There are actually six wave trains:
Figure 5.2 Propagation of a wave train in a polarization-preserving ring interferometer: (a) with one crossed-coupling point M;
and (b) with two symmetrical crossed-coupling points M and M′.
Two primary reciprocal wave trains with the same phase accumulated in the propagation along
the length L of the coil:
where β1 = 2π ⋅ n1/λ is the propagation constant of mode 1.
Two secondary wave trains that have propagated in mode 1 between the splitter and the coupling
points and that have been coupled in mode 2 for the rest of the propagation. Their respective
phases are:
ϕ 122 = β1LM + β2(LMM′ + LM′) (5.3)
(5.4)
where β2 is the propagation constant of mode 2, LM is the length between the splitter and M,
LMM′ is the length between M and M′, and LM′ is the length between the splitter and M′.
Two other secondary wave trains that have propagated in mode 1 between the splitter and the
coupling points and also between the two coupling points, and that have been coupled in mode 2
between the coupling points and the splitter at the output. Their respective phases are:
ϕ 112 = β1(LM + LMM′) + β2LM′ (5.5)
(5.6)
Both pairs of secondary wave trains have the same spurious phase difference:
∆ϕ s = (β1 − β2)(LM + LM′) (5.7)
Note that the propagation between the two coupling points does not yield any nonreciprocal
phase difference. The nonreciprocity is only related to the difference in accumulated
birefringence (β1 − β2)(LM − LM′) between the splitter and both symmetrical coupling points.
These two pairs of coherent wave trains are in the crossed mode 2 at the output and are
attenuated by the polarizer. The phase error signal is therefore coming from an intensity-type
effect, and it is reduced by the intensity rejection ratio of the polarizer:
(5.8)
where ρM and ρM′ are the intensity crossed-coupling ratio at M and M′. As ∆ϕ s may vary over
2π, the mean value of this phase error is zero, and its rms deviation is:
(5.9)
Note that there are additional wave trains that have experienced a crossed coupling twice, at the
input and at the output. They come back in the same mode 1 and have the same phase because of
reciprocity:
(5.10)
When there are several pairs of symmetrical coupling points, each pair creates two parasitic
interference waves, and because these waves are not coherent with each other, the spurious
intensity signals are simply added.
It is possible to evaluate the effect of a fiber coil with a simple model [6]. The random coupling
along the coil may be regarded as created by a discrete coupling point Mi for each depolarization
length Ld (Figure 5.3), with an intensity coupling h ⋅ Ld. With a coil length L, there are N = L/Ld
coupling points and N/2 pairs of symmetrical points. Each pair generates two pairs of secondary
wave trains (in each direction) that yield the same phase-bias error; therefore,
∆ϕ ei = 2∈2[hLdsin∆ϕ si] (5.11)
as hLd is the intensity ratio between the main wave trains and the secondary cross-coupled wave
trains. The phase differences ∆ϕ si are randomly spread over 2π and the total effect is the
summation of independent random variables ∆ϕ ei, which are centered about zero. Therefore, the
mean value of the total phase error ∆ϕ e is also zero and the square of its rms deviation is equal to
the sum of the square of the rms deviations of each variable:
(5.12)
We have
because the mean value of sin2∆ϕ si is 1/2; therefore,
Because L = N ⋅ Ld, and:
where h ⋅ L is the total coupling of polarization along the coil. The rms deviation of the
birefringence phase error of a fiber coil is not proportional to the polarization coupling ratio h ⋅
L in the coil, as might at first be thought. It is reduced proportionally to the square root of the
number N of depolarization lengths along the coil [6].
For example, with a typical depolarization length of 10 cm, N is equal to 104 in a 1-km coil.
Because the conservation is typically 20 dB/km, there are 104 couping points of 60 dB that yield
before filtering by the polarizer a total phase error (rms value) of only 10–4 rad, not 10–2 rad. A
polarizer of 40 dB of rejection is good enough to reduce this intensity type bias error below 10–8
rad. This 40-dB value must be compared with the 160 dB that would be required without
polarization conservation and depolarization effect.
Figure 5.3 Simple model of pairs of symmetrical discrete crossed-coupling points M i and M i′ for each depolarization length Ld
This result applies only to couplings that are continuously distributed along the fiber coil, the
random coupling being regarded as a stationary stochastic process. However, in practice, the
main limiting factor is not the coil itself but the localized couplings at the two ends of the coil. As
a matter of fact, in the previous example, the effect of the fiber coil is equivalent to a pair of
symmetrical points with 40 dB of polarization cross-coupling, even if the total conservation of
the coil is only 20 dB. To preserve this benefit, it is desirable to control the polarization crosscouplings of the coil ends at better than this 40-dB value.
With an all-fiber configuration, problems arise with the polarization conservation of the coil
coupler that is typically in the 20- to 30-dB range. There are other parasitic couplings at the
splices between the coupler leads and the coil ends, but it is possible to unbalance the lengths of
both coupler leads. If their dissymmetry is larger than the depolarization length (i.e., typically 10
cm), the coherence between these spurious waves is suppressed.
With an integrated optic circuit, the polarization coupling at the splitter (with a Y-junction in
particular) is negligible, as the strong birefringence of LiNbO3 provides excellent polarization
conservation. Parasitic couplings come from the coil ends butt-coupled on the circuit. A 40-dB
coupling is only a 0.5° misalignment of the principal axes; however, in practice, it is difficult to
have a rugged connection of a fiber without inducing stress that yields locally an additional
birefringence and thus a residual polarization cross-coupling typically equal to 20–25 dB. The
depolarization length in LiNbO3 is much shorter than in a fiber (about 0.7 to 2 mm instead of 10
to 30 cm, as the birefringence index difference is 7 × 10–2 instead of 5 × 10–4), but it is still too
long to get a sufficient imbalance between both branches of the Y-junction. Typical values are a
branch separation of 200 μm and a slant angle of 10° for the parallelogram-shape circuit; that is,
an imbalance of only 200 μm × tan10° = 35 μm compared to a depolarization length of 0.7 to 2
mm.
This shows that it is very important to accurately control the magnitude and the position of the
various localized polarization cross-couplings, even if they seem buried in the total system crosscoupling.
5.2.2 Comment About Length of Depolarization Ld Versus Length of Polarization
Correlation Lpc
As is explained in Section A.12, the contrast or visibility of interference fringes is equal to the
autocorrelation function γc of the centered spectrum. Most problems are usually analyzed,
independently of the exact bell shape of the function, with a half-width of the function called the
coherence time τc or the corresponding coherence length Lc = cτc. As outlined in Section 5.1,
their exact definition is not always precisely defined and it deserves to be recalled more
accurately.
The classical definition [3] is similar to normalized quadratic averaging, using γc as a
probability distribution:
(5.13)
where ∆Lop is the optical path length difference. The coherence length is thus the rms half-width
of γc or its half-width at 1σ, σ being the standard deviation. This definition is the most significant
in classical interferometry, because the problem is usually to define the length that preserves a
good interference contrast.
As we have already seen, the length that suppresses the contrast is also very important: it may
be called the decoherence length Ldc. Because, in theory, the bell-shaped autocorrelation function
reaches zero asymptotically at infinity, it should be infinite, but, in practice, a half-width at 4σ is
significant. This decoherence length Ldc is defined simply as:
(5.14)
where
is the mean wavelength of the broad spectrum and ∆λFWHM is its full width at half
maximum. For this distance, the interference contrast is reduced to few percentages.
When wave trains are used to analyze the effect, it has to be considered that Ldc is their full
width, as the full width of the autocorrelation of a pulse is double the full width of this pulse.
Applying these concepts to the problem of depolarization in a birefringent medium, it seems
more suitable to relate the depolarization length Ld to the decoherence length Ldc, with Ld = Ldc/
∆nb. In particular, to ensure that two crossed-polarization coupling points do not yield any
spurious signal, their imbalance has to be equal or larger than Ld with this last definition. If Lc/
∆nb is used, the signal is still 80% of the perfectly contrasted case. It seems more suitable to call
Lc/∆nb the length of polarization correlation Lpc, with:
Lpc = Ld /4
This is the length of propagation in the birefringent medium, which preserves a good degree of
polarization (i.e., a good statistical correlation between both orthogonal polarization
components).
The case of random coupling along the coil is more delicate. In Section 5.2.1 we considered the
number N of depolarization lengths Ld along the fiber, which yields a safe evaluation of the upper
limit of the phase error. However, we may want to reach a more precise quantification of the
effect, and, in particular, to define the actual half-width that has to be used. In any case, if Lpc
happens to be more suitable than Ld, this multiplies by four the number of averaged elements,
which gives a phase error that is only
times lower than previously found.
It is possible to use a refined model by dividing the fiber coil in elementary length segments Le
that are shorter than the depolarization length Ld but that remain longer than the autocorrelation
width of the stochastic crossed-coupling process in the fiber, which is assumed to be very short
(Figure 5.4). Taking two symmetrical elementary segments, they yield a phase-bias error:
Figure 5.4 Refined model with elementary segments Le.
∆ϕ eii = 2∈2hLesin∆ϕ sii (5.15)
The factor of 2 takes into account the input and output crossed-couplings, and h ⋅ Le is the
intensity crossed-coupling ratio along Le. However, an elementary segment i yields a spurious
signal with its symmetrical segment i, but also with the other segments j, which are around this
symmetrical segment. Compared to the symmetrical case, the error is reduced proportionately to
the autocorrelation function γc of the source:
∆ϕ eij = 2∈2hLeγcijsin∆ϕ sij (5.16)
where γcij is the value of γc for the actual path difference between the two slightly unbalanced
segments. Similar to the analysis of Section 5.2.1, the total phase error ∆ϕ e is a random variable
that is the sum of the independent variables ∆ϕ ij. Because ∆ϕ sij are uniformly spread over 2π,
they are all centered about a zero mean value, and the square of the rms deviation of ∆ϕ e is the
sum of the square of the rms deviation of all these independent variables:
(5.17)
For a given element i, we first have to sum over j. As a matter of fact, the various ∆ϕ eij are
independent, as Le is assumed to be longer than the autocorrelation width of the stochastic crosscoupling process in the fiber, and:
(5.18)
Replacing discrete summation with an integral summation, we have:
(5.19)
We have to introduce a new length of polarization correlation Lpc′ with:
(5.20)
and:
(5.21)
We have now to sum over i (i.e., the L/2Le segments over half the coil), and:
(5.22)
Compared to the analysis of Section 5.2.1, Lpc′ replaces Ld. In the particular case of a Gaussian
autocorrelation, Lpc′ may be computed easily, as
found that:
and
It is
(5.23)
This refined analysis shows that the length Lpc′ that yields the exact number of averaged
elements along the coil has an intermediate value between the depolarization length Ld and the
length of polarization correlation Lpc, but in any case the phase bias error induced by a fiber coil
with a stationary random cross-coupling remains very similar to the simple analysis of Section
5.2.1.
5.2.3 Amplitude-Type Effects
As described in Section 3.4.1, the main source of birefringence nonreciprocity with ordinary
fiber is caused by an amplitude-type effect where light that is misaligned at the input with respect
to the transmission mode of the polarizer is coupled back in the main mode at the output. It
carries a nonreciprocal phase error, but also interferes with the primary waves, which yields an
amplification of the spurious error term with a coherent detection process [7].
The use of birefringent fiber for the coil and also for the couplers and the fiber pigtail of the
source suppresses this effect. As a matter of fact, the primary waves always remain in the same
fast (or slow) mode, and light that is misaligned at the input accumulates some path difference
with respect to the primary waves in the birefringent input lead of the ring interferometer. When it
is cross-coupled in the coil, it cannot compensate for this previous path difference. The
birefringence of the input fiber lead and the one of the base branch of the Y junction destroy the
correlation between the primary input wave that has a polarization aligned with the polarizer and
the crossed input wave: this suppresses (to the first order) the amplitude-type effect [6, 8].
However, if there is no birefringence in the input lead, it is found that cross-coupling in the first
depolarization length of the coil yields an amplitude-type effect, since the crossed wave remains
coherent with the primary wave in this case [4, 9]. Even if the coupling is typically only 60 dB
along a depolarization length, this would require a rejection of 60 dB, assuming an input
misalignment of 20 dB, to limit the amplitude-type phase error below 10–7 rad.
It is seen that the birefringence of the input lead is as important as that of the coil to get full
advantage of the depolarization effect. The need of a birefringent input lead makes integrated
optics very desirable because of the strong natural birefringence of LiNbO3 [6]. It is necessary to
be careful to use the same fast (or slow) mode in the circuit and in the fiber to avoid path
compensation that could restore correlation between the crossed-polarization modes.
This simple analysis assumes that the primary waves are composed of a single wave train that
propagates in front of all the parasitic wave trains if the fast mode is used, or behind if the slow
mode is used, avoiding any coherent detection. A real system is actually more complex because
multiple crossed couplings have to be taken into account. The primary reciprocal waves are
composed of a first main wave train, but there are also additional smaller wave trains coming
from an even number of crossed-couplings along the system. They do not carry any
birefringence-induced phase error, but, contrary to the main primary wave trains, they may be
coherent with a spurious signal and serve as a local oscillator [6]. This effect is much smaller
than if it was the first main wave train, but it should be addressed when the best performance is
looked for.
The amount of even multiple crossed-couplings can be easily measured taking the spectrum of
the polarized light exiting the gyro and calculating by Fourier transform its coherence function.
Figure 5.5(a) displays a typical bell-shaped input spectrum with its coherence function in
logarithmic scale, and one sees a very fast decrease. Now Figure 5.5(b) displays the spectrum of
this same light but after propagation through the PM fiber coil of the gyro and filtering by the
polarizer. One sees a noisy spectrum and the calculated coherence function yields a plateau
instead of the fast decrease in the case of the bell-shaped input light.
Figure 5.5 Spectrum and related coherence function: (a) input light and (b) output light with spectral noise and a plateau of the
coherence function due to even multiple crossed couplings.
This plateau has a typical elevation of 3 × 10−3 and its length corresponds to the maximum
delay between the two orthogonal polarizations going through the coil. With ∆nb = 5 × 10–4, it is
50 cm for a coil length of 1 km.
The residual phase bias defect due to polarization nonreciprocities can be evaluated as the
product of:
The amplitude rejection ratio ∈ of the polarizer;
The amplitude polarization coupling coefficient
of the localized connections between the
fiber coil and the integrated optic circuit;
The elevation of the coherence plateau Cplateau due to the multiple even cross couplings in the
coil;
The degree of polarization Dp of the unpolarized source transmitted through components which
have a residual PDL (polarization dependent loss) or PDC (polarization coupling), as described
in Section 3.4.4.
Typically
Cplateau = 3 × 10–3, and Dp = 3 × 10–2 (PDL or PDC of 0.3
dB); then it requires a polarization rejection:
∈ = 10–2, that is 40 dB
for a bias defect of 10–7 rad needed for medium-grade performance. However, the required
rejection is ∈ = 10–4, that is, 80 dB for a bias defect of 10–9 rad needed for very high
performance.
In practice, a careful analysis has to be performed of the amount of crossed-coupling at the
various defect points (source pigtailing, fiber splices, coupler, integrated optic circuit pigtailing),
the birefringence delay between these points, and the actual coherence of the source to ensure that
any second-order amplitude-type error is sufficiently reduced by the polarizer.
Note that it is possible to further improve the decorrelation effect of the birefringence of the
input lead with an additional modulation of this birefringence as proposed by Carrara et al. [10]. It
can be performed with a phase modulator on the base branch of the Y junction of the MIOC.
However, such a method increases the complexity of the device, and the present polarizer
performance allows a very low phase error to be obtained without this additional modulator.
5.3 Use of a Depolarizer
Polarization-maintaining fiber is the optimal technical choice for high performance, but it
remains possible to use standard nonpolarization-maintaining fiber without polarization control
and without getting signal fading if a Lyot-type depolarizer is inserted into the system [11]. Such a
depolarizer is based on the loss of polarization correlation with propagation of a broadband
source in a birefringent medium (see Section B.7.4). This is similar to the depolarization of
parasitic crossed waves that we saw in Section 5.2, but it is now done on purpose. As a matter of
fact, when a linearly polarized broadband source is aligned at 45° of the principal axes of the
birefringent medium, the input wave train is split evenly along both axes and that, at the output,
the two wave trains do not overlap if the path is longer than the depolarization length Ld. To get
depolarization with any input state of polarization, a Lyot depolarizer is actually composed of a
first birefringent medium with a length LL and an index difference ∆nb, and a second one rotated
by 45° and with a length 2LL twice as large. The wave train is first split unevenly as a function of
the input state of polarization, and each secondary wave train is split again, but now evenly, along
the rotated axis of the second birefringent medium. At the output, the four wave trains do not
overlap, and there is the same decorrelated power along each axis (Figure 5.6) if LL is larger than
Ld. When such a depolarized light is sent through an ordinary fiber, it remains depolarized if the
residual fiber birefringence (particularly bending-induced birefringence, described in Section
B.5.2) does not compensate for the delay given by the depolarizer. Such depolarizers may be
fabricated with short lengths of high-birefringence polarization-maintaining fiber or with
integrated optic waveguides that have a natural crystal birefringence.
A first depolarizer has to be used between the source and the polarizer and a second one has to
be used in the coil, but the immediate drawback of this approach is an additional loss of 3 dB at
the input through the polarizer, and a second loss at the output. Furthermore, depolarization is not
perfect, especially when it is obtained with fibers that always have residual spurious crosscoupling, and extreme care must be taken to avoid path compensation that could restore the
contrast of amplitude-type error signals.
In practice, the use of depolarizers does not yield results that are as good as with polarizationmaintaining fibers. Its main benefit is in terms of the cost of the fiber coil, but, for the whole
system, the cost saving is not obvious because of the increased complexity brought by the
additional depolarizers.
Figure 5.6 Principles of a Lyot depolarizer.
5.4 Testing with Optical Coherence Domain Polarimetry
(OCDP)
5.4.1 OCDP Based on Path-Matched White-Light Interferometry
This analysis of polarization nonreciprocities with polarization-maintaining fibers or with
depolarizers has shown the importance of depolarization effects brought about by birefringence
when a low temporal coherence source is used. However, the location and the magnitude of the
various cross-coupling points have to be determined precisely, even if they seem buried in the
background cross-coupling of the system, to evaluate the residual bias error.
The usual polarization measurement techniques cannot differentiate a series of defects, but this
problem has been solved by applying techniques of path-matched white-light interferometry [12–
14]. By analogy with the well-known method of optical time domain reflectometry (OTDR) [15]
used to test the attenuation of ordinary fibers and its coherence domain counterpart, or OCDR
[16], this technique was called the optical coherence domain polarimetry (OCDP).
The principle of path-matched white-light interferometry is based on the use of a broadband
source (white light being historically the first one available) in an unbalanced interferometer.
When the path imbalance is much longer than the coherence length, the interference contrast
vanishes (see Section A.12). Considering wave trains, the path difference suppresses their overlap
at the output (Figure 5.7). However, passing through a second readout interferometer that has the
same path imbalance, the interference contrast is restored to one-half, because among the four
output wave trains, two overlap, as they have both propagated along one short path and one long
path in each interferometer.
Figure 5.7 Principle of white-light interferometry: propagation of a wave train in a pair of path-matched interferometers.
Now if one splitter in the first interferometer has a very low power-splitting ratio (∈2 and 1 −
∈2 with ∈2 << 1/2), the contrast of the restored interference is twice ∈, the amplitude ratio, and
not ∈2, the intensity ratio (Figure 5.8). The powerful wave train serves actually as a local
oscillator for the coherent detection process (see Section A.8.1) of the low-power wave train. For
example, a splitting ratio of 60 dB thus yields a contrast of 2 × 10–3 and not 10–6.
A polarization cross-coupling point in a high-birefringence fiber is equivalent to this weak
coupling. The main wave train propagates along the primary mode, and the cross-coupled wave
train propagates along the other, the fast mode being equivalent to the short path and the slow
mode being equivalent to the long path. An output polarizer aligned at 45° of the principal axes of
the birefringent fiber is equivalent to the 3-dB recombiner (Figure 5.9). Using a readout
interferometer, particularly a Michelson interferometer, the birefringence path imbalance in the
fiber may be compensated for, yielding a contrast recovery equal to twice the amplitude ratio of
the polarization cross-coupling, which makes this method of OCDP very sensitive. When several
coupling points are located in series along the fiber, they can be measured separately by scanning
one mirror of the Michelson interferometer.
Figure 5.8 White-light interferometry with a very low-power splitting ratio ∈2 of one splitter: the restored interference contrast
is the splitting ratio ∈ in amplitude.
The path difference ∆Lr in the readout interferometer gives the distance Lf between the point
that is analyzed and the fiber end:
∆Lr = ∆nbLf (5.24)
where ∆nb is the birefringence group index difference of the fiber. The spatial resolution is equal
to the depolarization length Ld, which is about 10 to 30 cm in a PM fiber, but, as we have already
seen, it is only 0.7 to 2 mm in a LiNbO3 integrated optic circuit.
This technique was first proposed to test the spatial distribution of the random coupling along
the fiber [12], which is very useful, since it is important to control the degradation of the
polarization conservation of the coil by the winding process, and in particular to check that it is
not due to a few localized cross-coupling points. However, the main interest of this technique is
the in situ control of the in-line components, like couplers, and of the connections [13, 14],
particularly the fiber connections on the integrated optic circuit, which, as we have seen, are
crucial for limiting the bias error, as they are symmetrical with respect to the splitter. It may also
be applied to the test of depolarizers.
Figure 5.9 Principle of optical coherence domain polarimetry (OCDP): the restored interference contrast is the coupling ratio ∈
in amplitude.
This OCDP testing is perfectly adapted to the case of the fiber gyro, because it measures
spurious polarization defects in exactly the same way as they induce phase error in a ring
interferometer. As an example, it can be used for nondestructive control of the fiber pigtail
alignment of a packaged superluminescent diode, despite its low degree of polarization. A
partially polarized source is usually considered to be the superposition of two orthogonal,
incoherent polarizations, but when tested by white-light interferometry, it should actually be
considered to be the superposition of a perfectly unpolarized source and a perfectly linearly
polarized source (Figure 3.33). For example, with a 4-to-l power ratio between the two axes, the
superluminescent diode is actually the superposition of a perfectly unpolarized source with twofifths of the power and a perfectly polarized source with three-fifths of the power. The
unpolarized component does not give any contrast recovery in the readout interferometer in the
same way that unpolarized light eliminates birefringence bias error in a fiber gyro (see Section
3.4.4), while the polarized component behaves like an ordinary polarized source: a misalignment
of the principal axes of the pigtail produces a polarization coupling that can be measured with the
value of the contrast that is recovered when the path difference of the readout interferometer
compensates for the birefringence of the fiber pigtail. Notice that the Michelson interferometer
used in this setup is also a suitable system for precisely measuring the coherence function of the
broadband source and, in particular, for evaluating its residual multimode laser structure [17]
(Figure 5.10).
Figure 5.10 Measurement of the effect of the residual multimode laser structure of a superluminescent diode with its
coherence function (log scale).
Another very interesting feature of this method is the measurement of polarizer rejection [13].
On entering a linearly polarized wave train at 45° of the principal axes of an integrated optic
polarizer on a birefringent substrate, there are, at the output, a main transmitted wave train and a
secondary wave train attenuated by the polarizer (Figure 5.11). An output polarizer at 45°
recombines both wave trains in the same polarization state, and a readout interferometer yields a
measurement of the amplitude extinction ratio of the integrated optic polarizer. Note that a high
rejection of the input and output polarizers of the setup is not required. The additional polarizers
are used only to split the input light into the two modes of the polarizer under test and to
recombine them at the output.
Figure 5.11 Propagation of a wave train in an integrated-optic polarizer fabricated on a birefringent substrate (case of a fast
transmitted TE mode, as with x-cut y-propagating LiNbO3 circuit).
It is even possible to differentiate the rejection and the cross-coupling of the fiber connections
when the circuit is pigtailed: polarized light has to be entered at 45° of the axes of the input
birefringent fiber lead of the IO circuit, and a polarizer has to be placed at 45° of the axes of the
output lead. The polarizer rejection is measured with a readout imbalance that corresponds to the
total birefringence of the two leads and the circuit, while the cross-coupling at each connection is
measured with an imbalance that relates to the birefringence of the corresponding lead (Figure
5.12). This also applies to all-fiber polarizers such as coiled polarizers.
Figure 5.12 OCDP testing of a pigtailed integrated-optic polarizer with: (1) the path difference of the wave train corresponding
to the first crossed coupling; (2) the path difference of the wave train corresponding to the second crossed coupling; and (3) the
path difference corresponding to the wave train attenuated by the IO polarizer (it is assumed that the IO polarizer transmits the
fast TE mode).
Finally, this technique can be extended to spatial modes and, in particular, to test the quality of
spatial filtering in the short length of single-mode fiber used at the input-output port of the ring
interferometer [13]. Light coupled in the fiber propagates in the fundamental LP01 mode, but if
the fiber is not perfectly single-mode, there is also some light in the higher-order LP11 mode. As
with polarization modes, there is a velocity difference and the main LP01 wave train does not
overlap in time with the attenuated LP11 wave train at the output. They must be recombined and
sent into a readout interferometer to measure the amplitude extinction ratio of the spurious LP11
mode. It can be clearly observed with optical coherence domain reflectometry (OCDR) [18] that
path-matched white-light interferometry equalizes group transit time difference and not that of
phase transit time. The discrepancy is negligible with stress-induced birefringence, but it may be
very different with spatial modes where there are dispersion effects.
OCDP testing does provide a very powerful tool for controlling and understanding the various
problems of polarization and birefringence. It allows one to optimize the compromise among
polarization rejection, polarization couplings, and depolarization, and to get a very low bias
error despite the limited performance of the components. In the early days of fiber gyros,
intensity coupling ratios were measured when testing the selectivity of components and
assembling the system, while the spurious signals were dependent on amplitude ratios because of
the high coherence of the laser source. Broadband sources and OCDP testing put the problem
back in the right perspective: amplitude coupling ratios are measured (and localized) during
mounting, which means high sensitivity for test and control, while the spurious effects are now
dependent on intensity ratios, which means reduced parasitic signals in the final setup.
Note finally that OCDR techniques, which are, as we have seen, very similar to OCDP, is a very
useful tool to accurately evaluate residual backreflections at splices and connections between
fiber and integrated optic circuit.
Known under the name optical coherence tomography (OCT) [19], OCDR techniques are, by
the way, well developed in medicine.
5.4.2 OCDP Using Optical Spectrum Analysis
This technique of OCDP was first based on path-matched interferometry and has yielded several
commercially available dedicated instruments [17]. However, a path-matched interferometer is
actually a scanning interferometer which the base of Fourier-transform spectroscopy (see Section
A.14).
In Fourier-transform spectroscopy, one measures the full coherence function of the tested light
and through Fourier transform (see Section A.12 on coherence), the spectrum is retrieved.
Conversely, a classical optical spectrum analyzer (OSA), grating-based in particular, measures
the spectrum and through (inverse) Fourier transform the coherence function is retrieved.
Because an OSA is now a very common multipurpose test instrument with the development of
wavelength domain multiplexing (WDM) telecoms, it is tempting to use it for OCDP instead of a
dedicated unipurpose test instrument.
Both approaches are theoretically equivalent but they have some practical differences. An OSA
has a limited spectral resolution, typically ∆λr = 0.01 nm, and then can test coupling only within
an optical distance λ2/∆λr (i.e., 250 mm). For testing couplings in a PM fiber with ∆nb = 5 × 10–
4, it corresponds to a length limited to λ2/∆λ ⋅ ∆n (i.e., 500m).
r
b
In terms of sensitivity, an OSA is limited by excess RIN because it filters a very narrow broad
spectrum. It requires usually an averaging of many scans to get a low noise, and because the
interference pattern, which is looked for, varies in temperature, it is not stable enough to get full
advantage of this averaging. In practice, a dedicated instrument allows one to measure contrast
down to 10–4 to 3 × 10–5 (i.e., coupling or rejection of −80 to −90 dB), while an OSA is limited
to 10–3 to 3 × 10–4 (i.e., −60 to −70 dB).
As seen in Section A.14, there is a last spectrum analysis technique, now available because of
WDM telecoms, the tunable laser. It has a very narrow line width and then a much higher spectral
resolution than an OSA. In addition, the probe laser line has no excess RIN as a broadband source
filtered in an OSA, as seen in Section A.2.
References
[1] Kaminow, I. P., “Polarization in Optical Fibers,” IEEE Journal of Quantum Electronics, Vol. QE-17, 1981, pp. 15–21.
[2] Rashleigh, S. C., et al., “Polarization Holding in Birefringent Single-Mode Fibers,” Optics Letters, Vol. 7, 1982, pp. 40–42.
[3] Born, M., and E. Wolf, Principles of Optics, New York: Pergamon Press, 1975.
[4] Burns, W. K., and R. P. Moeller, “Polarizer Requirements for Fiber Gyroscopes with High-Birefringence Fiber and Broad-Band
Source,” Journal of Lightwave Technology, Vol. LT2, 1984, pp. 430–435.
[5] Arditty, H. J., et al., “Integrated-Optic Fiber Gyroscope, Progresses Towards a Tactical Application,” Proceedings of OFS 2’84,
Stuttgart, VDE-Verlag, 1984, pp. 321–325.
[6] Lefèvre, H. C., et al., “Progress in Optical Fiber Gyroscopes Using Integrated Optics,” Proceedings of AGARD-NATO, Vol.
CPP-383, 1985, pp. 9A1–9A13.
[7] Fredricks, R. J., and R. Ulrich, “Phase-Error Bounds of Fibre Gyro with Imperfect Polariser/Depolarizer,” Electronics Letters,
Vol. 20, 1984, pp. 330–332.
[8] Jones, E., and J. W. Parker, “Bias Reduction by Polarization Dispersion in the Fibre-Optic Gyroscope,” Electronics Letters, Vol.
22, 1986, pp. 54–56.
[9] Burns, W. K., “Phase-Error Bounds of Fiber Gyro with Polarization-Holding Fiber,” Journal of Lightwave Technology, Vol.
LT4, 1986, pp. 8–14.
[10] Carrara, S. L. A., B. Y. Kim, and H. J. Shaw, “Bias Drift Reduction in Polarization-Maintaining Fiber Gyroscope,” Optics
Letters, Vol. 12, 1984, pp. 214–216.
[11] Böhm, K., et al., “Low-Drift Gyro Using a Superluminescent Diode,” Electronics Letters, Vol. 17, 1981, pp. 352–353.
[12] Takada, K., J. Noda, and K. Okamoto, “Measurement of Spatial Distribution of Mode Coupling in Birefringent PolarizationMaintaining Fiber with New Detection Scheme,” Optics Letters, Vol. 11, 1986, pp. 680–682.
[13] Lefèvre, H. C., “Comments About the Fiber-Optic Gyroscope,” SPIE Proceedings, Vol. 838, 1987, pp. 86–97.
[14] Takada, K., K. Chida, and J. Noda, “Precise Method for Angular Alignment of Birefringent Fiber Based on an Interferometric
Technique,” Applied Optics, Vol. 26, 1987, pp. 2979–2987.
[15] Barnoski, M. K., and M. S. Jensen, “Fiber Waveguides, a Novel Technique for Investigating Attenuation Characteristics,”
Applied Optics, Vol. 15, 1976, pp. 2112–2115.
[16] Youngquist, R. C., S. Carr, and D. E. N. Davies “Optical Coherence Domain Reflectometry: A New Evaluation Technique,”
Optics Letters, Vol. 12, 1987, pp. 158–160.
[17] Martin, P., G. Le Boudec, and H. C. Lefèvre, “Test Apparatus of Distributed Polarization Coupling in Fiber Gyro Coils Using
White Light Interferometry,” SPIE Proceedings, Vol. 1585, 1991.
[18] Kohlhaas, A., C. Frömchen, and E. Brinkmeyer, “High-Resolution OCDR for Testing Integrated-Optical Waveguides:
Dispersion-Corrupted Experimental Data Corrected by a Numerical Algorithm,” Journal of Lightwave Technology, Vol. 9,
1991, pp. 1493–1502.
[19] Fercher, A. F., K. Mengedoht, and W. Werner, “Eye-Length Measurement by Interferometry with Partially Coherent Light,”
Optics Letters, Vol. 13, 1988, pp. 186–188.
5.2 Analysis of Polarization Nonreciprocities in a Fiber Gyroscope
5.2 Analysis of Polarization Nonreciprocities in a Fiber Gyroscope
5.2 Analysis of Polarization Nonreciprocities in a Fiber Gyroscope
5.2 Analysis of Polarization Nonreciprocities in a Fiber Gyroscope
5.2 Analysis of Polarization Nonreciprocities in a Fiber Gyroscope
CHAPTER 6
Time Transience-Related Nonreciprocal Effects
6.1 Effect of Temperature Transience: The Shupe Effect
Because of reciprocity, the two counterpropagating paths are equalized in a ring interferometer,
but this is strictly valid only if the system is time-invariant. We have already seen (Section 3.2.2)
that biasing phase modulation can be generated with a reciprocal modulator placed at one end of
the coil acting as a delay line, which provides a filtering behavior with a sinusoidal transfer
function: sin[(π/2)⋅(f/fp)]. This also applies to parasitic phase shifts generated by the environment,
particularly with nonuniform temperature change as analyzed very early by Shupe [1]. The two
interfering waves do not see the perturbation at exactly the same time, unless it is applied in the
middle of the coil.
Considering that the phase perturbation has low frequencies compared to the proper frequency
of the coil, the gyro behaves like a dc rejection filter with the usual 6 dB per octave roll-off and
differentiation properties (Figure 6.1): the phase error is proportional to the temporal derivative
of the temperature, which may be very harmful, particularly during the warm-up.
Figure 6.1 Transfer function for phase perturbation about dc.
An elementary fiber segment of elementary curvilinear coordinate δz, accumulates a phase:
(6.1)
With a temperature change dT, this accumulated phase has a variation:
(6.2)
where the temperature dependence of the index of silica dn/dT is 8.5 × 10–6/°C, and the thermal
expansion coefficient αSiO2 is 0.5 × 10–6/°C as seen in Section B.1.5.
To simplify equations, we may define a coefficient αT of thermal dependence of the
accumulated phase:
(6.3)
which is about 9 × 10–6/°C, and:
(6.4)
At the coordinate z, the time delay ∆t(z) between both interfering waves is (Figure 6.2):
(6.5)
where L is the coil length. It is zero in the middle of the coil where z = L/2, and maximum at the
end where z = 0.
Figure 6.2 Effect of an asymmetrical perturbation.
If the temperature varies at a rate dT/dt, it yields an elementary phase difference error δϕ e in
the ring interferometer:
(6.6)
and then:
(6.7)
This elementary phase error δϕ e is proportional to the temporal derivative of temperature,
dT/dt, often written Ṫ (T dot). Taking, for example, a coil of 200m composed of 40 layers of 50
turns, and assuming that the first external layer is heated solely at a moderate rate of 0.01°C/s, a
phase error as high as 5 × 10–6 rad is yielded.
Note that a constant rate Ṫ generates a phase ramp which, as will be seen in Chapter 8, is used as
a feedback in the signal processing technique to linearize the scale factor of a fiber gyro.
These elementary phase errors have to be summed over the fiber coil length L, and the good
way to do it is to consider pairs of symmetrical elementary segments at the coordinates z and (L −
z) (Figure 6.3). For such a pair, the phase error generated in the interferometer is:
δϕ ep(z) = δϕ e(z) + δϕ e (L − z) (6.8)
which yields:
(6.9)
since
L − 2(L − z) = −(L − 2z)
This result is very important as it shows that, actually, the Shupe effect does not depend on T
dot but on the difference between the temporal derivatives of the temperature at z and at (L − z): it
is a delta T-dot effect.
This has a first implication because the difference of the derivatives of two functions is the
derivative of the difference between these two functions:
(6.10)
The elementary rotation rate error δΩe induced by the pair of symmetrical elementary
segments δz at z and L − z is proportional to d(T(z) − T(L − z))/dT. In inertial navigation the rate
is integrated to yield the accumulated angle of rotation. Because in a temperature change, the
temperature difference between z and L − z is zero before the change and back to zero at the end,
the Shupe effect does not yield an error on the accumulated angle that is calculated by integration,
even if it induces a transient error on the rate measurement; the average rate error is nulled out.
Figure 6.3 Effect of a symmetrical pair of asymmetrical temperature changes Ṫ (z) and Ṫ (L − z).
6.2 Symmetrical Windings
Equations (6.8) and (6.9) show that if the same temperature change Ṫ is experienced by
symmetrical segments at equal distance from the middle of the coil, the effect is canceled out, as
δϕ e(z) becomes opposite to δϕ e(L − z). This compensation is obtained with a symmetrical
winding as analyzed by Shupe [l]: the fiber is wound from the middle, and by alternating layers
coming from each half-coil length, this places symmetrical segments in proximity [Figure
6.4(a)].
Figure 6.4 Section view of symmetrical windings (one-half coil length is represented by + and the other is represented by −):
(a) dipolar and (b) quadrupolar.
Compared to ordinary winding, this dipolar winding reduces the Shupe effect by a factor
approximately equal to the number of layers [2]. An even better compensation, approximately
equal to the square of the number of layers, is obtained with a quadrupolar winding proposed by
Frigo [2] [Figure 6.4(b)]. With a pair of symmetrical layers in a dipolar winding, a radial heat
propagation always first reaches the layer of the same half-coil length, which yields a residual
transient sensitivity. With the quadrupolar winding, the layer order is reversed from pair to pair,
which further improves the compensation. To obtain the best results, the number of layers must be
a multiple of four and all the layers must have the same number of turns. Some additional benefit
is also obtained by thermally insulating the coil to slow down its rate of temperature change.
6.3 Stress-Induced T-Dot Effect
In practice, the Shupe effect is almost not experienced in a fiber gyro with a quadrupolar winding
as the variation of the difference of temperature, the delta T dot, is very small between the two
adjacent layers of the symmetrical points and also because of its thermal inertia the packaging
slows down the temperature change, which yields a good temperature uniformity in the coil.
However, (6.9), which describes the Shupe effect, is incomplete. It assumes that the coefficient αT
of thermal dependence of the accumulated phase is the same for the two elementary segments of a
pair of symmetrical points, which is not at all the case because of nonuniform temperatureinduced stresses in the coil that change the actual index dependence dn/dT as well as the actual
thermal expansion αSiO2. This was described quite early by Cordova and Swabian [3] and
analyzed in more details recently by Mohr and Schadt [4].
For a pair of symmetrical elementary elements δz at positions z and L − z, instead of (6.9) of a
pure Shupe effect, the elementary phase error δϕ ep(z) induced in the interferometer is actually:
(6.11)
which can be approximated by:
(6.12)
The first term ⟨αT⟩ ⋅ ∆Ṫ, where ⟨αT⟩ = (αT(z) + αT(L − z)/2 is the mean thermal dependence of
the phase at z and L − z, is the pure Shupe effect, while the second term ∆αT ⋅ ⟨Ṫ⟩, where ⟨Ṫ⟩ =
(Ṫ(z) + Ṫ(L − z))/2 is the mean temperature derivative, is the effect of temperature-dependent
stresses which induces a difference ∆αT(z) = αT(z) − αT(L − z) of the temperature dependence of
the phase between the position at z and the symmetrical one at (L − z).
To summarize, the elementary phase error created by a symmetrical pair of elementary
segments δz is the sum:
δϕ ep(z) ≃ δϕ Sh(z) + δϕ Tdot(z) (6.13)
with the pure Shupe effect:
(6.14)
which depends on the delta T dot, (∆Ṫ), and the effect of the temperature-induced nonuniformity
of the stress in the coil, which depends on the mean T dot, (⟨Ṫ⟩):
(6.15)
In practice, this T-dot effect δϕ Tdot is much larger than the pure Shupe effect δϕ Sh, which
depends on the delta T dot. This T-dot dependence is actually good news as it can be modeled
much more easily than the ∆T dot.
Knowing αT(z) along the whole fiber coil requires a good understanding of the complex
composite structure of the coil composed of the silica fiber, its polymer coating and the
supporting frame of the coil with usually a potting material [4].
Symmetrical windings like the quadrupolar winding are also very efficient because the
elementary segments of a symmetrical pair are close to each other, which drastically reduces
∆αT(z) = αT(z) − αT(L − z).
Finally, the two effects, Shupe and T dot, have a difference in temporal signature as outlined by
Mohr and Schadt [4]. To understand it, we are going to recall the basics of heat propagation,
which is actually a process of diffusion.
6.4 Basics of Heat Diffusion and Temporal Signature of the
Shupe and T-Dot Effects
There are three different mechanisms of heat transfer: conduction, convection, and radiation. In a
fiber gyro, one can consider, at least in the first approximation, that the main process is
conduction. Heat steady-state transfer, that is, the power flow through a material, is well known. If
there is a difference of temperature ∆T across a material, the heat power flow Q is the ratio of this
temperature difference ∆T and the thermal resistance RT:
(6.16)
Like the intensity I of an electrical current is the ratio between the voltage difference ∆V and the
electrical resistance R:
(6.17)
The unit of thermal resistance is K/W (kelvin per watt). Similarly to the electrical resistivity of
a material, there is a thermal conductivity K in W/K ⋅ m. A homogeneous material of section S (in
m2) and length L (in m) has a thermal resistance:
(6.18)
The general differential law is given by Fourier ’s law between the heat flow vector q and the
gradient vector of the temperature:
q = −K grad T = −K ∇ T (6.19)
Conductivity is the parameter involved in a steady state process like the insulation of a house
where you want to know how many watts are lost as a function of the outside temperature.
Now, for a homogeneous medium with no internal heat generation, the differential equation of
heat transfer is:
(6.20)
where D is the diffusivity of the material that is related to the conductivity K by:
(6.21)
where ρ is the density of the material and C is its specific heat.
Note that (6.20) has some similarity with a wave propagation equation like (see Section A.3):
(6.22)
In both case there is a Laplacian operator ∇2 of the spatial coordinates, but in a wave
propagation equation there is a second-order time derivative with the square of the wave velocity
v, while in the process of heat diffusion, there is only a first-order time derivative with the
diffusivity D that may look like a velocity but actually, there is a big difference: its dimension is
length to the square by time and not length by time as for a velocity. Typical values of diffusivity
are on the order of:
1,000 mm2/s for diamond;
100 mm2/s for aluminum;
10 mm2/s for titanium;
1 mm2/s for silica;
0.1 mm2/s for polymers.
The differential equation (6.20) of heat diffusion is not easy to solve, but one can get a good
understanding of this process of diffusion with the response to a step function in temperature in a
semi-infinite medium [5]. The mathematical result involves the error function, erf(x), drawn in
Figure 6.5:
Figure 6.5 Error function erf(x).
(6.23)
If the original temperature T0 is raised to T1 at the time t = 0, the temperature T(z, t) at a
distance z in a material of diffusivity D is:
(6.24)
Figure 6.6 displays the normalized temperature change (T(z, t) − T0)/(T1 − T0) with a
logarithmic abscissa for a normalized time t/τc, where τc = z2/D is the characteristic time.
Figure 6.6 Time response to a temperature step T1 − T0 at a distance z with a characteristic time τc = z 2/D.
One sees that the temperature does not change over a duration τc/16, as erf
=
erf 2 ≈ 1, and then increases to get to about (T1 + T0)/2 at τc, as erf
= erf (0.5) ≈ 0.5
The characteristic time follows a square law as a function of the distance z, and it is possible to
show that with a diffusion through several media in series, the resulting total characteristic time
τct is:
(6.25)
where τci = zi2/Di is the characteristic time of medium i, which has a diffusivity Di and a
thickness zi. In particular, two media in series with the same characteristic time τc1 and τc2 yields:
τct = 4τc1 = 4τc2 (6.26)
and when τc2 << τc1:
(6.27)
Following this simple model and considering that the navigation system housing has a
characteristic time τc1 typically equal to 60 minutes (because of its thermal inertia) and the coil
has a characteristic time typically equal to τc2 = 0.5 minute, a fiber element in contact with the
housing follows the characteristic time τc1 while an element far in the coil follows T′(t) with a
characteristic time τct =
≈ 1.2 τc1. The two temporal changes of temperature T(t)
and T′(t) are displayed in Figure 6.7 with their difference ∆T(t).
Figure 6.7 Temporal variations of the temperatures T(t) and T′(t) (solid lines) and of their temperature difference ∆T(t) (dotted
line).
Nothing happens until τc/16 and then temperatures T(t) and T′(t) start to rise to get to half of T1
− T0 around τc. The temperature difference ∆T(t) = T(t) − T′(t) behaves quite differently: it gets to
a plateau around τc/2 and then slowly decays.
This difference of behavior is outlined with the time derivatives. The delta T dot (∆Ṫ) that
drives the Shupe effect returns very fast to zero around τc/2, and then becomes negative and
slowly returns to zero again (Figure 6.8). The T dot (Ṫ) that drives the T dot effect due to
nonuniform temperature-dependent stresses also rises quickly, but its decay is much slower and
its value remains always positive (Figure 6.9). The maximum of Ṫ is about (T1 − T0) /τc and
happens at a time of about τc/6. In practice, the transience effect does follow the temporal
signature of the T dot, showing that the main effect is not the Shupe effect but the effect of
temperature-dependent stresses.
Figure 6.8 Normalized temperature difference and its normalized time derivative delta T dot.
This analysis was carried out considering a step function in temperature in a semi-infinite
medium. One has seen that the temperature rise is very slow above τc (Figure 6.6). At a time t =
10τc only 83% of the temperature step is reached, and it requires t = 1,000τc to reach 100%. In
practice this rise is much faster, as the temperature step function is applied all around the coil.
Figure 6.10 displays a typical temperature variation in this case of excitation. One sees that 100%
of the temperature step is reached much faster than in the case of the semi-infinite medium.
However, the behaviors are very similar between 0 and τc/2 when the maximum of the Shupe
effect and the T-dot effect take place.
Figure 6.9 Normalized temperature and its normalized time derivative T dot with a maximum of about (T1 − T0)/τc .
Figure 6.10 Temperature rise in a semi-infinite medium (dotted line) and in a practical coil surrounded by its housing (solid line).
6.5 Effect of Acoustic Noise and Vibration
Acoustic noise and vibrations are also a potential source of parasitic effects related to transient
phenomena. Fiber optics are actually used to make very sensitive hydrophones, with acoustic
pressure yielding phase change through the photoelastic effect [6]; but with the ring configuration
of the interferometric fiber gyro, the absolute effect is greatly reduced, since as we saw in Section
6.1, there is a dc rejection filtering with 6 dB per octave roll-off.
In any case, a perturbation at a given frequency may yield a spurious effect only at this
frequency and its harmonics: it does not affect the mean bias offset and does not reduce the
sensitivity to rotation if the spurious phase variation remains in the linear range of the sine
response, which is always the case in practice. Therefore, it is necessary to consider only
perturbations within the detection bandwidth of the system (i.e., a few kilohertz). In this frequency
range, the acoustic wavelength in the fiber is equal to several meters (sound velocity is 7 km/s in
silica), which makes the perturbation quasi-uniform over the coil dimension: symmetrical fiber
segments experience the same phase modulation, which makes the residual effect negligible. This
applies to external perturbations and to the proper fluctuations due to phonons, which are about 1
for a kilometer-long single pass at the absolute temperature of 300K, but
compensated for at low frequencies in the ring interferometer [7].
Vibrations are a more severe problem, and an adequate potting has to be used to ensure a good
coil ruggedness. An adequate mechanical design is also required to avoid specific mechanical
resonances within the system, but this is eased because of the solid-state configuration of the fiber
gyro.
References
[1] Schupe, D. M., “Thermally Induced Nonreciprocity in the Fiber-Optic Interferometer,” Applied Optics, Vol. 9, 1980, pp. 654–
655.
[2] Frigo, N. J., “Compensation of Linear Sources of Non-Reciprocity in Sagnac Interferometers,” SPIE Proceedings, Vol. 412,
1983, pp. 268–271.
[3] Cordova, F., and G. M. Swabian, “Potted fiber optic gyro sensor coil for stringent vibration and thermal environments,” U.S.
Patent 5,546,482, 1996.
[4] Mohr, F., and F. Schadt, “Error Signal Formation in FOGs Through Thermal and Elastooptical Environment Influences on the
Sensing Coil,” ISS Conference, Karlsruhe, 2011.
[5] Crank, J., The Mathematics of Diffusion, Oxford, U.K.: Clarendon Press, 1975.
[6] Yurek, A. M., “Status of Fiber-Optic Acoustic Sensing,” Proceedings of 8th Optical Fiber Sensors Conference, Monterey,
1992, pp. 338–341.
[7] Logozinskii, V. N., “Fluctuations of the Phase Shift Between Opposite Waves in a Fiber Ring Interferometer,” Soviet Journal of
Quantum Electronics, Vol. 11, 1981, pp. 536–538.
CHAPTER 7
Truly Nonreciprocal Effects
7.1 Magneto-Optic Faraday Effect
Even in a perfectly reciprocal system, the Sagnac phase shift is not the only effect to be truly
nonreciprocal. In particular, due to the magneto-optic Faraday effect, a longitudinal magnetic
field B modifies the phase of a circularly polarized wave by an amount determined by the Verdet
coefficient V of the medium. The sign of this phase shift depends on the left- or right-handed
character of the circular polarization and also on the relative direction of the field and light
propagation vectors. It is well known that, as seen in optical isolators (Section B.8.2), this phase
shift may manifest itself as a circular birefringence with a change θF in the orientation of a
linearly polarized light [Figure 7.1(a)], resulting from the opposite phase shifts of its
copropagating left- and right-handed circularly polarized components: θF = V ⋅ B ⋅ L, where V is
the Verdet constant of the medium and L is its length. It may also be detected as a phase difference
∆ϕ F in a ring fiber interferometer where identical circularly polarized waves counterpropagate
around the coil [Figure 7.1(b)]. As seen in Section A.13.2, this phase difference is the double of
the angle of Faraday rotation θF, and then:
Figure 7.1 Faraday effect: (a) rotation of a linear polarization and (b) phase difference between two counterpropagating circular
polarizations.
∆ϕ F = 2V ⋅ B ⋅ L (7.1)
At first, it seems that the total Faraday effect along a given path is proportional to the line
integral of the magnetic field vector B along this path. For a closed path, the result should be
different from zero according to the Ampere law only if the path encloses an electrical current. A
toroidal closed path configuration has been used to demonstrate an electrical current fiber sensor
[1], but a fiber-optic gyroscope should not be sensitive to environmental magnetic fields, because
of the absence of traversing electrical current. However, this is actually true only if the same state
of polarization is preserved along the fiber. The Faraday phase shift accumulated along an
elementary length vector dz is:
dϕ F = αp(z) ⋅ V ⋅ B ⋅ dz (7.2)
where αp(z) is a coefficient that depends on the state of polarization. It is equal to zero for linear
polarization and to ±1 for circular polarizations. It has an intermediate value for elliptical
polarization. The total phase difference ∆ϕ F between both counterpropagating waves is
represented by the relationship:
∆ϕ F = 2∫closed pathαp(z) ⋅ V ⋅ B ⋅ dz (7.3)
which may be different from zero even if the line integral ∫B ⋅ dz is equal to zero, because αp(z)
is not constant. This is due to polarization changes along the fiber arising from residual
birefringence [2–4]. A configuration using bending-induced birefringence to enhance sensitivity
to an external magnetic field and make a magnetometer with a ring interferometer has even been
demonstrated [5].
As described in Figure 7.2, a λ/4 loop transforms the input linear polarization (LP) in a righthanded (RH) circular polarization, and a λ/2 loop transforms this right-handed polarization in a
left-handed (LH) polarization. As the direction of propagation and the direction of rotation of the
circular polarization have been inverted, the Faraday phase shift is accumulated.
Figure 7.2 Configuration of a ring interferometer sensitive to magnetic field according to [5].
Assuming that the influence of the earth’s magnetic field Bearth were integrated constructively
along the whole fiber length L, the maximum nonreciprocal phase difference would be:
∆ϕ Fmax = 2 ⋅ V ⋅ Bearth ⋅ L (7.4)
The Verdet constant V has a λ–2 wavelength dependence and is equal for silica to 2 rad T –1m–1
(i.e., 0.2 rad G–1km–1) at 850 nm, and 0.6 rad T –1m–1(i.e., 0.06 rad G–1km–1) at 1,550 nm. Since
Bearth is typically 0.5G (or 50 microtesla), ∆ϕ Fmax would be as high as 0.2 rad, at 850 nm and
0.06 rad at 1,550 nm with a 1-km coil length. Experimentally, it was observed [2–4] that there is a
compensation factor of about 103 in a gyroscope using an ordinary fiber, which yields a
measurement error equivalent to approximately the earth rotation rate (i.e., 15°/h).
Notice that the Faraday effect is also given in scientific textbooks as a function of the H field.
Because in diamagnetic material like silica, B and H are proportional and the relative magnetic
permeability is very close to unity, the unit change of the Verdet constant V is obtained by
multiplying its B-value by μ0 = 4π × 10–7 Hm–1 (or T ⋅ A–1 ⋅ m); that is, the H-value of V is 2.5
× 10–6 rad A–1 for a wavelength of 850 nm and 0.75 × 10–6 rad A–1 at 1,550 nm.
The use of polarization-maintaining fiber, which has been seen to be very helpful to reduce
birefringence-induced nonreciprocities, should suppress the magnetic dependence, as a linear
polarization does not get Faraday phase shift. However, in practice, the residual Faraday phase
error is on the order of 1 to 10 μrad for 1G (100 μT). As analyzed by Hotate and Tabe [6], the
effect is not completely nulled out because of the residual rotation of the birefringence axes of
practical PM fibers. As discussed in Section B.6.1, their preforms experience very high stress,
which tends to give a helicoidal shape to the stressing rods, and stress-induced high-birefringence
fibers used to preserve polarization are drawn with a slowly changing orientation of their
principal axes [7].
Figure 7.3 Poincaré sphere describing the stable elliptical states of polarization in a high-birefringence fiber with rotated
principal axes.
When the principal axes of a linear-birefringence fiber are rotated, the eigen polarization
modes do not have a perfectly linear state of polarization. This can be viewed simply by
considering a Poincaré sphere (see Section B.5.3) defined with respect to a rest reference frame
that rotates with the principal axes at the twist rate tw (in radians per meter). In this rest frame, the
linear birefringence is represented with a stable equatorial vector ∆βl, but there is an additional
circular birefringence vector ∆βc aligned along the polar axis to take into account the change of
frame of reference (Figure 7.3). The magnitude of ∆βc is equal to tw, but it corresponds to the
opposite direction of rotation. The total birefringence ∆βt is obtained simply with the vectorial
sum ∆βt + ∆βc. The magnitude ∆βc is much smaller than ∆βl, otherwise the polarization would
not be preserved at all; therefore, the two stable orthogonal states of polarization are slightly
elliptical to correspond to the intersection of ∆βt with the Poincaré sphere. Going back to the
laboratory frame, the two states preserve the same constant ellipticity, but their minor and major
axes follow the rotation of the principal axes of the birefringence of the fiber. The linear
polarization of the PM fiber is dragged by the twist of the birefringence axes and becomes
slightly elliptical.
Note that the two stable elliptical states have opposite ellipticity.
In a ring interferometer using such a linear polarization maintaining fiber, it is possible to
consider that a magnetic field has a negligible dependence on the state of polarization that is the
same in both opposite directions. However, it modifies the phase of the counterpropagating waves
as a function of the coefficient αp equal to the ellipticity of the state; that is, the ratio ∆βc/∆βl =
−t/∆βl. The accumulated Faraday phase difference is therefore:
(7.5)
Assuming a circular coil of radius R, this yields:
(7.6)
where θB is the angle of the B field with respect to the reference axes. This formula is equivalent
to a synchronous demodulation of the twist rate tw(z) at a frequency (2πR)–1 and with an
integration time L.
The residual magnetic dependence is therefore coming from the spatial frequency components
of tw(z) equal to the inverse of a turn perimeter 2πR within a bandwidth equal to the inverse of the
total length of the coil. Assuming that tw(z) is a random function with a constant power density,
the usual result of detection of a white noise with a lock-in amplifier may be applied.
It is difficult to evaluate directly this power density, but it is possible to predict the effect of a
length increase of a known fiber: it is similar to a random walk process, and the magnetic
dependence of the phase error should increase as the square root of the length (which is
equivalent to an integration time). However, the phase error should be independent of the coil
diameter and the turn perimeter if the noise power density of the parasitic twist is uniform (i.e.,
frequency-independent). A good order of magnitude of the practical phase error induced by the
earth’s magnetic field (i.e., 0.5 G or 50 μT) is 0.5 to 5
(at λ = 850 nm), depending on
the quality of the fiber. Because the Sagnac sensitivity is proportional to the length and the
diameter of the coil, the equivalent rotation rate error due to an external magnetic field should
decrease proportionally to the coil diameter and to the square root of its length.
This phase error is coming from a random process, the parasitic twist rate of the highbirefringence fiber; but for a given gyro coil, this rate is stable over time, and the magnetic
dependence does not yield long-term drift or noise if the gyro keeps the same orientation with
respect to the magnetic field. The gyro has an axis and a coefficient of magnetic sensitivity that
are stable and can be modeled to compensate for the predictable bias offset. This sensitive axis
lies approximately in the plane of the coil, because the Faraday effect is nulled out for a magnetic
field orthogonal to the direction of propagation. In practice, there is an additional axial
dependence about tenfold smaller than the in-plane dependence.
Note that this magnetic dependence can be reduced by connecting the PM coil fiber at 45° of the
polarization axis of the MIOC. Both eigen polarization modes are used and, because they have
opposite ellipticity because of orthogonality, they have also opposite Faraday dependence that
cancels it out. However, this creates two strong polarization coupling points and could generate
bias if the polarizer is not good enough (as seen in Chapter 5).
If the application requires a low magnetic dependence, it is possible to get a further
improvement of about two orders of magnitude by shielding the sensing coil with a high
permeability material such as μ-metal. For the highest-grade applications, a double shielding
eliminates the sensitivity. Note that all other gyro technologies, RLG or mechanical, have also a
magnetic dependence with the same order of magnitude, and it is also solved with μ-metal
shielding.
Because of the λ–2 dependence of the Faraday effect, one could think that the use of a long
wavelength (1,300 nm or 1,550 nm) should reduce the phase error by a factor of 3 or 4, but the
linear birefringence of a PM fiber being inversely proportional to the wavelength increases the
polarization ellipticity for a given twist, and in practice the magnetic dependence keeps the same
order of magnitude of 1 to 10
Note that PM solid-core microstructured fiber based on shape birefringence (see Section B.10)
has less random twist and then a magnetic dependence about ten times lower than the one of
classical PM fibers based on a stress structure [8]. A hollow-core fiber would be even more
interesting, as only few percentages of the light propagate in silica where Faraday effect occurs
but, as seen in Section B.10, they have still significant attenuation, at least as of today.
As we have seen, polarization-maintaining fibers provide a better reduction of Faraday
nonreciprocity than ordinary fibers. However, it was shown that if an additional depolarizer is
placed between the polarizer and the coil coupler in addition to the coil depolarizer, the Faraday
nonreciprocity is also greatly reduced, even with an ordinary fiber coil [9].
7.2 Transverse Magneto-Optic Effect
As analyzed by Logozinskii [9], in addition to the Faraday effect, which depends on the magnetic
field component parallel to the fiber, there is also a smaller but significant magnetic dependence
which is orthogonal to the fiber. It is due to the fact that the fundamental LP01 mode of a singlemode fiber has a small longitudinal component in addition to its main transverse component; it is
actually a hybrid HE11 mode as seen in Section B.2. The effect of this small longitudinal
component is generally negligible excepted in few cases. We show in Section B.5.2 that this
longitudinal component explains the circular birefringence created by elastically twisting a fiber,
and it is also the reason of this transverse magnetic dependence. The derivation of [9] concludes
that this magnetic dependence is parallel to the coil axis and requires a polarization orthogonal to
this coil axis. It yields a nonreciprocal phase difference ∆ϕ B⊥ proportional to the number N of
turns with:
At 1,550 nm where the Verdet constant V = 6 × 10–5 rad ⋅ G–1 ⋅ m–1, it yields:
Considering a gyro with a coil of 1 km over a diameter of 10 cm (i.e. 3,200 turns), it yields a
dependence of 2.5 μrad/G.
In practice the PM axes rotate because of twisting along the coil and this transverse magnetooptic effect is reduced to one-half as there is no transverse magnetic effect if the polarization is
parallel to the coil axis. This yields about 1 μrad/G for a coil of 3,000 turns.
7.3 Nonlinear Kerr Effect
Another important case of truly nonreciprocal effect may arise due to the nonlinear optical Kerr
effect as analyzed by Ezekiel et al. [11]. Reciprocity is indeed based on the linearity of
propagation equation (see Section 3.1), but an imbalance in the power levels of the
counterpropagating waves can produce a small nonreciprocal phase difference because of
propagation nonlinearity induced by the high optical power density in the very small silica fiber
core. Slow variations in the splitting ratio of the power divider feeding the sensing coil may
therefore translate directly into bias drift. Experimentally, a power difference of 1 μW (e.g.,
arising from 10–3 splitting imbalance of a 1-mW source) gives a nonreciprocal index difference
as small as 10–15, but when integrated along a kilometer of fiber, this produces a phase
difference of a few 10–5 rad, at least three orders of magnitude above the theoretical sensitivity
limit. It could be reduced by simply reducing the power in the fiber, but this would increase the
relative influence of detection noise.
The Kerr-induced rotation-rate error results in fact from a complex four-wave mixing process
and not simply from an intensity self-dependence of the propagation constant of each
counterpropagating wave. It depends also on the intensity of the opposite wave [11, 12]. In a linear
medium, the electric polarization vector P is defined as (see Section A.5.1):
P = χe∈0E (7.7)
but when the wave has a high energy density (i.e., a large E field), an additional nonlinear term
depending on the third-order susceptibility
and the scalar square ⎪E⎪2 of the electric field
has to be taken into account and P becomes:
(7.8)
The relative dielectric permittivity ∈r = 1 + χe is changed to:
(7.9)
and the actual index of refraction
has an additional nonlinearity term δnNL:
(7.10)
In a ring interferometer where two fields E1 and E2 propagate in opposite directions, two
polarization vectors P 1 and P 2 have to be considered for each propagation direction. The former
relationship between the P and E vectors of a single wave applies, but now each
counterpropagating wave cannot be considered independently. The total polarization vector P 1 +
P 2 has to be related to the total field E1 + E2, and therefore:
(7.11)
A potential source of nonreciprocity comes from the ⎪E1 + E2⎪2 term, which represents the
intensity of the standing wave resulting from the interference between both counterpropagating
fields E1 and E2.
Assuming continuous monochromatic waves with the same linear state of polarization and the
same frequency ω and opposite propagation constant β and −β we have:
(7.12)
where
z
is
the
spatial
longitudinal
coordinate
along
the
fiber
coil.
As
this yields:
(7.13)
The first two terms of this relationship depend both on the sum of the field squares (i.e.,
intensities) of the two waves and therefore yield the same nonlinear index change for E1 and E2
in each opposite direction. However, the two last terms induce a nonreciprocity, as:
(7.14)
and similarly:
(7.15)
The effect of the terms at a spatial frequency of 3β or −3β is averaged out in the propagation,
but the two other terms at β and −β are phase-matched and yield a constant susceptibility change
as the waves propagate. Each polarization vector is actually:
(7.16)
This gives a different nonlinear change of the index of refraction for each opposite direction:
(7.17)
and a nonreciprocal index difference:
(7.18)
Assuming a uniform power distribution over the core area and a diameter of about 5 μm, this
Kerr-induced index difference may be evaluated from the value of χe(3) in silica as a function of
the power difference ∆P (proportional to ⎪E2⎪2 − ⎪E1⎪2) between both direction as [10]:
∆nK/∆P ≈ 2 × 10–15μW–1 (7.19)
This difference is very small, but as the Sagnac effect, it is integrated along the whole length L
of the fiber coil and gives rise to a significant phase difference ∆ϕ K. For a wavelength of 0.633
μm [10]:
∆ϕ K/L ⋅ ∆P ≈ 2 × 10–5 rad ⋅ μW–1 ⋅ km–1 (7.20)
This analysis shows that the Kerr effect nonreciprocity results solely from the formation of a
nonlinear index grating, due to the interference between the two counterpropagating waves within
the fiber, which yields a standing wave. As stated early on in [13], if the contrast of this standing
wave is washed out by some process, then nonreciprocity should decrease. This important point
explains why the use of a broadband source with a short coherence length greatly reduces the
Kerr nonreciprocity: the standing wave is contrasted only over a distance equal to the coherence
length Lc in the middle of the fiber coil (Figure 7.4), and therefore the effect of the nonreciprocal
index difference is integrated only along Lc and not along the whole fiber length L.
Figure 7.4 Contrasted standing wave in the middle of the fiber coil.
The cancellation of the Kerr nonreciprocity with a broadband source was originally explained
independently with the statistics of the light intensity fluctuation by Bergh et al. [13] and
Petermann [14]. As a matter of fact, this original explanation considered the case of an intensitymodulated wave, which yields a nonlinear index perturbation that depends on time t and position z
in the fiber:
(7.21)
An important feature of these equations, as we have already seen, is that the cross-effect of the
power of one wave is twice its self-effect. The use a square-wave intensity modulation of a
monochromatic source was first proposed to reduce the Kerr nonreciprocity by Bergh et al. [15].
In this case, the cross-effect is present only when both counterpropagating intensities are
coincident (Figure 7.5) (i.e., half the time), while the self-effect is present all the time.
Figure 7.5 Square-wave intensity modulation of the counterpropagating waves to reduce Kerr-induced nonreciprocity.
Therefore, the factor of 2 of the cross-effect is reduced to an averaged value of unity, which
effectively cancels out the nonreciprocity, as the mean phase perturbation becomes identical in
both directions.
Such a kind of compensation is not restricted to square waves, and it applies if the mean value
<I> of the modulated intensity is equal to its standard deviation
which can be also written as:
[13, 14],
2 <I>2 = <I2> (7.22)
By virtue of the central limit theorem, a polarized broadband spontaneous source has a random
intensity with an exponential probability distribution:
(7.23)
and this fulfills the requirement 2 <I>2 = <I2>, which ensures the absence of Kerr-induced
nonreciprocity.
In addition, as we saw in Section A.2.4, this statistic of random intensity fluctuation of a
spontaneous (also called “thermal”) source is preserved in the process of amplified spontaneous
emission (ASE).
However, this explanation gives the misleading feeling that the reduction of Kerr effect
nonreciprocity with a broadband source that happens to have the adequate statistics is only very
fortunate. However, following Bergh [16], the direct analysis of the nonlinear process shows
clearly that it is related to an interference phenomenon of the standing wave and that a lowcoherence source reduces the parasitic effect in the same way that is already very beneficial with
backreflection, backscattering, or polarization nonreciprocities, which also depend on parasitic
interference phenomena.
However, the similarity in terms of coherence between this nonlinear effect and the other
coherence-related linear effects is limited to the use of broadband source with continuous light
emission that destroys the contrast of the standing wave, but ensures that both counterpropagating
intensities are permanent in the fiber. A very short pulse can also limit the coherent effect of
backreflection, backscattering, and polarization nonreciprocities, but for the nonlinearity
problem each counterpropagating pulse would experience mostly the self-effect, which would
yield a nonreciprocity with a power imbalance. Furthermore, for the same mean power the
nonlinearity would be further increased, because it depends on the peak power, which is much
higher in the case of a pulse.
References
[1] Arditty, H. J., et al., “Current Sensor Using State-of-the-Art Fiber-Optic Interferometric Techniques,” Proceedings of IOOC,
Paper WL3, 1981.
[2] Böhm, K., K. Petermann, and E. Weidel, “Sensitivity of a Fiber-Gyroscope to Environmental Magnetic Fields,” Optics Letters,
Vol. 7, 1982, pp. 180–182.
[3] Schiffner, G., B. Nottbeck, and G. Schröner, “Fiber-Optic Rotation Sensor: Analysis of Effects Limiting Sensitivity and
Accuracy,” Springer Series in Optical Sciences, Vol. 32, 1982, pp. 266–274.
[4] Bergh, R. A., H. C. Lefèvre, and H. J. Shaw, “All Single-Mode Fiber Optic Gyroscope,” Springer Series in Optical Sciences,
Vol. 32, 1982, pp. 252–255.
[5] Bergh, R. A., H. C. Lefèvre, and H. J. Shaw, “Geometrical Fiber Configuration for Isolators and Magnetometers,” Springer
Series in Optical Sciences, Vol. 32, 1982, pp. 400–405.
[6] Hotate, K., and K. Tabe, “Drift of an Optical Fiber Gyroscope Caused by the Faraday Effect: Influence of the Earth’s Magnetic
Field,” Applied Optics, Vol. 25, 1986, pp. 1086–1092.
[7] Marrone, M. J., et al., “Internal Rotation of the Birefringence Axes in Polarization-Holding Fibers,” Optics Letters, Vol. 12,
1987, pp. 60–62.
[8] Li, J., et al., “Design and Validation of Photonic Crystal Fiber for Fiber-Optic Gyroscope,” ISS 2013, Karlsruhe, 2013, pp. 5.1–
5.12.
[9] Logozinskii, V. N., “Magnetically Induced Non-Faraday Nonreciprocity in a Fiber-Optic Gyroscope,” Journal of
Communications Technology and Electronics, Vol. 51, No. 7, 2006, pp. 836–840.
[10] Blake, J., “Magnetic Field Sensitivity of Depolarized Fiber Optic Gyros,” SPIE Proceedings, Vol. 1367, 1990, pp. 81–86.
[11] Ezekiel, S., J. L. Davis, and R. W. Hellwarth, “Intensity Dependent Nonreciprocal Phase Shift in a Fiberoptic Gyroscope,”
Springer Series in Optical Sciences, Vol. 32, 1982, pp. 332–336.
[12] Kaplan, A. E., and P. Meystre, “Large Enhancement of the Sagnac Effect in a Nonlinear Ring Resonator and Related Effects,”
Springer Series in Optical Sciences, Vol. 32, 1982, pp. 375–385.
[13] Bergh, R. A., et al., “Source Statistics and the Kerr Effect in Fiber-Optic Gyroscopes,” Optics Letters, Vol. 7, 1982, pp. 563–
565.
[14] Petermann, K., “Intensity-Dependent Nonreciprocal Phase Shift in Fiber-Optic Gyroscopes for Light Sources with Low
Coherence,” Optics Letters, Vol. 7, 1982, pp. 623–625.
[15] Bergh, R. A., H. C. Lefèvre, and H. J. Shaw, “Compensation of the Optical Kerr Effect in Fiber-Optic Gyroscopes,” Optics
Letters, Vol. 7, 1982, pp. 282–284.
[16] Bergh, R. A., “All-Fiber Gyroscope,” in Optical Fiber Rotation Sensing, Burns, W. K., (ed.), New York: Academic Press,
Chapter 5, 1994, pp. 137–170.
CHAPTER 8
Scale Factor Linearity and Accuracy
8.1 Problem of Scale Factor Linearity and Accuracy
The modulation-demodulation detection scheme described in Section 3.2.2 provides a very good
bias, as it preserves the reciprocity of the ring interferometer. However, if a high-performance
gyroscope must have a stable and low-noise bias, it also requires good accuracy over the whole
dynamic range and not only about zero. The measurement of interest is the integrated angle of
rotation and not simply the rate. Any past error will affect the future information. This constraint
implies the need for an accurate measurement at any rate (i.e., an accurate scale factor).
Furthermore, the intrinsic response of an interferometer is sinusoidal, while the desired rate
signal of a gyroscope should be linear.
This problem is solved with a closed-loop (or phase-nulling) signal processing approach as
proposed independently by Davis and Ezekiel [1] and Cahill and Udd [2]. The demodulated biased
signal (or open-loop signal) is used as an error signal that is fed back into the system to generate
an additional feedback phase difference ∆ϕ FB that is opposite to the rotation-induced phase
difference ∆ϕ R (Figure 8.1). The total phase difference ∆ϕ T = ∆ϕ R + ∆ϕ FB is servo-controlled
on zero, which provides good sensitivity, as the system is always operated about a high-slope
point. With such a closed-loop scheme, the new measurement signal is the value of ∆ϕ FB. This
yields a linear response with good stability, as this feedback value ∆ϕ FB is independent of the
returning optical power and of the gain of the detection chain. The measured value of the rotation
rate becomes:
Figure 8.1 Principle of closed-loop operation of an interferometric fiber gyro with a feedback phase difference ∆ϕ FB.
(8.1)
A closed-loop operation provides a stable and linear measurement of the optical phase
difference in the interferometer; however, the accuracy of the scale factor of the rotation rate
measurement also depends on the stability of the mean and on the geometrical stability of the
area of the coil.
We are going to detail the closed-loop operation methods to provide high scale-factor linearity
as well as the approaches to ensure in addition high scale-factor accuracy with coil area and
wavelength control.
8.2 Closed-Loop Operation Methods to Linearize the Scale
Factor
8.2.1 Use of a Frequency Shift
The first proposed arrangements of closed-loop operation used a frequency shift generated with
acousto-optic modulators (AOM), also called Bragg cells [1, 2]. The Sagnac effect may be
interpreted actually as a Doppler effect on the coil beam splitter (as seen in Section 2.1.1).
Therefore, a frequency shifter placed at the beginning of the coil may null out the Doppler shift
of the Sagnac effect. The absolute phase ϕ abs accumulated in the propagation is:
(8.2)
where f is the temporal frequency of the wave. Then a feedback frequency shift ∆fFB generates a
feedback phase difference ∆ϕ FB:
(8.3)
The new measurement is now ∆fFB, which is linearly dependent on the rotation rate Ω:
(8.4)
To get high sensitivity, the error signal is obtained with an additional phase modulator and the
usual modulation-demodulation scheme of the open-loop scheme [3].
However, to cover the whole dynamic range, this feedback frequency shift ∆fFB has to vary
around zero between plus and minus several hundreds of kilohertz. Because AOMs work only at a
high center frequency (typically 50 to 100 MHz) with a relative bandwidth of several percentages,
this requires the use of two cells to generate the feedback frequency shift ∆fFB by difference.
They can be placed in opposition at one end of the coil, the first cell generating an upshift and the
second one generating a downshift [Figure 8.2(a)], or they can be placed at both ends of the coil
with the same shift sign [Figure 8.2(b)]. One is operated at a constant center frequency fc, yielding
a constant wavelength shift ∆λc, and the other is operated at a controlled frequency fc + ∆fFB,
yielding a wavelength shift ∆λc + ∆λFB. When the gyro is at rest, the feedback frequency shift
∆fFB should ideally be zero, but the intrinsic single-mode reciprocity has been destroyed [4].
Considering first the symmetrical case [Figure 8.2(b)], the source emits a wavelength λ, but the
actual wavelength of the counterpropagating waves is:
Figure 8.2 Closed-loop scheme with two acousto-optic frequency modulators (AOM): (a) modulators in series and (b)
modulators at both ends of the coil.
λ between the splitter and the modulators at the input;
λ + ∆λc between both modulators;
λ + 2∆λc between the modulators and the splitter at the output.
Both counterpropagating waves preserve the same wavelength λ + ∆λc between both
modulators, but they have a difference of 2∆λc between the splitter and the modulators. This
wavelength difference between the opposite waves yields a nonreciprocal phase difference ∆ϕ 1
when they propagate between the splitter and the first modulator, and another nonreciprocal phase
difference ∆ϕ 2 with an opposite sign between the splitter and the second modulator. When the
distances between the splitter and each modulator are perfectly equal, the total nonreciprocal
phase difference ∆ϕ 1 + ∆ϕ 2 is nulled out, but a small imbalance ∆LAOM in the exact symmetry of
the two modulators yields a residual phase error:
(8.5)
With a typical center frequency fc of 100 MHz, this nonreciprocal phase error is as high as 2
μrad/μm.
When the two modulators are placed in opposition at one end of the coil [Figure 8.2(a)], a
similar result is obtained where ∆LAOM is now the distance between both modulators. There is
the same problem of stability, but there is also a constant bias offset related to the averaged value
of ∆LAOM.
Several solutions have been proposed to improve the mechanical stability of this assembly,
including a double Bragg cell in which both frequency shifters are combined in the same
component [5] and an integrated optic implementation [6], but this closed-loop approach with
acousto-optic frequency shifters is complex technologically and the power consumption is
relatively high. Furthermore, even with an improved stability, these approaches do not respect
reciprocity. Note that this sensitivity to modulator imbalance has been proposed as a sensing
method to detect any effect that affects this imbalance [7].
8.2.2 Use of an Analog Phase Ramp (or Serrodyne Modulation)
This stability problem of the pair acousto-optic frequency shifters is completely overcome by the
use of a linear phase ramp first proposed by Arditty et al. [8]. A frequency is the derivative of a
phase, and a phase ramp modulation ϕ PR(t) = ϕ̇t (where ϕ̇ is the time derivative, i.e., the slope)
that is applied with a phase modulator instead of a frequency shifter is equivalent to a frequency
shift ∆f = ϕ̇/2π:
(8.6)
Such a processing scheme, also called serrodyne modulation, allows one to work positively or
negatively about zero, depending on the sign of the ramp slope, thereby eliminating the former
need of a high center frequency, which has been shown to destroy the intrinsic reciprocity of the
ring interferometer. However, the ramp cannot be infinite, and, in practice, a sawtooth modulation
form has to be used with a very fast flyback at the reset (Figure 8.3). This requires a phase
modulator with a flat efficiency over a large bandwidth, which is one of the main technical
advantages of integrated optics (see Appendix C).
The effect on a gyro can be viewed directly by considering the group delay difference ∆τg
between the long and short paths that connect the phase modulator and the splitter. Like the case of
the biasing modulation (see Section 3.2.2), the same phase ramp feedback modulation ϕ PR(t) is
applied on the two opposite waves, but because of the delay ∆τg, this generates a feedback phase
difference ∆ϕ PR(t) with:
∆ϕ PR(t) = ϕ PR(t) − ϕ PR(t − ∆τg) (8.7)
∆ϕ PR(t) is equal to ϕ̇ ⋅ ∆τg during the ramp and to ϕ̇ ⋅ ∆τg − ϕ RS during a time ∆τg after the
flyback. The value ϕ RS is the height of the phase reset. This reset induces an error unless ϕ RS is
equal to 2π rad (or a multiple of 2π), the period of the interferometer response [8–12].
Figure 8.3 Analog sawtooth phase ramp modulation ϕ PR and induced feedback phase difference ∆ϕ PR.
Let us assume that when the processing loop is closed, the slope ϕ̇ is adjusted to compensate for
the rotation-induced phase difference ∆ϕ R and that the total phase difference ∆ϕ T is nulled out:
∆ϕ T = ∆ϕ R + ∆ϕ PR = 0 (8.8)
that is:
∆ϕ R = −ϕ̇ ⋅ ∆τg (8.9)
After the reset, ∆ϕ T becomes equal to ϕ RS instead of zero. Assuming, for simplicity, a biased
sine response of the interferometer (Figure 8.4), the signal is zero when ∆ϕ T = 0, but becomes
sin∆ϕ RS during the time ∆τg after each reset. The gating out of this spurious signal has been
proposed [9], but it has been shown that this is actually a very convenient error signal for
checking the phase modulator efficiency with a second feedback loop activated at each reset [8,
10].
Figure 8.4 Effect of the reset of the phase ramp.
With such a second processing loop, the reset is precisely controlled on 2π rad, and therefore
the counting of the positive and negative resets provides an accurate measurement of the angle of
rotation. The slope ϕ̇ is proportional to the rate Ω, as ∆ϕ R = −ϕ̇∆τg:
(8.10)
Because ∆τg = n ⋅ L/c, we have:
(8.11)
Each reset corresponds precisely to 2π rad, and also to an angular increment θinc, defined by:
(8.12)
where T is the period of the sawtooth modulation, and as
for any slope, we have:
(8.13)
This incremental value is independent of the coil length L, but it is inversely proportional to the
coil diameter D. For a wavelength λ = 850 nm, the product θinc ⋅ D is equal to 255 arcsec ⋅ mm,
and for a wavelength λ = 1,550 nm, it is 465 arcsec ⋅ mm. Note that the frequency of these 2π
increments is the same as the one obtained with the direct frequency shift feedback and as the one
naturally generated in a laser gyro (see Section 2.2.1).
An important feature of this phase ramp processing technique is that the control of the 2π reset
does not have to be very accurate, as it has only a third-order dependence on the actual scale
factor accuracy as outlined by Kay [10]. To understand this result, let us consider Figure 8.5 in
that, starting from an ideal phase ramp, there is a small decreasing ∈m in the gain of the
modulation chain. The actual modulation ϕ PRa is related to the ideal modulation ϕ PRi by:
Figure 8.5 Effect of a change of the ideal gain of the modulation chain.
ϕ PRa = (1 − ∈m)ϕ PRi (8.14)
the actual induced difference ϕ PRa is:
∆ϕ PRa = (1 − ∈m)∆ϕ PRi (8.15)
The actual phase ramp undercompensates for the rotation rate-induced phase difference ∆ϕ R
and the total actual phase difference ∆ϕ Ta becomes:
∆ϕ Ta = ∆ϕ R + ∆ϕ PRa = ∈m∆ϕ R (8.16)
However, if the ramp slope is decreased, the reset height is also decreased with the same
coefficient ∈m, and during a time ∆τg after the reset, the actual phase ramp overcompensates
∆ϕ R:
∆ϕ Ta = 2π − ∈m(2π − ∆ϕ R) (8.17)
Therefore, over a full period T of the sawtooth modulation, and assuming a biased sine
response of the open-loop error signal, the signal is:
sin[∈m∆ϕ R] during T − ∆τg (8.18)
−sin[∈m(2π − ∆ϕ R)] during ∆τg
If these phase errors are small, the sine response may be linearized (ignoring third-order
terms) and the mean signal becomes:
(8.19)
As:
∆ϕ R = −ϕ̇∆τg (8.20)
(8.21)
∆ϕ R ⋅ T = 2π∆τg (8.22)
the mean error signal ⟨S⟩ remains equal to zero independently of the exact value of the reset, if
∈m is small enough to keep the operating points in the linear parts of the sine response.
This analog phase ramp feedback scheme looks very attractive, but it requires a very short and
stable flyback time to yield high-scale factor stability and linearity [10, 12]. Typically, a stability
of 10 ppm requires a flyback time of less than 1% of the transit time through the coil (i.e., less
than a few tens of nanoseconds).
8.2.3 Use of a Digital Phase Ramp
This problem of analog phase ramp flyback is very simply solved with a digital approach
proposed by Arditty et al. [4, 13, 14]. Instead of a continuous ramp, the digital phase ramp
generates phase steps ϕ S with a duration equal to the delay ∆τg through the coil. The induced
phase difference ∆ϕ FB is then constant and equal to the step value ϕ S. These phase steps and the
resets can be synchronized with the square-wave biasing modulation, which is preferably
operated at the proper frequency (see Sections 3.2.2 and 3.2.3), that is, the half-period is equal to
this same transit time ∆τg (Figure 8.6). The amplitude ϕ S of the phase step is set by the phasenulling feedback loop to be opposite to the rotation-induced phase difference ∆ϕ R:
ϕ s = −∆ϕ R (8.23)
and this value ϕ S of the phase step provides a linearized readout of the phase difference induced
by the rotation rate.
Figure 8.6 Digital phase ramp with steps and resets synchronized with the square-wave biasing modulation.
The real magic of the digital phase ramp technique is the fact that the use of digital logic and a
D/A (digital/analog) converter naturally yields an adequate synchronized reset with the automatic
overflow of the converter for any value of the step, thus making implementation of this powerful
technique very simple (Figure 8.7). A digital register contains the digital value Ds of the phase
step with a dynamic range that can be very high (more than 25 bits), and a digital integrator
generates the digital value DR of the staircase ramp. The analog driving voltage of the phase
modulator is produced with a D/A converter and a buffer amplifier. With N bits, they transform a
digital number D into an analog voltage over a dynamic range between zero and (2N − 1)VLSB,
where VLSB is the driving voltage corresponding to the least significant bit (LSB). When DR
becomes higher than (2N − 1), the automatic overflow yields a voltage equal to (DR − 2N)VLSB.
If the gain of the modulation chain is adjusted such that:
2NVLSB = 2Vπ (8.24)
where Vπ is the voltage that generates a π-rad phase shift, as detailed in Appendix C. The
overflow automatically generates a reset that is equivalent to the 2π reset of the analog ramp and
does not produce any scale-factor error.
Figure 8.7 Generation of the digital phase ramp.
Note that this automatic overflow may be used with the sole ramp, but also with the digital sum
of the ramp and the square-wave modulation (Figure 8.8). This allows the use of a push-pull
connection of the two modulators of the Y-junction, which reduces their effective nonlinearity
(see Section 3.3.4). With the analog approach, the electronic circuitry that generates the sawtooth
modulation is so delicate that it is preferable to apply this feedback modulation on one modulator
and apply the biasing modulation on the second modulator with an independent circuit, which
makes the setup more complicated.
Figure 8.8 D/A overflow when the phase ramp and the phase modulation are added digitally.
The resets, as the steps, are synchronized with the clock time ∆τg. This eases the control of the
exact 2π value of 2NVLSB with a second feedback loop that is activated at each flyback and is not
disturbed by the transients of the square-wave biasing modulation, as it is also synchronized with
∆τg.
Let us take, for example, the case of a rotation-induced phase difference ∆ϕ R equal to −4π/5
rad (Figure 8.9). With an analog ramp, the resets are equal to 2π and have a periodicity of 2.5 ⋅
∆τg, and then they overlap with the biasing modulation that has a periodicity of 2 ⋅ ∆τg. However,
with the digital phase ramp, the step value is 4π/5 rad and the first reset waits for the third clock
time ∆τg while the second reset happens only after 2 ⋅ ∆τg. These resets are no longer periodic,
but they are now synchronized with the clock time ∆τg and the biasing modulation.
Figure 8.9 Nulling phase ramp for a rotation-induced phase difference of −4π/5 rad: (a) analog case and (b) digital case.
Note that, strictly speaking, the resets of the digital ramp are not equal to 2π. This value 2π is
the difference between the value of where the next step would go if the number of bits were not
limited and the value of where it is actually going because of the overflow.
Another very important point is that the digital phase ramp does not require a large number of
bits for the D/A converter, though the actual dynamic range between 2π rad and a resolution of
10–8 rad is as high as 30 bits. With an N-bit converter, the least significant phase step ϕ LSB is:
(8.25)
The exact value of the required phase step ϕ S is contained within:
mϕ LSB ≤ ϕ s < (m + 1)ϕ LSB
where m is an integer smaller than 2N. The value ϕ S is stored in the rate register, which must have
a large enough number of bits, but only the N most significant bits of the ramp are used in the
D/A to drive the phase modulator. Instead of a series of identical steps, the D/A converter
generates a ramp composed of m′ steps m ⋅ ϕ LSB, which undercompensates for the rotation, and m
″ steps (m + 1)ϕ LSB, which overcompensates for the rotation. On average, the feedback phase
difference < ∆ϕ FB > is such as:
(8.26)
that is, m′ and m″ are such as:
(8.27)
This phase step averaging yields the same averaging of the actual interference signal if the
maximum instantaneous error (i.e., ϕ LSB) remains in the linear part of the sine response. With 10
bits, for example, ϕ LSB is equal to 2π/210 = 6 × 10–3 rad, which corresponds to a rate as high as
several thousands of degrees per hour; but the residual nonlinearity of the sine response, (ϕ LSB −
sinϕ LSB)/ϕ LSB ≈ ϕ LSB2/6, remains below 10 ppm. The averaging also applies if ϕ s is smaller
than ϕ LSB: there is no dead zone. For example, if ϕ s = [1/(m′ + 1)]ϕ LSB, there will be no step
during m′ clock times and just one step during one clock time (Figure 8.10).
Another averaging process is also very useful for relaxing the requirement of linearity of the
D/A converter. A converter defect generates a phase error, and the value of the phase ramp ϕ sj
after j steps may be different from jϕ s; but on average the feedback phase difference is over m
steps:
(8.28)
The only condition is that the linearity error of the converter remains in the linear part of the
interferometer sine response. The usual specification of D/A converters is a linearity error of less
than one LSB; therefore, the previous condition about ϕ LSB applies also for this second
averaging process. In a practical device, drift down to 10-4°/h can be reached with ϕ LSB
corresponding to 10°/h: five orders of magnitude.
Figure 8.10 Actual phase ramp when the required phase step ϕ s is smaller than ϕ LSB (case where ϕ s = ϕ LSB/4).
Besides these very useful relaxed constraints on the characteristics of the converter, the digital
phase ramp technique also has several basic advantages over its analog counterpart. The true rate
measurement is the digital value Ds of the phase step ϕ s, which is stored in a register of the
digital logics. The clock time driving the electronics has to be approximately matched to the
transit time ∆τg through the coil, to limit the width of the transient pulses, but the step value is not
directly related to ∆τg. When ∆τg is changing, this slightly modifies the width of the transient
spikes that are gated out, but the value of the feedback step ϕ s remains unchanged. This is a clear
advantage over the analog approach, in which ∆τg is part of the scale factor through the value of
the index n. With the digital ramp and a stable electronic clock, the scale factor has only the basic
dependence of the Sagnac effect on the geometrical area of the coil (i.e., a temperature coefficient
of 10–6/°C with silica), instead of an index dependence, which has a temperature coefficient of 8.5
× 10–6/°C.
Furthermore, the usual readout of these analog ramping techniques is the count of the 2π resets,
which yields, as we have already seen, an angular increment value θinc = nλ/D. With the digital
approach, in which the rate is stored in a register, it is easy to generate any submultiple value of
θinc without waiting for the 2π resets and particularly θinc/2N′ using N′ bits in parallel for the
output. This is very important for stabilization and pointing applications where very small
increments are required.
It is even possible to tailor the value of the increment to avoid any quantization error within the
measurement
bandwidth.
Assuming,
for
example,
a
typical
noise
of
(i.e., a random walk of 1
= 0.03
), the increment may be adjusted to 0.03 arcsec, while θinc is on the order of few arcseconds to
avoid any quantization error within a 1-kHz bandwidth.
The value of the rate is stored in a digital register and can be used directly with a parallel
interface, but this would require a very large number of bits. The I-FOG is intrinsically a rate
gyroscope. To generate 2π-increments or subincrements with a logic integration of the rate value
simplifies the interface and yields a rate-integrating gyroscope; but the difference between a rate
device and a rate integrating device is simply a matter of signal processing logic and interface
protocol and is not a basic difference.
Note that the digital phase ramp approach allows the dynamic range to be easily extended to
several fringes. The phase step value is stored in a register that may correspond to more than ±π
rad, and the overflow of the D/A converter automatically limits the range of actual phase
modulation to less than 2π.
This analysis demonstrates that the digital phase ramp technique is a highly efficient closedloop method for linearizing the scale factor of an interferometric fiber gyroscope. It has
fundamental advantages and does not require stringent performance for the electronic
components.
8.2.4 All-Digital Closed-Loop Processing Method
The first implementation of the digital phase ramp technique used an analog demodulator to get
the sinusoidal rate signal that serves as the feedback error signal of closed-loop processing
(Figure 8.11) [4, 13, 14]. However, such a method is difficult to implement over 160 dB of
dynamic range of an operating FOG because of the intrinsic offset drift of the analog
demodulation circuit and of the A/D (analog/digital) converter that digitizes the integrated error
signal. Typical variations are on the order of 10-4 to 10-5/°C of the output voltage range.
Compensation and modeling of this thermal drift are possible, but they increase the complexity of
the electronics, yielding higher cost and lower reliability.
Figure 8.11 Original implementation of digital phase ramp feedback with an analog demodulation.
This problem can be completely overcome with an all-digital closed-loop approach proposed
by Arditty et al. [15, 16]. The value of the modulated output signal is converted directly into
digital form at each half-period, and the converter value of the second half-period is simply
subtracted digitally from that of the first half-period. Such a digital demodulation is a calculation
and then is intrinsically free of any source of electronic long-term drift. In particular, drift of the
analog input offset of the A/D converter is canceled out by the subtraction of the odd and even
samples with the demodulation. However, one might think that an A/D converter with a very large
number of bits is required to limit the dead zone of the LSB: 180 dB of dynamic range is
equivalent to 30 bits, as we have already seen. Fortunately, this crude analysis is not true. Because
the sampling frequency is typically in the megahertz range, the bandwidth of the analog output
signal is quite large, and thus a great deal of white noise is present (Figure 8.12). Signalprocessing theory shows that sampling the analog signal with an LSB just smaller than the σ
value of the noise is sufficient. Thus, digital integration yields the same noise reduction as with
analog filtering, without any dead zone or spurious offset.
This can be simply understood with a hand-waving argument, by considering an analog signal
with a noise extending over a range of several bits of the quantization circuit (Figure 8.13). If the
mean value of the analog signal is zero, there will be as many digital samples above zero as there
are below. If the mean value is slightly positive, although the variation is much smaller than the
LSB, it is still possible to measure it because, on average, there will be slightly more digital
samples above zero than below. There is no degradation of the mean signal if the σ value of the
noise is bigger than the LSB.
Figure 8.12 Actual signal with its noise.
In practice, the noise is typically
or, in relative optical power value,
With an analog bandwidth of 1 MHz, the σ value of the noise becomes 3 × 10-4
of the π/2 biasing power; that is, 12 bits are enough to convert the analog signal over the entire
dynamic range of power variation, without any dead zone. The use of a digital integrator yields
the same noise reduction as lowpass analog filtering, but without any electronic source of long-
term drift.
Figure 8.13 Digital quantization of an analog signal with an additional noise.
This digital demodulation scheme is naturally compatible with digital feedback schemes such
as the digital phase ramp. This makes it even more efficient, because the error signal is servocontrolled to zero. With its noise extending over ±4σ, it is spread only over 3 bits, and this avoids
the effect of A/D converter nonlinearity which is directly translated into scale-factor nonlinearity
when digital demodulation is used with an open-loop scheme [17]. All-digital closed-loop
processing yields a simple implementation that will be particularly adequate for integration. The
functions are:
Optical detection;
Analog gating of the transient pulses and filtering;
A/D conversion of the gyro signal;
Digital logic driven with a common clock;
D/A conversion of the feedback phase modulation.
The entire logic circuitry may be implemented on a single logic circuit. Its functional
schematics are (Figure 8.14):
Subtraction of odd and even samples to demodulate the error signal;
Digital integration of this demodulated error signal;
Storage of the rotation rate value in a register;
Second integration of the rate value to generate the digital ramp;
Digital addition of the square-wave biasing modulation.
Figure 8.14 Functional schematics of the logic circuit used in the all-digital closed-loop approach.
This all-digital approach may also be used for the second feedback loop which controls the
gain of the phase modulation chain as proposed by Arditty and al. [15], particularly with the 2π
reset. The second error signal is obtained by comparing the digital value of the sampled detector
signal just before and just after the reset. A digital integrator is also used to close this second
loop, and a second D/A converter controls the reference voltage of the first D/A that generates the
ramp or the gain of the buffer amplifier (Figure 8.15). This second D/A converter operates about
dc to compensate for the long-term drift of the phase modulator response, but there is no
stringent requirement of quantization error for this converter either. As we have already seen for
the analog ramp (Section 8.2.2), the scale factor error induced by an imperfect 2π reset is only a
third-order effect. This also applies to the digital ramp, but with the additional advantage of
avoiding the requirement of a very fast flyback, as the transients are synchronized and can be
gated out [16]. A control of the 2π reset within 1,000 ppm (i.e., 10 bits) is sufficient to ensure a
linearity better than 10 ppm.
Figure 8.15 Functional schematics of the all-digital approach with a second feedback loop for the control of the modulation
amplitude.
It is possible to directly view the reason why the digital ramp technique tolerates many defects
(such as imperfect 2π reset control, quantization, nonlinearity of the electronic drive,
nonlinearity of the phase modulator response) without degrading the scale factor performance.
The actual phase ramp ϕ PRa is the sum of an ideal phase ramp ϕ PRi and a defect ϕ PRd. The
induced phase difference is:
∆ϕ PRa = ∆ϕ PRi + ∆ϕ PRd (8.29)
As with any phase modulation, the defect ∆ϕ PRd of the phase difference is given by the delay
∆τg through the coil:
∆ϕ PRd = ϕ PRd(t) − ϕ PRd(t − ∆τg) (8.30)
As the mean value of a difference is the difference of the mean values, the mean defect <
∆ϕ PRd > is:
< ∆ϕ PRd > = < ϕ PRd(t) > − < ϕ PRd(t − ∆τg) > (8.31)
Over a phase ramp period, mean values < ϕ PRd(t) > and < ϕ PRd(t − ∆τg) > are perfectly equal
because of reciprocity; therefore:
< ∆ϕ PRd > = 0 (8.32)
If the defect ∆ϕ PRd of the phase difference remains in the linear part of the biased sine
response of the interferometer, the mean error of the interference signal is also averaged out to
zero as ∆ϕ PRd. This applies to most defects except imperfect transients that yield instantaneous
errors much larger than this linear range, but with the digital ramp they can be synchronously
gated out.
This simple result stating that the averaged error is zero is absolutely fundamental and explains
why digital phase ramp feedback yields scale factor linearity in the parts per million range
despite defects in the electronic chain and the response of the MIOC phase modulator that are
much higher.
In summary, the all-digital closed-loop processing method brings a drift-free high-linearity
phase measurement without requiring a large number of bits for the various converters, because
the quantization errors as well as the component defects are canceled out through averaging
processes. While reciprocity is a fundamental concept of creating a good optical system, the alldigital closed-loop approach is just as important from the signal processing standpoint: it also
allows a perfect device to be created from imperfect components. It does appear to be the ideal
processing technique for the interferometric fiber-optic gyroscope, combining performance,
simplicity, and the potential for optimal circuit integration.
8.2.5 Control of the Gain of the Modulation Chain with Four-State Modulation
As just seen, the gain of the modulation chain driving the phase ramp can be controlled with its
2π resets. However, the frequency of these resets depends on the rotation rate and the stability of
this gain-control feedback loop is therefore delicate.
This can be overcome with the four-state modulation [18]. Instead of the two-state biasing
modulation described in Section 3.2.2 where the phase difference of the ring interferometer is
modulated between +ϕ b and −ϕ b, the biasing modulation uses four states ϕ b, aϕ b, −ϕ b, and −aϕ b,
where the coefficient a is such as ϕ b + aϕ b = 2π. This is obtained with a four-state phase shift
modulation in the phase modulator of +ϕ b/2, +aϕ b/2, −ϕ b/2, and −aϕ b/2 with a duration ∆τg/2
for each state, instead of +ϕ b/2, and −ϕ b/2 with a duration ∆τg (∆τg is the difference of group
transit time between the long and short paths that connect the modulator and the splitter as seen in
Section 3.2.2).
At rest, the four modulation states yield the same signal (Figure 8.16):
Figure 8.16 Four-state modulation at rest (ϕ b = 3π/4, aϕ b = 5π/4, a = 5/3, in this example).
P(0, −ϕ b) = P(0, ϕ b) = P(0, aϕ b) = P(0, −aϕ b) = P0/2(1 + cosϕ b) = P0/2(1 + cosaϕ b) (8.33)
Since
cosϕ b = cosaϕ b (8.34)
when
ϕ b = 2π − aϕ b (8.35)
and in rotation, there are:
P(∆ϕ R, ϕ b) = P0/2[1 + cos(∆ϕ R + ϕ b)]
P(∆ϕ R, aϕ b) = P0/2[1 + cos(∆ϕ R + aϕ b)] (8.36)
P(∆ϕ R, −ϕ b) = P0/2[1 + cos(∆ϕ R − ϕ b)]
P(∆ϕ R, −aϕ b) = P0/2[1 + cos(∆ϕ R − aϕ b)]
Both states ϕ b and −aϕ b yield the same value as well as both states −ϕ b and aϕ b, and the
difference between these two pairs of states yields the same signal as with a two-state modulation
[see (3.10)]:
Figure 8.17 Four-state modulation with a rate signal at the proper frequency fp = 1/(2 ∆τg) (with ∆ϕ R = π/8 in this example).
∆P(∆ϕ R, ϕ b) = P0sinϕ bsin∆ϕ b (8.37)
As one can see on Figure 8.17, the modulated rotation signal does keep a frequency equal to the
proper (or eigen) frequency, fp = 1/2 ⋅ ∆τg, which as seen in Section 3.2.3 is key for best
performance.
Now, when the gain of the modulation chain is increased by (1 + α), the four states become (1 +
α) ⋅ ϕ b, (1 + α) ⋅ aϕ b, −(1 + α) ⋅ ϕ b, and −(1 + α) ⋅ aϕ b. As seen in Figure 8.18, it yields an error
signal at 2 ⋅ fp, twice the proper frequency. This Vπ signal can be demodulated independently of
the rotation signal that is at fp and used as an error signal to control the amplitude of the biasing
modulation. The Vπ control loop works at the constant high frequency 2 ⋅ fp, contrarily to the one
of the 2π resets of the phase ramp which depends on the rotation rate.
Since the phase ramp is generated through the same modulation chain (digital register, D/A
converter, analog amplifier, and integrated-optics phase modulator), as the biasing modulation,
its amplitude is calibrated, as well as its 2π resets.
The simple way to generated this four-state biasing modulation, is to have a first two-state
square-wave modulation ±π at fp and a second two-state square-wave modulation ±((a − 1)/(a +
1))π in quadrature. In the case of a = 5/3 where the four-states of ∆ϕ m(t) are 3π/4, 5π/4, −3π/4,
and −5π/4, it yields a first modulation ±π and a second modulation ±π/4 in quadrature. It is
obtained with a biasing phase shift ϕ m(t) of ±π/2 and ±π/8 in quadrature generated in the phase
modulator. The digital phase ramp is then added with its 2π resets. The total amplitude of the
phase shift modulation ϕ m(t) is 2π + 5π/4 < 4π.
Figure 8.18 Four-state modulation with a change (1 + α) of the gain of the modulation chain (a = 5/3 and α = 1/6 in this
example). The Vπ signal is at 2 × fp = 1/∆τg, twice the proper frequency.
With an N-bit D/A converter which has a dynamical range of (2N − 1) VLSB, where VLSB is the
voltage generated by the 0th bit (least significant) bit, there are:
A 0 − π square-wave phase shift modulation generated by a digital modulation of the (N − 2)th
bit, yielding 2N−2 VLSB;
A 0 − ϕ b′ square-wave modulation generated by the digital value of ϕ b′((N − 4)th bit for ϕ b′ =
π/4), yielding 2N-4VLSB;
A phase ramp with 2π resets generated by the reset of the (N − 1)th bit.
8.2.6 Potential Spurious Lock-In (or Deadband) Effect
We saw with the laser gyro (Section 2.2.1) that there is a problem of lock-in at low rates which
yields a deadband (or dead zone) about zero.
As a fiber gyro being a passive interferometer and not an active resonator, it was expected that
it should not face it. This is actually true with an open-loop design, but a closed-loop operation
may yield such a lock-in problem (Figure 8.19) as seen for example in [19, 20].
It is due to any parasitic effect that yields an error bias signal Se(ϕ PR) that varies with the phase
ramp ϕ PR. When the ramp gets to a value ϕ PR0 such as Se(ϕ PR0) is opposite to the rate signal
SR, the total signal ST = SR + Se(ϕ PR0) becomes zero and then the feedback loop does not need to
generate any ramping anymore, so the closed-loop output signal becomes blocked on zero.
There are many potential sources of lock-in: electronic coupling, Michelson interferometer
due to residual coherent back-reflections at the interfaces between the MIOC and the coil fiber
ends, lack of polarization filtering, or improper biasing modulation frequency.
Figure 8.19 Transfer functions for an ideal fiber gyro (solid line) and for a fiber gyro with deadband (dashed line), due to lock-in
effect.
Such ramp-dependent parasitic signals may be evaluated using an open-loop scheme and
adding a constant ramp. Each kind of defect yields a specific temporal signature over the reset
ramp period. For example, the Michelson interferometer gets twice the phase ramp modulation
and is then scanned over 4π radian for a 2π phase ramp: it yields two periods of a sinus over one
period of the reset ramp.
It is possible to find many patents and articles that treat modulation techniques to reduce such
lock-in effects (see, for example, [21] and its references), but with careful electronic and optical
design, all these effects can be sufficiently reduced to avoid any problem of lock-in without
additional modulation. Deadband should not be viewed as a limit to high-performance fiber
gyroscopes.
8.3 Scale Factor Accuracy
8.3.1 Problem of Scale Factor Accuracy
Assuming one has achieved a perfect measurement of the Sagnac phase difference ∆ϕ R with the
phase-nulling closed-loop scheme, the scale factor remains, as we have seen, related to the area
of the coil and to the wavelength of the source. The coil area A should have a variation of a part
per million per degree Celsius considering silica thermal expansion: it is 0.5 ppm/°C and the
relative variation of the area of the circle is the double of one of its perimeters. In practice, there
is an additional length increase due to the expansion of the polymer coating that puts the fiber
under tensile stress as seen in Section B.1.5. The total relative length change is typically 4 to 5
ppm/°C for an 80 μm fiber with a coating diameter of 160 μm. It yields a thermal dependence of
the area:
This requires modeling for high-grade applications.
Wavelength stability is more difficult to solve. For example, semiconductor diode sources,
which are popular sources for medium accuracy FOGs, have a typical wavelength drift of about
400 ppm/°C with temperature and of about 40 ppm/mA with driving current, even without taking
into account additional factors such as aging or the feedback effect of the light returning to the
source. Temperature control of the source and a stabilized driver are sufficient for a medium
accuracy in the 100-ppm range.
Erbium-fiber ASE sources (see Section A.2.9) have a much better stability, but highperformance applications require a direct wavelength control to be able to reach the parts per
million range.
8.3.2 Wavelength Dependence of an Interferometer Response with a Broadband
Source
A first fundamental question is to precisely define the effective wavelength involved in the scale
factor when the broadband source needed for high performance has a relative spectrum width of
several percentages, which is several orders of magnitude larger than the parts per million
stability that is looked for. As explained in Section 2.3.1, the rotation-induced phase difference
∆ϕ R may be expressed with an equivalent path length difference ∆LR that is perfectly wavelengthindependent:
(8.38)
with
(8.39)
The intrinsic unbiased interference response is:
(8.40)
where C(∆LR) is the coherence function of the source (see Section A.12.1) as measured in a
scanning interferometer in a vacuum like a Michelson interferometer, and where λcent is the
central wavelength of the spectrum. However, this simple result applies only if the spectrum is
symmetrical with respect to the optical spatial frequency σ (i.e., the inverse of the wavelength λ),
with the central frequency corresponding to the maximum power. In practice, gyro source spectra
have a significant asymmetry, and, as discussed in Section A.12.4, the unbiased interference
response is, in the most general case:
(8.41)
where γce is the coherence function of the even component of the asymmetrical spectrum and
is the mean wavelength. Compared to the simple symmetrical case, there is an additional term
γco(∆LR)sin(2π∆LR/λ), which takes into account the odd component of the spectrum. When the
gyro is operated on the central fringe, as it is usually the case, this additional term is negligible in
practice; however, if it is operated on a wider dynamic range, a shift of the actual mean
wavelength is yielded as the phase difference increases, as the zero crossings of the variable term
of the interferometer response are not precisely periodic anymore.
This can be seen directly by regarding the interferometer as a filter with a transmission T(λ or
σ), depending on the wavelength λ (or the spatial frequency σ) for a given path difference ∆LR:
(8.42)
A broad power spectrum P(σ) yields the integrated response:
(8.43)
For the zero crossing points of the variable cosine part of the response, the mean frequency
is such that the product of P(σ) with cos[2π∆LR(σ − ) has a null integral (Figure 8.20). If the
spectrum is symmetrical (in terms of frequency), the problem is simple and the mean value is the
central value, but when it is asymmetrical, it is more complex and depends on the fringe order. In
particular, the mean value is not equal to the value σ max, which corresponds to the maximum
intensity. A linear mean, defined with the product of the spectrum with a linear function instead of
a sine (Figure 8.21), is a good approximation for gyros working around the zero order between
±π rad [22]. This linear mean is actually equivalent to a center of gravity.
Figure 8.20 Interferometric definition of the mean spatial frequency
of a broad spectrum.
8.3.3 Effect of Phase Modulation
So far, we have considered the unbiased response of the interferometer; however, as we have
already seen, an I-FOG has to be operated with a biasing phase modulation and a phase-nulling
feedback. Therefore, the wavelength dependence of the modulators may also have to be taken into
account.
In the open-loop case, the demodulated biased response at a given wavelength is always a sine
independent of the modulation depth; therefore, even with the wavelength dependence of the phase
modulator, the total response with a broad spectrum is also a perfect sine. The amplitude of this
open-loop response depends slightly on the spectrum, but, in practice, other sources of scale
factor error are much more important.
Figure 8.21 Definition of the linear mean of an asymmetrical broad spectrum.
With the phase-nulling scheme using Bragg modulators (see Section 8.2.1), there is a feedback
frequency shift ∆fFB, which is wavelength-independent; therefore, the scale factor has the same
spectrum dependence as in the unbiased case. With analog or digital phase ramps (see Sections
8.2.2 and 8.2.3), the problem is slightly more delicate. A phase modulator based on the elastooptic effect as a piezoelectric modulator, or on the electro-optic Pockels effect as in integrated
optics, is actually a path-difference modulator. A given driving voltage Vd yields a given pathdifference that is, to first order, wavelength-independent. The modulation is mainly due to an
index change, which is almost wavelength-independent. This yields, for a given voltage ramp or
step, a feedback phase difference ∆ϕ PR inversely proportional to the wavelength:
(8.44)
where ∆LPR is the equivalent path length difference induced by the ramp. Therefore, a rotation
also yielding a path difference effect leads to a phase difference:
(8.45)
Both phase differences ∆ϕ R and ∆ϕ PR have the same wavelength dependence, and phase ramp
feedback should be globally wavelength-independent: for any wavelength, the driving voltage
yields the same feedback path length difference ∆LPR, which compensates for the rotationinduced path-length difference ∆LR, which is also wavelength independent. However, this would
be true only if the effect of the reset is gated out. In this case, other causes of drift of the
modulator efficiency or of the gain of the driving electronics (temperature in particular) would
then become predominant. We have seen (Section 8.2.2) that a small change of the gain of the
phase modulation chain has only a third-order effect on the scale factor accuracy when the reset is
close to 2π and when the signal after the reset is taken into account. A wavelength change is
equivalent to a gain change, as ∆ϕ FB = 2π∆LFB/λ. Therefore, a wavelength change does not
modify the mean effect of the feedback ramp while it modifies the rotation-induced phase
difference ∆ϕ R to the first order. Then the basic wavelength dependence of the Sagnac effect is
retrieved with the phase ramp feedback when the effect of the reset is not gated out. Despite this
drawback, a controlled 2π reset or a four-state modulation are preferable, since the wavelength
may be controlled independently, while other sources of modulation efficiency drift (gain of the
electronic chain or LiNbO3 modulator response) are very difficult to control. For example the
thermal dependence of the Vπ of a LiNbO3 phase modulator is as high as −800 ppm/°C.
8.3.4 Wavelength Control Schemes
Among the problems to be solved to get a high-performance fiber gyro, wavelength control was
the last one to be addressed, and publications on this subject are not very numerous compared to
the total literature on the FOG. No definitive answer has yet been given to this problem, in
contrast to the problems of optical architecture with reciprocity, those of the various parasitic
noises and drifts with a broadband source or those of the signal processing scheme with biasing
modulation and phase nulling feedback. This is also more of an engineering problem, the
solutions of which are usually kept confidential, rather than a basic theoretical analysis that may
be published. Nevertheless, some design concepts have been described that directly control the
linear mean value involved in the scale factor of the gyro.
In addition to temperature control or modeling of the source spectrum, the simplest approach is
using a narrow optical filter in front of the detector at the output. Because each emission
wavelength has independent behavior, this approach is equivalent to using a source with a
spectrum equal to the product of the emission spectrum and the transmission of the filter.
Assuming that the reference filter has a stable transmission, the stability of the actual spectrum is
improved by a factor equal to the square of the ratio ρf between the widths of the source spectrum
and of the filter response (Figure 8.22). Interference filter, for example, may be as narrow as 5
nm, while superluminescent diodes have a typical width of 15 nm, that is, ρf ≈ 3. This would yield
a tenfold improvement in actual mean wavelength stability. However, there is the drawback of a
detected power reduced by a factor equal to ρf, but this degrades the theoretical signal-to-noise
ratio only by
With an erbium-doped fiber source, it is possible to avoid this power reduction by replacing the
broadband mirror of a double-path ASE source (Section B.9) by a wavelength selective mirror
[23] and in particular a fiber Bragg grating (see Section B.7.5).
Figure 8.22 Filtering of an unstable broad spectrum with a narrower reference filter.
Some other approaches control the wavelength with the light tapped out at the input by the
source splitter. A proposed scheme is to use a miniaturized grating spectrometer to spread the
source spectrum spatially [24]. Two detectors connected with an opposite polarity directly
provide the linear mean of the wavelength because they have an additional triangular mask that
simulates the product by a linear function.
The use of an additional narrow reference source was also proposed [25] to stabilize the path
difference ∆Lcont of a control interferometer on an integral number m of the reference
wavelength λref, and to adjust the broad spectrum to have the path difference of the stabilized
interferometer equal to another integral number m′ of the mean wavelength of the broadband
source:
(8.46)
This method is very accurate with a laboratory setup, but it is difficult to make a compact
control device for practical applications. However, it is possible to get a similar interferometric
control of the source mean wavelength with a reference wavelength using the wavelength
dependence of LiNbO3 modulators [26], which was described in Section 8.3.3. In particular the
Vπ error signal of the four-state modulation explained in Section 8.2.5 is a very sensitive signal
of mean wavelength drift. It can detect a change as small as 0.1 ppm.
In any case, all of these schemes require a stable reference: a stable reference filter, a stable
reference wavelength, or a stable reference spectrometer. A stability of 1 ppm in real operation is
a difficult but reasonable engineering goal.
Another interesting possibility is to use the propagation dispersion in fibers. The group transit
time τg through the coil has a wavelength dependence (see Section 3.2.3 and Appendix A), and this
transit time variation may be detected with an accurate measurement of the proper (or eigen)
frequency fp of the coil using a square wave modulation with an asymmetrical duty cycle [22]
(see Section 3.2.3). However, such a technique does not work at 1.3 μm where silica shows zero
dispersion, and it also requires the use of a reference filter to differentiate the transit time
variation due to wavelength from the strong temperature dependence.
A similar transit time dependence may be obtained with the frequency shifting feedback scheme
[27]; but, as we have seen, this approach does not provide very good performance because it
destroys the reciprocity of the interferometer (see Section 8.2.1).
8.3.5 Mean Wavelength Change with a Parasitic Interferometer or Polarimeter
The mean wavelength involved in the gyro scale factor is the mean wavelength of the light
actually detected at the output of the interferometer and not the mean wavelength of the broadband
source entering the interferometer.
In Section A.8.3, it is shown that an interferometer or a polarimeter has a channeled spectral
response; therefore, any spurious interferometer or polarimeter in the gyro setup modifies the
output spectrum and its mean wavelength.
A typical example is the spurious polarimeter between a polarized SLD and the polarizing
MIOC connected with a PM fiber. There is always a parasitic polarization crossed coupling at a
fiber end because of the stress induced by its holder. It is typically −20 dB and, at best, −30 dB.
Therefore, the polarized SLD light is partially coupled in the crossed polarization mode of its PM
pigtail. This light is eliminated by the polarizing MIOC but some of it is coupled back in the main
transmitted polarization mode at the second fiber end connected to the MIOC, and it interferes
with the high-power wave that is always propagated in the main polarization mode. It yields a
channeled spectral transmission:
(8.47)
where σ is the spatial frequency (σ = 1/λ) and ρM and ρM′ are the crossed polarization coupling
coefficients in intensity at the fiber ends, that is, 0.01 for −20 dB and 0.001 for −30 dB.
As it is shown in Section A.8.3, the contrast
response is the amplitude ratio
of the interferometric channeled
and not the intensity ratio. Even with two couplings of
−30 dB, that is, −60 dB for the spurious wave, the contrast C remains 2 × 10–3, and with −20-dB
couplings, it is as high as 2 × 10–2.
Figure 8.23 Mean frequency (or wavelength) change
with a spurious channeled polarimeter, in the case when FWHM
= FSR/2 (dotted lines for the source spectrum and the channeled spectral transmission, solid line for the resulting spectrum)
general case with a contrast C case where C = 1 and then
.
When the free spectral range (FSR) of this spurious polarimeter is much larger or much
smaller than the full width at half maximum (FWHM) of the source spectrum, it does not modify
the mean wavelength/frequency but there is an important effect when:
FWHM ≈ FSR/2 (8.48)
This case is described in Figure 8.23(a), with a maximum 1 + C of T(σ) at
a minimum 1 − C at
maximum variation
approximatively:
and
The original source spectrum is modified, yielding a
of the mean spatial frequency
that can be calculated to be
(8.49)
It yields a maximum relative scale factor error:
(8.50)
This result is simply shown in Figure 8.23(b) describing the case where the interferometer
contrast C is 1: the mean frequency shift
is then ∆σ FWHM/4. Note that this case of C = 1 is
realized with a Lyot depolarizer, as seen in Section B.7.4.
With a temperature change, the FSR varies, which yields a shift of the channeled transmission
as seen in Section A.8.3. When the mean wavelength is on a maximum or a minimum of T(σ),
there is no wavelength error because of symmetry, but with this shift, the mean frequency
oscillates between
Care must be taken to avoid this worst-case condition not to get fringes on the scale factor that
can be as high as several hundreds of parts per million.
With a typical SLD at 850 nm with an FWHM of 20 nm, this worst case corresponds then to a
FSR of 40 nm, in wavelength, that is, a path unbalance ∆Lop0 = λ2/∆λFSR according to Appendix
A that is 18 μm. This correspond to a length L of 36 mm of a PM fiber with a birefringence index
∆nb = 5 × 10-4, as ∆Lop0 = L ⋅ ∆nb.
As seen in Figure 8.24 that displays the residual mean frequency error
as a function of the
ratio FSR/FWHM with a maximum for FSR = 3 FWHM, one has to ensure that FSR < FWHM/5 to
avoid defect. It corresponds to a PM fiber length of 36 cm in the previous example.
Figure 8.24 Mean spatial frequency error
the spurious polarimeter (or interferometer).
as a function of the ratio FSR/FWHM in logarithmic scale. C is the contrast of
References
[1] Davis, J. L., and S. Ezekiel, “Techniques for Shot-Noise-Limited Inertial Rotation Measurement Using a Multi-Turn Fiber
Sagnac Interferometer,” SPIE Proceedings, Vol. 157, 1978, pp. 131–136.
[2] Cahill, R. F., and E. Udd, “Phase-Nulling Fiber-Optic Laser Gyro,” Optics Letters, Vol. 4, 1979, pp. 93–95.
[3] Davis, J. L., and S. Ezekiel, “Closed-Loop, Low-Noise Fiber-Optic Rotation Sensor,” Optics Letters, Vol. 6, 1982, pp. 505–
507.
[4] Lefèvre, H. C., et al., “Integrated Optics: A Practical Solution for the Fiber-Optic Gyroscope,” SPIE Proceedings, Vol. 719,
1986, pp. 101–112.
[5] Auch, W., “The Fiber-Optic Gyro—A Device for Laboratory Use Only?” SPIE Proceedings, Vol. 719, 1986, pp. 28–34.
[6] Shimizu, H., R. Ishikawa, and R. Kaede, “Integrated-Optical Frequency Modulator for Fiber-Optic Gyroscope,” Electronics
Letters, Vol. 22, 1986, pp. 334–335.
[7] Michal, R. J., E. Udd, and J. P. Theriault, “Derivative Fiber-Sensors Based on Phase-Nulling Optical Gyro Development,” SPIE
Proceedings, Vol. 719, 1986, pp. 150–154.
[8] Arditty, H. J., C. Puech, and M. Papuchon, “Device for measuring a phase shift which is not reciprocal produced in a ring
interferometer,” U.S. Patent #5,056,919, Foreign Application Priority of November 1980, 1991.
[9] Kim, B. Y., and H. J. Shaw, “Gated Phase-Modulation Approach to Fiber-Optic Gyroscope with Linearized Scale Factor,” Optics
Letters, Vol. 9, 1984, pp. 375–377.
[10] Kay, C. J., “Serrodyne Modulator in a Fibre-Optic Gyroscope,” IEE Proceedings, Part J--Optoelectronics, Vol. 132, 1985,
pp. 259–264.
[11] Elberg, A., and G. Schiffner, “Closed-Loop Fiber-Optic Gyroscope with a Sawtooth Phase-Modulated Feedback,” Optics
Letters, Vol. 10, 1985, pp. 300–302.
[12] Kurokawa, A., et al., “Evaluation of a Sawtooth Generator in a Closed-Loop Fiber-Optic Gyroscope,” Proceedings of OFS
6/’89, Paris, Springer Proceedings in Physics, Vol. 44, 1989, pp. 107–114.
[13] Lefèvre, H. C., et al., “Double Closed-Loop Hybrid Fiber Gyroscope Using Digital Phase Ramp,” Proceedings of OFS 3/’85,
San Diego, OSA/IEEE, Postdeadline Paper 7, 1985.
[14] Lefèvre, H. C., et al., “Progress in Optical Fiber Gyroscopes Using Integrated Optics,” AGARD-NATO Proceedings, Vol. CPP383, 1985, pp. 9A1–9A3.
[15] Arditty, H. J., et al., “Fiber-Optic Gyroscope with All-Digital Processing,” Proceedings of OFS 6/’89, Paris, Springer-Verlag
Proceedings in Physics, Vol. 44, 1989, pp. 131–136.
[16] Lefèvre, H. C., et al., “High Dynamic Range Fiber Gyro with All-Digital Processing,” SPIE Proceedings, Vol. 1367, 1990, pp.
72–80.
[17] Auch, W., M. Oswald, and D. Ruppert, “Product Development of a Fiber-Optic Rate Gyro,” Proceedings of Symposium Gyro
Technology, DGON, Stuttgart, 1987, pp. 3.0–3.19.
[18] Lefèvre, H. C., and P. Martin, “Optical-fiber measuring device gyrometer, central navigation and satisfying systems,” U.S.
Patent #5,141,316, 1992.
[19] Buschelberger, J., and M. Kemmler, “Closed Loop Fiber Gyro Triad,” Fiber-Optic Gyros: 15th Anniversary Conference,
SPIE Proceedings, Vol. 1585, 1991, pp. 89–97.
[20] Pavlath, G. A., “Closed-Loop Fiber Optic Gyros,” Fiber-Optic Gyros: 20th Anniversary Conference, SPIE Proceedings, Vol.
2837, 1996, pp. 46–60.
[21] Egorov, D. A., et al., “Study of Dead Zones of Fiber-Optic Gyros,” Gyroscopy and Navigation, Vol. 2, No. 4, 2011, pp. 197–
207.
[22] Lefèvre, H.C., “Comments About the Fiber-Optic Gyroscope,” SPIE Proceedings, Vol. 838, 1987, pp. 86–97.
[23] Gaiffe, T., et al., “Light source with stabilized broadband and associated optical filter gyroscope,” U.S. Patent #5,894,488,
1999.
[24] Schuma, R. F., and K. M. Killian, “Superluminescent Diode (SLD) Wavelength Control in High-Performance Fiber-Optic
Gyroscopes,” SPIE Proceedings, Vol. 719, 1986, pp. 192–193.
[25] Chou, H., and S. Ezekiel, “Wavelength Stabilization of Broadband Semi-Conductor Light Sources,” Optics Letters, Vol. 10,
1985, pp. 612–614.
[26] Lefèvre, H., and P. Martin, “Optical fiber measuring device using phase difference,” U.S. Patent #5,337,142, 1994.
[27] Udd, E., and R. F. Cahill, “From Conception to the Field: Fiber-Optic Gyro Development at McDonnell-Douglas,” SPIE
Proceedings, Vol. 719, 1986, pp. 17–23.
CHAPTER 9
Recapitulation of the Optimal Operating
Conditions and Technologies of the I-FOG
9.1 Optimal Operating Conditions
The interferometric fiber-optic gyroscope (I-FOG) is a ring interferometer that uses a multiturn
fiber coil to enhance the Sagnac effect induced by rotation with respect to inertial space. This
yields a difference ∆tR of propagation time between the two counterpropagating waves that is
proportional to the rotation rate Ω and the area A of the sensing coil (see Chapter 2):
(9.1)
It is measured as a phase difference ∆ϕ R is the interferometer:
∆ϕ R = ω ⋅ ∆tR (9.2)
∆ϕ R = 2π ⋅ f ⋅ ∆tR (9.3)
(9.4)
where ω is the angular frequency, f is the temporal frequency, and λ is the wavelength of the
source of the interferometer. This result is usually given as:
(9.5)
where L is the length of the coil and D is its diameter.
The sensitivity of an I-FOG may be tuned with the size of the coil as summarized in Table 9.1,
where A is the total area of the sensing coil.
The optimal operating conditions can be summarized as follows:
Use a single-mode reciprocal configuration with a truly single-mode filter (single spatial mode
and single polarization) at the common input-output port of the interferometer. This ensures that
the paths of both opposite waves are perfectly equalized and that only truly nonreciprocal effects
as rotation yield a phase difference (see Chapter 3).
Table 9.1 Accuracy Versus Coil Size
D
L
A = LD/4
Medium accuracy
3 cm 100m 0.75 m2
High accuracy
10 cm 1 km 25 m2
Ultra-high accuracy 30 cm 10 km 750 m2
Use a modulation-demodulation biasing scheme with a reciprocal phase modulator at the end of
the fiber coil. The interferometer behaves like a delay line filter because of the transit time
through the coil, which yields a high sensitivity operation point without degrading the
reciprocity. Best performance is obtained at the proper (or eigen) frequency fp that matches the
half-period of the modulation to the coil transit time. The product fp × L is about 100 kHz ⋅ km.
The combination of this processing scheme with a reciprocal configuration yields the minimum
configuration (see Section 3.2).
Use a broadband optical source to take advantage of its short coherence length. This destroys the
interference contrast of the various parasitic waves generated in the system by backreflection and
backscattering (see Chapter 4), or polarization cross-coupling (see Chapter 5). It also suppresses
the effect of Kerr nonlinearity, which is also related to an interference phenomenon (see Section
7.3).
Use polarization-maintaining (PM) fibers that provide a very beneficial depolarization effect on
the crossed-polarized waves because of their intrinsic birefringence and of the low coherence of
the broadband source (see Chapter 5). This relaxes the constraint of the ultra-high polarizer
rejection. The accurate analysis of these polarization problems is obtained with optical
coherence domain polarimetry (OCDP), based on path-matched white-light differential
interferometry or spectrum analysis (see Section 5.4). These PM fibers also reduce the influence
of the nonreciprocal magneto-optic effects (see Sections 7.1 and 7.2).
Use a symmetrical winding (quadrupolar in particular) to reduce sensitivity to temperature
transients (see Section 6.2).
Use a closed-loop processing scheme applying a phase-nulling feedback to linearize the Sagnac
phase measurement ∆ϕ R. Among the possible techniques, the all-digital closed-loop approach
provides fundamental advantages. It combines a digital phase ramp feedback, which yields very
good scale factor linearity over the whole dynamic range because of averaging effects, a second
servo-loop for gain control, and a digital demodulation, which is intrinsically free of any source
of electronic bias drift (see Section 8.2). Such a digital phase ramp technique requires a
wideband phase modulator.
Use wavelength control to stabilize, measure, or model the source mean wavelength and get an
accurate scale factor of the rotation rate measurement (see Section 8.3).
Figure 9.1 summarizes the optimal architecture that combines a Y-coupler configuration, using
a multifunction integrated-optic circuit, with all-digital closed-loop processing electronics
generating a staircase digital phase ramp.
Figure 9.1 Optimal architecture of the I-FOG with a Y-coupler configuration and an all-digital closed-loop processing scheme.
It is composed of:
A broadband source: superluminescent semiconductor diode (SLD) for medium performance
and Er-doped fiber ASE source for high performance.
A multifunction integrated-optic circuit (MIOC) based on proton exchange in LiNbO3 and
composed of a Y-junction and a pair of phase modulators. Very good polarization filtering is
ensured by the single-polarization propagation in proton-exchanged waveguides, and LiNbO3
phase modulators do have the wide band required by the phase ramp.
A polarization-maintaining (PM) fiber coil with a quadrupolar winding.
A source coupler (or circulator) to send the returning light to a detector.
A detector that converts the optical power returning from the interferometer into an electrical
signal.
An analog-digital (A/D) convertor that samples this signal.
A processing logic that generates the rotation rate measurement.
A digital-analog (D/A) convertor that drives the phase modulators with the biasing modulation
and the phase-ramp feedback modulation.
Figures 9.2(a, b) are photographs of the first fiber-gyro using the Y-coupler configuration and
the digital phase ramp in the mid-1980s.
Figure 9.2 Photographs of an early prototype (1985) of Thomson-CSF (now Thales) Central Research Laboratory, using an
SLD at 850 nm: (a) close view showing the MIOC and the coupler of the Y-coupler configuration and (b) test setup with the
phase-ramp trace on the oscilloscope.
9.2 Broadband Source
9.2.1 Superluminescent Diode
The first laboratory experiments in the late 1970s were performed with He-Ne gas lasers (Figure
9.3), which were the only available sources that could be efficiently coupled into a single-mode
fiber, as their Gaussian emission mode is matched to the pseudo-Gaussian fundamental mode of
the fiber (see Section B.4.1).
In the early 1980s, the development of semiconductor emitting diodes, particularly for
telecommunications, made these compact solid-state light sources working with a low driving
voltage the ideal choice for practical devices. These diodes are mainly two kinds: surface light
emitting diodes (LEDs) and laser diodes (LDs) [1]. They use III-V semiconductor junctions,
AlGaAs-GaAs for a wavelength in the 800- to 850-nm range, or InGaAsP-InP for the 1,300 and
1,550-nm ranges.
Figure 9.3 Early breadboard of a fiber gyro, in the late 1970s, from Thomson-CSF (now Thales) Central Research Laboratory,
using an He-Ne laser as the source.
As seen in Section A.2, these two sources were not optimal for gyro applications: LEDs based
on spontaneous emission have an adequate spectrum width, but they cannot be coupled efficiently
into a single-mode fiber, because the emission area is large (50 to 100 μm in diameter) compared
to the fiber core; LDs may be coupled efficiently because the wave, generated in a narrow stripe
(a few micrometers) by stimulated emission, is spatially coherent, but the spectrum is composed
of the narrow emission peaks of the modes of the Fabry-Perot laser cavity. To get simultaneously
a good spatial coherence and a low temporal coherence, laser diodes have to be modified to make
superluminescent (or superradiant) diodes, abbreviated SLDs (or SRDs). The lasing effect is
suppressed by decreasing the reflectivity of the mirror facets with an antireflection coating at the
emission output and an absorbing region at the other diode end (Figure 9.4) [2–6]. The use of an
angled stripe [7], which works on a similar principle as the angled edge of the integrated circuit,
has also been demonstrated (see Section 4.1.1).
Figure 9.4 Superluminescent diode with a rear absorbing region.
The gain of a semiconductor diode is very high, and even without cavity feedback, the output
power may be almost as high as that of a laser. Along a single pass, the first spontaneous
emission photons are amplified by stimulated emission, and the output wave has a spatial
coherence similar to that of a laser, because it is also generated in a narrow stripe, yielding an
efficient coupling into a single-mode fiber. However, the multimode laser structure of the
spectrum is greatly reduced, and SLDs behave like quasi-broadband sources. At 850 nm, the full
width at half maximum (FWHM) of the spectrum is on the order of 20 nm.
Practical devices, which are typically 300 to 500 μm long, are hermetically packaged in a
rugged casing with a fiber pigtail soldered in front of the emission window of the diode. The
typical coupled power is few milliwatts (the coupling efficiency is 10% to 20%) for a driving
current of 50 to 150 mA. A polarization-maintaining fiber may be used for pigtailing with
adequate orientation of its birefringence axes. SLDs are partially polarized, with 70% to 80% of
the power in the horizontal polarization parallel to the junction.
The main problem of SLDs is their poor spectrum stability: the mean wavelength has a drift of
about 400 ppm/°C with temperature and 40 ppm/mA with driving current. A temperature control
with a Peltier element and a stable driver yield stability in the range of 100 ppm; but higher
performance requires a direct wavelength control.
9.2.2 Rare-Earth Doped Fiber ASE Sources
To overcome the problem of SLD wavelength stability, work was devoted in the early 1990s to
the development of alternative broadband sources based on rare-earth doped fiber [8]. As with
SLDs, rare-earth doping provides a very high amplification gain, and high-power broadband
emission may be obtained over a single pass with amplified spontaneous emission (ASE) without
requiring a cavity feedback (see Section A.2.4). These wideband fiber sources were first called
super-fluorescent fiber sources, but today the term ASE is mainly used. Super-radiance, superluminescence, super-fluorescence, and ASE are actually similar [9].
Rare-earth energy levels are much more stable than those of semiconductors and greatly
improve the wavelength stability. They can be pumped with compact high-power laser diodes.
Two dopants are particularly efficient: neodymium (Nd), with an emission around 1,060 nm and a
pump of 800 nm [8, 10], and erbium (Er), with an emission around 1,550 nm and a pump of 980
nm or 1,480 nm [11, 12]. Only erbium is used today because it takes advantage of the huge
industrial development of the erbium-doped fiber amplifier (EDFA) that has revolutionized
telecommunications since the 1990s.
These Er-fiber ASE sources are almost ideal for high-performance fiber gyroscopes, however,
the subject is not straightforward, and to obtain a very good wavelength stability depends on
various parameters, particularly pump wavelength, pump power, and light feedback [13].
Several features of these erbium sources must be pointed out:
Their emitted light is unpolarized, which is very beneficial for reducing polarization
nonreciprocities (see Section 3.4.4). It allows one also to use the ordinary (non-PM) single-mode
fiber for the source coupler.
Their output power may be very high (up to a few hundreds of milliwatts) and is naturally
matched to single-mode fiber, which is useful for getting low photon noise or sharing a single
source for three gyro axes.
Their spectrum is very asymmetrical (Figure 9.5) and care must be taken to carefully evaluate
the mean wavelength (see Sections 8.3.2 and A.12.4).
Figure 9.5 Asymmetrical spectrum of an erbium-doped fiber ASE source extending from 1.52 to 1.57 μm. The
mean wavelength is 1.54 μm.
Their spectrum width is narrower that the one of an SLD, and the decoherence function is
reduced to 0.03 for Ldc = 350 μm (Figure 9.6).
This narrow width also creates a significant excess RIN. This problem is addressed next, as well
as the ways to compensate for it.
Figure 9.6 Coherence function of an erbium ASE source with logarithmic scales. The decoherence length Ldc =
350 μm.
9.2.3 Excess RIN Compensation Techniques
As discussed in Section A.2.2, a spontaneous emission source suffers from excess relative
intensity noise (excess RIN, or simply RIN), and an ASE source (see Section A.2.4) keeps the
excess RIN of its seed spontaneous emission.
Theory shows that the power spectral density (PSDRIN) of the excess relative intensity noise is
simply the inverse of the spectrum temporal frequency width ∆fFWHM:
PSDRIN ≈ 1/∆fFWHM (9.6)
For an SLD with a mean wavelength λ = 850 nm and ∆λFWHM-SLD = 20 nm, the frequency
width is ∆fFWHM-SLD = c ⋅ ∆λFWHM-SLD/λ2 = 8 THz, and then:
PSDRIN-SLD = 1.25 × 10–13/Hz (i.e. −129 dB/Hz) (9.7)
As seen in Section A.2.1, this value of excess RIN corresponds to the theoretical photon noise
of a power of 4 μW, which is in practice the returning power of a fiber gyro using such an SLD.
For an erbium-fiber ASE source, the spectrum is very asymmetrical, but it may be possible to
define an equivalent RIN width ∆λFWHM-Er = 20 nm, yielding ∆fFWHM-Er = 2.5 THz, and then:
PSDRIN-Er = 4 × 10–13/Hz (i.e. −124 dB/Hz) (9.8)
This is significantly higher than the one of an SLD, and in addition an erbium source provides
much more power. Then the related photon noise is in the range of −140 dB/Hz, that is, 40 times
lower than the PSD of the RIN and six times in terms of relative σ values (the square-root of the
PSD).
It can be even worse (−120 dB/Hz), when an internal spectral filtering of the erbium source is
performed to ensure a good wavelength stability and get a high scale-factor accuracy as discussed
in Section 8.3.4.
However, excess RIN is not a fundamental noise like photon noise: when a light beam is split,
the photon noises of both outputs are uncorrelated, but their RINs are identical. It is possible to
compensate for the RIN by measuring it with a reference and subtracting it from the signal as
proposed by Moeller and Burns [15].
There is also a very simple method of RIN reduction by operating the gyro at a bias point close
to a black fringe, that is, a phase bias close to π instead of π/2 [16]. The sensitivity is proportional
to the slope of the raised cosine response curve, that is, sinϕ b (where ϕ b is the phase bias) while
the excess RIN is proportional to the actual power on bias (i.e., the response (1 + cosϕ b)).
Working, for example, at a 0.9π bias instead of π/2, the sensitivity is reduced by a factor of
sin(0.9 × π)/sin(π/2) = 0.3, while intensity noise experiences a reduction six times higher, since [1
+ cos(0.9 × π)]/[1 + cos(π/2)] = 0.05. Furthermore, as seen in Section 2.3.2, this slightly
improves the theoretical signal-to-photon-noise ratio.
Such a technique of reduction of excess RIN allows in practice to get very close to the
theoretical photon noise and to obtain a phase noise about
random performance (ARW) performance as good as 7 × 10–5°/
diameter of 170 mm [17].
It yields an angular
with a coil of 3 km over a
9.3 Sensing Coil
As explained previously, the best performances are obtained with a sensing coil made of stressinduced high-birefringence polarization-maintaining fiber. An elliptical core PM fiber (see
Section B.5.1) has a higher attenuation and a lower polarization conservation, and is rarely used.
Furthermore, a symmetrical quadrupolar winding reduces the effect of temperature transience.
As detailed in Sections 6.1 to 6.4, the pure Shupe effect and the T-dot effect are related to
temperature-dependent stresses induced in the coil, which requires a careful mechanical design of
the holding of the coil.
Some other practical specifications are also needed. In particular, high-NA fiber with a highly
doped core is suitable for avoiding bending loss in very compact coils: with an NA of 0.16, coil
diameter may be as small as 20 mm without additional attenuation.
Careful winding and adequate potting material are also important to ensure good pointing
accuracy: the sensitivity axis is parallel to the equivalent area vector A defined by the line integral
∫(1/2)r × dr along the fiber (see Section 2.1.1). To get the specified performance in a three-axis
unit, the axis stability, in radian, must be equal to that of the scale factor; that is, for example, 10–5
rad is required with 10 ppm.
The fiber diameter should be as small as possible to limit the volume of the coil. Gyro fibers
have a typical cladding diameter of 80 μm instead of the standard 125 μm of telecommunication
fibers. The same applies to the protective coating, which should be as thin as possible. However, it
must minimize microbending to avoid loss or degradation of the polarization conservation,
which actually limits its minimum thickness. The coil characteristics must be conserved over the
entire temperature range of the operation: two-layer coatings with a soft inner layer have been
developed to solve this problem.
To avoid an increase of attenuation under radiation in military or space applications, the use of
long wavelengths (1,300 or 1,550 nm) is preferred.
In addition, single or even double μ-metal shieldings are required to eliminate the residual
magnetic dependence (see Section 7.1).
A last very important point is the fiber reliability. This problem is very complex [18, 19], but
the basic ideas may be outlined. The fiber surface contains very small intrinsic flaws due to the
basic structure of silica, and larger extrinsic flaws due to dust or particles included in the fiber
during the drawing process. When the fiber is under tensile stress, the size of the flaws is
increased, which may eventually cause breakage of the fiber. When wound in coils, the fiber is
placed under tensile stress at the outside. The related strain is equal to the ratio between the fiber
diameter and the coil diameter. To ensure a good reliability, the whole fiber length has to be
proof-tested at a high strain level (typically 0.5% to 2%, while ideal silica may withstand strain up
to 10%) for a few seconds to check that the sample does not contain weak points that would
otherwise induce breakage. Based on the Weibull model of the weakest link in a chain, the failure
probability over the expected lifetime of the gyro may be evaluated as a function of the proof-test
level and of the fiber characteristics [18]. The quality of practical fibers ensures a very good
reliability for coils in the 10-cm range that experience a strain of about 0.1%. However, for small
coils (about 2 to 3 cm), high-strength fiber would have to be used to achieve a very long lifetime.
Note that proof-test level is also expressed in terms of stress instead of strain, which are related
by the Young modulus of silica ESiO2 = 70 GPa in SI units; using pounds per square inch,
remember that 1% of strain corresponds to a stress of 100 kpsi. Another useful order of
magnitude is that 1% of strain is induced by a force of 10N (1 kilo) for a 125-μm fiber, and 4N
(400 grams) for an 80-μm fiber.
9.4 The Heart of the Interferometer
As described in previous chapters (particularly Chapter 3), the main subject of concern has been
the heart of the interferometer, composed of a source beam splitter, a polarizer, a spatial singlemode filter, a coil beam splitter, and phase modulators. Because bulk-optic components require
delicate alignments to couple light into single-mode fiber, research has been focused on a
rugged, all-guided approach, and the users of integrated optics have had the advantage of getting
wideband phase modulators.
Optimal simplicity is obtained with a hybrid approach: the Y-coupler configuration (see Section
3.3.4) widely used since the 1990s [20]. A multifunction integrated-optic circuit (MIOC) combines
a Y-junction for the coil splitter, wideband phase modulators, and preferably the polarizer, while
a coupler is used for the source splitter, with its lead acting as the spatial filter on the common
input-output port of the interferometer. This source splitter can be replaced by a circulator (see
Section B.8.3) to get higher returning power and lower noise.
A very critical component is the polarizer, and a proton-exchanged LiNbO3 circuit (see Section
C.3) that guides only one state of polarization is considered as the optimal approach [21], even if
experimental results [22] demonstrated that it is possible to get a rejection of 60 dB with a
metallic overlay on a Ti-indiffused waveguide. An in-line fiber polarizer using a metallic overlay
also [23] is another interesting alternative.
Remember that OCDP (see Section 5.4) is an essential technique for evaluating accurately the
rejection of the polarizing element, and the amount of polarization crossed couplings.
9.5 Detector and Processing Electronics
Finally, it is necessary to be careful when choosing the detector so as not to degrade the
performance of the optical system, which would normally be limited by photon shot noise (see
Section 2.3.2). Semiconductor PIN (Positive-Intrinsic-Negative) junction photodiodes are ideal
because of their very high quantum efficiency: the number of primary electrons generated is very
close to the number of input photons; the flow of electrons has about the same fundamental shot
noise as the theoretical value for the flow of photons.
As seen in Section A.2.1, the relative power spectral density (PSD) of the shot noise of an
electrical current I:
(9.9)
and the one of an optical power P is:
(9.10)
where q is the charge of the electron, and hf is the energy of the photon. Considering the flow of
incoming photons
and the one of outgoing electrons
, it yields:
(9.11)
(9.12)
The quantum efficiency
is typically 80% yielding a responsibility of 0.68 A/W at 850
nm and 1 A/W at 1,550 nm. It degrades the PSP by 20%, but it is only 10% for the root mean
square value.
For 850 nm, a silicon (Si) photodiode has to be used, while indium gallium arsenide (InGaAs)
is optimal for 1,300 and 1,550 nm. Both materials provide low dark current and short response
time that would not be the case with germanium in particular, and they can be considered as a
quasi-perfect detector.
However, an electrical current I cannot be measured directly; it has to be converted into a
voltage V with a load resistor Rload. This yields the additional thermal noise of this load, with a
voltage rms value:
(9.13)
where k = 1.38 × 10–23 J ⋅ K–1 is the Boltzmann constant and Tabs is the absolute temperature in
Kelvin (i.e., about 300K).
If the load resistance Rload is increased, the conversion factor increases linearly while the
thermal noise increases only as the square root, which improves the signal-to-thermal-noise ratio.
However, the improvement is limited by the saturation voltage or the gain-bandwidth product of
the amplifier which has to be used. This dictates the maximum load resistance that may be used
for the required bandwidth and with the actual detected power. In practice, thermal noise of the
detector is not the limiting factor. As seen earlier, it is the excess RIN of the broadband source,
but it can be compensated for; high-performance fiber gyros have a noise close to their
theoretical photon shot noise: typically 0.1 to 0.3
in terms of equivalent phase noise
in the interferometer.
In early experiments when the returning power was low, photomultiplier tubes and avalanche
photodiodes were used. In both cases, there is a direct amplification process of the photocurrent
that reduces the relative effect of the thermal noise of the load resistor, but it degrades the
quantum efficiency, which increases the actual relative shot noise.
With the progress of components in terms of attenuation, present returning optical powers are
high, and a PIN diode is the optimal choice.
Note in addition that the detection unit requires a very careful electronic design to avoid ground
loop and electromagnetic coupling problems. The biasing modulation voltage is typically on the
order of few volts and the primary current in the detector is typically few tens of microamperes
for few tens of microwatts of returning optical power. To limit bias error due to electronic
coupling to below 10–8 rad, the coupled current at the modulation frequency has to remain below
10–13 ampere, that is, about a single electron per microsecond, which is typically the sampling
time. This problem of electromagnetic coupling applies also to the driving current of an SLD;
however, with an Er-source this constrain is relaxed since a modulation of the pumping laser
diode is filtered out in the process of erbium amplification above a frequency of few kilohertz,
while the biasing modulation-demodulation is done at the proper frequency that is typically 100
kHz to 1 MHz.
For the digital processing electronics, we already saw in Sections 8.2.3 and 8.2.4 that the
required number of bits of the A/D and D/A convertors is limited because of averaging effects. In
practice, 12 bits are sufficient for medium-grade performance, and even ultra-high performance
does not require more than 16 bits.
The logic circuit can use Application Specific Integrated Circuit (ASIC) technology but FieldProgrammable Gate Array (FPGA) technology is usually privileged because it significantly
reduces development time and cost.
Figure 9.7 Photograph of an I-FOG prototype of Photonetics (now iXBlue) in 1989. One can see the MIOC, the source
coupler, the sensing coil with its μ-metal shielding, and the all-digital processing electronics using an FPGA circuit. This
prototype used an SLD at 850 nm and a 200m-long PM fiber coil with a mean diameter of 30 mm.
Figure 9.7 displays a photograph of the first gyro prototype combining the Y-coupler optical
configuration and an all-digital processing electronics in 1989.
References
[1] Fukuda, M., Reliability and Degradation of Semiconductor Lasers and LEDs, Norwood, MA: Artech House, 1991.
[2] Lee, T. P., C. A. Burrus, and B. I. Miller, “A Stripe-Geometry Double-Heterostructure Amplified-Spontaneous Emission
(Superluminescent) Diode,” IEEE Journal of Quantum Electronics, Vol. QE-9, 1983, pp. 820–821.
[3] Wang, C. S., et al., “High Power Low Divergence Superradiant Diode,” Applied Physics Letters, Vol. 41, 1982, pp. 587–589.
[4] Wang, C. S., et al., “High-Power Long-Life Superluminescent Diode,” SPIE Proceedings, Vol. 719, 1986, pp. 203–207.
[5] Kwong, N. S. K., et al., “High Power, High Efficiency Window Buried Heterostructure GaAlAs Superluminescent Diode with an
Integrated Absorber,” Applied Physics Letters, Vol. 151, 1987, pp. 1879–1881.
[6] Kwong, N. S. K., N. Bar-Chaim, and T. Chen, “High-Power 1.3 μm Superluminescent Diode,” Applied Physics Letters, Vol. 54,
1989, pp. 298–300.
[7] Niesen, J., et al., “Recent Development in 0.83 μm Superluminescent Diodes at TRW,” SPIE Proceedings, Vol. 719, 1986, pp.
208–215.
[8] Liu, K., et al., “Superfluorescent Single-Mode Nd: Fiber Source at 1060 nm,” Proceedings of OFS’88, New Orleans, LA,
1988, pp. FDD5-1–FDD5-4.
[9] Auzel, F., S. Hubert, and D. Meichenin, “Very Low Threshold CW Excitation of Superfluorescence at 2.72 μm in Er3+ ,”
Europhysics Letters, Vol. 7, 1988, pp. 459–462.
[10] Fesler, K. A., et al., “Behavior of Broadband Fiber Sources in a Fiber Gyroscope,” SPIE Proceedings, Vol. 1171, 1989, pp.
346–352.
[11] Morkel, P. R., “Erbium-Doped Fibre Superfluorescent Source for the Fibre Gyroscope,” Proceedings of OFS’89, Paris,
Springer Proceedings in Physics, Vol. 44, 1989, pp. 143–148.
[12] Wysocki, P. F., et al., “1.55 μm Broadband Fiber Sources Pumped Near 980 nm,” SPIE Proceedings, Vol. 1373, 1990, pp.
66–77.
[13] Wysocki, P. F., et al., “Spectrum Thermal Stability of Nd- and Er-Doped Fiber Sources,” SPIE Proceedings, Vol. 1373, 1990,
pp. 234–245.
[14] Morkel, P. R., R. I. Laming, and D. N. Payne, “Noise Characteristics of High-Power Doped-Fibre Superluminescent Sources,”
Electronics Letters, Vol. 26, 1990, pp. 96–98.
[15] Moeller, R. P., and W. K. Burns, “Low noise fiber gyroscope system which includes excess noise subtraction,” U.S. Patent
#5,331,404, 1994.
[16] Lefèvre, H. C., et al., “Integrated Optics: A Practical Solution for the Fiber-Optic Gyroscope,” SPIE Proceedings, Vol. 719,
1986, pp. 101–112.
[17] Paturel, Y., et al., “One Nautical Mile per Month FOG-Based Strapdown Navigation System: A Dream Already Within
Reach,” Inertial Sensors and Systems 2012, Karlsruhe, paper 12, 2012.
[18] Miyajima, Y., “Studies on High-Tensile Proof Tests of Optical Fibers,” Journal of Lightwave Technology, Vol. 1, 1983, pp.
340–346.
[19] Kurkjian, C. R., J. T. Krause, and M. J. Matthewson, “Strength and Fatigue of Silica Optical Fibers,” Journal of Lightwave
Technology, Vol. 7, 1989, pp. 1360–1370.
[20] Ezekiel, S., and E. Udd, (eds.), “Fiber Optic Gyro: 15th Anniversary Conference,” SPIE Proceedings, Vol. 1585, 1991.
[21] Suchosky, P. G., T. K. Findakly, and F. L. Leonberger, “LiNbO3 Integrated Optical Components for Fiber-Optic Gyroscopes,”
SPIE Proceedings, Vol. 993, 1988, pp. 240–243.
[22] Szafraniec, B., C. Laskoskie, and D. Ang, “High-Performance Ti-Indiffused LiNbO3 Multifunction Chip for Use in Fiber Optic
Gyros,” SPIE Proceedings, Vol. 1585, 1991, pp. 393–404.
[23] Johnstone, W., S. Carey, and B. Culshaw, “Developments in the Characterisation and Performance of Optical Fibre Polarising
Devices Using Thin Metal Films,” SPIE Proceedings, Vol. 1585, 1991, pp. 365–370.
[24] Dandridge, A., and H. F. Taylor, “Noise and Correlation Effects in GaAlAs Broadband Sources,” IEEE Journal of Lightwave
Technology, Vol. 5, 1987, pp. 689–693.
CHAPTER 10
Alternative Approaches for the I-FOG
10.1 Alternative Optical Configurations
The minimum configuration (see Section 3.2) using a single-mode filter at the common inputoutput port of the ring interferometer and a phase modulation-demodulation to bias the response
is now almost universally used. In the early days of gyro research, other schemes were proposed,
such as the use of an acousto-optic modulator to split or recombine both counterpropagating
waves [1, 2], or the use of a polarization splitter and a quarter-wave plate to passively bias the
signal [3]. These schemes are not significantly simpler than the minimum configuration, and their
performances are limited, as they do not fully respect reciprocity.
Among these nonreciprocal approaches, one solution, the use of a [3 × 3] coupler proposed by
Sheem [4], is still pursued [5, 6] because of its great simplicity. It does not yield a very good
performance, but it is an interesting scheme for a very low-cost fiber gyro. It is based on the
intrinsic phase shift induced in the evanescent wave coupling of a [3 × 3] coupler.
Figure 10.1 Fiber gyro using a [3 × 3] coupler.
We have already seen indeed (Section 3.3.1) that a [2 × 2] coupler yields a π/2 phase shift for
the coupled wave. At the reciprocal port of the interferometer, both waves have experienced the
same coupling phase shift and are in phase, while at the nonreciprocal port, one wave has
experienced twice this π/2 phase shift and the other one has experienced none, which yields a π
phase difference. In a [3 × 3] coupler, each coupled wave experiences a 2π/3 phase shift, and by
connecting the coil on the two coupled output ports (Figure 10.1), both counterpropagating waves
experience the 2π/3 phase shift at the input, while, at the output, one is transmitted and the other is
coupled with a second 2π/3 phase shift. Therefore, they interfere, in absence of rotation, with a
2π/3 phase difference at one output free port of the interferometer and with a −2π/3 phase
difference at the other free port. By taking the power difference between these two ports, a sine
biased signal is obtained, but with a very simple electronic circuit.
Figure 10.2 A [3 × 3] coupler: (a) uniform excitation and (b) phasor diagram of output amplitude.
It is possible to easily check that the phase shift has to be 2π/3. Let us consider a symmetrical [3
× 3] coupler where all the ports are equivalent [Figure 10.2(a)]. They can be made with a fusedtapered technique [6] similar to the one used in a [2 × 2] coupler (see Section B.7.1). Assuming
that three waves are entering the coupler in phase and with the same input power, because of
symmetry, the power must be equal in the three ports at the output. At each port there is
interference between three waves with the same modulus
of their amplitude (where
Ain is the input amplitude at the three ports). The transmitted wave does not experience any phase
shift, while both coupled waves have the same additional phase shift. Considering a phasor
diagram [Figure 10.2(b)], the output amplitude Aout is the vectorial sum of a transmitted
amplitude At and two coupled amplitudes Ac and Ac′. The moduli of these three amplitudes are
equal. Simple geometrical rules of equilateral triangles show that if the phase shift of Ac and Ac′
is 2π/3, the modulus
equals
(i.e., the modulus
of each input wave).
This gyro configuration is very attractive in terms of simplicity, but it faces the same problem
as the use of the nonreciprocal free port of a [2 × 2] coupler (see Section 3.3.1): the coupler loss
induces a spurious phase difference and there is a problem of polarization nonreciprocity, as a
polarizer cannot be used at the common input-output port. This last situation is improved with the
use of an unpolarized source (see Section 3.4.4) [5], but this approach remains limited to lowperformance applications.
10.2 Alternative Signal Processing Schemes
10.2.1 Open-Loop Scheme with Use of Multiple Harmonics
As we have already seen in Section 8.2, closed-loop schemes using phase ramp feedback provide
by far the best performance of scale factor, but they require the use of wideband integrated optic
phase modulators. The all-fiber approach (see Section 3.3.3) yields very good sensitivity, but
wideband phase modulation has not been demonstrated in a practical all-fiber form.
Numerous signal processing schemes compatible with an all-fiber configuration avoid the use
of integrated optics, even if they are not preferred anymore. A first solution uses the usual
modulation-demodulation technique with a sine wave (see Section 3.2.2), but also considers the
various harmonic components of the detector signal [7, 8]. The first and other odd harmonics
provide a biased sine signal of the rotation-induced phase difference ∆ϕ R, while the even
harmonics provide a cosine signal, and through calculation it is possible to recover the value of
∆ϕ R. Such an open-loop approach is limited in practice to a typical scale factor accuracy of 0.1%
to 1% because of the imperfect stability of the gain of the modulation-demodulation chain.
10.2.2 Second Harmonic Feedback
An all-fiber gyro is not bound to an open-loop scheme. In particular, the first proposed closedloop scheme compatible with an all-fiber piezoelectric phase modulator was the use of the
second-harmonic feedback [9, 10]. As seen in Section 3.2.3 and Figure 3.10, an additional secondharmonic phase modulation yields an unbalanced biasing modulation, which may be used to
compensate for the rotation-induced phase difference. Such an approach yields a nonlinear
response, and performances depend on the stability of the phase modulator, which is difficult to
control accurately.
10.2.3 Gated Phase Modulation Feedback
The gated phase modulation approach is conceptually derived from the harmonic feedback
scheme. We have already seen that any phase modulation ϕ m(t) yields a modulation ∆ϕ m(t) of the
phase difference, with ∆ϕ m(t) = ϕ m(t) − ϕ m(t − ∆τg) (see Sections 3.2.2 and 8.2.2). Because the
mean value of a difference is the difference of the mean values, the mean modulation < ∆ϕ m > of
the phase difference is:
⟨∆ϕ m(t)⟩ = ⟨ϕ m(t) − ϕ m(t − ∆τg)⟩ = ⟨ϕ m(t)⟩ − ⟨ϕ m(t − ∆τg)⟩ (10.1)
and both mean values ⟨ϕ m(t)⟩ and ⟨ϕ m(t − ∆τg)⟩ are perfectly equal because of reciprocity;
therefore:
< ∆ϕ m > = < ϕ m > − < ϕ m > = 0 (10.2)
This particularly applies to the case of the phase ramp (Section 8.2.2); however, because of the
reset, the amplitude of ∆ϕ PR is then much larger than the linear part of the sine response of the
interferometer, and the average of the detector signal is no longer equal to the null value of the
average modulation of the phase difference < ∆ϕ PR >. When the amplitude of ∆ϕ m remains in
this linear range, its mean feedback effect should be zero as the mean value of the phase
modulation. Harmonic feedback is working because the demodulation at the fundamental
frequency takes into account only one half-period of the second harmonic component. The
opposite half-period is in quadrature with respect to the demodulation at the fundamental
frequency and can be considered as “gated out” (Figure 10.3). Therefore, the mean effect of the
gated second harmonic modulation is not zero, even if its mean value does equal zero.
For a lower frequency of feedback modulation, an equivalent effect is obtained with an actual
electronic gating of the detector signal over one half-period of feedback modulation [11]. This
yields a nonlinear response similar to the harmonic feedback case and with the same problem of
control of modulator stability.
It is possible to linearize the response with a combined phase modulation feedback [12] using
the sum of a low-frequency sine modulation and a specific amount of its second harmonic
component, which flattens the feedback modulation during the time when the detector is turned on
(Figure 10.4).
Another principle of gated modulation feedback may also be implemented with a high-
frequency square-wave modulation when an integrated-optic modulator is used [13]. This has the
advantage of a linear response without an accurate control of harmonic ratio, in contrast to the
previous case.
All these schemes suffer from the instability of the phase modulator response. They do not have
the equivalent of the 2π-reset of the phase ramp (see Sections 8.2.2 and 8.2.3) to control it. Like
the situations discussed in Section 8.3.2, these methods are, in principle, wavelength-independent,
but the modulator response drift does not allow this fact to be taken advantage of in practice.
Figure 10.3 Actual gating of one half-period of the second harmonic with demodulation at the fundamental frequency.
10.2.4 Heterodyne and Pseudo-Heterodyne Schemes
Heterodyne techniques are commonly used in interferometry to avoid the problem of the basic
nonlinearity of the cosine response. A frequency shifter is placed on one arm of the
interferometer, which yields a beating of the output signal, because two waves interfere with
different frequencies. The output signal becomes (see Appendix A)
(10.3)
where ∆f is the frequency difference and ∆ϕ is the optical phase difference induced by the path
imbalance of the interferometer. This value ∆ϕ is then measured linearly with an electronic phase
meter, which compares the phase of the interference beating and the reference phase of the
modulation voltage of the frequency shifter.
Several schemes have been proposed to apply this technique to a ring interferometer [2, 14, 15],
but if a frequency shifter is used, both counterpropagating paths must be separated to apply the
shifting on only one wave. This destroys reciprocity, and even if some common-mode rejection
may be used [14, 15], it does not provide a bias stability comparable to the one obtained with a
reciprocal configuration. Note that the frequency shift feedback described in Section 8.2.1 is not a
heterodyne technique, despite the use of an acousto-optic frequency modulator: both
counterpropagating waves experience the frequency shift, and they interfere with the same
frequency without beating.
Figure 10.4 Combined phase modulation for gated feedback.
To avoid this problem of nonreciprocity induced by separating the counterpropagating waves,
the implementation of a heterodyne scheme with an integrated optic phase modulator in a typical
configuration (i.e., placed at one end of the fiber coil) was proposed [16]. A sawtooth electrical
generator is sent into an electrical integrator and the output voltage is then applied to the phase
modulator. As seen in Section 8.2.2 a sawtooth modulation is equivalent to a frequency shift, and,
as seen in Section 6.1, the ring interferometer responds to phase modulation as a differentiator.
Therefore, the combination of frequency shifting, integration, and differentiation results in a
constant frequency shift between both counterpropagating waves, and the optical phase in the
interferometer may be measured with an electronic phase meter, as in any heterodyne scheme.
However, the value of this frequency shift depends on the amplitude of the sawtooth and on the
gain of the electrical integrator, which modifies the scale factor. This method requires hardware
comparable to that needed in phase ramp schemes (see Sections 8.2.2 and 8.2.3) without obtaining
the same performance.
However, pseudo-heterodyne techniques have also been proposed [17, 18], which use an allfiber piezoelectric phase modulator and are therefore compatible with an all-fiber gyroscope. It is
based on the use of a large amplitude of phase modulation that scans several fringes. Contrary to
the usual demodulation technique (see Section 3.2.2), which compares the value of the detector
signal at the peaks of the phase modulation, the pseudo-heterodyne technique analyzes the phase
of the modulated detector signal with a gating when the optical phase modulation has a high
slope. This yields the equivalent of a heterodyning frequency upshift for the positive slope and a
downshift for the negative slope, since the frequency is the derivative of the phase. When the
interferometer is balanced, both beatings are symmetrical; but when there is an additional optical
phase difference, these two beatings are shifted in opposite directions (Figure 10.5), and by
comparing their phases, the value of the optical phase difference can be retrieved. This technique
requires careful control of the amplitude of the phase modulation, which limits its practicability.
10.2.5 Beat Detection with Phase Ramp Feedback
An alternative approach to the simple phase ramp feedback (see Section 8.2.2) was proposed [19]
that uses additional couplers to superimpose a Mach-Zehnder interferometer on the Sagnac
interferometer. The phase ramp used to null out the phase difference in the gyro generates an
interference beating in the Mach-Zehnder interferometer, and the counting of the beats yields an
incremental angular measurement of the rotation.
Figure 10.5 Principle of the pseudo-heterodyne scheme.
If it were possible to generate infinite ramps, this approach would be interesting because the
beats yield a direct calibration of the ramp slope. However, the ramp modulation has to be reset
because of the limited range of driving voltage, and it requires control of the 2π-reset, as in the
case of the simple serrodyne scheme. Therefore, this approach has no real advantage, particularly
because it requires a much more complex integrated optic circuit (Figure 10.6) [20].
Figure 10.6 Integrated optic circuit for beat detection.
10.2.6 Dual-Phase Ramp Feedback
Phase ramp feedback techniques require the use of integrated optics to get the high modulator
bandwidth required by the resets. However, an alternative dual phase ramp feedback, using a
triangular waveform instead of a sawtooth, has been proposed by Bergh [21]. This approach,
which has no fast reset, is compatible with the all-fiber piezoelectric modulator.
At rest, the positive slope of the triangular wave form ϕ PR(t) induces a π rad phase difference
∆ϕ PR while the negative slope induces a −π rad phase difference [Figure 10.7(a)]. In rotation, the
feedback loop keeps the system locked on ±π, and it yields a difference of duration between the
positive and negative slopes [Figure 10.7(b)], which is proportional to the rotation rate. This dual
ramp technique may be also implemented in a digital form.
This approach requires a larger phase modulation amplitude than the more conventional phase
ramp technique, but has some interest, particularly since it is compatible with the all-fiber
configuration of the fiber gyroscope.
10.3 Extended Dynamic Range with Multiple Wavelength
Source
Most fiber gyros work over an unambiguous dynamic range of ±Ωπ, which corresponds to a
Sagnac phase difference of ±π rad (see Section 2.3.1). However, it is possible to work over
several fringes, even with a broadband source, because the contrast is preserved over the
coherence length that corresponds to many wavelengths, but there is an ambiguity. For
applications where the gyro is turned on while being in the unambiguous range, it is possible to
count the fringes that are passed and to keep a valid measurement over an extended dynamic
range.
Now, if several wavelengths are used, the phase measurement varies with the wavelength, and it
is possible to recognize the fringe order, which increases the true unambiguous dynamic range
[22].
Figure 10.7 Dual phase ramp: (a) at rest and (b) in rotation.
References
[1] Udd, E., and R. F. Cahill, “Compact Fiber-Optic Gyro,” Springer Series in Optical Sciences, Vol. 32, 1982, pp. 189–194.
[2] Hotate, K., et al., “Rotation Detection by Optical Heterodyne Fiber Gyro with Frequency Output,” Optics Letters, Vol. 7, 1982,
pp. 331–333.
[3] Jackson, D. A., A. D. Kersey, and A. C. Lewin, “Fibre Gyroscope with Passive Quadrature Detection,” Electronics Letters, Vol.
20, 1984, pp. 399–401.
[4] Sheem, S. K., “Fiber-Optic Gyroscope with [3 × 3] Directional Coupler,” Applied Physics Letters, Vol. 37, 1980, pp. 869–871.
[5] Burns, W. K., R. P. Moeller, and C. A. Villaruel, “Observation of Low Noise in a Passive Fibre Gyroscope,” Electronics Letters,
Vol. 18, 1982, pp. 648–650.
[6] Poisel, H., and G. F. Trommer, “Low Cost Fiber Optic Gyroscope,” SPIE Proceedings, Vol. 1169, 1989, pp. 361–372.
[7] Böhm, K., et al., “Direct Rotation-Rate Detection with a Fibre-Optic Gyro by Using Digital Data Processing,” Electronics
Letters, Vol. 19, 1983, pp. 997–999.
[8] Frigo, N. J., “A Constant Accuracy, High Dynamic Range Fiber Optic Gyroscope,” SPIE Proceedings, Vol. 719, 1986, pp. 155–
159.
[9] Kim, B. Y., et al., “Harmonic Feedback Approach to Fiber-Gyro Scale Factor Stabilization,” Proceedings of OFS l/’83, IEE,
London, Vol. 221, 1983, pp. 136–137.
[10] Kim, B. Y., et al., “Response of Fiber Gyros to Signals Introduced at the Second Harmonic of the Bias Modulation Frequency,”
SPIE Proceedings, Vol. 425, 1983, pp. 86–89.
[11] Kim, B. Y., and H. J. Shaw, “Gated Phase-Modulation Feedback Approach to Fiber-Optic Gyroscope,” Optics Letters, Vol. 9,
1984, pp. 263–265.
[12] Kim, B. Y., and H. J. Shaw, “Gated Phase-Modulation Approach to Fiber-Optic Gyroscope with Linearized Scale Factor,”
Optics Letters, Vol. 9, 1984, pp. 375–377.
[13] Page, J. L., “Multiplexed Approach for the Fiber Optic Gyro Inertial Measurement Unit,” SPIE Proceedings, Vol. 1367, 1990,
pp. 93–102.
[14] Culshaw, B., and I. P. Giles, “Frequency Modulated Heterodyne Optical Fiber Sagnac Interferometer,” Journal of Quantum
Electronics, Vol. QE-18, 1982, pp. 690–693.
[15] Ohtsuka, Y., “Optical Heterodyne Detection Schemes for Fiber-Optic Gyroscopes,” SPIE Proceedings, Vol. 954, 1988, pp.
617–624.
[16] Eberhard, D., and E. Voges, “Fiber Gyroscope with Phase-Modulated Single-Sideband Detection,” Optics Letters, Vol. 9,
1984, pp. 22–24.
[17] Kersey, A. D., A. C. Lewin, and D. A. Jackson, “Pseudo-Heterodyne Detection Scheme for the Fibre Gyroscope,” Electronic
Letters, Vol. 20, 1984, pp. 368–370.
[18] Kim, B. Y., and H. J. Shaw, “Phase-Reading, All-Fiber-Optic Gyroscope,” Optics Letters, Vol. 9, 1984, pp. 378–380.
[19] Goss, W. C., “Fiber Optic Gyro Development at the Jet Propulsion Laboratory,” SPIE Proceedings, Vol. 719, 1986, pp. 113–
121.
[20] Minford, W. J., et al., “Fiber Optic Gyroscope Using an Eight-Component LiNbO3 Integrated Optic Circuit,” SPIE
Proceedings, Vol. 1169, 1989, pp. 304–322.
[21] Bergh, R. A., “Dual-Ramp Closed-Loop Fiber-Optic Gyroscope,” SPIE Proceedings, Vol. 1169, 1989, pp. 429–439.
[22] Kersey, A. D., A. Dandrige, and W. K. Burns, “Two-Wavelength Fibre Gyroscope with Wide Dynamic Range,” Electronics
Letters, Vol. 22, 1986, pp. 935–937.
CHAPTER 11
Resonant Fiber-Optic Gyroscope (R-FOG)*
11.1 Principle of Operation of an All-Fiber Ring Cavity
As described in Section 2.2.2, the resonant fiber-optic gyroscope, or R-FOG, uses a recirculating
ring resonant cavity [1] to measure the Sagnac effect and increase the shot noise limited
sensitivity of the rotation measurement. The fundamental photon shot-noise limited sensitivities
of the R-FOG and I-FOG can be represented by (11.1a) and (11.1b) for ease of comparison. The
photon shot noise limited sensitivity of the R-FOG is approximately given by
(11.1a)
(11.1b)
where δΩ is the uncertainty in rotation rate caused by the presence of shot noise, c and λo are the
speed of light and wavelength in vacuum, respectively, L and D are the length and diameter of the
fiber coil, respectively, and ηD and Nph are the quantum efficiency of the photodetector and the
number of detected photons per second, respectively [1].
As seen in (11.1a) and (11.1b), the sensitivity of an R-FOG is comparable to the sensitivity of an
I-FOG that has a fiber length F/2 times larger (where F is the finesse of the passive ring cavity).
Thus, for a finesse of 100, theoretically, the shot-noise-limited sensitivity of the resonator is
expected to be 50 times greater for the same coil length, diameter, and optical power.
Alternatively, the greater sensitivity afforded by the cavity finesse could be traded for the use of
F/2 less fiber to achieve the same sensitivity as the I-FOG. This possibility of using a shorter
fiber length has looked attractive, but the serendipity of the two-wave interferometer does not
apply to the resonant approach, and the R-FOG faces much more difficult technical challenges,
particularly because it requires the use of a very coherent light source, and the various parasitic
effects cannot be reduced as simply nor as efficiently as in the case of the I-FOG, where a
broadband low-coherence source is a very good solution to these problems.
The principle of a ring cavity is very similar to that of a Fabry-Perot cavity (see Section A.9.2).
There is multiple beam interference between the recirculating waves instead of the reflected
waves. An all-single-mode-fiber configuration uses low-loss fiber couplers instead of mirrors [2,
3]. Assuming a lossless propagation and two similar couplers (Figure 11.1), the finesse F of the
cavity is:
Figure 11.1 All-fiber ring cavity.
(11.2)
where C is the identical low coupling ratio of the two couplers, which replaces, in the formula,
the low mirror transmissivity T of ordinary Fabry-Perot interferometers. There is a periodic
spectral response (in spatial frequency σ, that is, inverse of wavelength λ). In particular, the
transmission response is:
(11.3)
where nLr is the optical length of one circulation and the Airy coefficient m is equal to 4(1 −
C)/C2. There is a resonant effect, and the light is transmitted when the optical length nLr of one
round trip along the ring cavity is equal to an integral number of wavelengths. Like the FabryPerot cavities, the periodicity of the response is called the free spectral range ∆σ free = 1/nLr. It
can be expressed in terms of temporal frequency ∆ffree = c ⋅ ∆σ free = c/nLr or in terms of
angular frequency ∆ωfree = 2π∆σ free = 2πc/nLr. The full width at half maximum of the narrow
transmission peaks is simply related to the free spectral range, with:
∆σ FWHM = ∆σ free/F (11.4)
Assuming that the source frequency σ 0 = 1/λ0 is matched on a resonance peak of the cavity, the
transmitted power varies as a function of the rotation rate as:
(11.5)
where ∆LR′(Ω) = LrDΩ/2c is the length change (in one direction) of the cavity due to rotation. It
is possible to define a free range ∆Ωfree [Figure 11.2(a)] in terms of rate with:
∆LR′(∆Ωfree) = λ0 (11.6)
that is,
(11.7)
Therefore, the full width at half maximum ∆ΩFWHM of the rate response is:
(11.8)
The equivalent of ∆ΩFWHM for the two-wave interferometer is the rate that corresponds to
±π/2 rad [Figure 11.1.2(b)]; that is, Ωπ, defined in Section 2.3.1 as:
(11.9)
Figure 11.2 Comparison between (a) resonator response (transmission port) and (b) interferometer response (reciprocal port).
It can be seen that the resonance response curve (for one direction) is equivalent to the response
of a two-wave interferometer, which would have a coil length L that is F/4 times larger than the
length Lr of the ring cavity. As will be seen later, the sensitivity is actually multiplied by two by
comparing the resonances between the opposite directions, which divides the dynamic range by
two.
Figure 11.3 Single-coupler fiber ring cavity.
It is also possible to use the reflection port (Figure 11.1), where the notch response, a resonance
dip, is complementary to the transmission response composed of resonance peaks. It is important
to remember that a resonant cavity is perfectly contrasted if both mirrors (or couplers) have the
same transmissivity (or the same coupling) and if the cavity has no loss. In practice, there is
always a residual attenuation, but it is still possible to get a perfect contrast for high finesse by
using a single-coupler ring with a coupler coupling ratio equal to the loss (Figure 11.3). This loss
is thus equivalent to a loss resulting from a coupling in a second coupler, and the notch response
is then perfectly contrasted. Note that such a single-coupler cavity may be realized without any
fiber splice along the resonant path with the use of a high coupling ratio and low transmission in
the coupler [4] (Figure 11.4).
To analyze the resonance of the ring cavity, the source spectrum width has to be narrower than
the width of the response peaks. This corresponds to a coherence length longer than the coil
length multiplied by the finesse. Therefore, the R-FOG requires a very long source coherence of
a few kilometers, as the coil length is typically a few tens of meters and the finesse is on the order
of 100. In terms of temporal frequency, this corresponds to a very narrow line width, on the order
of 10 to 100 kHz.
Figure 11.4 (a) Spliced and (b) unspliced single-coupler cavity.
11.2 Signal Processing Method
The principle of the signal processing method of the R-FOG has some similarities to the case of
the I-FOG. In a first step, a modulation-demodulation scheme is used to get an open-loop biased
signal that is a derivative of the unmodulated response (Figure 11.5), and in a second step this
signal is used as the error signal of a closed-loop processing unit that linearizes and stabilizes the
scale factor [5, 6].
However, in contrast to the I-FOG, the response is not automatically centered on an extremum
for zero rotation rate. The source frequency and the cavity length have first to be matched in
resonance in one direction, and the rotation is detected in the opposite direction, where the
sensitivity is then doubled. The dithering modulation required to get a biased demodulated signal
is performed by modulating the cavity length with a phase modulator placed inside the cavity [5,
6], or by modulating, outside the cavity, the input light frequency [7], or directly the source
frequency, particularly with the driving current of a semiconductor laser [8]. Note that a sine
modulation of the frequency may be obtained with a sine modulation of a frequency shifter, but
also with a sine modulation of a phase modulator. We have already seen that a frequency is the
derivative of a phase, and a sine-modulated phase yields by differentiation a (co)sine modulation
of the frequency.
This biasing modulation-demodulation is not defect-free, in contrast to that of the I-FOG,
because it does not use the rejections of nonlinearities and spurious intensity modulation brought
by the use of the coil as a delay line filter at the proper frequency (see Section 3.2.3).
Furthermore, the interferometer response is perfectly symmetrical because it is an
autocorrelation function, while the resonance peaks may carry some dissymmetry, especially
because of coupler loss [9]. Therefore, it is important to make the processing systems of the two
opposite directions symmetrical to get a good common mode rejection of their defects.
Figure 11.5 Biasing modulation-demodulation of the resonant peak response.
The complete system is composed of a modulation of the cavity length or of the light
frequency. The two counterrotating signals are demodulated, and one is used as an error signal to
keep the system on the resonance peak, while the opposite path is used as an error signal, which
applies an additional frequency shift through a closed-loop processing circuit (Figure 11.6). The
value of this frequency shift is used as the rotation rate signal. It corresponds to the difference ∆fR
of resonance frequency, which is induced by the rotation rate Ω between both counterrotating
paths (see Section 2.2.2):
Figure 11.6 Architecture of a resonant fiber-optic gyroscope.
∆fR = DΩ/(nλ) (11.10)
where D is the coil diameter, n the index of the fiber, and λ the source wavelength. The frequency
shifting required to close the two processing loops was originally performed with bulk acoustooptic Bragg cells [5, 6], but it can also be done with sawtooth serrodyne modulation (see Section
8.2.2) applied on integrated-optic phase modulators [10], which preserves the ruggedness of an
all-guided approach. Alternatively, it can be done by using separate lasers in the two directions,
provided the relative phase jitter is sufficiently low [2]. It can be seen that if the R-FOG is using a
shorter fiber coil than the I-FOG, this advantage is counterbalanced by a higher complexity, as the
number of components (couplers and modulators) is nearly doubled in comparison to the
minimum configuration of the two-wave ring interferometer. While the additional technical
complexity of the R-FOG system must be overcome, the advancement of highly integrated
photonic circuits may reduce the size and cost of the additional components, making the R-FOG
configuration more attractive.
However, note that, for the same sensitivity, the unambiguous dynamic range of the R-FOG is
larger than that of the I-FOG. Their sensitivities are equivalent when ∆ΩFWHM is equal to Ωπ,
but the range of the resonator is ±∆Ωfree/2, which is F/2 wider than the interferometer range of
±Ωπ.
11.3 Reciprocity of a Ring Fiber Cavity
11.3.1 Introduction
Reciprocity has been seen to be a fundamental feature of the I-FOG (see Chapter 3), and it is also
possible to define a reciprocal configuration of an R-FOG. In principle, a configuration using the
reflection ports of the cavity, with an in-line polarizer on the two single-mode fiber leads [11], as
shown in Figure 11.7, is reciprocal, provided that each input wave to the resonator excites one and
the same polarization state within the resonator. With such a configuration, each couple of
counterpropagating waves that recirculates the same number of times along the cavity follows
exactly the same path, including the leads, in opposite directions: the two counterpropagating
waves accumulate exactly the same phase, and they have exactly the same attenuation because of
reciprocity. Because this is valid for all the recirculating couples, the resonance responses of a
cavity using such a reciprocal configuration are perfectly identical in both of the opposite
directions.
Figure 11.7 Reciprocity of a ring cavity.
However, the practical R-FOG case generally consists of two polarization states within the
resonator [12, 13], that is, two polarization eigenstates [14], that reproduce themselves with a
certain loss and round-trip phase shift, within the ring resonator. The objective is to launch light
by exciting the same polarization eigenstate in both directions and compare the clockwise (cw)
and counterclockwise (ccw) resonance frequencies to derive a rotation measurement, as these
states have the same resonance frequency in a nonrotating frame. In practice, the two input waves
will not be perfectly aligned to excite, or couple all their energy into, a single eigenstate within
the resonator. This imperfection is not identical, in general, for the cw and ccw input waves.
Because of this, most implementations similar to that of Figure 11.7 do not offer a guarantee of
perfect symmetry between counterrotating waves, or error-free operation. In contrast, the
reciprocal configuration of the I-FOG, the polarizer may be imperfect, but both cw and ccw
waves are equally aligned or misaligned from the axes of the fiber in the coil, and both travel
through the same polarization filter prior to detection. The reason why cw and ccw input wave
polarization states for the R-FOG are not necessarily the same is due to the existence polarization
cross-coupling in each input path prior to reaching the resonator. Hence, the input polarization
states of the cw and ccw waves may excite or may overlap with the desired polarization eigenstate
of the resonator to a different degree. As will be seen later, this produces unequal resonance
asymmetries for cw and ccw resonance dips in the configuration using the reflection port. This
problem can be addressed by using the transmission ports of an all-fiber ring resonator [12]. It is
possible to consider that the second coupler is just tapping off a small amount of the intensity of
the light which resonates inside the cavity and that parasitic phase shifts in the tapping coupler do
not affect the measurement of this intensity.
11.3.2 Basic Reciprocity Within the Ring Resonator
The preceding discussion prompts the question of whether the light propagation within the
resonator is inherently nonreciprocal, or if potential nonreciprocity is due to unequal and
imperfect excitation and detection of the eigenstates of the resonator in the two directions. Figure
11.8 shows one example of a fiber ring resonator to illustrate the concept of reciprocity within the
loop itself. We use the resonator configuration of similar to that of Figure 11.1 with provisions
for birefringent fiber, a couple of polarization dependent loss points (representing polarizers, for
example), and a couple of polarization cross-coupling points. We assume the resonator is not
rotating.
Following Figure 11.8 and the Jones matrix formalism described in Section A.10, we can
represent the round trip propagation of light around the ring in the cw direction, starting at point
A as:
Figure 11.8 Fiber ring resonator with polarization crosstalk, birefringence, and polarization dependent loss.
[Ecw]o = a0[M1][C1][M7][R3(θ3)][M6][P2][M5][R2(θ2)]
[M4][C2][M3][R1(−θ1)][P1][R1(θ1)][M2][Ecw]i (11.11)
In (11.11), the electric field starting at point A is given by the column vector,
(11.12)
the field at point A after round-trip traversal of the ring resonator is given by the column vector,
(11.13)
where the matrix [Hcw] represents the total loop transmission matrix in the cw direction and the
matrices Mj are the Jones representation for propagation through a length lj of birefringent fiber,
that is,
(11.14)
where nx and ny are the indices of refraction of the principal axes of the fiber. The matrices P 1
and P 2 represent polarization dependent loss elements, such as polarizers with a transmission
coefficient of unity for one axis and a transmission coefficient of ε1 and ε2, respectively, for the
other axis. For the purpose of illustration P 1 is assumed to be misaligned from the principal axes
of the fiber by an angle θ1 relative to the principal axes of the fiber. The polarizer represented by
P 2 is assumed to be aligned with the principle axes of the fiber. P 1 and P 2 are given by
(11.15)
and
(11.16)
R1(θ1) and R1(−θ1) represent, respectively, the rotation of axes into, and out of, the frame of
the principal axes of P 1,
(11.17)
Polarization cross-coupling points, possibly due to splice misalignments, are represented by
rotation matrices R2(θ2) and R3(θ3) given by
(11.18)
The couplers of Figure 11.18 are represented in (11.9) by matrices C1 and C2 given by
(11.19)
where C1 and C2 are the intensity coupling coefficients of couplers C1 and C2. The couplers are
assumed to have polarization-independent coupling ratios. The polarization independent
roundtrip loss is represented by the scalar constant a of (11.11).
By comparison, the roundtrip propagation of light within the resonator in the ccw direction is
represented as
[Eccw]o = a0[M2][R1(−θ1)][P1][R1(θ1)][M3][C2][M4][R2(−θ2)]
[M5][P2][M6][R3(−θ3)][M7][C1][M1][Eccw]i (11.20)
where the matrix [Hccw] represents the total loop transmission matrix in the ccw direction and
where the electric field starting at point A in the ccw direction is given by the column vector
(11.21)
where the field at point A after round-trip traversal of the ring resonator in the ccw direction is
given by the column vector
(11.22)
Note that a cross-coupling angle of θ (electric field magnitude ⎪sinθ⎪) for cw propagation is
represented as R(θ) and for ccw propagation by R(−θ) = RT(θ) where the superscript T represents
the matrix transpose. Because the diagonal matrices at transposes of themselves, (11.20) can be
rewritten as
[Eccw] = a0[M2]T[R1(θ1)]T[P1]T[R1(−θ1)]T[M3]T[C2]T[M4]T[R2(θ2)]T
[M5]T[P2]T[M6]T[R3(θ3)]T[M7]T[C1]T[M1]T[Eccw]i = [Hcw]T[Eccw]i (11.23)
Hence,
[Hccw] = [Hcw]T (11.24)
It should be noted that (11.24) would not hold if there we true nonreciprocal effects like rotation
rate that would have to be added explicitly to (11.11) and (11.20) to affect propagation in each
direction differently.
The significance of (11.24) can be understood by the following. Because the resonant
eigenstates of the resonator are those that reproduce themselves around the resonator and
constructively interfere, the polarization eigenstates in the cw direction represented by column
vectors [e1]cw and [e2]cw and in the ccw directions by [e1]ccw and [e2]ccw must satisfy the
following characteristic equations
(11.25)
(11.26)
where λ1cw, λ2cw, λ1ccw, and λ2ccw are the eigenvalues representing the round-trip phase shift
and loss of each of the eigenstates,
(11.27)
and
(11.28)
where αjcw and αjccw are the round-trip loop transmission amplitude coefficients that are the
square root of one minus the intensity loss for each eigenstate, respectively, and the ϕ jcw and
ϕ jccw are the round-trip phase shift for each eigenstate. The round-trip phase shift determines the
resonance frequency for each eigenstate.
Because the eigenvalues of a matrix are the same as the eigenvalues of the transpose of a
matrix, the two eigenstates in the cw direction have the same eigenvalues as that of the ccw
direction.
Hence, the resonance frequencies of the resonator of Figure 11.8 and many similar ones
satisfying (11.24) are the same, and thus, light propagation in them is reciprocal for the same
eigenstate. Note that Figure 11.8 is an arbitrary example for illustration purposes, and that (11.24)
is valid for other combinations of polarization rotations, polarizers, birefringent elements, and
couplers as represented in (11.14) through (11.19).
11.3.3 Excitation and Detection of Resonances in a Ring Resonator
As discussed in the beginning of this section, the main problem of reciprocity stems from the
unequal excitation of the two polarization states within a ring resonator, not from propagation of
light in a single eigenstate within the resonator. A reciprocal condition would be obtained if both
cw and ccw input waves only excited one, and the same state (those that have the same
eigenvalue). In general, light incident on the ring in the two directions is imperfect, exciting, or
overlapping with the desired eigenstate for the most part, but to some degree with the unwanted
eigenstate. This degree of imperfection is not necessarily equal in the two directions. As a result
of exciting an unequal superposition of states, polarization and birefringence behavior in the ring
resonator can cause rotation-equivalent errors in the R-FOG. Light that is detected in practice is a
superposition of light from two states, and the consequential effects of the undesired state must be
mitigated or minimized. Because single-mode fiber transmits two polarization states with slightly
different velocities, the ring resonator has two polarization eigenstates (as discussed in the
previous section) and therefore two resonance frequencies in each direction. The resonance
frequencies correspond to propagation of light in these two polarization modes [4, 12–14]. One
way of viewing the polarization evolution around the loop for a particular eigenstate is the
following. Along one round trip, from one point in the cavity back to the same point, each
polarization eigenstate evolves, depending on the birefringence and the PDL along the path. As
the light propagates along the resonator fiber, these two states change as a function of the
birefringence, but at a given point in the ring they reproduce themselves; that is, each return to
their original state after one round trip at each point along the ring. In time, that eigenstate at any
given point may vary if the birefringence changes, but each of the two eigenstates is defined by
that path of polarization evolution that reproduces itself around the ring. This condition is
necessary to get resonance by multiple beam interference of the various recirculating waves, as
they must be in the same state of polarization to be combined constructively.
Thus, for simplicity, the cavity has two sets of resonances corresponding to its two polarization
modes that propagate over one round trip along different optical lengths, n1Lr and n2Lr, because
of birefringence. By reciprocity, as illustrated in the prior section, these two sets are identical in
optical path length and loss in both of the opposite directions (when the system does not rotate). It
seemed at first that an R-FOG could work with ordinary single-mode fiber by using only the
resonance of one polarization mode [7]. However, a precise measurement of the central
resonance frequency cannot tolerate a small dissymmetry of the peak or notch response. As
discussed earlier, in principle, there is no resonance asymmetry for either the resonance dip in
reflection or the resonance peak in transmission, provided that the light waves incident on the
cavity are perfectly matched to one single polarization eigenstate at the entrance to the resonator.
In practice, there is always some component of the input light that is not aligned with the input
state of the resonator. This parasitic input light can cause resonance asymmetry in in two ways.
The first case of imperfect polarization alignment to consider is that in which the parasitic light
significantly excites the second (unwanted) resonant eigenstate within the ring. In this case, the
second resonance frequency is sufficiently close enough the desired resonance frequency that the
excitation of the undesired resonance tail actually produces and intensity variation across the
main resonance signal. If the parasitic component of polarization is the same for both input
waves, the asymmetry is identical in both of the opposite directions [11]. However, if the parasitic
input polarization component is not equal in both directions the resonance asymmetry is not
identical. This causes a bias when using the resonance-detection signal processing technique
described earlier. Another problem with the use of single-mode fiber is that polarization fading
may occur for the resonance dips and peaks. It is not possible to take advantage of depolarization
effects, as in the case of I-FOG that works with a broadband source (see Chapter 5). One method
used to avoid polarization fading and to reduce the influence of the crossed-mode resonance, that
is, make the polarization eigenstate inside the resonator more stable (and easier to couple to), as
well as the make the input light polarization more stable, has been to use highly birefringent
polarization maintaining fiber [11, 15].
With polarization conservation using polarization maintaining fiber, most of the optical power
remains in one mode of the resonator, but there is still some light in the crossed mode because
parasitic polarization cross-coupling occurs, particularly in the coupler. In this case, there is still
a small resonance dip in addition to the main signal. Furthermore, the position of this parasitic
dip shifts as a function of temperature with respect to the resonance response of the main mode
[15]. This yields an unstable dissymmetry of the resonance peak (Figure 11.9).
Figure 11.9 Dissymmetry of the response induced by the crossed-polarization response.
As a matter of fact, the resonances of the two polarization modes have a periodicity equal to the
free spectral ranges nsLr and nfLr, where ns and nf are, respectively, the indexes of the slow mode
and of the fast mode. With stress-induced high-birefringence fibers, the index difference ns − nf
has a typical variation of 10–3/°C, and when a temperature change induces a change of (ns − nf)Lr
equal to the source wavelength λ, the small parasitic dip shifts over the free spectral range of the
main resonance signal. With a resonator length of 10m and a birefringence beat length of 1 mm,
this is obtained with a temperature change of only 0.1°C. Note that polarization-maintaining fibers
using an elliptical core have a much lower temperature dependence (about 10–5/°C instead of 10–
3/°C), but, in practice, they are not often used because of a lower polarization conservation.
To avoid this problem of resonance of the crossed polarization mode, placing a polarizing
element inside the cavity has been proposed [14]. Another, more subtle solution has been
described [16, 17] and demonstrated [12]. The resonator loop is still made of polarizationmaintaining fiber, but a splice with a 90° rotation of the principal axes of the fiber is added inside
the cavity (Figure 11.10).
Figure 11.10 Polarization-preserving cavity with a 90° rotation of the fiber principal axes.
With this configuration, despite the two polarization modes of the fiber, both the cavity modes
experience equal time on each axis, thus eliminating the mode-crossing issue discussed above. A
complete cavity round trip is now composed of propagation along the cavity in the fast mode and
a second propagation in the slow mode. The optical length of the cavity becomes (nf + ns)Lr and
the free spectral range becomes 1/(nf + ns)Lr. This idea is similar to the Moebius ring, which has
only one side, while a tape has two sides [private conversation with W. Schröder, Fachhochschule
Offenburg, 1991].
This technique uses a maximal separation of resonance frequencies to reduce the effect of the
second resonance’s tail under the desired resonance. Another way of greatly attenuating or
virtually eliminating the issue with excitation of the second polarization eigenstate and the
resonance crossing issue is to make the resonator with single-polarization fiber [14]. This greatly
increases the loss of the second polarization eigenstate. Depending on the extinction ratio of the
fiber, this greatly attenuates the intensity of the second resonance line shape so that its tail does
not cause asymmetry in the region of the desired resonance line shape. Historically, this fiber has
been more difficult to use because of bend sensitivity and loss; however, recent results have
demonstrated a resonator with sufficient finesse to demonstrate ARW performance below 0.008°/
, which was nearly shot-noise-limited for the modest detected light levels in the setup [18].
The second issue with an imperfect polarization input state is due to the electric field of the
light rejected by the resonator, but mixes in the detected signal, as was pointed out by Schröder et
al. [11]. Consider R-FOG configuration in which detection is implemented from the reflection
port of the resonator as shown in Figure 11.11.
Figure 11.11 Illustration of mixing of rejected light with signal light at reflection port detector.
The main signal light is first passed through a polarizer of finite extinction ratio, ε, and
launched toward the resonator, denoted as Exi. Prior to reaching the resonator, a small component
of light, ka, is cross-coupled from the x-axis to the y-axis at point a, given by
Eya = kaExi (11.29)
Assuming, in this case, that the resonator eigenstate at its input is x-polarized, the y-polarized
component of light reaching the resonator input is reflected, that is, its energy is not coupled into
the resonator. It then proceeds towards the detector, first encountering another small degree of
cross-coupling kb back to the x-axis at point b, and then passes along the transmission axis of a
polarizer prior to being detected. The electric field of this parasitic light can be then be
represented by
Ex′= kakbeiθExi (11.30)
where the phase shift ϕ contains the common phase shift between the parasitic light and the
detected signal light as well as a differential phase shift ∆ϕ between the two light waves due to a
different polarization path traveled in the input/output leads and the phase shift through resonance
experienced by the signal light, given by ∆ϕl and ∆ϕ r, respectively. Figure 11.12(a) shows the
shape of a symmetric resonance dip due to the signal light ⎪Ex⎪2 component only impinging
upon the detector (ka = kb = 0).
Figure 11.12 (a) Resonance dip of the signal light. (b) Phase shift, ∆ϕ r, of electric field of signal light in region about resonance.
(c) Interference term ∆I of signal light with parasitic light. (d) Total detected resonance dip intensity with resonance asymmetry.
The corresponding phase shift of the signal light, as the frequency is swept through resonance,
∆ϕ r, is shown in Figure 11.1(b), which assumes the resonator reflected signal light changes by +π
to −π as it crosses through the resonance center.
The cross-polarized light represented by (11.29) does not have the same phase response, as it is
off-resonance. Assuming ka < 1, kb < 1, its intensity at the photodetector can be neglected in
comparison to the interference term it produces with the signal light, denoted as ∆I, and given by:
∆I = 2⎪Ex⎪⎪Ex′⎪cos(∆ϕ l + ∆ϕ r) (11.31)
Figure 11.12(c) shows the interference term of the detected intensity in a region about
resonance for the example of ∆ϕl = −π/2. When added with the detected signal light intensity
⎪Ex⎪2 the total line shape is asymmetric, and shown in Figure 11.12(d). The shape of the
asymmetry will vary as ∆ϕl varies.
This error is a consequence of imperfect matching of the input light energy into a single
eigenstate and mixing the nonresonant light (or light rejected by the resonator) with the signal
light at the detector. This may be reduced by minimizing the magnitude of mismatched light to the
resonator and by employing a second coupler to detect the resonances in transmission [19], as is
shown in Figure 11.1. In the latter case, parasitic light that is rejected at the reflection port is not
mixed with signal light being detected in transmission. This type of transmission arrangement has
been demonstrated using single polarization fiber [18] in transmission which addresses the
problem of the intensity-type error of a second resonance associated with PM fiber approaches.
11.4 Other Parasitic Effects in the R-FOG
In addition to these problems of reciprocity, the R-FOG faces various parasitic effects similar to
the ones encountered in the I-FOG. Solutions to these problems are usually derived from what
was proposed for the I-FOG, except, as we have already seen, the fundamental drawback that a
broadband source cannot be used because an R-FOG requires a source with a very long
coherence length.
This makes Rayleigh backscattering and the Kerr effect severe obstacles to high performance.
The effect of Rayleigh backscattering may be reduced by various techniques of phase or
frequency modulation [5, 6, 20, 21] to avoid, in the detection band, spurious interference signals
between the primary waves and the backscattered waves, but this increases significantly the
complexity of the system. The problem of the nonlinear Kerr effect is even worse, since the
optical power stored in the cavity is much higher than the input power, and for the same relative
power imbalance between both counterpropagating waves, an R-FOG has a Kerr sensitivity
enhanced by about one-third the finesse compared to an equivalent I-FOG using a coherent
source. Square-wave intensity modulation techniques proposed for the I-FOG to reduce the Kerr
sensitivity (see Section 7.2) can also be used for the R-FOG [22], but the precision of modulation
required is very difficult to get and the Kerr effect is still regarded as the main limitation to low
drift for the R-FOG [23].
The other parasitic effects that are not related to coherence are not significantly different
between the R-FOG and the I-FOG. Transient related effects (see Chapter 6) are reduced in the RFOG [3], as the fiber coil is shorter, but this problem is mitigated to a high degree in the I-FOG
with quadrupolar winding and adequate potting of the fiber coil. The Faraday effect (see Section
7.1) also induces a rate error in the R-FOG if the polarization-preserving fiber has a twist
variation matched to the perimeter of one turn [24]. While the R-FOG scale factor stability has the
same dependence on wavelength stability found in the I-FOG, the R-FOG uses a monochromatic
source. If the other issues attendant with the monochromatic source can be adequately reduced,
there are opportunities for possibly greater wavelength stability. Because the R-FOG source is
spectrally a delta function, its interferometric wavelength is trivially defined, whereas, in the IFOG, variations in the shape of broadband spectrum can cause a shift in the gyro’s effective
operating wavelength.
Despite better shot-noise-limited theoretical performance the R-FOG still faces the difficult
technical problems [25], and published results do not indicate performance levels close to those
of the best I-FOGs to date. Recently, considerable technical progress has been reported in the
areas of both angle random walk (ARW), and bias stability. In the ARW area, greater
understanding of the effect of laser frequency noise has been reported [26, 27]. It was appreciated
early, by Ezekiel and Balsamo [2] that differential laser frequency noise could disrupt the signal
to noise performance of a passive resonator gyro, so a single laser was employed. Moreover,
differential frequency noise in the low-frequency range where rotation rate is measured was
recognized to be an issue. However, Ma et al. [26] reported that frequency noise at frequencies
well above the rotation-rate detection band posed a limitation to performance. Specifically, laser
frequency noise at even harmonics of the modulation frequency was translated to the lowfrequency gyro output range via the modulation and demodulation process. Sanders et al. [27]
also recognized that high frequency laser noise was an issue, and proposed a solution using
optical filtering and laser stabilization, which was recently demonstrated to give an ARW of
0.008°/
with a fiber coil length of 19m on an 11.5-cm diameter spool. By using polarizing
fiber for the ring to address polarization errors, they reported a bias stability of 0.1°/hr (1σ) for a
2-hour timeframe [28].
The reported performance of the R-FOG still remains well below that of the I-FOG, and
advancement of the R-FOG may rely on new technologies. One emerging technology discussed
in connection with its benefits to the I-FOG in [29–30], hollow core optical fiber [31] (see Section
B.10), may have an even greater impact on R-FOGs [32] than I-FOGs. The common benefits are a
reduction of roughly six times in the Shupe effect and roughly 250 times in the magneto-optic
Faraday effect. However, the impact of removal of glass in the light path promises to reduce the
Kerr effect in the R-FOG by over two orders of magnitude, an effect that was already eliminated
by the broadband source in the I-FOG. Similarly, the threshold for stimulated Brillioun scattering
(SBS) in the fiber coil should be greatly increased, allowing very high power levels within the
sensing coil.
Hollow-core fiber ring resonators capable of being employed for an R-FOG have been
demonstrated [32–33]. In one method, to complete the resonator loop, free space optical mirrors
were used, in an arrangement shown in Figure 11.13 instead of using fiber optic couplers of
Figure 11.1. Because the light traveling within the fiber is effectively in free space, low-loss
coupling between free space optics and the hollow core fiber were demonstrated. The total loss
attributed to free space to fiber coupling was on the order of only 3%. The fiber length and loss
were 0.9m and −20 dB/km, respectively. When combined with the other losses around the ring,
the finesse was 42, suitable for R-FOG application. Other means of coupling and completing the
loop have been proposed [34] and demonstrated, as well as initial R-FOG demonstrations with
them [33]. However, this fiber is in a relatively early stage of development and its performance
(e.g., loss or polarization) and cost need to approach that of conventional fibers to take maximum
advantage of this promising development.
Note that, based on this principle of passive ring cavity, an even more ambitious approach was
pursued with an integrated-optic single-loop ring cavity, which could have made possible the
fabrication of a fully integrated optical gyro by planar mass-duplication techniques [35–41], but
the technological challenges to be faced are very difficult, and this approach is not presently
believed to be competitive.
There has also been a proposal [42] to make an active fiber resonator similar to a ring laser
gyro, but in a fiber form, using amplification by stimulated Brillouin scattering. Backscattering
also induces lock-in around zero rotation rate, but this novel subject requires more R&D to
advance it and evaluate its advantages.
Figure 11.13 Transmission ring resonator using hollow core fiber.
Acknowledgments
The author of this chapter wishes to acknowledge the work of Mr. Lee K. Strandjord and Dr.
Tiequn Qiu of Honeywell International for helpful discussions regarding R-FOG performance
considerations and error mechanisms and for their insightful and expert review of this chapter.
References
[1] Ezekiel, S., and H. J. Arditty, “Fiber-Optic Rotation Sensors,” in Fiber-Optic Rotation Sensors and Related Technologies, New
York: Springer-Verlag, 1982, pp. 2–26.
[2] Ezekiel, S., and S. R. Balsamo, “Passive Ring Resonator Laser Gyroscope,” Applied Physics Letters, Vol. 30, 1977, pp. 478–
480.
[3] Schupe, D. M., “Fiber Resonator Gyroscope: Sensitivity and Thermal Non-Reciprocity,” Applied Optics, Vol. 20, 1981, pp.
286–289.
[4] Stokes, L. F., M. Chodorow, and H. J. Shaw, “All Single-Mode Fiber Resonator,” Optics Letters, Vol. 7, 1982, pp. 288–290.
[5] Sanders, G. A., M. G. Prentiss, and S. Ezekiel, “Passive Ring Resonator Method for Sensitive Inertial Rotation Measurements in
Geophysics and Relativity,” Optics Letters, Vol. 6, 1981, pp. 569–571.
[6] Meyer, R. E., et al., “Passive Fiber-Optic Ring Resonator for Rotation Sensing,” Optics Letters, Vol. 8, 1983, pp. 644–646.
[7] Carroll, R., et al., “The Passive Resonator Fiber Optic Gyro and Comparison to the Interferometer Fiber Gyro,” SPIE
Proceedings, Vol. 719, 1986, pp. 169–177.
[8] Ohtsu, M., and S. Araki, “Using a 1.5 μm DFB InGaAsP Laser in a Passive Ring Cavity-Type Fiber Gyroscope,” Applied
Optics, Vol. 26, 1987, pp. 464–470.
[9] Youngquist, R. C., L. F. Stokes, and H. J. Shaw, “Effect of Normal Mode Loss in Dielectric Waveguide Directional Couplers
and Interferometers,” Journal of Quantum Electronics, Vol. QE-19, 1983, pp. 1888–1896.
[10] Sanders, G. A., et al., “Resonator Fiber-Optic Gyro Using LiNbO3 Integrated Optics at 1.5 μm,” SPIE Proceedings, Vol. 985,
1988, pp. 202–210.
[11] Schröder, W., et al., “Progress in Fiber Gyro Development for Attitude and Heading Reference Systems,” SPIE Proceedings,
Vol. 719, 1986, pp. 162–168.
[12] Strandjord, L. K., and G. A. Sanders, “Resonator Fiber Optic Gyro Employing a Polarization-Rotating Resonator,” SPIE
Proceedings, Vol. 1585, 1991, pp. 163–172.
[13] Lamouroux, B. F., B. S. Prade, and A. G. Orszag, “Polarization Effects in Optical-Fiber Ring Resonators,” Optics Letters, Vol.
7, 1982, pp. 391–393.
[14] Iwatsuki, K., K. Hotate, and M. Higashiguchi, “Eigenstate of Polarization in a Fiber Ring Resonator and Its Effect in an Optical
Passive Ring Resonator Gyro,” Applied Optics, Vol. 25, 1986, pp. 2606–2612.
[15] Sanders, G. A., et al., “Evaluation of Polarization Maintaining Fiber Resonator for Rotation Sensing Applications,” Proceedings
of OFS 5, New Orleans, LA, 1988, pp. 409–412.
[16] Sanders, G. A., R. B. Smith, and G. F. Rouse, “Novel Polarization-Rotating Fiber Resonator for Rotation Sensing Applications,”
SPIE Proceedings, Vol. 1169, 1989, pp. 373–381.
[17] Mouroulis, P., “Polarization Fading Effects in Polarization-Preserving Fiber Ring Resonators,” SPIE Proceedings, Vol. 1169,
1989, pp. 400–412.
[18] Strandjord, L. K., et al., “Resonator Fiber Optic Gyro Progress Including Observation of Navigation Grade Angle Random
Walk,” Proceedings of OFS-22, October 2012.
[19] Sanders, G. A., L. K. Strandjord, and T. Qiu, “Optical Resonator Gyro and Method for Reducing Resonance Asymmetry
Errors,” U.S. Patent 7522284.
[20] Iwatsuki, K., K. Hotate, and M. Higashiguchi, “Effect of Rayleigh Backscattering in an Optical Passive Ring Resonator,”
Applied Optics, Vol. 23, 1984, pp. 3916–3924.
[21] Hotate, K., K. Takiguchi, and A. Hirose, “Adjustment-Free Method to Eliminate the Noise Induced by the Backscattering in an
Optical Ring-Resonator Gyro,” IEEE Photonics Technology Letters, Vol. 2, 1990, pp. 75–77.
[22] Iwatsuki, K., K. Hotate, and M. Higashiguchi, “Kerr Effect in an Optical Passive Ring-Resonator Gyro,” Journal of Lightwave
Technology, Vol. LT4, 1986, pp. 645–651.
[23] Hotate, K., and K. Takiguchi, “Drift Reduction in an Optical Passive Ring-Resonator Gyro,” SPIE Proceedings, Vol. 1585,
1991, pp. 116–127.
[24] Hotate, K., and M. Murakami, “Drift of an Optical Passive Ring-Resonator Gyro Caused by the Faraday Effect,” Proceedings
of OFS 5, New Orleans, LA, 1988, pp. 405–408.
[25] Sanders, G. A., “Critical Review of Resonator Fiber Optic Gyroscope Technology,” Proceedings of SPIE Conference Critical
Reviews of Optical Science and Technology, Vol. CR44, September 1992, pp. 133–159.
[26] Ma, H., et al., “Laser Frequency Noise Limited Sensitivity in a Resonator Optic Gyroscope” 15th OptoElect. and
Communication Conf. (OECC2010) Technical Digest, Vol. 8P-70, 2010.
[27] Sanders, G. A., T. Qiu, and L. Strandjord, “Light-Phase-Noise Error Reducer,” U.S. 8,009,296 B2, 2011.
[28] Qiu, T., et al., “Performance of Resonator Fiber Optic Gyroscope Using External-Cavity Laser Stabilization and Optical
Filtering,” 23rd International Conference on Fiber Optic Sensors (OFS23), Santander, Spain, SPIE Proceedings, Vol. 9157,
Paper #153, 2014.
[29] Kim, H. K., et al., “Fiber-Optic Gyroscope Using an Air-Core Photonic-Bandgap Fiber,” Proceedings of the OFS-17, May
23–37, 2005.
[30] Dangui, V., et al., “Phase Sensitivity to Temperature of the Fundamental Mode in Air-Guiding Photonic-Bandgap Fibers,” Opt.
Express, Vol. 13, No. 18, 2005.
[31] Cregan, R. F., et al., “Single-Mode Photonic Band Gap Guidance in Air,” Science, Vol. 285, 1999, pp. 1537–1539.
[32] Sanders, G. A., L. K. Strandjord, and T. Qiu, “Hollow Core Fiber Optic Ring Resonator for Rotation Sensing,” Proceedings of
the 18th International Conference on Optical Fiber Sensors, Cancun, Mexico, 2006.
[33] Terrel, M., M. Digonnet, and S. Fan, “Resonant Fiber Optic Gyroscope Using an Air-Core Fiber,” Journal of Lightwave
Technology, Vol. 30, No. 7, April 2012.[35] Haavisto, J., and G. A. Pajer, “Resonance Effects in Low-Loss Ring Waveguides,”
Optics Letters, Vol. 5, 1980, pp. 510–512.
[34] Stabley, J. S., et al., “System and method for fiber-based resonator coupling,” U.S. Patent 7,802,928 B2, 2010.
[36] Walker, R. G., and C. D. W. Wilkinson, “Integrated Optical Ring Resonators Made by Silver Ion-Exchange in Glass,” Applied
Optics, Vol. 22, 1983, pp. 1029–1035.
[37] Honda, K., E. M. Garmire, and K. E. Wilson, “Characteristics of an Integrated Optics Ring Resonator Fabricated in Glass,”
Journal of Lightwave Technology, Vol. 2, 1984, pp. 714–719.
[38] Naumaan, A., and J. T. Boyd, “Ring Resonator Fabricated in Phosphosilicate Glass Film Deposited by Chemical Vapor
Deposition,” Journal of Lightwave Technology, Vol. 4, 1986, pp. 1294–1303.
[39] Connors, J. M., and A. Mahapatra, “High Finesse Ring Resonators Made by Silver Ion Exchange in Glass,” Journal of
Lightwave Technology, Vol. 5, 1987, pp. 1686–1689.
[40] Bismuth, J., et al., “Low-Loss Ring Resonators Fabricated from Silicon Based Integrated Optics Technologies,” Electronics
Letters, Vol. 27, 1991, pp. 722–723.
[41] Adar, R., et al., “Measurement of Very Low-Loss Silica on Silicon Waveguides with a Ring Resonator,” Applied Physics
Letters, Vol. 58, 1991, pp. 444–445.
[42] Smith, S. P., F. Zarinetchi, and S. Ezekiel, “Fiber Laser Gyros Based on Stimulated Brillouin Scattering,” SPIE Proceedings,
Vol. 1585, 1991, pp. 302–308.
* This chapter was revised by Dr. Glen Sanders.
CHAPTER 12
Conclusions
12.1 The State of Development and Expectations in 1993
When the first edition of this book was published in 1993, the potential of FOG technology was
becoming ascertained, and results provided a good prospect for future applications. The 15th
Anniversary Conference on Fiber Optic Gyros held in September 1991 [1] gave a very complete
update of the activity in the various companies, universities, and research centers working on the
subject around the world.
In terms of architecture, some general trends were outlined:
Most companies were using the Y-coupler configuration with a closed-loop processing scheme
based on phase modulation feedback (usually phase ramp) applied on a multifunction integratedoptic circuit: Honeywell [2], Litton (today Northrop Grumman) [3], Smith Industries [4] in the
United States, JAE [5], Mitsubishi [6] in Japan, British Aerospace [7], Litef (today NG Litef) [8],
and Photonetics (today iXBlue) [9] in Europe. Alcatel-SEL (today SAGEM) [10] in Germany
developed early on a product based on the Y-coupler configuration, but was still using an openloop approach.
The all-fiber open-loop approach was pursued by Honeywell for its first products [11] and also
by Hitachi [12] in Japan.
Instead of using separate single-axis gyros, there was a tendency to make three-axis
measurement units with the sharing of a single source (Litton [today NG] [3] and Litef [today NG
Litef] [8]) or with the multiplexing of this source (Smith Industries [4]). These tactical-grade
inertial measurement units (IMU) were using, in addition, MEMS silicon accelerometers.
Most companies were using polarization-maintaining fiber, except Smith Industries [4], which
preferred the depolarized approach with an ordinary fiber coil.
The [3 × 3] coupler approach was pursued by MBB Deutsche Aerospace [13] for very low-cost,
low-performance applications.
The only companies that were still studying the resonant fiber gyro were Honeywell [14] and
British Aerospace [7].
The I-FOG technology was becoming mature, and a company like Alcatel-SEL (today SAGEM)
[10] had delivered a few hundred gyro products of medium accuracy (10°/h range) over the past
few years, and other companies like Hitachi [12], Honeywell [14], and Litton (today NG) [3] were
starting production. I-FOG technology was going to be used first for Attitude and Heading
Reference Systems (AHRS) for aircraft, which requires a gyro accuracy of 1 to 10°/h. Honeywell
was selected to provide a fiber gyro-based AHRS for the new regional airliner Dornier 328 and
to be part of the Secondary Attitude Air Data Reference Unit (SAARU) of the new Boeing 777
[15]. This was a redundant backup system of the primary flight control using laser gyros. Fibergyro-based AHRSs were also going to be used for military applications. Smith Industries won a
C/AHRS (Magnetic Compass and AHRS) contract from the U.S. Air Force for the development of
a fiber gyro system that will retrofit many previous mechanical reference systems [16]. The
advantages outlined for these contracts were lower cost and lower maintenance due to the solidstate configuration of the I-FOG.
I-FOGs for such tactical-grade applications (i.e., 1 to 10°/h) typically used 100m to 200m of
fiber wound around a 30- to 60-mm diameter. The best bias performance [3, 5, 6, 9, 14] was
below 1°/h (i.e., a few tenths of a microradian in terms of measured optical phase difference),
with a dynamic range of 500 to 1,500°/s and a scale factor accuracy better than 100 ppm. Higher
performance was also pursued with larger fiber coils (0.5 to 2 km around 8- to 10-cm diameter).
Bias stability in the inertial-grade range (0.01°/h) had been demonstrated [2, 3, 5, 6]. Prototypes
were then ruggedized to withstand difficult environments, particularly the temperature range of
−55°C up to 90°C, and vibration of more than 10g rms.
After 15 years of research and development, the fiber gyro started to be recognized as a crucial
technology for many applications of inertial guidance and navigation. Its low-mass, solid-state
configuration brings unique technical advantages: reliability and lifetime, ability to withstand
shocks and vibration, high dynamic range, large bandwidth, nearly instantaneous startup, and low
power consumption. It was also understood that its principle yields a very useful design versatility
to optimize the performance of a specific application by changing the length or the diameter of
the sensing coil while keeping the same opto-electronic components and assembly techniques.
The main applications were foreseen in the medium accuracy range (0.1 to 10°/h) with compact
units (30 to 50 mm in diameter): AHRS for airplanes or helicopters, tactical guidance for missiles
or smart ammunitions, but also bore-hole survey, robotics, and even guidance systems for
automobiles. It was also seen as a very good candidate for navigation systems aided by GPS
(global positioning system). With the progresses of the technology, it was foreseen as a
significant competitor in the high-performance navigation-grade (0.01°/h) domain, particularly
in space applications, where lifetime expectancy, low power consumption, and low disturbance of
the surrounding structure are fundamental characteristics.
Based on a new technology using components that can be mass-produced, the fiber gyro was
expected to bring significant cost reduction, which will further extend the field of inertial
guidance techniques.
12.2 The Present State of the Art, Two Decades Later
12.2.1 FOG Versus RLG
Two decades later, these expectations have been fulfilled and even exceeded. About 500,000 FOG
axes have been produced from industrial-rate-grade (10 to 100°/h) to inertial-grade performance
(0.001 to 0.01°/h) and even strategic-grade performance (better than 0.001°/h), taking advantage
of the design versatility brought by adapting the area of the coil to the application over at least
three orders of magnitude. RLG technology is much more limited from this standpoint: a cavity
perimeter longer than 30 cm makes it difficult for the laser gyro to operate with the required
single longitudinal mode (see Section A.9.1) and conversely the low He-Ne gain limits the
minimum perimeter to 5 to 10 cm. The RLG tuning range that depends on the area (the square of
the perimeter) is only one order of magnitude.
Honeywell, for example, which is completely dominating the RLG market with an estimated
share of 75%, has only two main product families: a high-performance inertial-grade line based
on GG1320-axis (0.003°/h bias,
5-ppm scale factor) using a zerodur block
and a 20-cm perimeter, and a tactical-grade line based on GG1308 axis (1°/h bias,
300-ppm scale factor) using a low-cost BK7 block and a 8-cm perimeter.
However, if the expectations for the FOG have been even exceeded, the RLG remains a very
strong competitor. It is interesting to compare the difference of attitude of the two clear leaders of
the inertial domain with respect to FOG technology. They share more than half the estimated
global IMU market of $1.75 billion [17], knowing that the gyro triad is one-half to two-thirds of
the cost of an IMU:
Honeywell, which controls more than one-third of this global IMU market, dominates by far the
RLG part (a 75% share). They made the right technological choices in the development phase
and built a very efficient manufacturing tool that allowed them to drive the RLG cost down,
especially for their tactical-grade HG-1700 line based on GG1308 axes (300,000 IMU
manufactured). They have an important R&D effort on FOG but they are not pushing FOG
products, except for the specific space market [18] where the RLG is not adapted.
Northrop Grumman (with its subsidiary NG LITEF), has about one-fifth of this IMU market, and
was not very successful in the RLG domain. They have pushed FOG technology where they are
the clear leader with 50% of the market. They have made a big success in the tactical-grade
market with the LN 200 IMU (30,000 produced since the early 1990s) as well as with the μ-FORS
from NG LITEF (60,000 axes produced). They are also entering significantly the inertial-grade
market with their LN-251 FOG-based navigation systems (2,000 produced since 2001, including
their LN-270 land navigation version) [19].
What to conclude is then difficult, but there has been a drastic change of view point over the last
decade, and the fiber gyro is not seen anymore, as limited to tactical-grade applications, but on
the contrary, as the only technology able to reach strategic-grade performance in a strap-down
configuration.
12.2.2 FOG Manufacturers
As just seen, the leader of the FOG market is Northrop Grumman (with its subsidiary NG Litef)
with products in the tactical, intermediate. and inertial grades. They use the closed-loop
configuration with a multifunction integrated-optic circuit (MIOC).
Two other important manufacturers, KVH Industries in the United States (with more than
50,000 axes delivered) and Fizoptika in Russia (with 100,000 axes), use the open-loop all-fiber
approach, which does not perform as well as the closed-loop solution but is lower cost and
sufficient for industrial and tactical grades.
Serving these medium grade markets, there are also IAI Tamam in Israel, Sagem Navigation in
Germany, and Optolink in Russia. Honeywell, as we saw, concentrates on the space market with
inertial and even strategic grade products. iXBlue (formerly iXSea, which was a spin-off of
Photonetics), my company in France, is focused on the high end of the market (bias of 0.05°/h
down to 0.0005°/h) and has delivered nearly 5,000 FOG-based inertial systems (AHRS,
Gyrocompass, INS). Its MARINS product is the only true strategic-grade strap-down INS on the
market (better than 1 nautical mile per day). iXBlue also cooperates with Astrium for space
products. Other manufacturers are L3 Communications and Emcore in the United States, Autoflug
in Germany, Hitachi, JAE and Tokyo Keiki in Japan, Al Cielo in Israel, CASC China Aerospace
and FARCI in China, and Civitanavi in Italy.
12.3 Trends for the Future and Concluding Remarks
Some inertial markets are quite conservative (mechanical gyros are still produced), but it is clear
that the FOG will continue to increase its share and become dominant in the high end, because of
its unique qualities.
Now the future of FOG in the tactical-grade market may be more at risk. It will depend on the
performance that MEMS gyros will be able to reach. Their bias stability is in the tens of degrees
per hour today, but progress is fast. Noise/ARW of few
and a minimum of Allan
deviation (see Section 2.3.4) better than 0.1°/h have been demonstrated. However, MEMS gyros
have an important rate random walk (RRW) and rate ramp that start quite fast: the correlation time
is several minutes only. A MEMS gyro does not take advantage of reciprocity as optical gyros.
Technologically, the optimal design rules of the FOG have been clear for years, but there is
still some possible improvement with the new concept of microstructured fibers (see Section
B.10), in particular, the hollow-core version [20]. Since light propagates mainly in the air,
parasitic effects related to silica are greatly reduced. This applies to Shupe and stress-related Tdot effects (seen in Chapter 6) and magneto-optic and Kerr effects (seen in Chapter 7). However,
the attenuation of these hollow-core fibers is still significant but progress is expected. A solidcore microstructured fiber brings less improvement but has some interest [21].
Finally, to conclude, the fiber-optic gyroscope is a fascinating subject for the physicist as well
as for the signal processing specialist. The serendipity of the device, which has brought simple
solutions to problems apparently complex, will continue to amaze. It is reasonable to expect to
measure a path difference of 10–10 micrometer after several kilometers of propagation, a
relative value close to 10–20.
This is perfectly summarized by the words of Anthony Lawrence about the FOG in his book
about Modern Inertial Technology [22]: “Nature is rarely that cooperative.”
References
[1] Ezekiel, S., and E. Udd, (eds.), “Fiber Optic Gyros: 15th Anniversary Conference,” SPIE Proceedings, Vol. 1585, 1991.
[2] Liu, R. Y., T. F. El-Wailly, and R. C. Dankwort, “Test Results of Honeywell’s First Generation High-Performance
Interferometric Fiber-Optic Gyroscope,” SPIE Proceedings, Vol. 1585, 1991, pp. 262–275.
[3] Pavlath, G. A., “Production of Fiber Gyros at Litton Guidance and Control Systems,” SPIE Proceedings, Vol. 1585, 1991, pp.
2–6.
[4] Page, J. L., “Multiplexed Approach for the Fiber Optic Gyro Inertial Measurement Unit,” SPIE Proceedings, Vol. 1367, 1990,
pp. 93–102.
[5] Sakuma, K., “Fiber-Optic Gyro Production at JAE,” SPIE Proceedings, Vol. 1585, 1991, pp. 8–16.
[6] Hayakawa, Y., and A. Kurokawa, “Fiber-Optic Gyro Production at Mitsubishi Precision Co.,” SPIE Proceedings, Vol. 1585,
1991, pp. 30–39.
[7] Malvern, A. R., “Progress Towards Fibre Optic Gyro Production” SPIE Proceedings, Vol. 1585, 1991, pp. 48–64.
[8] Böschelberger, H. J., and M. Kemmler, “Closed-Loop Fiber Optic Gyro Triad,” SPIE Proceedings, Vol. 1585, 1991, pp. 89–97.
[9] Lefèvre, H. C., et al., “Fiber Optic Gyro Productization at Photonetics,” SPIE Proceedings, Vol. 1585, 1991, pp. 42–47.
[10] Auch, W., M. Oswald, and R, Regener, “Fiber Optic Gyro Production at Alcatel-SEL,” SPIE Proceedings, Vol. 1585, 1991,
pp. 65–79.
[11] Blake, J., et al., “Design and Test of a Production Open Loop All-Fiber Gyroscope,” SPIE Proceedings, Vol. 1169, 1989, pp.
337–346.
[12] Kajoka, H., et al., “Fiber Optic Gyro Productization at Hitachi,” SPIE Proceedings, Vol. 1585, 1991, pp. 17–29.
[13] Hartl, E., et al., “Low Cost Passive Fiber Optic Gyroscope,” SPIE Proceedings, Vol. 1585, 1991, pp. 405–416.
[14] Weed, G., et al., “Fiber Optic Gyro Productization at Honeywell, Inc.,” Fiber Optic Gyro 15th Anniversary Conference, Paper
1585-01, SPIE, oral presentation only, 1991.
[15] Fiber Optics News, December 16, 1991, pp. 3–4.
[16] Military and Commercial Fiber Business, January 10, 1992, p. 3.
[17] www.yole.fr, announcement of the third update of the market report “Gyroscopes and IMUs for Defense, Aerospace &
Industrial,” 2012.
[18] Sanders, S., et al., “Fiber Optic Gyros in a High-Reliability Inertial Reference Unit for Commercial Satellites, OFS 22nd SPIE
Proceedings, Vol. 8421, paper 842106, 2012.
[19] Pavlath, G. A., “Fiber Optic Gyros: Past, Present, and Future,” OFS 22nd SPIE Proceedings, Vol. 8421, paper 842102, 2012.
[20] Blin, S., et al., “Reduced Thermal Sensitivity of a Fiber-Optic Gyroscope Using an Air-Core Photonic-Bandgap Fiber,” J.L.T.,
Vol. 25, No. 3, 2007, pp. 861–865.
[21] Li, J., et al., “Design and Validation of Photonic Crystal Fiber for Fiber-Optic Gyroscope,” ISS 2013, Karlsruhe, 2013, pp.
5.1–5.12.
[22] Lawrence, A., Modern Inertial Technology: Navigation, Guidance, and Control, 2nd ed., New York: Springer-Verlag, 1998.
APPENDIX A
Fundamentals of Optics for the Fiber Gyroscope
A.1 Basic Parameters of an Optical Wave: Wavelength,
Frequency, and Power
An optical wave is an electromagnetic wave like radio waves. The physical measurand that is
propagating is a mix of an electrical E field and a magnetic B field which are orthogonal to the
direction of propagation (Figure A.1). The wave is said transverse and it yields the important
question of polarization and birefringence that will be detailed later. With acoustic waves in a
fluid, the physical measurand is a compression in the direction of propagation: the wave is said
longitudinal. Because the physical measurand is along a unique direction, there is no question of
polarization or birefringence. Shear acoustic waves in a solid are, however, transverse like
optical waves.
As any wave, optical waves have a spatial period, the wavelength λ, and a temporal period T,
related by the velocity of propagation v, with:
v = λ/T (A.1)
is often used the temporal frequency f, the inverse of the period T, with:
f = 1/T (A.2)
and
v = λ ⋅ f (A.3)
and is also used the spatial frequency σ, the inverse of the wavelength λ, with:
σ = 1/λ (A.4)
and
v = f/σ (A.5)
In a vacuum, it is well known that the velocity of an optical wave is the speed of light:
c = 2.998 × 108 m/s ≃ 300,000 km/s (A.6)
A useful value is also
c ≃ 300 mm/ns (i.e., 1 foot per nanosecond) (A.7)
Optical measurements were done in the past only geometrically with distances, and only the
wavelength was used, with a visible range extending from the infrared (IR) limit of 0.75 μm (or
750 nm) to the ultraviolet (UV) limit of 0.4 μm (or 400 nm).
Figure A.1 Electromagnetic wave propagating along the z axis with E and B fields orthogonal to the direction of propagation.
Today, the tendency is to utilize more often temporal frequencies with the influence of optical
communications. Some problems are simpler to understand in the spatial domain but, for other
ones, it is easier in the temporal domain. A conversion Table A.1 with the visible and the most
current optical communication wavelengths in the near-IR range yields the wavelength λ (in
nanometers), the spatial frequency σ (in inverse millimeters, mm–1), the temporal period T (in
femtoseconds), and the temporal frequency f (in terahertz).
Table A.1 Conversion Table
Near IR
λ 1,500 nm
1,300 nm
Visible
850 nm
750 nm
600 nm
Near UV
400 nm
300 nm
σ 667 mm –1 770 mm –1 1,176 mm –1 1,333 mm –1 1,667 mm –1 2,500 mm –1 3,333 mm –1
T 5 fs
4.35 fs
2.85 fs
2.5 fs
2 fs
1.33 fs
1 fs
F 200 THz
230 THz
350 THz
400 THz
500 THz
750 THz
1000 THz
Note an interesting coincidence with the two limits of the visible range exchanging their values
of wavelength λ in nanometers and temporal frequency f in terahertz.
An important characteristic of an optical wave is that its frequency is so high (hundreds of
terahertz) that its amplitude cannot be measured directly like it is with radio waves. One can
measure only the mean power, proportional to the square of the amplitude. It used the term
quadratic, or square-law, detection. The amplitude is, as we saw, a combined electromagnetic
field, but the power is mainly carried by the E field, and the B field can be usually neglected.
Today, the fastest detectors reach the range of 100 GHz, but it is still more than three orders of
magnitude below the wave frequency. To retrieve the information of phase of the very highfrequency optical wave, interferometry is used as will be seen later.
Table A.2 Correspondence Between Watts and dBm
1W
100 mW
10 mW
1 mW 100 μW
10 μW
1 μW
+30 dBm +20 dBm +10 dBm 0 dBm −10 dBm −20 dBm −30 dBm
Optical sources used in fiber optics have power in the range of tens to hundreds of milliwatts.
Powers are also expressed in decibels relative to 1 mW, noted dBm. Expressed in milliwatts, the
power P of a wave of a dBm is 10(a/10) mW and a = 10 log (P/1 mW), which yields the
correspondence as in Table A.2.
Decibels are also used for gain and attenuation. They are based on logarithms that replace
multiplication by addition and power by multiplication. This is the opportunity to recall the beauty
of logarithms as stated by their inventor, John Napier, in the title of his publication, “Mirifici
logarithmorum canonis descriptio” (Description of the Marvelous Canon of Logarithms). The
decibels in Table A.3 are similar to Table A.2 for dBm.
Table A.3 Gain and Attenuation in Decibels
Gain
× 10 3
× 10 2
Attenuation
× 10 1
+30 dB +20 dB 10 dB
10 0 = 1 × 10 –1 × 10 –2 × 10 –3
0 dB
−10 dB −20 dB −30 dB
Now the value of 3 dB corresponds to a 1.998-fold power increase (i.e., twofold with a very
good approximation), and the value of 5 dB corresponds to
reasonable approximation).
Values in decibels can be deduced simply as shown in Table A.4.
(i.e., threefold with a
Table A.4 Correspondence for 0 dB to 10 dB
Decibel Value Equal to
Multiplication Coefficient Equal to
0 dB
1
+1 dB
10 dB − 3 dB − 3 dB − 3 dB 5/4 = 1.25
10/(2 × 2 × 2) = 10/8
+2 dB
3 dB − 1 dB
2/(5/4) = 8/5
+3 dB
1.6
2
+4 dB
10 dB − 3 dB − 3 dB
2.5
10/(2 × 2) = 10/4
+6 dB
3 dB + 3 dB
4
2 ×2
+7 dB
10 dB − 3 dB
5
10/2
+8 dB
5 dB + 3 dB
6
3 ×2
+9 dB
3 dB + 3 dB + 3 dB
8
2 ×2 ×2
+5 dB
+10 dB
10
Values in decibels can also be deduced for small values (see Table A.5).
Table A.5 Correspondence for +2 dB to -2 dB
Decibel Value Multiplication Coefficient Percentage
+2 dB
8/5 = 1.6
+60%
+1 dB
5/4 = 1.25
+25%
+0.5 dB
1/0.9 = 1.11
+11%
+0.1 dB
1.02
+2%
−0.1 dB
0.98
−2%
−0.5 dB
−10%
−1 dB
4/5 = 0.8
−20%
−2 dB
5/8 = 0.625
−37.5%
Now if light behaves clearly like a wave as observed in interferometers, it also behaves like
particles. This is the wave-particle duality. Quantum-mechanic physicists talk also of waveparticle complementarity. With light power detection, this is the particle nature of light which as
to be taken into account. A light beam is actually a flow of photons with an energy equal to the
product of the wave temporal frequency f by Planck constant (h = 6.63 ⋅ 10–34 J ⋅ s). At 1,500 nm
(i.e., 200 THz), it yields a photon energy of 1.3 × 10−19 J. A power of 13 μW is actually a flow of
1014 photons per second.
Optical sources can be single wavelength/frequency and they are then called monochromatic
(i.e., single color). This is the case of the laser gyro using helium-neon (He-Ne) that emits at 633
nm (about 500 THz) in red. They can be also broadband or broad spectrum, and this is the case
with fiber-gyro sources. The first fiber gyros used semiconductor diode sources emitting around
850 nm, but today erbium-doped-fiber sources emitting around 1,550 nm are mainly utilized for
high performance.
In a transparent medium, an optical wave is slowed down and propagates at a velocity c/n,
where n is the index of refraction. At 1,550 nm, silica (SiO2 glass), which is used for optical
fiber, has an index of around 1.45; lithium niobate (LiNbO3) crystal, which is used in an
integrated-optic circuit, has an extraordinary index of about 2.14; semiconductors used in sources
have an index as high as 3.5. Note that the actual wavelength λm in a material is shorter than the
wavelength λ in a vacuum:
λm = λ/n (A.8)
However, the temporal frequency remains always the same. It is summarized in Table A.6.
Table A.6 Wavelength as a Function of the Index of Refraction
Wavelength Wavelength Wavelength Wavelength Wavelength
in LiNbO3 in Semiconductor
Temporal Frequency in a Vacuum in Air
in Silica
(n = 1)
(n = 1.0003) (n ≃ 1.45) (n ≃ 2.14) (n ≃ 3.5)
350 THz
850 nm
≃ 850 nm
585 nm
395 nm
240 nm
200 THz
1500 nm
≃ 1500 nm
1070 nm
700 nm
430 nm
A.2 Spontaneous Emission, Stimulated Emission, and
Related Noises
A.2.1 Fundamental Photon Noise
The fundamental noise of light is photon noise. It is applied to photons, the classical shot noise of
the measurement of the flow of uncorrelated discrete particles that is found, in particular, with an
electrical current composed of a flow of discrete electrons. When a flow Ṅ = dN/dt of
uncorrelated particles is measured over a time tm, it yields a number of detected particles:
N = Ṅ ⋅ tm (A.9)
with an uncertainty equal to
As we already saw, an optical power of 13 μW corresponds to a flow of 1014 photons per
second for a wavelength of 1,500 nm. With a measurement time of 1 second (i.e., a detection
frequency bandwidth ∆fd of 1 Hz), the 1014 detected photons are measured with an uncertainty of
(i.e., a relative uncertainty of 10–7). To reduce this relative uncertainty by one
order of magnitude to 10–8, requires a flow of 1016 photons per second (i.e., 1.3 mW), two
orders of magnitude in optical power.
It is a Poisson distribution process, but when a very large number of particles are detected, it
approaches a normal Gaussian distribution. The standard (or root-mean-square [rms]) deviation
σ Ṅ of this counting follows:
(A.10)
For an electrical current I, composed of discrete electrons of charge q, it yields:
(A.11)
For an optical power P, composed of discrete photons of energy h ⋅ f, where f is the wave
frequency equal to c/λ, it yields a photon noise:
(A.12)
This can be also expressed in relative noise power spectral density PSD (with Hz−1 as the unit):
(A.13)
Note these PSD are constant in terms of frequency: shot noise and photon noise are white
noises. These PSD values are expressed as 10–x/Hz, or with decibels as (−10x) dB/Hz.
A.2.2 Spontaneous Emission and Excess Relative Intensity Noise (Excess RIN)
Spontaneous emission is the most common process of emission, with excited atoms or ions
spontaneously emitting photons with an energy equal to the difference between the excited level
and the ground state. These emitted photons are not correlated and spontaneous emission has a
broad spectrum. The various frequency components of this broad, spontaneously emitted,
spectrum have a random relative phase, and all these frequency components beat randomly
together. It yields random power (or optical intensity) fluctuations called excess relative intensity
noise (excess RIN or RIN).
The emission frequency spectrum has a full width at half-maximum ∆fFWHM =
about the mean frequency or mean wavelength
As the power detection process is quadratic,
it yields rectification and then power measurement noise about dc over the frequency range
∆fFWHM. Theory shows that the power spectral density of this excess RIN is simply the inverse
of the spectrum frequency width ∆fFWHM:
PSDRIN = 1/∆fFWHM (A.14)
For example, at 1,500 nm/200 THz, a spectrum width of 0.5% (i.e., 7.5 nm/1 THz = 1012 Hz)
has a RIN power spectral density simply equal to PSDRIN = 1/1 THz = 1/1012 Hz = 10–12 Hz–1
(i.e., −120 dB/Hz), that is, in terms of relative σ value:
(A.15)
In fiber gyros, the returning power is in the range of tens of microwatt, that is, the theoretical
photon noise PSDPh is about 10–14/Hz, that is, 100 times smaller in power density and 10 times
smaller in σ value than the RIN. Reduction of the effect of excess RIN is an important point for
high-performance fiber gyros.
Table A.7 Photon Noise Power Density as a Function of Power
10 −12 /Hz
(−120 dB/Hz)
PSDPh
10 −14 /Hz
(−140 dB/Hz)
10 −16 /Hz
(−160 dB/Hz)
(relative σ value)
for P at 1,500 nm
0.26 μW (−36 dBm) 26 μW (−16 dBm) 2.6 mW (+4 dBm)
(200 THz )
for P at 850 nm
(350 THz )
0.46 μW (−33 dBm) 46 μW (−13 dBm) 4.6 mW (+7 dBm)
In addition, spontaneous emission sources have, by principle, no spatial coherence since the
various emission points of the extended source are uncorrelated. This lack of spatial coherence
makes their coupling to a single-mode fiber inefficient.
A typical spontaneous emission source is a light emission diode (LED). They can be coupled
with a reasonable efficiency is a large-core multimode fiber but the efficiency is very poor with a
small-core single-mode fiber.
For fiber gyros requiring broadband sources, but also efficient coupling in a single-mode
fiber, spontaneous emission sources are not adapted. However, as we shall see later, amplified
spontaneous emission (ASE) sources provide, at the same time, the required broad spectrum of
spontaneous emission and an efficient coupling into single-mode fibers.
Finally, spontaneous emission sources are usually not polarized. They can be considered as the
sum of two uncorrelated sources with orthogonal states of polarization and the same power.
A.2.3 Resonant Stimulated Emission in a Laser Source
In addition to spontaneous emission, excited atoms or ions of a source also follow a process of
stimulated emission. An incoming photon, corresponding to the energy between excited and
ground states, stimulates the emission of a new photon that has the same phase, frequency,
polarization, and direction as the ones of this incoming photon. It yields a direct optical
amplification process. It is the basic of laser emission, laser being the acronym of light
amplification by stimulated emission of radiation.
Actually, a laser is a resonator, and not, strictly speaking, an amplifier. As any resonator in
electronics or mechanics, a laser is an amplifier with a feedback of its output to its input. When
the product gain × feedback is higher than 1 at a frequency, as any resonator, the laser resonates at
this specific frequency. The spontaneous photons act as a seed, and recirculate through stimulated
emission in the amplifying cavity.
A laser can be made single-frequency. A pure single-frequency (i.e., harmonic wave) has an
amplitude:
A(t) = A0sin(2πft + ϕ 0) (A.16)
It can be written as:
A(t) = A0sinϕ(t) (A.17)
where ϕ(t) is the phase which follows an affine function as a function of time:
ϕ(t) = 2πft + ϕ 0 (A.18)
The frequency f is proportional to the derivative of ϕ(t):
(A.19)
An actual single-frequency laser wave has a residual frequency/phase noise, yielding the line
width, which can be very narrow; however, this frequency/phase noise has no effect on the
amplitude or its square, the optical intensity or power. A single-frequency laser wave does not
suffer from the excess RIN mechanism of spontaneous sources, despite its very narrow frequency
band. In first approximation, it should be viewed as a constant amplitude/power wave with chirp
noise on frequency/phase.
The theoretical Schawlow-Townes line width is very small, in the millihertz range, but in
practice, lasers experience residual noise because of thermal and mechanical fluctuations of the
cavity. Line width depends on the kind of laser. It is usually in the megahertz range but can go
down to the hertz range for the ultra-stable lasers.
These thermal and mechanical fluctuations yield also some intensity noise, but laser PSDRIN is
much smaller than the one of broadband spontaneous emission sources. Laser RIN can be as
small as −160 to −170 dB/Hz (i.e., four to five orders of magnitude lower than the typical −120
dB/Hz of fiber-gyro sources).
Laser sources are also spatially coherent and, then, can be coupled efficiently into a singlemode fiber. They are also usually polarized.
They could appear as ideal sources for the fiber gyro because of their low RIN and their high
coupling efficiency to single-mode fibers, but they lack the broad spectrum and the related low
temporal coherence which is key for high performance.
A.2.4 Amplified Spontaneous Emission (ASE)
As we saw, spontaneous emission sources have the required low temporal coherence but they lack
the high spatial coherence needed for efficient coupling into a single-mode fiber. However,
resonant stimulated emission laser sources have the required high spatial coherence but they lack
the required low temporal coherence. The best of both worlds is found with amplified
spontaneous emission (ASE) sources.
An ASE source is a single-spatial-mode waveguide amplifier. Because of waveguidance, broad
spectrum spontaneous emission is partly guided in the waveguide instead of being emitted in all
directions as in free space. This partly guided broadband spontaneous emission is then amplified
by stimulated emission. This amplification taking place in the waveguide yields high power in a
single spatial mode and, at the same time, preserves the broad spectrum of the seed spontaneous
emission.
ASE is an abbreviation from telecommunications, in particular from erbium-doped (Er-doped)
fiber amplifier (EDFA), which revolutionized this domain in the 1990s. An EDFA without input
signal is actually a very efficient ASE source.
Before the invention of EDFA, fiber gyros used super-radiant diodes (SRDs), superluminescent diodes (SLDs), or edge-emitting light emitting diodes (E-LEDs). They all are
actually ASE sources based on semiconductor waveguide amplification instead of rare-earth-
doped-fiber amplification, but they work on the same basic principle of seed guided broad
spectrum spontaneous emission, which is amplified by stimulated emission in a single-spatialmode guided optical amplifier.
Early work on rare-earth-doped ASE sources used the term “super-fluorescent,” but “super”
radiance, “super” luminescence, or “super” fluorescence are all actually seed spontaneous
emission amplified by stimulated emission. Such ASE sources are quasi-ideal for the fiber gyro
because of their high power and low temporal coherence, but they keep the excess RIN of the seed
spontaneous emission.
However, as will be seen, excess RIN is not as fundamental a noise as photon shot noise. When
a photon-noise-limited wave is split, it yields two photon-noise limited waves, but both noises are
uncorrelated. As we saw, excess RIN comes from the random intensity power beating between the
various frequency components of the broad spectrum. When a wave limited by excess RIN is
split, it yields two waves carrying the same excess RIN. Then it is possible to use one port as a
RIN reference and to subtract it from the useful measurement of the other port.
Finally, like spontaneous emission sources, ASE sources may be unpolarized. This is the case
with Er-doped fiber sources. Such an unpolarization is very useful, as we shall see, to reduce bias
drift in a fiber gyro.
A.3 Propagation Equation in a Vacuum
As already discussed, an optical wave in a vacuum is an electromagnetic wave. Going to more
mathematics, it is a solution of the differential propagation equation, which is derived from
Maxwell equations of electromagnetism in a vacuum:
(A.20)
(A.21)
(A.22)
(A.23)
where the E vector is the electric field, the B vector is the magnetic field, the j vector is the
electric current density, ρ is the electric charge density, ∈0 is the dielectric permittivity of a
vacuum (∈0 = 8.854 × 10–12 F⋅m–1), and μ0 is the magnetic permeability of a vacuum (μ0 = 4π
× 10–7 H⋅m–1).
Note that the equation of conservation of charge, ∇ ⋅ j + ∂ρ/∂t = div j = ∂ρ/∂t = 0, is included in
Maxwell’s equations, since it can be deduced by applying ∇ ⋅ ∇×= div ⋅ curl = 0 to the MaxwellAmpere law M4. Furthermore, equations M1 and M2 express that the E and B fields are derived
from an electromagnetic scalar potential V and an electromagnetic vector potential A, with:
(A.24)
(A.25)
These equations use the divergence (div) and the curl (curl) operators on a vector function U of
the spatial coordinates, as well as the gradient operator (grad) on a scalar function U. The
notations are unified with the vector differential operator ∇, the divergence being the scalar
product ∇ ⋅ U, which gives a scalar function, the curl being the vector product ∇ × U, which gives
a vector function, and the gradient being the product of the vector operator ∇ with a scalar U,
which gives a vector function.
(A.26)
Another operator, the Laplacian operator, is of importance in vector analysis: it is a scalar
differential operator equal to the scalar square of the vector differential operator ∇. It is denoted
by ∇2 and applied to a vector function U, it gives a vector function ∇2U. In Cartesian coordinates:
(A.27)
The propagation equation in a vacuum is deduced from ∇ × (∇ × U) = ∇(∇ ⋅ U) − ∇2U or curl ⋅
curl U = grad div U − ∇2U, which yields:
(A.28)
A simple solution to this propagation equation is the monochromatic (i.e., single-color, singlefrequency, single-wavelength) sinusoidal plane wave that can be expressed in Cartesian
coordinates (x, y, z and time t) (Figure A.1). The term “harmonic solution” is also used. For
easing the calculations, complex exponential notations are used, but it must be kept in mind that
the true field is only the real part.
E = Eoxei(ωt-k0z)
E = Eoyei(ωt-k0z) (A.29)
In these equations ω, which is called the angular frequency, and k0, which is called the wave
number in a vacuum, are used to simplify notations. At a given position z, the electromagnetic
field oscillates sinusoidally with a temporal frequency ƒ, with ω = 2πf, and a temporal period T =
1/f. At a given time t, the electromagnetic field has a sinusoidal spatial distribution along the
direction of propagation z, with a spatial period λ, the wavelength in a vacuum, with k0 = 2π/λ.
The inverse of the wavelength λ is called the spatial frequency and is denoted σ = 1/λ.
The wave propagates at a velocity c in a vacuum, as it may be written as:
(A.30)
with c = ω/k0 = (∈0μ0)–1/2. Its measured value is c = 2.998 × 108 m ⋅ s–1 and:
c = λ/T = λ ⋅ f = f/σ (A.31)
An electromagnetic plane wave in a vacuum is a transverse wave where the E and B field
vectors are orthogonal to the direction of propagation. The magnitudes E and B of these two
fields are proportional, and their ratio is equal to the light velocity:
(A.32)
They oscillate in phase and are localized in two orthogonal planes. The electromagnetic field
has the same value at any point in a plane that is perpendicular to the direction of propagation.
Such planes are called the planar wavefronts of the plane wave.
This constant ratio E/B = c implies that the effect of the electromagnetic field of an
electromagnetic wave on matter is due mostly to the electric component. The electromagnetic
force fEM applied on a particle of charge q is the sum of an electric force fE and a magnetic
force fM:
fEM = fE + fB
fE = qE
fM = q(vp × B) (A.33)
where vp is the speed of the particle. Because, in matter, vp << c, this implies that fM << fE.
Therefore, most problems of electromagnetic waves are treated by considering only the electric
field.
The electromagnetic spectrum extends from radio waves to X-rays and gamma radiations. As
we already saw, optical waves correspond to the visible range (i.e., a vacuum wavelength between
400 and 750 nm) and to the near-ultraviolet and near-infrared ranges. An important characteristic
of these optical waves is their very high frequency: for a vacuum wavelength λ of 1 μm, the wave
frequency ƒ is 300 THz (i.e., 300,000 GHz). No electronic system can follow such a fast vibration,
and optical detectors can only measure the mean power of the wave. It is not possible to measure
directly the E or B field of the wave.
The power flow across unit area (i.e., the power density) is called the optical intensity Iopt of
the wave and is proportional to the mean value of the scalar square of the E field:
Iopt = c ⋅ ∈0 < E ⋅ E* > (A.34)
where E* is the complex conjugate of E, and the brackets < > denote temporal averaging. Optical
detectors are often called square-law or quadratic detectors.
When the exact physical or vectorial nature of the optical wave is not needed, the analysis of an
optical scheme can be carried out with a scalar quantity, the complex wave amplitude A:
A = A0ei(ωt–k0z) (A.35)
The wave power density, also called wave optical intensity, is the mean value of the scalar
square of the complex amplitude:
Iopt = < A ⋅ A* > = A02 (A.36)
A.4 State of Polarization of an Optical Wave
An optical wave in a vacuum is a wave denoted by TEM (transverse electromagnetic), as the E
and B fields are both in a plane 0xy orthogonal to the direction of propagation 0z. At a given
position 0 in space, a possible solution of the electric field (Figure A.2) is:
Figure A.2 Linear polarization aligned along the x-axis or the y-axis for propagation along the z-axis.
E = E0xeiωt (A.37)
Such a sinusoidal oscillation is said to be linearly polarized along the x-axis. The E vector
always remains parallel to E0x. Another possible solution is:
E = E0yeiωt (A.38)
The wave is then linearly polarized along the perpendicular y-axis.
Figure A.3 Linear polarization aligned along any axis considered as the sum of two orthogonal linear polarizations in phase.
The set of all the possible solutions is a two-dimensional linear space. The general solution is a
linear combination with complex coefficients:
E = (E0x + E0yei∆ϕ )eiωt (A.39)
where ∆ϕ is the phase difference between both orthogonal components E0x and E0y.
When ∆ϕ is equal to 0 or π rad, the wave is linearly polarized along an intermediate direction
(Figure A.3), but generally the wave is elliptically polarized. The extremity of the E vector
follows an ellipse with the frequency f = ω/2π (Figure A.4).
Figure A.4 Elliptical polarization considered as the sum of two orthogonal linear polarizations with a phase difference ∆ϕ.
When two linear polarizations with the same amplitude are combined in quadrature (i.e., with a
phase shift of ±π/2 rad), a circular polarization results (Figure A.5):
Figure A.5 Circular polarization considered as the sum of two orthogonal linear polarizations in quadrature.
E = [E0x = E0ye±iπ/2]eiωt (A.40)
where the magnitudes of the E0x and E0y vectors are equal: E0x = E0y.
The tip of the E vector follows a circle. There are two states of circular polarization which turn
in opposite directions depending on the sign of the ±π/2 phase shift. They are called the righthanded and left-handed circular states of polarization.
Because the set of polarization states is a two-dimensional linear space, it can be analyzed with
any orthonormal basis of eigenvectors. Two perpendicular linear polarizations are a possible
orthonormal basis, but the two circular polarizations are also an orthonormal basis. An elliptical
polarization can be expressed with linear polarizations [Figure A.6(a)]:
Figure A.6 Decomposition of an elliptical polarization: (a) with two perpendicular linear polarizations in quadrature and (b) with
two opposite circular polarizations.
E = (E0L + E0seiπ/2)eiωt (A.41)
where E0L is the peak value of the field along the large axis, and E0s is the peak value of the field
along the small axis. However, it can also be decomposed with circular polarizations [Figure
A.6(b)]:
(A.42)
where the magnitudes E of the various E vectors are given by:
(A.43)
(A.44)
with E0m > E0c, and where E0m is the magnitude of the rotating Em field of the main circular
component and E0c is the magnitude of the rotating Ec field of the other crossed circular
component, and there are:
E0L = E0m + E0c (A.45)
E0s = E0m − E0c (A.46)
Note that when the quality of a given state of polarization has to be measured, the
intensity/power ratio is usually measured between the component in this given state of
polarization and the component in the orthogonal state of polarization.
With a linear state of polarization, a polarizer is used to select the main polarization
component, and it is rotated by 90° to select the crossed orthogonal polarization. The ratio
between both measured powers directly gives the ratio between the power Pc of the crossed linear
state and the power Pm of the main linear state.
With a circular state of polarization, a polarizer is used to check that the transmitted power
remains equal to half the input power when the polarizer is rotated. When the circular state is not
perfect, a maximum power Pmax is measured along the major axis of the polarization that is not
perfectly circular and a minimum power Pmin is measured along the minor axis. Pmax is
actually the power of the linear E0L component of the elliptical polarization and is proportional
to E0L2, while Pmin is the power of E0s and is proportional to E0s2. The contrast of this
variation is:
(A.47)
and since E0L = E0m + E0c, E0s = E0m − E0c, and E0c/E0m << 1:
(A.48)
where Pm is now the power of the main circular state and Pc is the power of the crossed circular
state. Then the contrast of this variation is proportional to the ratio of the amplitudes (or fields) of
both orthogonal circular components, that is, the square root of the ratio of the powers.
If, for example, the ratio Pc/Pm = 1%, the contrast is as high as 20%. This does not mean that
this main circular state of polarization is not as well preserved as the main linear state of
polarization in the previous case. In both cases, the intensity or power ratio is Pc/Pm and the
amplitude or field ratio is
but in the linear case the experiment yields a
power ratio measurement of the defect, while in the circular case the experiment yields an
amplitude ratio measurement, which is much more sensitive.
A.5 Propagation in a Dielectric Medium
A.5.1 Index of Refraction
In a transparent dielectric medium, Maxwell equations can be averaged spatially to eliminate the
strong variations of the electromagnetic field between the discrete charged particles of the
material, and to be able to consider a macroscopically continuous material. The first two Maxwell
equations M1 and M2 that are independent of the charge density ρ and the current density j are
preserved in the averaging:
(M1) ∇ ⋅ B = div B = 0 (A.49)
(A.50)
For the last two Maxwell equations M3 and M4, the macroscopic effect of the charges of the
medium can be taken into account with an electric polarization vector P and a magnetic
polarization vector M. Note that polarization has, in this case, a completely different meaning
from the state of polarization, it is the polarization vector of the medium, while the state of
polarization relates to the wave. A derived electric field D and a derived magnetic field H may be
defined as follows:
D = ∈0 E + P (A.51)
(A.52)
The last two Maxwell equations become:
(M3) ∇ ⋅ D = div D = ρfree (A.53)
(A.54)
where ρfree is the free charge density and jfree is the free current density vector.
In an isotropic linear dielectric (i.e., nonconducting) medium, ρfree = 0, jfree = 0, P is
proportional to E, and M is proportional to B:
P = χe∈0E (A.55)
(A.56)
where χe and χm are the dielectric and magnetic susceptibilities, respectively. The D and H
derived fields are also respectively proportional to the E and B fields:
D = ∈r∈0E (A.57)
(A.58)
where ∈r is the relative dielectric permittivity and μr is the relative magnetic permeability. We
have:
∈r = 1 + χe (A.59)
(A.60)
Note that χe, χm, ∈r, and μr are dimensionless values.
The propagation equation in a dielectric medium becomes:
(A.61)
The plane wave harmonic solution is:
E = E0xei(ωt–kmz)
B = B0yei(ωt–kmz) (A.62)
The wave velocity v = ω/km in a medium becomes:
v = (∈rμr∈0μ0)–1/2 = c(∈rμr)–1/2 (A.63)
The index of refraction n of the material being defined by the ratio between the velocity in a
vacuum and the velocity v in the medium, this yields:
(A.64)
Even if this may look obvious, as we already saw, it is important to recall that when a lightwave
propagates in a vacuum and in various media, its temporal frequency f and its related angular
frequency ω = 2πf remain unchanged, but the actual wavelength λm and its inverse, the actual
spatial frequency σ m, in the medium are not equal anymore to the values λ and σ that the same
wave would have in a vacuum. We have:
λm = λ/n (A.65)
σ m = n ⋅ σ (A.66)
The wave number km in the medium is:
(A.67)
where k0 is the wave number that the same wave would have in a vacuum.
As it can be seen, the important characteristic of a light beam that remains invariant is its
temporal frequency, even if it is the habit to define a wave with its wavelength in a vacuum
because a length of 1 μm is a quantity easier to grasp and to measure than a frequency of 300 THz
(3 × 1014 Hz) or a temporal period of 3 femtoseconds (3 × 10–15 s).
A.5.2 Chromatic Dispersion, Group Velocity, and Group Velocity Dispersion
The relative dielectric permittivity ∈r depends on the frequency of the wave and at optical wave
frequencies, the relative magnetic permeability μr is equal to unity. The index of refraction and
the wave velocity then depend only on ∈r and are frequency (or wavelength) dependent:
(A.68)
Note that sometimes the term dielectric impermeability η is used for the inverse of the
dielectric permittivity ∈r with η = 1/∈r = 1/n2.
A material is composed of electric particles that are bound together with electromagnetic
forces. It can be considered as a complex set of elastic mechanical oscillators. When the light
frequency corresponds to a resonance frequency of the material, mechanical resonances are
induced by the synchronized electromagnetic field of the wave, which is then absorbed. In the
transparent region, the light frequency does not correspond to any resonance, but the oscillators
vibrate and the macroscopic effect described by ∈r depends on the respective value of the light
frequency and of the resonance frequencies of the material, which yields a frequency dependence
of ∈r and of the index n.
This effect is called chromatic dispersion, as its first well-known application is the spatial
decomposition of white light in prismatic colors. White light is composed of all the visible
wavelengths, and since the angular deviation by a prism is proportional to the index n, the
spectrum becomes spread angularly as a function of n(f) (or n(λ)). This function n(f) (or n(λ)) is
given by the Sellmeir equation. Figure A.7 displays it for silica, the base material of optical fiber.
Figure A.7 Index of silica versus wavelength.
Chromatic dispersion yields two other important phenomena. If the optical wave is modulated
in amplitude, frequency, or phase, the modulated signal does not propagate at the velocity v =
ω/km, often called phase velocity, but at the group velocity vg:
The difference is proportional to the first-order derivative dn/dλ.
(A.69)
Going to more mathematics, these formulae can be derived considering that any
nonmonochromatic wave is the integral summation of all its frequency components. At a given
spatial position, the wave amplitude A(t) is:
(A.70)
Its frequency components a(f) are defined inversely by Fourier transform:
(A.71)
Using the angular frequency ω instead of the temporal frequency f to simplify equations by
avoiding the 2π coefficient gives:
(A.72)
When this wave propagates in a dispersive medium, each frequency component propagates
independently at its own velocity, and the total wave amplitude A(t, z) propagates accordingly to:
(A.73)
The propagation argument [ωt − km(ω)z] may be decomposed in the sum of a mean term [ t −
km( )z] that depends on the mean frequency and a variable term [∆ωt − ∆km(∆ω)z] where ∆ω
= ω − and ∆km(∆ω)z = km(ω) − km( ) The wave amplitude becomes:
(A.74)
The second term has the same propagation argument as the one of a monochromatic wave and
represents the propagation of the carrier at a velocity equal to the mean phase velocity v = km(
).The first term M(t, z) is actually the modulation that propagates on the monochromatic
carrier, and:
(A.75)
The modulation term M(t, z) is the integral summation of frequency components ∆ω that
propagate respectively at a velocity ∆ω/∆km(∆ω). If this velocity is constant over the spectrum,
∆ω/∆km is equal to the first-order derivative dω/dkm, and the second-order derivative d2ω/d2km
is zero. The modulation term M(t, z) propagates without any change at the same velocity as the
one of all its frequency components, the group velocity vg = dω/dkm, because it can be written as:
(A.76)
The value of vg is independent of the spectrum width, and this result is very general. It applies
for any kind of function M (i.e., amplitude, phase, or frequency modulation signals), and it also
applies if the spectrum is broad because of the natural emission of the source.
Now if ∆ω/∆km is not constant over the spectrum (i.e., d2ω/d2km ≠ 0), the modulation term
M(t, z) propagates at the mean group velocity vg( ) = dω/dkm( ), and there is an additional
second-order effect of signal temporal spreading, which limits the transmission bandwidth. It is
called group velocity, or group delay, dispersion. The limitation induced by this effect over the
intrinsic spectrum broadening due to a signal modulation of a monochromatic source is in
practice negligible, but it has to be taken into account when the natural emission spectrum of the
source is not narrow enough.
Let us give orders of magnitude to get a better understanding: the mean light frequency is 300
THz (for a 1-μm wavelength), and the spectrum broadening due to electronically controlled
signal modulation is at maximum several tens of gigahertz (i.e., at maximum, about 10–4 of the
central frequency), while the natural emission spectrum width of the source may be as much as
10–3 to 5 × 10–2 of the central frequency (i.e., a width of 1 to 50 nm in wavelength). In this case,
the modulation term M(t, z) is a complex combination of the effect of a modulated signal S(t, z)
and a random term Mr(t, z) that takes into account the effect of the broad emission spectrum of
the source. One may consider that each emission frequency ωe of the source carries the signal
that propagates on this frequency at the group velocity vg(ωe). The total signal is actually
decomposed in elementary signals that propagate at a different velocity, comprised in ∆vg = vg[
e + (∆ωe/2)] − vg[ e − (∆ωe/2)], where e is the mean frequency of the emission spectrum and
∆ωe is the width of the emission spectrum. These velocity differences yield, over a propagation
length L, a temporal spreading ∆τs of the elementary signals:
(A.77)
since the group propagation time is τg(ωe) = L/[vg(ωe)]. Expressed in terms of index n and mean
emission wavelength
(A.78)
where ∆λe is the width of the emission spectrum, in wavelength.
For example, for silica used in optical fibers, with
∆τs/L ⋅ ∆λe = 100 ps/nm⋅km (A.79)
However, because the second-order derivative d2n/dλ2 = 0 around 1,300 nm in silica (Figure
A.7), this wavelength is the optimal choice for large-bandwidth telecommunications that require
minimum signal spreading. At 1,550 nm, the wavelength with the lowest attenuation, the group
velocity dispersion is 17 ps/nm⋅km.
Note that when d2n/dλ2 = 0, it is often said that there is no dispersion. More precisely, it should
be said that there is no group velocity dispersion. The reason for this is that even with a secondorder derivative d2n/dλ2 = 0, it may still be possible to get chromatic dispersion in a prism,
because this depends on the first-order derivative dn/dλ.
A.5.3 E and B, or E and H?
The basic electromagnetic field is the set of the electric E field and the magnetic B field.
Relativity shows that E and B are connected through Lorentz transformation for moving frames
of reference. The derived fields D and H are only auxiliary fields that allow the effect of a
medium to be treated macroscopically. This is particularly seen in the electromagnetic theory of
the Sagnac effect detailed in Appendix D.
However, for historical reasons, many optics and radiowave textbooks that are mostly devoted
to motionless systems are considering H as the basic magnetic field instead of B. The advantage
of the (E, H) approach is that it offers a good analogy with other wave phenomena. As a matter of
fact, it is usually possible in a wave to consider a first physical measurand that is a potential and a
second physical measurand that is a flow. An argument to explain wave propagation is the
consideration that a variation of the potential induces a variation of the flow. Then this variation
of the flow induces a variation of the potential and so on. The power P of the wave is the product
potential by flow, and the impedance Z of the propagation medium is the ratio potential over flow.
In electric circuits, the potential is the voltage V and the flow is the intensity of the electric
current i, and the power and the impedance are:
P=V⋅i
Z = V/i (A.80)
For acoustic waves, the potential is the variation of pressure ∆p, the flow is the actual matter
flow fmatter , and:
(A.81)
With electromagnetic waves, the dimension of the E field is voltage × length–1 and the
dimension of the H field is current × length–1. The impedance Z is defined by:
(A.82)
and has the same unit as V/i (i.e., the unit is the ohm (Ω)). In particular, for waves in a vacuum,
there is the impedance of a vacuum:
(A.83)
It can be shown that in a medium with an index n, the impedance becomes:
(A.84)
and partial reflection on a dielectric interface may be interpreted as an impedance mismatch. The
E field would be the potential and the H field would be the flow. Furthermore, the product
potential by flow does have a power significance, as it corresponds to the Poynting vector Π,
which is equal to the power flow per unit area. In a vacuum:
Π = μ0–1(E × B) (A.85)
Then
Π = E × H (A.86)
As we can see, this analogy using E as a potential and H as a flow has some interest, but the
general tendency nowadays in physics is to consider the (E, B) couple as the basic
electromagnetic field. The fact that the E/B ratio of an electromagnetic wave in a vacuum is equal
to the velocity of light is, after all, a more fundamental result than the fact that the E/H ratio has
the same unit as the one of electrical resistance.
Furthermore, note that so far we have used four physical constants of electromagnetism that are
not independent:
∈0 is the dielectric permittivity of a vacuum.
μ0 is the magnetic permeability of a vacuum.
c is the light velocity in a vacuum.
Z0 is the impedance of a vacuum.
If relativistic arguments are predominant, then there are actually only two basic physical
constants:
∈0, because without motion of charges there is only an electric field;
c, because it is the basic constant of relativity, which explains transformation by motion.
As we have already seen, the magnetic B field is deduced from the electric E field through the
relativistic Lorentz transformation, which is using c. Then μ0 should not be a basic constant: it
should be deduced from μ0 = 1/∈0c2, even if, for historical and practical reasons, μ0 is a basic
constant in the international unit system equal to 4π × 10-7 because of the definition of the
ampere. Similarly, Z0 is also a secondary physical constant deduced from Z0 = 1/∈0c. Now,
secondary does not mean that it is not useful: historically, it was fundamental to verify that the
measured value of the light velocity c equals
with ∈0 and μ0 being defined
independently in electrostatic and magnetostatic experiments.
As an aside, it is interesting to note that if c is considered a fundamental constant of physics,
using (ct) as a temporal coordinate with the dimension of a length and (cB) as a magnetic field
with the dimension of an electric field, Maxwell equations in a vacuum, and without charge or
current, appear very symmetric:
(M1) ∇ ⋅ (cB) = div (cB) = 0 (A.87)
(A.88)
(M3) ∇ × E = div E = 0 (A.89)
(A.90)
A.6 Dielectric Interface
A.6.1 Refraction, Partial Reflection, and Total Internal Reflection
The basic laws of reflection and refraction at an interface between two dielectric media can be
formulated very simply in geometrical optics (Figure A.8):
The partially reflected and refracted rays lie in the plane of incidence (i.e., the plane that is
orthogonal to the interface and that contains the incident ray).
The angle of partial reflection θ2′ is equal to the angle of incidence θ2.
The angle of refraction θ1 is given by the Snell law:
n1 sinθ1 = n2 sinθ2 (A.91)
where n1 and n2 are the respective indexes of the dielectric media. Note that these indexes are
called indexes of refraction, as, historically, it is refraction that was the first observed
phenomenon. It is only later that it was found out that the index is the ratio between light velocity
in a vacuum and light velocity in the medium.
These laws give the angular deviation of the rays, but the calculation of the power splitting
ratio at the interface, called Fresnel coefficients, require a more complicated analysis that uses the
boundary conditions of the electromagnetic field at the interface.
Figure A.8 Reflected and refracted rays on a dielectric interface: (a) from low index n2 to high index n1 and (b) from high index
n1 to low index n2.
For normal incidence (i.e., θ1 = θ1′ = θ2 = 0), the power (or optical intensity) ratio between the
incident beam and the reflected beam, called Fresnel reflectivity RF is:
(A.92)
The power ratio between the refracted beam and the incident beam, called transmissivity TF, is:
(A.93)
It can be seen that:
In particular, at the interface between silica fiber (n1 = 1.45) and air (n2 = 1), RF = 0.04, and TF
= 0.96.
Between silica and lithium niobate, as the relative index ratio 2.15/1.45 = 1.48 is similar to the
one between air and silica, there are the same 4% reflection at normal incidence.
Note that the transmissivity and the reflectivity have the same values in both directions (i.e.,
when the incident beam is coming from the low index medium or from the high index medium).
This is a first case of the very basic principle of reciprocity of light propagation.
Also note that, at this normal incidence, the phase of the wave is unchanged by partial reflection
on the low-index medium, but it experiences a π phase shift in the partial reflection on the highindex medium.
For other incidences, the result is more complicated and depends on the state of polarization,
which may be parallel to the plane of incidence, and then denoted by the subscript // or p, or
which may be perpendicular to the plane of incidence, and then denoted by a subscript ⊥ or s
(from the German “senkrecht”). Several important results can be pointed out (Figure A.9):
Figure A.9 Fresnel reflection coefficients for an index ratio of 1.45 for the s-polarization and the p-polarization, in the lowindex medium.
The reflectivity and the transmissivity remain about constant in a cone of about 10° to 20°
around normal incidence.
The reflectivity for an s-polarized wave increases monotonically up to unity for a 90° incidence
in the low index medium.
The reflectivity for a p-polarized wave decreases to zero for an incidence angle called the
Brewster angle, and then increases with a π phase change up to unity.
At Brewster incidence, the reflected beam is perpendicular to the refracted beam (Figure A.10):
n1 sinθ1B = n2 sinθ2B
θ1B =θ1B′ = 90° − θ2B (A.94)
and since sin(90° − θ) = cosθ
(A.95)
Figure A.10 Brewster incidence.
Between silica (n1 = 1.45) and air (n2 ≈ 1), there are: θ1B = 34.6° and θ2B = 55.4°.
There is also a critical refraction angle θc in the high n1 index medium, for a 90° refraction
angle in the low n2 index medium, for which the reflectivity is 1 (Figure A.11):
Figure A.11 Total internal reflection on a dielectric interface with critical angle θc in the high-index medium.
n2 sin90° = n1 sinθc (A.96)
and
(A.97)
Between silica and air there is: θc = 43.6°.
If a beam coming from the high index medium has an incidence angle larger than the critical
angle θc, the light cannot be refracted anymore in the low index medium and it is entirely
reflected. This phenomenon is known as total internal reflection (TIR), and it is the basis of
dielectric waveguidance.
By reciprocity of transmission, and conservation energy (RF + TF = 1) seen from the high
index medium, Fresnel reflection coefficients have the same value as the ones for the
corresponding refraction angle in the low-index medium. Above the critical angle θc, the TIR
domain, it becomes 1 (Figure A.12).
Figure A.12 Fresnel reflection coefficients for an index ratio of 1.45 for the s-polarization and the p-polarization in the highindex medium.
In this TIR domain, the phase of the fully reflected wave follows a complex variation from 0
phase shift at θc to a π phase shift for grazing incidence (θ1 = 90°).
Note the difference of value between p and s polarizations in the intermediate domain (Figure
A.13).
Figure A.13 p and s phase shifts induced by total internal reflection (index of 1.45)
A.6.2 Dielectric Waveguidance
A light beam being coupled through the edge of a dielectric plate, it will propagate inside the
plate because its incidence on the interfaces yields multiple total internal reflections (Figure
A.14).
Figure A.14 Dielectric waveguide with its numerical aperture
.
The sine of the maximum input angle θmax in a vacuum yielding total internal reflection is called
the numerical aperture (NA), defined by:
(A.98)
This numerical aperture NA is the most important characteristic of a waveguide.
The s (or ⊥) polarization, that is perpendicular to the plane of incidence, is parallel to the
interface between n1 and n2. The electrical E field remains always orthogonal to the incidence
plane and to the axis of the waveguide. This s (or ⊥) polarization is often called TE for
transverse electric (Figure A.15):
Figure A.15 s or ⊥ or TE polarization, orthogonal to the waveguide axis.
However, the p or // polarization yields an additional longitudinal component of the E field
with respect to the waveguide axis. This polarization is not perfectly transverse anymore, it is
hybrid, and it could have been called HE for hybrid electric. However, it is usually called TM for
transverse magnetic, since the magnetic B(or H) field is perpendicular to the E field, and remains
orthogonal to the axis of the waveguide (Figure A.16).
Figure A.16 p or // or TM polarization: the E field has a longitudinal component but the B field remains transverse.
A.7 Geometrical Optics
A.7.1 Rays and Phase Wavefronts
A lot of optical problems, particularly optical imaging, can be treated without considering the
wavelength that is very small macroscopically. They use the laws of geometrical optics, which
consider that the optical energy propagates along light rays. In a plane wave, these rays are
parallel to the direction of propagation and perpendicular to the planar wavefronts (also called
phase fronts) [Figure A.17(a)]:
Another important solution of the propagation equation is the spherical wave:
(A.99)
where r is the radial coordinate and r0 is a normalization radius. The wavefronts are concentric
spheres. In geometrical optics, such a wave is described with rays that are radii of the spherical
wavefronts [Figure A.17(b)].
Figure A.17 (a) Plane wave and (b) spherical wave.
Note that the amplitude of the wave has to decrease in 1/r, since the conservation of energy
requires that the optical intensity (the power density equal to the square of the amplitude)
decreases in 1/r2 as the wavefront area increases in r2.
A.7.2 Plane Mirror and Beam Splitter
The plane mirror is the simplest and most commonly used optical imaging component everybody
is familiar with, but it is the opportunity to define some vocabulary.
When a spherical diverging beam, coming from a real source object, is reflected on a plane
mirror, the analysis of the reflected wave is simply done with a virtual image, that is symmetrical
with respect to the geometrical plane defined by the mirror [Figure A.18(a)]. Conversely an
inverse converging wave yields a virtual object, and a real symmetrical reflected image.
Figure A.18 (a) Real object and virtual reflected image. (b) Virtual object and real reflected image.
In an imaging instrument, the object is what there is without the instrument, and the image is the
transformation of the object by the instrument. They are said to be real if light actually exists, and
said to be virtual if light does not really exist at this geometrical position.
The most common mirrors are metallic mirrors. A metal being conductive, the incident E field
generates electrical current that reemits the reflected wave. An ideal metal with infinite
conductivity would reflect 100% of the incoming power, but, in practice, the reflection is at best
99%.
With a very thin layer of metal, one gets a beam splitter with partial reflection and partial
transmission, which is called a half-silvered mirror.
Note that, contrary to metallic reflection, total internal reflection at a dielectric interface is
lossless; therefore, making low-loss dielectric waveguides is possible.
A.7.3 Lenses
The divergence of spherical waves may be modified with lenses, particularly convergent lenses.
Rays are deviated by refraction on the spherical faces of the lens. Lenses are defined by their
focal length f and when the distance d between the source point and the lens is equal to this focal
length f, the divergent spherical wave is transformed into a plane wave (Figure A.19).
Figure A.19 Transformation of a divergent spherical wave into a plane wave by a convergent lens of focal length f.
When the distance d is larger than f, the wave becomes convergent and a real image of the
source is refocused at a distance d′ (Figure A.20), with:
(A.100)
Note this equation yields d = d′ when d = 2f. This is the 2f − 2f configuration. Because of
reciprocity, in the reverse operation the light rays follow the same path in opposite direction.
Figure A.20 Transformation of a divergent spherical wave by a convergent lens into a converging wave. Note the case in
which d = d′ = 2f.
These convergent lenses have convex faces, and conversely it is possible to make divergent
lenses with concave faces.
Spherical mirrors behave like lenses with an index of −1. This minus sign implies that concave
mirrors are convergent, and convex mirrors are divergent.
Simple lenses have spherical surfaces, which created aberrations. In fiber optics, such lenses
are used to collimate or focus divergent beams getting out of fibers or waveguides, so coming
from a single-point source. Therefore, they do not face off-axis field aberration like microscope
objectives or camera lenses, which image extended objects.
One faces only spherical aberration and it is solved today very simply with aspherical lenses. In
addition, with simple geometrical considerations of homothety, aberrations that are defined as
wavefront defects with respect to wavelength are reduced linearly when the focal length is
shortened.
Numerous fiber-optic components are actually millimeter-scale bulk-optic benches, using such
low-aberration, short-focal-length lenses to collimate or focus beam exiting pigtail fibers (Figure
A.21).
Figure A.21 Compact fiber-optic component using millimeter-scale optical bench and short focal length lenses.
A.8 Interferences
A.8.1 Principle of Two-Wave Interferometry
As we have already seen, the frequency of optical waves is so high that it is not possible to
measure directly the modulated electromagnetic field, particularly its phase. However, an indirect
power or optical intensity measurement is possible through the phenomenon of interferences. In
an interferometer, the input light wave is split, then propagates along two different paths, and is
recombined at the output. The total field of the interference wave is the vectorial sum of the two
fields that have propagated along both paths:
E = E1 + E2
E1 = E10ei(ωt+ϕ1) (A.101)
E2 = E20ei(ωt+ϕ2)
where ϕ 1 and ϕ 2 are the respective phases of the fields E1 and E2.
These phases come from the delay accumulated in the propagation along both geometrical
paths L1 and L2 with:
(A.102)
where λm1 and λm2 are the actual wavelengths in the media:
(A.103)
where λ is the wavelength in a vacuum and n1 and n2 are the respective indexes of refraction of
the media.
It is often expressed as:
(A.104)
where Lop1 = n1 ⋅ L1 and Lop2 = n2 ⋅ L2 are called the optical paths.
The accumulated phase is proportional to the number of wavelengths along the path. One may
use the actual wavelength in the medium and the actual geometrical path, or alternatively the
wavelength in a vacuum and the optical path.
The optical intensity of the interference wave is proportional to the temporal averaging of the
scalar square of the field:
Iopt = c ⋅ ∈0 < E ⋅ E* > (A.105)
and since:
(A.106)
The general formula of interferences is then:
(A.107)
where I1 and I2 are the optical intensities (i.e., the power density) of the interfering waves, and
where ∆ϕ = ϕ 1 − ϕ 2 is their phase difference induced by the difference ∆Lop between both optical
paths n1L1 and n2L2:
(A.108)
In interferometers using single-mode fibers, the two waves having the same power density
distributions, one can use the total powers, with:
(A.109)
There is a maximum power Pmax when both waves are in phase (∆ϕ = 2mπ, m being an
integer); that is, the two field magnitudes are added:
(A.110)
and
that is,
(A.111)
and there is a minimum power Pmin when both waves are in phase opposition (∆ϕ = (2m + 1)π);
that is, the two field magnitudes are subtracted:
(A.112)
and
that is,
(A.113)
Note that these results assume that both fields have the same state of polarization. If they have
orthogonal states of polarization, the interferences are suppressed, as the scalar product E1 ⋅ E2*
= E1* ⋅ E2 = 0.
Interferometry is a very sensitive method for measuring various parameters, because the whole
dynamic range of measurement between Pmax and Pmin is scanned for a change of π in the
phase difference, which is induced by a change of λ/2 of the optical path difference (i.e., a change
of less than a micrometer).
Two cases are of particular interest:
1. P1 = P2; then [Figure A.22(a)]:
P(∆ϕ) = 2P1 (1 + cos∆ϕ) (A.114)
Figure A.22 Response of an interferometer: (a) perfect contrast; (b) low contrast with a fringe visibility of
.
The interference is said to be perfectly contrasted, as Pmin = 0 and the contrast C or fringe
visibility V is defined by:
(A.115)
1. P1 >> P2; then [Figure A.22(b)]
(A.116)
The contrast C or fringe visibility V is:
(A.117)
with P(∆ϕ) = P1(1 + Vcos∆ϕ).
That is, the relative peak-to-peak relative variation of the interference signal is:
It is proportional to the square root of the power ratio (i.e., the field or amplitude ratio) and not
the power or intensity ratio. For example, a power ratio of 10–4 still yields a fringe visibility of 2
× 10–2.
It can be summarized by considering a high-power wave with a normalized power and
normalized amplitude of 1, interfering with a low-power wave with a power ε2 and an amplitude
ε. Their interference yields:
(1 ± ε)2 ≈ 1 ± 2ε (A.118)
This effect is used in the coherent detection scheme, where a high-power local oscillator of
power PL interferes with a low-power signal PS. Instead of measuring the low-power PS directly,
one measures the interference signal:
(A.119)
The low-power signal PS is amplified by a coefficient
the high-power PL of the local oscillator.
through its interference with
A.8.2 Most Common Two-Wave Interferometers: Michelson and Mach-Zehnder
Interferometers, Young Double-Slit
Among the most well-known interferometers is the Michelson interferometer (Figure A.23), in
which the light is separated by a beam-splitting plate, back-reflected on two mirrors, and
recombined on the beam-splitting plate. The scanning of one mirror along a distance d allows the
optical path difference ∆Lop to be changed.
When light propagates in a vacuum:
∆Lop = ∆L = 2d
Note that the general formula for interference, in which the power P may be larger than the
sum P1 + P2, does not violate the principle of conservation of energy. The input power Pin is
split in two waves of respective intensities:
R Pin and T ⋅ Pin
where R is the splitter reflectivity and T is the splitter transmissivity, with R + T = 1. These two
waves are split again when they come back on the beam-splitter. At the free output port, there is
interference between:
Figure A.23 Michelson interferometer with a scanning mirror.
R ⋅ T ⋅ Pin and R ⋅ T ⋅ Pin
and
Pfree = 2R ⋅ T ⋅ Pin + 2R ⋅ T ⋅ Pin cos∆ϕ (A.120)
At the common input-output port there is interference between:
R2Iin and T2Iin
and
Pcommon = R2Pin + T2Pin + 2R ⋅ T ⋅ Pin cos∆ϕ′ (A.121)
Because of conservation of energy, the two output ports must be complementary:
Pfree + Pcommon = Pin (A.122)
with
Pfree = 2R ⋅ T ⋅ Iin + 2R ⋅ T ⋅ Iin cos∆ϕ
Pcommon = R2Pin + T2Pin + 2R ⋅ T ⋅ Iin cos∆ϕ′ (A.123)
Since
R2 + T2 + 2R ⋅ T = (R + T)2 = 1 (A.124)
this implies that ∆ϕ = ∆ϕ′ = π. This π difference is due to the intrinsic π/2 phase shift induced by
reflection. At the common input-output port, both waves have experienced a transmission and a
reflection and they both carry this π/2 reflection phase shift, while at the free port one wave
experienced two reflections and carries a double π/2 phase shift and the other one experienced
two transmissions, without π/2 phase shift.
Note that the fringe visibility is always unity at the free port of a Michelson interferometer
where both interfering waves have the same power, but that it depends on the balance of the
splitter at the common port.
This result of complementary outputs respecting the principle of conservation of energy can be
generalized to any lossless interferometer.
Another well-known interferometer is the Mach-Zehnder interferometer (Figure A.24). Light is
separated by a first beam-splitter, propagates along two different paths, and is recombined on a
second beam-splitter.
Figure A.24 Mach-Zehnder interferometer.
The above results for the Michelson and Mach-Zehnder interferometers implies the ideal case
where the various splitters and mirrors are perfectly at 45° or 90° of the input waves. In practice,
a slight angular misalignment yields a spatial display of fringes (Figure A.25).
Figure A.25 Mach-Zender interferometer with angular misalignment of the output splitter yielding fringes.
One can consider the planar wavefronts orthogonal to the rays (see Figure A.26).
If there is an angle θ between the two wavefronts exiting the interferometer output port, the
phase difference varies spatially. The periodicity i of the spatial fringes is λ/θ (Figure A.27).
Such a spatial display of interference fringes is also observed with Young’s double-slit
interferometer, which was actually, in 1803, the first demonstration of the wave nature of light
when, at that time, the corpuscular theory of light was predominant in the physicist community.
Figure A.26 Interference fringes between tilted wavefronts.
Figure A.27 Young double-slit interferometer (distances and angle are magnified in the circles).
Assuming a distance a between the two slits, waves coming from these two slits are in phase in
the direction perpendicular to the slit plane as well as in any direction such that the angle θ is
(Figure A.27) an integer multiple of λ/a.
It yields a sinusoidal interference fringe pattern with a periodic fringe spacing:
A.8.3 Channeled Spectral Response of a Two-Wave Interferometer
So far, we have considered an interferometer as a system that uses a monochromatic source with
a given wavelength λ0 (and a given spatial frequency σ 0 = 1/λ0) and that has a periodic response
as a function of the optical path length difference ∆Lop:
(A.125)
It is very important to understand that an interferometer can be alternatively considered as a
wavelength (or frequency) filter. For a given path difference ∆Lop0, the response as a function of
the wavelength λ (or the spatial frequency σ) is:
(A.126)
Instead of spatial frequency σ and path difference ∆Lop, one can use the temporal frequency f
and the transit time difference ∆τ between both arms of an unbalanced interferometer, and (A.126)
becomes:
(A.127)
The interference fringes appear like channels in the spectral response of the interferometer
(Figure A.28).
Figure A.28 Channeled spectral response of an unbalanced interferometer.
These channels have a period of ∆σ FSR = 1/∆Lop0 in the spatial frequency domain and a
period of ∆fFSR = 1/∆τ in the temporal frequency domain. These periods ∆σ FSR or ∆fFSR are
called the free spectral range (FSR). For example, a path unbalance of 3 mm yields a spatial FSR
∆σ FSR = 0.3 mm−1. The transit time difference ∆τ being 0.01 ns, it yields a temporal FSR ∆fFSR
= 100 GHz.
The ratio between the free spectral range and the optical frequency is the inverse of the number
of wavelengths in the path difference.
(A.128)
For example, a path difference of ∆Lop = 3 mm is 2,000 wavelengths of 1.5 μm, therefore the
free spectral range is 1/2,000 of the corresponding optical frequency of 200,000 GHz. The
product ∆fFSR ⋅ ∆Lop is equal to about 300 GHz.mm in a vacuum. In an interferometer using
silica fiber with an index of 1.45, it is about 200 GHz⋅mm (i.e., 200 kHz⋅km). Note this is the
double of the relationship between the proper frequency and the coil length of a fiber gyro.
Now, when there is a slight change of the interferometer unbalance, this yields an inverse slight
change of the free spectral range:
(A.129)
Observed about the very high frequency of the optical wave, it is seen mainly as a shift of the
channeled response, like when one compresses an accordion. Going back to the numerical
example above, an increase of a half wavelength (i.e., 0.75 μm) of the 3-mm path difference yields
a downshift of a half FSR (i.e., 0.167 mm−1 or 50 GHz) of the periodic channeled spectral
response.
Note that the free spectral range (FSR) may be expressed in terms of wavelength with:
(A.130)
However, in this wavelength domain, the FSR is not constant anymore and it depends on λ.
As already seen, one has to remember that the contrast or fringe visibility, is the amplitude ratio
between the interfering waves and not their intensity (power) ratio.
This problem of channeled spectral response of a low-visibility unbalanced interferometer is
key to understand the question of mean-wavelength instability of a fiber gyro.
A.9 Multiple-Wave Interferences
A.9.1 Fabry-Perot Interferometer
The Michelson and Mach-Zehnder interferometers are, as well as Young slits, two-wave
interferometers with a raised cosine response. Some other interferometers are multiple-wave
interferometers, and their response is also periodic, but not (co)sinusoidal anymore. They behave
like resonators with narrow response peaks. Such interferometric resonators are often called
optical cavities. Their behavior is not straightforward, and it is an opportunity to detail the
fundamental phenomenon of resonance, which has numerous applications in optics.
A good example of such a multiple-wave interferometer is the Fabry-Perot interferometer,
which is composed of two parallel high-reflectivity mirrors (Figure A.29). At first, it seems that
the added reflectivities of both mirrors should completely reflect the incident light, but there is a
resonance effect when the thickness d of the interferometer is equal to a multiple number of halfwavelengths, and for specific resonance wavelengths λr the light is fully transmitted.
This can be explained with the following arguments. Let us assume that the reflectivity R of
each mirror is 99%. When an input power Pin is sent into the Fabry-Perot interferometer, 1% of
Pin is transmitted inside the cavity. On the second mirror, 0.01% of Pin is transmitted and 0.99%
remains trapped inside the cavity. This trapped light oscillates back and forth, losing 1% of its
own power at each reflection. At the output, there is a series of transmitted waves with a
respective intensity iti, which exponentially decays from 0.01%, the value of the first transmitted
wave. These multiple waves interfere together, and for a resonance wavelength λr they are all in
phase. Thus, the amplitude At of the transmitted wave is the sum of the interfering amplitude ati of
all the transmitted waves and, therefore, this is not the power that is the sum of the interfering
powers.
Figure A.29 Fabry-Perot interferometer.
With mirrors with 1% of transmissivity, we can consider that at the output there are 100 waves
of power pt = 10–4 Pin, but of amplitude at = 10–2 Ain, since the amplitude is the square root of
the power. The constructive interferences of 100 waves having an amplitude equal to 10–2 Ain
give a total transmitted amplitude At equal to the input amplitude Ain: the specific resonance
wavelengths are fully transmitted despite the two high-reflectivity mirrors, since on resonance the
amplitudes and not the intensities of the interfering waves are added.
It is interesting to notice that, when the light is fully transmitted at resonance, the power stored
inside the cavity is 1/(1 − R) times higher than the input/output power (Figure A.30). In a laser
gyro where R = 0.9999, it yields a 104-fold in the cavity.
Figure A.30 Power Pout/(1 − R) stored in a Fabry-Perot interferometer at resonance.
Calculations show precisely that the transmitted power of a Fabry-Perot interferometer is
(Figure A.31):
Figure A.31 Transmission and reflection responses of a Fabry-Perot interferometer as a function of the phase difference
where R is the reflectivity of the mirrors (the transmissivity T = 1 – R).
(A.131)
where ∆ϕ = 2π∆Lop/λ is the phase shift induced by a complete path ∆Lop = 2d inside the cavity.
The parameter m, called the Airy coefficient, is defined by:
(A.132)
Because of conservation of energy, the reflected power Pr and the transmitted power Pt are
complementary, as with two-wave interferometers:
(A.133)
As we have already seen, a two-wave interferometer may be considered as a wavelength filter.
For a multiple-wave Fabry-Perot interferometer, this filtering property also exists and is, in fact,
one of its main applications. With a vacuum cavity and for a given thickness d0, the transmission
wavelength response is
(A.134)
or, with respect to the spatial frequency σ = 1/λ,
(A.135)
or, with respect to the temporal frequency f:
(A.136)
where ∆τ is the transit time of a single pass.
A Fabry-Perot interferometer has a periodic frequency response like a two-wave
interferometer. The period is also called the free spectral range (FSR). Note that this FSR of a
cavity is equal to the FSR of a two-wave interferometer which has a path unbalance equal to the
path length of a single pass in the cavity.
Figure A.32 Filtering transmission of a Fabry-Perot interferometer as a function of spatial frequency σ (inverse of wavelength
λ) and temporal frequency, f.
The filtering characteristic of a Fabry-Perot interferometer is given by the full width at half
maximum (FWHM) ∆σ FWHM (or ∆fFWHM) of the filtering transmitted peaks, and by the free
spectral range (FSR) ∆σ FSR = 1/(2d0) (or ∆fFSR = 1/∆τ) between two response peaks (Figure
A.32). The ratio ∆σ FSR/∆σ FWHM in spatial frequencies, or ∆fFSR/∆fFWHM in temporal
frequencies, depends only on the reflectivity of the mirror. It is called the finesse F (finesse means
“sharpness” in French) of the Fabry-Perot cavity, and:
(A.137)
For example, with R = 0.99 and T = 0.01, the finesse F is 300.
Filtering width and free spectral range may alternatively be given in terms of wavelength, with:
(A.138)
(A.139)
but, as with two-wave interferometers, the wavelength response obtained by inversing the
frequency is not perfectly periodic anymore.
Note that a very important application of the Fabry-Perot interferometer is its use as a resonant
cavity to make a laser source with an amplifying medium. In this particular case, it is called an
active resonator instead of a passive resonator for the basic device. The emission wavelengths λe
of a laser correspond to the transmission peaks of the passive cavity and then are submultiples of
the optical length ∆Lop = 2nd of the cavity (where n is the index):
∆Lop = m ⋅ λe (m is an integer) (A.140)
If the gain curve is wide, the laser will have multiple longitudinal modes since several
wavelengths λe are emitted simultaneously with a spectral difference equal to the FSR. If the gain
curve is narrow and gets to the free spectral range of the cavity, only one single
frequency/wavelength is emitted, and the laser is called a single-longitudinal mode. This is the
case of a laser gyro with a cavity length of 20 cm yielding ∆fFSR = 1.5 GHz while the He-Ne gain
curve has a width of 1 GHz.
A.9.2 Ring Resonant Cavity
A ring resonant cavity, active as it is the case with a ring-laser gyroscope, or passive as it is the
case with a resonant fiber-optic gyroscope (R-FOG), is very similar to a Fabry-Perot
interferometer.
Instead of oscillating back and forth between both Fabry-Perot mirrors, light recirculates many
times around the ring cavity.
As a Fabry-Perot interferometer, a ring resonant cavity has a finesse and a free spectral range
and the power stored in the cavity at resonance is 1 (1 − R) times the input-output power. In the
case of laser gyro where R can be as high as 99.99%, the power stored in the cavity is 104-fold
the output power.
A.9.3 Multilayer Dielectric Mirror and Bragg Reflector
It is well known that metals reflect light but with, at best, a loss of few tenths of percent. To get
ultra-high reflectivity, multilayer dielectric coatings are used.
Like a Fabry-Perot interferometer, a multilayer dielectric mirror (Figure A.33) is a resonant
multiple-wave interferometer. Such a mirror is a stack of alternate transparent glass layers of
high index nh and low index nl with the same optical thickness:
nhdh = nldl (A.141)
At each interface there is a low Fresnel power reflection RF = (nh − nl)2/(nh + nl)2, but as with
a Fabry-Perot, it is the various reflected amplitudes that are summed and not the reflected powers
(or optical intensities).
Assuming, for example, nh = 1.6 and nl = 1.5, RF is only 0.1% in power but 3% in amplitude.
With about 30 alternate layers, these 3% amplitude reflections sum up at resonance to 100%.
The reflected resonant wavelength λR is equal to the optical path back and forth through a pair
of alternate layers.
λR = ∆Lop = 2[nhdh + nldl] (A.142)
At first, one could think that resonance could also be obtained for the path through a single
layer:
λR = 2nhdh = 2nldl (A.143)
but Fresnel reflection coefficients show that there is a π phase shift in the reflection at the
interfaces between low index medium and high index medium, which creates destructive
interference with the reflections at the high-low interfaces that do not experience this π phase
shift.
Figure A.33 Multilayer dielectric mirror (k is an integer).
These resonant multilayer dielectric mirrors are often called Bragg reflectors, because they
work on the same principle as X-ray Bragg reflections on a crystal lattice. Laser-gyro mirrors
are made with this technique. They reach a reflectivity as high as 99.99% with very low loss (less
than few 10–5).
A very similar principle is used in fiber Bragg gratings (FBG) (Figure A.34). An ultraviolet
lateral grid illumination of the core of the fiber induces a permanent periodic index increase. It
yields a sinusoidal variation of the index, similar to the square-wave variation of the index of a
multilayer dielectric mirror. The reflected wavelength is then:
λR = 2 ⋅ nSiO2 ⋅ ΛBragg (A.144)
where nSiO2 is the index of silica (1.45) and ΛBragg is the period of the grid illumination. For a
reflected wavelength of 1,550 nm, the period Λ is 535 nm which is easy to obtain with UV
illumination around 250 nm.
Figure A.34 Fiber Bragg grating (FBG): λ R = 2 nSiO2 ΛBragg.
A mathematical analysis shows that the reflected spectrum is the Fourier transform of the index
modulation. This offers the flexibility to achieve various spectral characteristics, with apodized
and chirped gratings. Fiber Bragg gratings are also used as reflectors for fiber lasers.
A.9.4 Bulk-Optic Diffraction Grating
A last case of multiple-wave interference is the bulk-optic diffraction grating. First gratings were
made with a transmissive grid. They can be considered as multiple-slit interference. Instead of the
sinusoidal interference fringes of the Young double-slit interferometer, one gets narrow
interference fringes (Figure A.35), but the period i of these fringes follows the same law as the
one of Young double-slit interferometer:
(A.145)
It is very similar to the case of a two-wave Michelson interferometer and a multiwave FabryPerot interferometer, which have the same law for free spectral range, but a sinusoidal
interferometer response in the two-wave case and periodic interference peaks in the multiplewave case.
Now, since the fringe period is proportional to the wavelength λ, a grating spreads a broad
spectrum (see Figure A.36). There is a central interference peak, called the zero-order of the
grating, where all wavelengths interfere in phase, but on other fringes in the higher grating
orders, wavelengths are spread.
Figure A.35 Transmissive diffraction grating with a peak response instead of a sinusoidal response.
To increase the wavelength selectivity, one has to reduce the slit period a and to increase the
number of slits. It has actually a resonance effect as in a Fabry-Perot interferometer.
Figure A.36 Spectrum spreading in a transmissive grating.
Gratings work also in the reflective mode, with very narrow grooves and a metallic overlay.
The zero-order corresponds to the residual specular reflection and has to be made as small as
possible. In addition, if the groove period is small enough, there is only a first-order mode that
avoids losing power in the higher-order modes. The grating equation is derived with simple
geometry (Figure A.37) where θin is the input incidence angle and θout(λ) is the output angle that
depends on wavelength λ:
a[sin[θout(λ)] + sinθin] = λ (A.146)
The path difference between the reflections on two adjacent grooves has to be equal to λ to get
constructive interference.
Figure A.37 Reflective grating.
By differentiation of this equation, one gets:
(A.147)
yielding:
(A.148)
Note finally that the rainbow color spreading of white light with a reflective grating is
commonly observed with CDs and DVDs since the recording is done with a spiral of small pits
with a micrometer width that acts as the grooves of a grating. However, it is not observed on a
Blu-ray disk because the pits are smaller than visible wavelengths.
A.10 Diffraction
A.10.1 Fresnel Diffraction and Fraunhofer Diffraction
Diffraction is a fundamental phenomenon related to the wave nature of light propagation and
showing the limit of geometrical ray optics. It is not possible to isolate one ray in an optical wave.
When light is selected by a small filtering hole, it can be observed that it is diverging with a mean
angle θD, which increases as the hole size ah decreases (Figure A.38):
Figure A.38 Diffraction of a wave by a filtering hole.
(A.149)
Diffraction has to be viewed as multiple interference between a continuum of an infinite
number of waves. It is explained by considering that, according to Huygens’ principle, each point
of the hole reemits a spherical wave and that all these spherical waves interfere together. The
amplitude of the diffracted wave at a given point is the integral summation of all the amplitudes
of these waves that interfere.
Diffraction at a finite distance, known as Fresnel or near-field diffraction, is very complicated
mathematically, but diffraction at an infinite distance (or in the focal plane of a lens that is making
an image of infinity), known as Fraunhofer or far-field diffraction, is much simpler to solve. In a
given direction θ, the waves interfere with a respective path length difference (Figure A.39):
Figure A.39 Calculation of the Fraunhofer diffraction.
∆Lop = x ⋅ sinθ (A.150)
The total diffracted amplitude Ad in a direction θ is then:
(A.151)
where Af is the amplitude of the spatially filtered wave.
Hence, the diffracted amplitude Ad(sinθ) is the Fourier transform of the spatially filtered
amplitude Af(x/λ). The relation θD = λ/ah corresponds to the fact that the product of the widths of
a pair of Fourier transforms is constant.
This interferometric analysis of diffraction can be explained with the following arguments: in a
direction angle such as θ > θD = λ/ah, each spherical wave emitted in the filtering hole has a
corresponding wave with a path difference of λ/2, and then both waves interfere destructively.
Note that diffraction is a phenomenon intrinsically related to the wavelength; therefore, in a
medium, the same effect occurs, but as a function of the actual wavelength λm = λ/n in the
medium.
In the case of a slit (i.e., a gate function), the amplitude diffraction pattern is a sinc function (sin
x/x) because of Fourier transform, and the optical intensity pattern is a sinc squared function
(Figure A.40) with periodic zero values at angles equal to λ/ah and a width at half maximum
θFWHM ≃ λ/ah. A square hole has the same behavior but in two dimensions.
Figure A.40 Fraunhofer diffraction by a slit: amplitude (dashed curve) and intensity (solid curve).
For a circular hole of diameter d, the calculation involves Bessel function instead of a sinc and
yields an Airy pattern, with the first zero being at 1.22 λ/d, that is, slightly larger than the
corresponding λ/ah of slit or square hole.
If strictly speaking, Fraunhofer diffraction takes place at an infinite distance, it remains a good
approximation above a distance of ah2/λ. Below this value, one is in the Fresnel region (Figure
A.41).
Figure A.41 Fresnel and Fraunhofer regions for diffraction, assuming a slit width (ah) equal to N wavelengths (λ).
This can be summarized as a diffracting aperture with a size ah of N ⋅ λ, which yields a
diverging Fraunhofer diffraction angle 1/N, and the limit between the near-field Fresnel region
and the far-field Fraunhofer region is N2 ⋅ λ.
A.10.2 Knife-Edge Fresnel Diffraction
Because of the very small size of optical wavelengths, Fresnel diffraction is rarely encountered in
practice, besides diffraction by a knife edge. The mathematics remain tedious, but the result can
be shown graphically. Instead of a clear-cut shadow that one might expect with ray propagation,
there are fringes in the geometric line-of-sight region and a semi-darkness decay in the
geometric shadow region (Figure A.42).
Figure A.42 Fresnel diffraction by a “knife” edge as a function of a normalized abscissa
X (X =
.
At the theoretical limit between the two regions, the optical intensity is reduced to 25%. The
abscissa X is normalized by
where D is the distance and λ is the wavelength, with
These 25% can be understood simply: without the knife-edge, the amplitude is the sum of two
equal amplitudes coming from two symmetrical half-spaces; with the knife-edge, there is only
one, and the amplitude is reduced to 1/2, yielding 25% for its square, the optical intensity.
Note an unattended result: if far-field Fraunhofer diffraction follows a linear dependence as a
function of distance since it is related to an angle, knife-edge Fresnel diffraction follows a
square-root dependence. With λ = 1 μm and D = 2m, the normalized abscissa X is 1 for 1 mm, but
it is a square-root law dependence, and at a distance 100 times shorter (i.e., D = 20 mm), the
normalized abscissa is 1 for 0.1 mm (i.e., only 10 times smaller). We shall see that Fresnel
diffraction by a knife edge has to be taken into account with stray light in an integrated-optic
circuit.
Note also that the decay is not very fast as can be seen on Figure A.43 with a logarithmic scale
for transmission. For a normalized abscissa X = −5, the transmission is still 2 × 10–3 (i.e., only
−27 dB).
Figure A.43 Logarithmic scale of knife-edge diffraction with a normalized abscissa
(0.25) on the geometric shadow, -20 dB (0.01) for –2X, –2, and –27 dB (0.002) for –5X.
. Attenuation is –6 dB
A.11 Gaussian Beam
The far-field Fraunhofer diffraction pattern is the Fourier transform of the spatially filtered
amplitude distribution; a Gaussian distribution is of particular interest because the Gaussian
function is invariant under Fourier transform. Such Gaussian beams are generated in particularly
in gas lasers and, as will be seen, the mode of a single-mode fiber is quasi-Gaussian.
A Gaussian beam that propagates along the Oz axis has a Gaussian radial amplitude distribution
A(r) at z = 0 (Figure A.44):
(A.152)
and an optical intensity distribution that is the square of the amplitude distribution:
(A.153)
The term w0, called the waist, is the radius at 1/e in amplitude and at 1/e2 in intensity since:
A(w0)/A0 = 1/e (A.154)
I(w0)/I0 = 1/e2
From the beam waist at z = 0, light is diffracted but keeps a Gaussian amplitude distribution:
Figure A.44 Gaussian beam.
(A.155)
with
(A.156)
This formula may be simplified in two cases:
1. λz/πw02 << 1 (or z << πw02/λ), which is the domain of near-field Fresnel diffraction, and w(z)
is about constant w(z) ≈ w0, and the phase front remains planar. This is the only case where nearfield Fresnel diffraction is simple.
2. λz/πw02 >> 1 (or z >> πw02/λ), which is the domain of Fraunhofer far-field diffraction, and
w(z) is proportional to z:
(A.157)
The Gaussian beam diverges with a full angle θD at 1/e2 in intensity:
(A.158)
(A.159)
and it is found that:
(A.160)
as stated before, if we consider that ah = 2w0 is about the width of the aperture. The phase front
becomes spherical and centered on the original waist at z = 0.
To summarize a Gaussian beam with a full width 2w0 equal to N wavelengths λ (Figure A.45):
Figure A.45 Approximate near-field parallelism and far-field divergence of a Gaussian beam.
Below a distance of about N2λ, it keeps the same width Nλ and a plane phase front.
Above this distance of about N2λ, it diverges with a full angle of about 1/N, and a spherical phase
front centered on the original waist.
Gaussian beams are important in a single-mode optical fiber because the fundamental mode can
be approximated by a Gaussian profile, as will be seen in Appendix B.
A.12 Coherence
A.12.1 Basics of Coherence
In the case of the ideal monochromatic plane wave, the phase of the wave at a given point can be
deduced from the phase at any other point:
The phase is the same at any point on the same phase front, transverse to the direction of
propagation. This perfect transverse correlation is called spatial coherence.
The phase difference ∆ϕ between two different phase fronts can be deduced from the distance d
between both phase fronts, with ∆ϕ = 2πnd/λ. This perfect longitudinal correlation is called
temporal coherence.
In practice, this ideal case is not possible, and a wave has a limited coherence. The complete
theory of coherence is complicated and involves the mathematical apparatus of stochastic
processes. However, we are going to analyze the problem of temporal coherence, which is very
important for the interferometric fiber-optic gyroscope. However, the fiber gyro uses singlemode waveguide, and spatial coherence is automatically ensured because all the points of a mode
have a perfect phase correlation in the transverse direction.
When a broad-spectrum source is sent into an interferometer, it is observed that there is good
interference contrast about a null path difference, and as this path difference increases (in absolute
value), the fringe contrast (or fringe visibility) starts to decrease and finally vanishes completely
(Figure A.46). When the path difference increases both interfering waves start to lose their
correlation, and their phase difference varies as a function of time, which averages out the cos∆ϕ
term of the formula of interferences.
The distance from which the fringe contrast starts to decrease is called the coherence length and
the distance from which this contrast vanishes is called the decoherence length.
Figure A.46 Interference signal with a broad spectrum source.
One way to understand the phenomenon is to decompose the broad spectrum into separate
wavelengths. Each wavelength λi creates its own interference pattern with a period λi. About zero
path difference, all the interference patterns coincide, but as the path difference increases, they
lose their coincidence because they have slightly different periods (Figure A.47). Because some
wavelengths are on a bright fringe and other ones are on a dark fringe, on average, the total
intensity becomes constant, and the contrast disappears. The coherence length Lc and the
decoherence length Ldc are inversely proportional to the spectrum width ∆λ.
Another way to see this question is to consider the channeled spectral response of an
unbalanced interferometer (see Figure A.28). It is equivalent to say that the path difference ∆Lop
of the unbalanced interferometer is longer than the coherence length Lc, or to say that the free
spectral range (FSR) of the channeled spectral response is smaller than the broad spectrum width
[full width at half maximum (FWHM)].
Figure A.47 Interference signals with different wavelengths.
The actual output spectrum is the product of the channeled response by the original source
spectrum (note that the two outputs are complementary) (Figure A.48).
Figure A.48 Pair of complementary channeled spectra at the two outputs of an unbalanced interferometer.
When the path length difference ∆Lop varies the channels of the spectrum are translated
laterally, and if the FSR is much smaller than the spectrum width, the mean optical powers
measured at both outputs of the interferometer remain constant.
With a perfect single-frequency/single-wavelength source, the interference power is:
P = P0[1 + cos(2π ⋅ ∆L/λ)] (A.161)
With a broad-spectrum source that has a mean wavelength
∆λFWHM, the interference power becomes:
, and a full width at half-maximum
P = P0[1 + C(∆L)cos(2π ⋅ ∆L/ )] (A.162)
Where the envelope C(∆L) giving the decreasing in fringe visibility is called the coherence
function. One calls the coherence length Lc, the length that preserves a good fringe visibility:
C(Lc) ≈ 0.8 (A.163)
One can also define a decoherence length Ldc with:
(A.164)
and above this distance, C(∆L) is greatly reduced. In the approximation of a Gaussian-shape
spectrum, C(L) is also Gaussian and C(Ldc) » 0.03 (Figure A.49).
Figure A.49 (a) Broad spectrum with a mean wavelength
and a width ∆λ FWHM (b) Corresponding coherence
functionC(∆L) with a decoherence length
.
Note finally, as we already saw, that in addition of spatial coherence and temporal coherence,
there is also a polarization coherence: actually two waves cannot interfere if their states of
polarization are orthogonal, and these states have to be identical to get full contrast of their
interferences.
A.12.2 Mathematical Derivation of Temporal Coherence
The exact result may be derived rigorously with Fourier transforms. Let us consider the wave
amplitude A(t), which varies as a function of time t at a given spatial position. As we have already
seen, the frequency components a(f) of this amplitude A(t) may be defined by the Fourier
transform:
(A.165)
and, inversely, the wave amplitude A(t) is the integral summation of all its frequency components:
(A.166)
In general, the frequency components a(f) are complex and may be written with a positive real
modulus α(f) and a phase term eiϕ(f):
(A.167)
When this wave is sent in an interferometer with two 50-50 (in intensity) splitters, there are
interferences between the wave A(t) and itself, but they are shifted by a temporal delay τ, A(t − τ).
Considering that light propagates in a vacuum along both paths, the temporal delay τ is related to
the geometrical path difference ∆L by:
∆L = c ⋅ τ (A.168)
To respect energy conservation, each 50-50 splitter reduces the intensity by 1/2 and the
amplitude by
Therefore, after passing two splitters, there are interferences between 2(A(t))
⋅ A(t)/2 and (A(t − τ)/2). The intensity I of the interference wave is:
(A.169)
(A.170)
(A.171)
where the brackets < > denote temporal averaging. The term <A(t) ⋅ A*(t − τ)> is proportional to
the autocorrelation function Γ(τ) of the function A(t), defined in signal processing theory as:
(A.172)
Then the intensity I of the interferences is proportional to:
(A.173)
The basic autocorrelation theorem, also called the Wiener-Khinchin theorem, states: “if A(t)
has the Fourier transform a(f), then its autocorrelation function
has a real and positive Fourier transform equal to the power spectral density ⎪a(f)⎪2 = α2(f).”
Therefore:
(A.174)
and, inversely:
(A.175)
In practice, the power spectral density α2(f) is centered about a mean frequency , and a
centered spectrum αc2 may be defined by (Figure A.50):
Figure A.50 Intensity spectrum as a function of temporal frequency f: (a) actual spectrum α2(f) and (b) centered spectrum
αc 2(f).
(A.176)
Then
(A.177)
and
(A.178)
where Γ c(τ) is the inverse Fourier transform of the centered intensity spectrum αc2(f).
If the spectrum α2(f) is symmetrical about the mean frequency
the centered spectrum is an
even real function and its inverse Fourier transform Γ c(τ) is also an even real function [i.e., Γ c(τ)
= −Γ c(τ) and Γ c(τ) = Γ c*(τ)]. Since:
(A.179)
where Iin is the intensity of the input wave. The intensity I of the interference wave is:
(A.180)
It is possible to define a normalized centered autocorrelation function:
(A.181)
where C(0) = 1 and
And finally:
(A.182)
In terms of spatial coordinates, there is:
(A.183)
where the path length difference ∆L = cτ and the mean wavelength
Then, the effect of a broad spectrum is to yield a visibility decrease of the cosinusoidal
modulation of the interference fringes, with a decrease of C(τ) as ⎪τ⎪ increases (or a decrease of
C(∆L) as ⎪∆L⎪ increases). The fringe visibility V actually becomes:
V(τ) = C(τ) (A.184)
where C(τ), called the coherence function of the source, is the normalized inverse Fourier
transform of the centered intensity spectrum αc2 (Figure A.51).
Figure A.51 Centered spectrum αc 2(f) and coherence functionγ c (τ) (assuming Gaussian functions).
The half-width of the C(τ) function is called the coherence time τc of the input wave A(t). This
half-width is classically defined as the root-mean-square (rms) half-width of C(τ), or half-width at
1σ [1]:
(A.185)
When the centered intensity spectrum αc2(f) and its normalized Fourier transform C(τ) are
Gaussian, calculations are simple and give:
(A.186)
since:
(A.187)
and
(A.188)
Then the coherence time τc is the time that yields (Figure A.44):
C(τ) = e–1/4 ≈ 0.8
Note that the half-width at 2σ (i.e., 2τc) is the half-width at 1/e, since
C(τc) = e–1 (A.189)
It is also possible to define an rms half-width of the intensity spectral density α2(f):
(A.190)
In the case where α2(f) is Gaussian, there is:
(A.191)
and
(A.192)
When
1. Therefore,
are a pair of Fourier transforms, the product of their rms half-widths is
(A.193)
This result has been derived exactly for the Gaussian spectrum, but it remains approximately
true for other spectrum shapes. The coherence length Lc is defined by:
(A.194)
and since
where ∆λ is the rms half-width of the spectrum in wavelength:
(A.195)
As we have seen, the coherence time τc and the coherence length Lc correspond to the
maximum temporal or spatial delay that preserves a good fringe visibility:
V(τc or Lc) = C(τc or Lc) ≈ 0.8 (A.196)
As can be seen in coherent noise analysis in the fiber gyro, the time or length above which the
wave loses its coherence is more of interest. A good definition of this decoherence time τdc is the
inverse of the full width at half maximum ∆fFWHM of the spectrum (Figure A.51):
The decoherence length Ldc is then:
(A.197)
Since
Going back to the simple case of Gaussian functions, it is found that:
(A.198)
and then:
C(τdc) = 0.03
which means that for τdc (or Ldc) the contrast has been reduced to 3%.
The coherence time τc is the half-width at 1σ, and the decoherence time τdc defined by 1/
∆fFWHM is actually the half-width at 3σ to 4σ (the half-width at 1/e being the half-width at 2σ).
These results derived exactly for Gaussian functions may be approximately extended to any
bell-shaped spectrum. However, to solve certain parasitic effects in the fiber gyro, it is necessary
to know precisely the whole coherence function of the source and not simply its half-width.
A.12.3 The Concept of a Wave Train
The very important result of this analysis is the fact that the autocorrelation function and the
fringe visibility depends on the Fourier transform of the power spectral density ⎪a(f)⎪2 = α2(f)
and not on the amplitude spectral density a = αeiϕ . As is usually stated, the process of
autocorrelation loses the information about the phase of the spectral components of A(t).
This means that waves, with the same amplitude modulus α(f) but a different phase ϕ(f) of their
frequency components, yield the same fringe visibility. In particular, a broad-spectrum source has
frequency components with a random phase, while a pulse with pure intensity modulation has
frequency components that have the same phase; but both yield the same fringe visibility if they
have the same power spectrum α2(f). Very often, the temporal coherence behavior of a broadspectrum source is explained by considering wave trains of duration equal to the decoherence
time τdc and a length equal to the decoherence length Ldc. They are actually image pulses with
pure intensity modulation; but as the phase is lost in the autocorrelation, they would yield the
same interference contrast as the broad-spectrum source. This allows one to analyze simply the
effect in the time domain, and, particularly when a wave train is sent in an unbalanced
interferometer, there are two wave trains at the output. If the path imbalance ∆Lop is larger than
the wave-train length Ldc, the two output wave trains do not overlap and cannot interfere (Figure
A.52).
Figure A.52 Propagation of a wave train in an interferometer with an imbalance of ∆Lop. When ∆Lop > Ldc , the two output
wave trains do not overlap and then cannot interfere.
A.12.4 The Case of an Asymmetrical Spectrum
These results are simple if the spectrum is symmetrical, which yields an even-centered spectrum.
When the spectrum is not symmetrical, as is the case with the broadband sources used in the fiber
gyro (superluminescent diode or rare-earth doped fiber source), the analysis is more
complicated. The centered spectral density αc2 has to be decomposed in an even density αce2(f)
and a residual odd density αco2(f) (Figure A.53):
(A.199)
The mean frequency is of course defined as the frequency that minimizes αco(f). The even
component αce2(f) does yield through Fourier transform an even and real autocorrelation
function Γ ce(τ), but the residual odd component αco2(f) yields through Fourier transform an odd
and purely imaginary autocorrelation function i ⋅ Γ co(τ). The interferences become:
(A.200)
where Ce(τ) and Co(τ) have been accordingly normalized to:
(A.201)
(A.202)
wit
Γ(τ) = Γ ce(τ) + iΓ co(τ) (A.203)
and
Γ(0) = Γ ce(0)
since, as an odd function:
Γ co(0) = 0
Compared to the case of a symmetrical spectrum, there is an additional term
that is an even function of τ, since it is the product of two odd functions: Co
and sine. Even with an asymmetrical spectrum, the interference remains symmetrical with respect
to zero. However, because of this additional term, the zero crossings of the fringe modulation
have lost the periodicity of the cosine term for
π/2, 3π/2, and so on. This effect is usually
negligible in an interferometric fiber gyro that works on the central fringe about zero phase
difference, but it may have to be taken into account with certain spectrum stabilization schemes.
Figure A.53 Decomposition of an asymmetrical spectrum and related coherence functions.
Note that this analysis has been carried out with respect to the frequency spectrum, which has
the same shape if we consider the temporal frequency or the spatial frequency σ = f/c. In practice,
the spectrum is often given with respect to the wavelength λ, which is inversely proportional to
the frequencies, λ = 1/σ = c/f; and a symmetrical frequency spectrum does not strictly give a
symmetrical wavelength spectrum (nor, inversely, a symmetrical wavelength spectrum does not
strictly give a symmetrical frequency spectrum), particularly with the large relative width (a large
percentage) of the broadband sources desired in fiber-gyro applications (Figure A.54).
Figure A.54 Problem of spectrum symmetry: (a) case of symmetry with respect to spatial frequency σ and (b) related
asymmetry with respect to wavelength λ (inverse of σ).
A.12.5 The Case of Propagation in a Dispersive Medium
This analysis has yielded the definition of the coherence time τc (and decoherence time τdc),
which is a characteristic of the source spectrum. To observe this effect, an interferometer has
been used in a vacuum, where the temporal delay τ is related simply to the difference of
geometrical length L between both paths of this interferometer:
∆L = c ⋅ τ (A.204)
The spatial equivalent of the temporal quantities τc and τdc have been defined as the coherence
length Lc = c ⋅ τdc and the decoherence length Ldc = cτdc, which are also a characteristic of the
source.
Now when light propagates in a medium, the result is more complicated because of dispersion
effects. As we saw in Section A.3, the amplitude A may be decomposed as the product of:
A monochromatic wave with an angular frequency equal to the mean angular frequency
that propagates at the phase velocity v = /km( );
A modulation term M that propagates at the group velocity vg = dω/dkm.
and
Temporally, there is:
(A.205)
and taking into account spatial propagation, there is:
(A.206)
It can be shown that the autocorrelation function of M is actually the inverse Fourier transform
of the centered spectrum αc:
< M(t)M*(t − τ) > = Γ c(τ) (A.207)
Going back to the concept of a wave train, it can be shown that a wave train is a sinusoidal wave
modulated in amplitude with an envelope Mwt(t) which has an autocorrelation function also equal
to Γ c(t). The wave train amplitude is
(A.208)
Now the following formula is still valid, as it has been derived only with temporal coordinates:
(A.209)
However, when light propagates in a medium, the simple relation ∆L = c ⋅ τ is not valid
anymore. The temporal delay τ does not correspond to the same spatial delay for the
autocorrelation terms γce and γco and for the fringe terms cosine and sine. Then, in an
interferometer with one path in a medium 1 with a geometrical length L1 and the other path in a
medium 2 with a geometrical length L2, the actual temporal delay is different in both cases. It is:
for the autocorrelation terms γce and γco, since they are derived from M(t), which
propagates at the group velocity vg. This delay is called the group temporal delay, denoted by τg.
It is also the propagation time of the envelope of a wave train.
for the fringe terms, as they are derived from eiϖt, which propagates at the phase
velocity v. This delay is called the phase temporal delay, denoted by τϕ , with c ⋅ τϕ = n1( )L1 −
n2( )L2.
Therefore, in the most general case, the interference intensity is:
(A.210)
In practice, material dispersion yields only a small difference between τg and τϕ , but
interferometric measurements are very sensitive, and this may yield spurious effects. For
example, this problem is avoided in Michelson interferometers with an additional compensating
plate, which cancels out the effect of dispersion of the material supporting the 50-50 splitting
coating.
In a fiber, τg may be significantly different from τϕ , as will be seen in Appendix B; however,
the Sagnac effect, which does not depend on matter, is not sensitive to these dispersion effects.
A.13 Birefringence
A.13.1 Birefringence Index Difference
As we have seen, in an isotropic linear dielectric medium, the derived electric field D is
proportional to the electric field E:
D = ∈r∈0E (A.211)
Gas, liquids, and amorphous solids like glass, and in particular fused silica, are isotropic
because their structure is random and they do not have any predominant axis of orientation.
Conversely, crystals have an ordered lattice with predominate axes, and the permittivity ∈r may
depend on the orientation of the field. They have three principal axes x, y, and z that are
orthogonal. The permittivity depends on the principal axes:
Dx = ∈rx∈0Ex
Dy = ∈ry∈0Ey (A.212)
Dz = ∈rz∈0Ez
Each value ∈ri corresponds to an index value
Crystals may be classified into three groups:
with i = x, y, or z.
Crystals of the cubic system like diamond, for example, where nx = ny = nz. They are optically
isotropic and behave like amorphous glasses.
Uniaxial crystals that have two ordinary axes with nx = ny = no, and one extraordinary axis with
nz = ne ≠ no. The birefringence index difference, denoted ∆nb, is ∆nb = ne − no. The
birefringence is said to be positive when ∆nb > 0 (i.e., ne > no), and negative when ne < no. The
extraordinary z-axis is often called the C-axis of the uniaxial crystal. Lithium niobate (LiNbO3)
used in integrated optics is such a uniaxial crystal, and has a negative birefringence.
Biaxial crystals that have different indexes for each axis: nx ≠ ny ≠ nz ≠ nx.
The birefringence ∆nb of crystals is typically in the range of 0.01 to 0.1. It is 0.07 in LiNbO3.
We shall see that polarization-maintaining (PM) fibers are birefringent, but their birefringence is
smaller: ∆nb is around 5 × 10–4.
When a wave is linearly polarized along a principal axis, it propagates with an index of
refraction
(with i = x, y, or z) that corresponds to the axis of the electric E field, since,
as we have seen, the effect of the magnetic B field of the wave on matter is usually negligible. For
example, a wave that is linearly polarized along the x-axis and that propagates along the y- or zaxis has a velocity c/nx.
Propagation along an intermediate direction is complicated, but birefringence problems that we
will have to analyze in fiber optics are simpler because they are concerned only with propagation
along one principal axis and the polarization lies in the perpendicular plane of the two other axes.
The main effect of birefringence is to modify the state of polarization during the propagation
when the wave is not linearly polarized along a principal axis.
A.13.2 Change of Polarization with Birefringence
Let us consider a birefringent plate with the x-axis and z-axis parallel to the interface and the yaxis parallel to the direction of propagation (Figure A.55). When an input state of polarization is
sent through this plate, it will be modified at the output. The input state Ein has to be decomposed
along the principal x-axis and z-axis of the plate:
(A.213)
where ∆ϕ in is the phase difference between both components at the input. Each component E0x
and E0z propagates without change of polarization. At the output, the state of polarization
becomes:
(A.214)
The phase difference between both components is at the output:
(A.215)
where d is the thickness of the plate and ∆nb = nz − nx.
Figure A.55 Change of polarization due to propagation in a birefringent medium.
The change of the state of polarization along propagation through the plate is periodic. The
phase difference increases linearly with respect to the thickness d, and the input state of
polarization is retrieved when the accumulated phase difference is 2π rad. The spatial period of
the change is called the birefringence beat length Λ, with:
Two kinds of birefringent plates are particularly useful:
Half-wave plates where d = mΛ + Λ/2 and ϕ out − ϕ in = 2mπ + π (m being integral);
Quarter-wave plates where d = mΛ + Λ/4 and ϕ out − ϕ in = 2mπ + π/2.
The effect of a half-wave plate is to generate an output state of polarization symmetrical to the
input state with respect to the principal axes (Figure A.56).
Figure A.56 Change of polarization with a half-wave plate (Λ/2).
This symmetry reverses the direction of rotation of an elliptical or circular state of
polarization. Note that, strictly speaking, a half-wave plate does not rotate a linear state of
polarization, but symmetrizes it with respect to the principal axes. However, the rotation of a halfwave plate does rotate a linear state of polarization just as the rotation of a mirror rotates an
image.
The effect of a quarter-wave plate is to change the ellipticity of the state of polarization. In
particular, when the principal axes are parallel to the axes of the ellipse, it yields a linear
polarization. Inversely, a linear polarization yields an elliptical polarization aligned along the
principal axes (Figure A.57). In the particular case of a linear polarization at 45° of the principal
axes, this yields a circular polarization.
Mathematically, this linear transformation of the state of polarization is a product of matrixes,
known as the Jones formalism. The state of polarization can be represented by a 1 × 2 column
matrix, and the effect of the plate is a multiplication by a 2 × 2 square matrix. Decomposing the
polarization along the principal axes is actually the usual mathematical decomposition along the
eigen-axes of the 2 × 2 matrix, which yields a diagonal matrix that becomes much easier to
handle. The effect of the birefringent plate can be written as:
[Eout] = [M] × [Ein] (A.216)
}where the column matrices [Eout] and [Ein] are called the Jones vectors:
(A.217)
The mathematical interest of this matrix formalism is due to the fact that the propagation
through several plates can be calculated with a matrix product:
[Eout] = [M] × [Ein] (A.218)
with
[M] = [Mm] × … × [M2] × [M1] (A.219)
Note that the phase delay created by birefringence is inversely proportional to the wavelength,
and therefore a change of polarization through a birefringent medium may be modified by
wavelength variation, even with a stable medium.
Figure A.57 Change of polarization with a quarter-wave plate (Λ/4).
So far, we have described birefringent effects related to orthogonal principal axes of a crystal,
which is actually called linear birefringence. There is also a phenomenon of circular
birefringence, sometimes called optical activity. With circular birefringence, the eigenstates of
polarization that propagates without change are the two circular states of polarization. Then an
input state of polarization has to be decomposed with its two circular components. Each circular
component propagates with its own index of refraction, and their recombination yields the output
state of polarization. The effect of a circular birefringence with an index difference ∆nc is to
rotate any input state of polarization with an angle θ (Figure A.58) equal to half the phase
difference change between both circular components:
(A.220)
(A.221)
Note that the ellipticity of the state of polarization remains unchanged by circular
birefringence.
A.13.3 Interference with Birefringence
A birefringent crystal, placed between two polarizers, creates a polarimeter that behaves like an
interferometer (Figure A.59). The input wave is polarized by the input polarizer, and can be
decomposed into the states of polarization corresponding to both orthogonal principal
birefringence axes. Each polarization propagates with its own index, which yields a phase
difference at the output. Both orthogonal output polarizations are then projected on the axis of the
output polarizer, often called analyzer, and they interfere. The lower-index fast polarization is
equivalent to the short path of an interferometer, and the higher-index slow polarization is
equivalent to the long path. The output power follows the classical formula of interference:
(A.222)
but now with ∆Lop = ∆nb ⋅ d, where d is the thickness of the birefringent plate.
Figure A.58 Change of polarization with circular birefringence.
The interference fringes are perfectly contrasted when the angle between the polarizers and the
birefringence axes is 45°, which is equivalent to 50-50 splitter and combiner.
Decoherence effects also take place with a broadspectrum source when the path difference ∆nb
⋅ d is larger than the decoherence length
The depolarization length Ldp is also used with:
the output wave is depolarized.
Ldp = Ldc/∆nb (A.223)
The wave gets depolarized when the crystal thickness d is larger than Ldp.
A.14 Optical Spectrum Analysis
Spectral analysis is a key measurement in optics. The most common measurement technique is the
use of a bulk diffraction grating is a monochromator (single-color) configuration. The huge
market of wavelength-division multiplexing (WDM) telecommunication has yielded the advent of
performing optical spectrum analyzers (OSA), especially in the 1,550-nm window. Resolution
can be as good as 10 picometers (pm), that is, less than 10-5 of the wavelength.
For fiber gyros, these OSAs are very useful to calculate the mean wavelength of a broad
spectrum, as well as its coherence function that is its Fourier transform, as we already saw. Laserline analysis requires higher resolution and it is often performed with a scanning Fabry-Perot.
Spectrum analysis techniques are all based on interferometry, and it is useful to understand that
the theoretical resolution of a setup is, at best, the inverse of the number of wavelengths in the
difference between the shortest and the longest paths, which is the setup. It is simply observed with
fiber Bragg grating: the grating period being about 0.5 μm, a 5-mm grating (i.e. 104 periods), has
at the best a relative wavelength selection width of 10–4 (i.e., 0.15 nm at 1,550 nm).
Figure A.59 Perfectly contrasted polarimeter with input polarization and output analyzer at 45° of the principal axes of the
birefringent plate.
This argument about the number of wavelengths being the inverse of the resolution explains the
advantage of Fourier transform spectroscopy. It makes use of a scanning Michelson
interferometer, and the spectrum is retrieved from the Fourier transform of the recorded
interference pattern, as we saw earlier. Because the scanned path unbalance can be very long
(meter range), it yields a very high resolution. Such a technique is used for optical coherencedomain reflectometry (OCDR) and optical coherence domain polarimetry (OCDP).
A last technique, now available because of WDM telecoms, is tunable-laser spectroscopy. The
resolution is then the line width of the tunable laser, which is much narrower than the resolution
of a passive interferometric spectrometer. It can be as good as the megahertz range (i.e., less than
10–8 of the analyzed frequency of 200 THz).
As we see, spectral resolution is usually given with length unit (nm, pm…) when it is measured
with interferometric techniques depending on wavelength. A laser-line width is often measured
with electrical signal spectrum analysis of power beating, and it is expressed in temporal
frequency units, even if one could say 1.3-GHz resolution instead of 10 pm, or 0.007-pm line
width instead of 1 MHz.
Reference
[1] Born, M., and E. Wolf, Principles of Optics, Oxford, U.K.: Pergamon Press, 1975.
Selected Bibliography
Bass, M., ed., Handbook of Optics, 3rd ed., New York: McGraw-Hill, 2009.
Bracewell, R., The Fourier Transform and Its Applications, New York: McGraw-Hill, 1965.
Derickson, D., (ed.), Fiber-Optic Test and Measurement, Hewlett-Packard Professional Books, 1998.
Feynman, R., Lectures on Physics, Reading, MA: Addison-Wesley, 1965.
Kogelnik, H., and T. Li, “Laser Beams and Resonators,” Proceedings of the IEEE, Vol. 54, 1966, pp. 1312–1329.
Refractiveindex.info.
APPENDIX B
Fundamentals of Fiber Optics for the Fiber
Gyroscope
B.1 Main Characteristics of a Single-Mode Optical Fiber
B.1.1 Attenuation of a Silica Fiber
An optical fiber is composed of a core with an index of refraction n1 slightly higher than n2, the
one of its surrounding cladding (Figure B.1). This yields propagation in the core because of total
internal reflection (TIR) at the core-cladding interface, as seen in Section A.6.2.
For low attenuation, the privileged fiber material is silica, the amorphous (or glassy) structure
of silicon dioxide (SiO2). Silicon dioxide has also a crystal-lattice structure, quartz, and therefore
the term “fused quartz” is sometimes used for silica.
The attenuation is due to three main factors (Figure B.2):
Figure B.1 Optical fiber with its core of index n1 and its surrounding cladding of index n2 < n1.
Rayleigh scattering loss due to dipolar radiation and inhomogeneities in amorphous glass-like
silica. This yields an attenuation per unit length αR proportional to λ–4.
Infrared absorption tail of silica that limits the very low attenuation range to below 1,700 nm. At
a 2-μm wavelength, attenuation reaches 10 dB/km, and silica fibers remain useable on short
distance but above 2 μm it increases drastically.
Effect of OH impurities that bring residual water absorption peaks at 1,390, 1,240, and 950 nm.
With this attenuation, there are three transmission windows that are commonly used:
Wavelength of 850 nm, with an attenuation around 2 dB/km. It is used for medium-accuracy fiber
gyros where the sensing coil length is few hundreds of meters.
Wavelength of 1,300 nm, with an attenuation going down to 0.4 dB/km. It is used in
telecommunications, but rarely for fiber gyro.
Wavelength of 1,550 nm, with an attenuation going down to 0.2 dB/km. It is used for highaccuracy fiber gyros where the sensing coil length can go up to several kilometers. It takes
advantage of erbium-doped fiber source. Such a source is derived from erbium-doped fiber
amplifier (EDFA) that revolutionized optical-fiber communication in the 1990s.
Note that there is also a neodymium YAG window around 1,060 nm with an attenuation of 1
dB/km, but it is not used very often.
Figure B.2 Attenuation of a silica fiber (log scale).
B.1.2 Gaussian Profile of the Fundamental Mode
As for any dielectric waveguide (see Section A.6.2), the important parameter of an optical fiber is
its numerical aperture (NA) defined as:
A single-mode (SM) fiber carries only one mode, the fundamental mode, for a wavelength
higher than the cutoff wavelength λc of the second-order mode. It is related to NA by:
λc = 2.6 a NA
where a is the core radius.
Figure B.3 Pseudo-Gaussian profile of the fundamental mode of a single-mode fiber (dotted line for normalized power density
(P) and solid line for normalized amplitude (A)).
This fundamental mode has a quasi-Gaussian transverse power density, and the formalism of
free-space Gaussian beam (see Section A.11) is used with a waist wo, the radius at 1/e2 in optical
power density (or optical intensity), that is, at 1/e in amplitude (with e = 2.72) (Figure B.3). Is also
used the mode field diameter (MFD) with MFD = 2wo.
The fundamental mode has theoretically no cutoff, but the practical range of use of a singlemode fiber is from λc to about 1.5λc where the mode starts to widen very fast, which makes the
fiber very lossy with bending. In this practical range of use, the mode diameter/radius is about
linear as a function of the wavelength λ:
This yields:
It is interesting to notice that in this practical single-mode regime (λc < λ < 1.5λc) the mode size
depends only on the wavelength λ and the numerical aperture NA, and not on the core
radius/diameter.
At the output of the single-mode fiber, this pseudo-Gaussian mode follows the usual diffraction
law of free-space Gaussian beam (see Section A.11), with a full divergence angle θD at 1/e2 in
power density (1/e in amplitude):
Since in the practical range of use of a single-mode fiber (λc < λ < 1.5λc) the MFD is
proportional to λ, diffraction yields a constant divergence angle which depends only on the
numerical aperture NA (Figure B.4). With MFD ≈ 0.84λ/NA, it yields:
θD ≈ 1.5 NA
If the mode diffracts in a medium of index n instead of a vacuum, this angle θD is reduced to
θD′:
θD′ ≈ 1.5 NA/n
Now, placing this single-mode fiber output at the focal point of a convergent lens yields a
collimated Gaussian beam with a beam diameter 2w at 1/e2 in power:
2w = θD ⋅ f
where f is the focal length of the lens.
Standard telecom fiber, the ITU Standard G-652, has a pure silica cladding with n2 = 1.444 at
1,550 nm. The core is said to be doped with germanium (Ge). It might be more adequate to say
that it is a mixture of GeO2 glass (germania) with an index of 1.59 at 1,550 nm and SiO2 glass
(silica) with an index of 1.444, knowing that a glass should be viewed as an infinitely viscous
liquid; and liquids are mixed, not doped. With few percents of GeO2, the core index is raised by
∆n = n1 − n2 = 0.005, yielding a numerical aperture NA = 0.12. Standard G-652 fiber has a core
diameter of 8 μm, yielding a cutoff wavelength λc = 1,250 nm and an MFD of 11 μm at 1,550 nm.
Figure B.4 Diverging far-field output beam of a single-mode fiber, with a full divergence angle θD = 1.5 NA, and a collimated
beam with a diameter 2w = θD ⋅ f.
Gyro coil fibers have a higher NA that improves guidance and avoids bending loss. This
requires a higher percentage of GeO2 in the core, which slightly increases attenuation. Respective
parameters are compiled in Table B.1.
Table B.1 Main Characteristics of Telecom and Gyro Fibers
∆n
Telecom fiber
NA
Core
MFD at θ
diameter 1,550 nm D
0.005 0.12 8 μm
Gyro coil fiber 0.01 0.17 6 μm
Attenuation
at 1,550 nm
11 μm
0.18 rad 0.2 dB/km
8 μm
0.25 rad 0.5 to 1 dB/km
B.1.3 Beat Length and h Parameter of a PM Fiber
A single-mode fiber is a single-spatial-mode fiber, but it is a two-mode fiber in terms of
polarization. There is a residual birefringence and the two orthogonal states of polarization have
a residual difference of propagation velocity.
The fiber-gyro uses fibers that preserve the same state of polarization. They are usually called
polarization-maintaining (PM) fibers, but are sometimes called polarization-holding or
polarization-preserving fibers. Linear polarization maintenance is obtained by creating a strong
linear birefringence with a stress structure. There is a stress-induced birefringence index
difference and the effect is often expressed in terms of beat length L, the fiber length that yields
one wavelength of path difference between both (slow and fast) polarization modes (see Section
A.13.2):
A typical value for ∆nb is 5 × 10–4, i.e. 1/2,000 (remember that n1 − n2 ≈ 10–2), which yields a
beat length of Λ equal to 2,000 wavelengths λ (see Table B.2).
Fiber birefringence yields polarization maintenance but there is a residual crossed-polarization
coupling that grows linearly in power as a function of length. It is expressed in term of hparameter, h standing for holding. The typical value for h-parameter is 3 to 1 × 10–5/m,
sometimes expressed as a polarization extinction ratio (PER) of 15 to 20 dB over 1 km.
Table B.2 Beat Length Λ as a Function of Wavelength λ
λ 633 nm
850 nm 1,300 nm 1,550 nm
Λ 1.25 mm 1.7 mm 2.6 mm
3 mm
B.1.4 Protective Coating
A silica fiber is mechanically protected with a two-layer plastic polymer coating: a soft inner
primary coating to limit the transmission of external stress, and a harder outer secondary coating
to ease handling of the fiber (Figure B.5). A typical coating diameter is 250 μm for telecom fiber
with a 125-μm cladding diameter. Now, to limit the volume of sensing coils, the gyro fiber has a
typical cladding diameter of 80 μm with a coating diameter of 170 μm. These polymer coatings
have an index of refraction higher than the one for the cladding. This allows spurious light in the
cladding to leak in the coating and to be then attenuated, since coatings have attenuation in the
range of 1 dB/cm.
Figure B.5 Fiber with its two-layer coating.
B.1.5 Temperature Dependence of Propagation in a PM Fiber
Temperature dependence of the various parameters of a fiber is obviously of importance for
practical applications. For silica index nSiO2, there are values are seen in Table B.3.
Table B.3 Thermal Dependence of Silica Index
n SiO2 dn SiO2 /dT
At 850 nm
1.452 8.6 × 10 –6 /° C
At 1,550 nm 1.444 8.4 × 10 –6 /° C
Silica expansion coefficient αSiO2 = dL/L ⋅ dT is 0.5 × 10–6/°C, expressed also as 0.5 ppm/°C.
The optical path nSiO2 ⋅ L then follows by logarithmic differentiation:
that is, 6.45 ppm/°C at 850 nm and 6.35 ppm/°C at 1,550 nm.
Such a relative thermal variation of the optical path is found in the reflection wavelength
ΛBragg of a Bragg grating (see Section A.9.3), but also in the transit time τ through a fiber since
the phase index and the group index follow very similar thermal variations:
Conversely, for the proper frequency fp of a fiber gyro related to the inverse of transit time
through the coil:
Now if silica is a very stable material thermally, a silica fiber is protected, as we saw, by a
polymer coating which has a high expansion coefficient, typically 50 to 100 ppm/°C. As the
temperature increases, the coating tends to expand, but its expansion is blocked by the fiber which
is placed under tensile stress. The actual expansion of the coated fiber is related to the transverse
area, Young modulus, Poisson ratio, and expansion coefficient of silica and the ones of the
coating polymers.
A gyro-coil fiber with a cladding diameter of 80 mm and a double-layer-coating diameter of
170 μm has typically and expansion coefficient: αF = 4 to 5 ppm/°C, that is, about 10 times higher
than the expansion of silica αSiO2 = 0.5 ppm/°C. This additional thermal expansion also changes
the variation of proper frequency:
instead of −6.35 ppm/°C for a bare fiber.
Finally, as we saw, birefringence of PM fiber is created with a stressing structure made of glass
with higher expansion coefficient than silica. The stress being induced during the cooling phase
of this structure over 800°C to 1,500°C, depending on the softening temperature Tg of the glass
used in the stress-structure, the relative thermal dependence of the birefringence is conversely:
d ∆nb/(∆nb ⋅ dT) ≈ −1.2 to −0.7 × 10–3/°C
B.2 Discrete Modal Guidance in a Step-Index Fiber
As we already saw, a step-index fiber is composed of a cylindrical core with an index of
refraction n1 and a radius a, and of a surrounding cladding with a lower index n2 < n1 (Figure
B.1). Going to more mathematics, because of the boundary conditions at the core-cladding
interface, there is a discrete number of eigensolutions of the general propagation equation that
are guided in the fiber. These discrete eigensolutions are called the modes of the fiber, which is
multimode. They can be written as:
Emi(x, y, z, t) = E0mi(x, y)e–(iωt–βiz)
Bmi(x, y, z, t) = B0mi(x, y)e–(iωt–βiz) (B.1)
where x and y are the transverse spatial coordinates, and z is the longitudinal spatial coordinate
corresponding to the direction of propagation.
Contrary to the case of the plane wave, each mode has specific transverse distributions of the
fields E0mi(x, y) and B0mi(x, y) that tend to zero far from the core. The propagation phase term
e–i(ωt–βiz) depends on the angular frequency w and on a specific mode propagation constant βi
that depends on ωand that is comprised between the wave number k2 in the cladding and the wave
number k1 in the core:
(B.2)
An equivalent index neqi of the mode i is often used with:
(B.3)
The modes are denoted by TE for transverse electric or TM for transverse magnetic, or EH and
HE for hybrid electromagnetic, when they have a longitudinal component Ez or Bz in addition to
the usual transverse components (Ex, Ey) and (Bx, By) of a TEM wave in free space: TE modes
are purely transverse for the electric field and have a longitudinal magnetic component;
conversely, TM modes have a longitudinal electric component, and EH and HE modes have a
longitudinal component for both E and B fields.
In practice, the index difference ∆n = n1 − n2 is small: the relative index difference ∆ = ∆n/n2 is
usually on the order of 0.2% to 1%. This yields a negligible longitudinal component, and the
modes, then denoted by LP for linearly polarized, can usually be regarded as transverse waves.
The fundamental interest of this modal decomposition is the fact that the set of modes is an
orthonormal basis of eigenvectors of the ensemble of all the possible solutions of the
propagation equation. This ensemble is, from the mathematical viewpoint, a linear space with a
scalar product.
The most familiar linear space is the three-dimensional geometrical space. Any vector U can be
decomposed on the orthonormal basis of eigenvectors (a1, a2, a3):
(B.4)
The orthonormal basis is orthogonal; that is, the scalar product of two different eigenvectors is
zero:
ai ⋅ aj = 0 if i ≠ j
and it is also normal; that is, all the scalar square of the eigenvectors are equal:
a12 = a22 = a32 = ai2
The norm or magnitude or modulus U of a vector U is defined as the square root of its scalar
square, and:
There are two important results that may look obvious, or at least very familiar, in a
geometrical space (Figure B.6), but that are extremely useful in other linear spaces, for which it is
not as straightforward:
Figure B.6 Coordinates x 1 and x 2 of a vector U in a 2-D geometrical orthonormal linear space. There are a1 ⋅ a2 = 0 and a12
= a22.
The coordinate xi is:
(B.5)
The square of the modulus is:
(B.6)
In particular, it is very convenient to consider that the ensemble of complex functions f(x),
which are said to be square integrable, is a linear space. The infinite integral
is
convergent (which is the definition of a square integrable function) and may be regarded as the
generalized scalar square < f⎪f > of the function. A generalized scalar product may be defined
with:
(B.7)
It is possible to find an orthonormal basis of eigenfunctions fi of this linear space, any function
f being accordingly decomposed to:
(B.8)
with
and
The fact that the linear space of the functions f has an infinite dimension does not change the
generality of the above results, and this definition of the scalar product may be extended to a
function of several variables with a multiple integral.
Going back to the fiber, the ensemble of the solutions of the propagation equation is also a
linear space, which is the sum of the nonguided solutions and the sum of the guided solutions.
The dimension of the ensemble of the nonguided solutions is infinite, but the dimension of the
ensemble of the guided solutions is finite: it is equal to the discrete number of modes in the fiber.
Any solution E of the propagation equation may be decomposed with:
(B.9)
where emi are the normalized guided modes and eri are the nonguided modes that are radiated.
The guided modes emi(x, y, z, t) are eigenvectors, and, therefore, their generalized scalar
products < emi⎪emj > are null. Eliminating the z and t dependence, these generalized scalar
products yield, for the transverse field distributions em0i(x, y), the overlap integrals, which are
also null for orthogonal modes:
(B.10)
The coordinate xi of E on the mode emi is defined with generalized scalar products:
(B.11)
similar to the definition of the coordinate of a vector in geometry with:
xi = (U ⋅ ai)/ai2 (B.12)
With overlap integrals, the (z, t) dependence is eliminated:
(B.13)
Furthermore, similar to
in geometry, there is also:
(B.14)
From the physic standpoint, this last equation shows that the total power of the wave E equals
the sum of the total powers in each mode, as we might expect. As a matter of fact, the square of
the electric field is proportional to the intensity of the wave (i.e., the spatial density of power), and
the overlap integral, which is an infinite integral of the power density over the transverse plane
xy, yields the total power.
This relation applies only to the total powers of the modes. The problem is very different for
the local power density in the core. It is the square of the sum of the amplitudes of these modes
and is the result of interferences between the various modes. Specifically, there are places without
light because of destructive interferences. With a large number of modes, this yields a speckle
pattern.
Note that overlap integrals using the usual scalar product em0i ⋅ em0j* are valid with the
assumption of transverse LP modes when the index difference is small. Generally, where there are
longitudinal field components, the term em0i ⋅ em0j* has to be replaced by the vector product
em0i × bm0j. This makes calculations more complicated, but the basic principle of orthogonality
of the modes is preserved.
B.3 Guidance in a Single-Mode (SM) Fiber
B.3.1 Amplitude Distribution of the Fundamental LP01 Mode
The calculation of the modes is usually carried out with a normalized frequency V that is defined
with:
(B.15)
This can be written:
(B.16)
where
also:
is the numerical aperture of the fiber as defined in Appendix A. There is
(B.17)
The important result is that when 0 < V < 2.405 a step index fiber is in the single-mode regime,
where only the fundamental spatial mode (symbolized HE11 in the most general case and LP01 in
the low ∆n approximation) can be guided. The value 2.405 is the first zero of the J0 Bessel
function. The cutoff wavelength λc is defined by:
(B.18)
that is
λc = 2.6 a NA
and
The fiber is single-mode for λ > λc. Note that an ideal single-mode fiber can guide any state of
polarization with the same propagation constant: the two-dimensional linear space of the
polarization modes is said to be degenerated, but in practice there is always a residual difference.
A single-mode fiber is actually a single-spatial-mode fiber, but it is a two-mode fiber in terms of
polarization.
The exact definition of the modes require the use of Bessel and modified Bessel functions, but
the fundamental mode may be described approximately with a Gaussian distribution as free-space
Gaussian beams described in Section A.11, the wave amplitude being (Figure B.7):
Figure B.7 Pseudo-Gaussian amplitude of the fundamental LP 01 mode of a single-mode fiber (case where λ = 1.2λ c ).
(B.19)
where
is the radial coordinate.
The mode radius w0 (at 1/e in amplitude and 1/e2 in power density, or optical intensity) is for
0.8λc < λ < 2λc (Figure B.8):
(B.20)
Is often used the mode field diameter (MFD) with:
MFD = 2w0
An even simpler linear approximation is usually accurate enough in the practical range of use
of a single-mode fiber (i.e.,λc < λ < 1.5λc):
(B.21)
which yields w0 ≈ 0.42λ/NA and MFD ≈ 0.84 λ/NA.
It is interesting to notice that, in this practical single-mode regime, the mode radius/diameter is
independent of the core radius, and depends only on the numerical aperture NA and the
wavelength.
Figure B.8 Fundamental LP 01 mode radius w0 as a function of wavelength λ, with a linear dependency between λ c and 1.5 λ c .
Above 1.5λc, the LP01 mode starts to widen very fast, and the fiber becomes lossy with
bending.
B.3.2 Equivalent Index neq and Phase Velocity vϕ of the Fundamental LP01 Mode
The equivalent index neq varies continuously from n2 for an infinite wavelength to n1 for a null
wavelength:
When the wavelength is very large, the mode is very wide and sees mainly the cladding and its
index n2.
When the wavelength is very short, the mode is confined in the core and sees mainly the core and
its index n1.
A normalized propagation constant b(V) is often used with:
(B.22)
When the relative index difference ∆ is small, we can write:
β ≈ k2(1 + b ⋅ ∆)
or
neq ≈ n2(1 + b ⋅ ∆) (B.23)
The phase velocity vϕ of the mode is then:
(B.24)
For 0.6 < λc/λ < 1, the value of b(V) can be approximated by (Figure B.9):
(B.25)
Then, at the limit of the single-mode regime,
(B.26)
(B.27)
B.3.3 Group Index ng of the Fundamental LP01 Mode
The wavelength dependence of the propagation constant b(λ) [and of the equivalent index neq(λ)]
yields dispersion effects due to guidance in addition to the proper chromatic dispersion of the
material. In particular, while the phase velocity is vϕ = ω/β, modulated signals propagate at the
group velocity vg:
Figure B.9 Normalized propagation constant b(λ) and equivalent index neq(λ) of the fundamental LP 01 mode (n1 is the core
index and n2 is the cladding index).
(B.28)
and
(B.29)
with
(B.30)
where dn2/dλ is due to the material, and (db/dλ) (n2∆) is due to the guidance.
One can define a group index ng due to guidance with:
remembering that the equivalent index is:
This group index ng is significantly different from neq as displayed on Figure B.10. In the
practical domain of use of a single-mode fiber (λc < λ < 1.5 λc, that is, 0.7 < λc/λ < 1), ng is about
the core index n1, while the equivalent phase index neq is close to the cladding index n2.
Figure B.10 Guidance group index ng of the fundamental LP 01 mode (neq is displayed with the dotted curve).
In addition of guidance, group velocity depends also on material dispersion. For silica, the
group index ngSiO2 is significantly different from the index of refraction as summarized in
Table B.4.
Table B.4 Index and Group Index of Silica
Wavelength λ
850 nm 1,300 nm 1,550 nm
Index of refraction n SiO2 1.452
1.446
1.444
Group index n gSiO2
1.461
1.462
1.468
B.3.4 Case of a Parabolic Index Profile
This analysis has been derived for a perfect step-index fiber; however, with some specific
manufacturing processes, there is some grading of the index profile. Evaluation of the
characteristics of such fibers is usually done with the use of an equivalent step-index fiber, which
provides simply approximated values of cutoff and mode diameter. In particular, a parabolic
index profile of maximum radius amax and maximum relative index difference ∆max may be
approximated with an equivalent index step having a radius ae = 0.8 amax and a relative index
difference ∆e = 0.75 ∆max (Figure B.11).
B.3.5 Modes of a Few-Mode Fiber
Now, when the fiber is used below its cutoff (λ < λc or V > 2.4), there are higher-order guided
modes. The first one to appear is the antisymmetric LP11 mode that becomes guided in addition
to the fundamental LP01 mode. This second-order mode has an odd distribution with respect to
one transverse coordinate. The wave amplitude may be approximately described using the
normalized derivative of a Gaussian function (Figure B.12):
Figure B.11 Equivalent index step of a parabolic index profile.
Figure B.12 Amplitude and intensity distributions of the second-order LP 11 mode.
(B.31)
This mode is composed of two lobes where the amplitude (or field) has opposite signs (or a π
rad phase difference). The mode is said to be antisymmetrical; the intensity (or power density),
which is the square of the amplitude (or field), is identical and always positive for both lobes. The
field es0 is maximum and equal to e0 for x = w1 and y = 0. The value w1 is the half-width at the
maximum.
Figure B.13 Spatial degeneracy of the LP 11 mode.
The LP11 mode is degenerated in terms of polarization as the LP01 mode, but it also has a
spatial degeneracy, since the lobes may be aligned along any transverse axis (Figure B.13).
When V > 3.8, that is, 1/λ > 1.6/λc or λ < 0.63λc, there are also the LP21 and LP02 modes. Their
equivalent indexes are displayed in Figure B.14.
The LP21 mode has four lobes with a double antisymmetry and spatial degeneracy as the LP11
mode. The LP02 has a secondary ring with a sign change with respect to the central part (Figure
B.15). As will be seen in Appendix C, these LP11 and LP21 antisymmetric modes are used to
understand the behavior of light in integrated-optics circuits.
Figure B.14 Equivalent indexes of a few-mode fiber.
Figure B.15 Power distribution of the first modes: signs + and − indicates sign change on the amplitude, and the LP 11 and
LP 21 modes have a spatial degeneracy.
B.4 Coupling in a Single-Mode Fiber and Its Loss
Mechanisms
B.4.1 Free-Space Coupling
As we already saw in Section B.1.2, at the output of a single-mode fiber, the pseudo-Gaussian
fundamental mode with a radius w0 at 1/e2 is diffracted to form a free-space diverging pseudoGaussian beam, called the far field, with a divergence angle θD at 1/e2:
(B.32)
and since, in the practical range of use (λc < λ < 1.5λc), wo/a ≈ 1.1λ/λc, the divergence angle θD
is about constant:
θD ≈ 1.5 NA (B.33)
Considering the inverse propagation, a free-space Gaussian beam converging with this same
angle θD creates a focused Gaussian spot with the same radius w0, and can be fully coupled in the
fiber if the core is centered on the input beam. To get such a converging beam, a parallel
Gaussian laser beam with a beam diameter w0 at 1/e2 has to be focused with a convergent lens
that has a focal length f such as (Figure B.16):
2w/f = θD ≈ 1.5 NA (B.34)
B.4.2 Misalignment Coupling Losses
Misalignments decrease the coupling ratio that can be calculated with the overlap integral between
the input wave and the fundamental mode. The input wave Ein may be decomposed on the set of
eigenvectors, which comprises a unique guided mode, the normalized fundamental mode ef, and
radiating modes erj:
(B.35)
Figure B.16 Coupling of a Gaussian free-space beam in a single-mode fiber.
The coordinate xf is calculated by the generalized scalar products:
(B.36)
The power or intensity coupling ratio C is the ratio between the generalized scalar square of (xf
⋅ ef), which is proportional to the power of the coupled wave, and the generalized scalar square of
Ein which is proportional to the power of the input wave:
(B.37)
(B.38)
The phase term ei(ωt–βz) may be eliminated in these generalized scalar products, and the
power coupling ratio is defined with the overlap integrals of the transverse field distributions:
(B.39)
To go back to the analogy with the three-dimensional geometrical space, the fundamental mode
is equivalent to an eigenvector ai, and the input wave is equivalent to a vector U. Coupling light
into the fiber is equivalent to projecting U on the axis of the eigenvector ai to get a projected
vector:
(B.40)
The coupled power is equivalent to the square of the length of Up, and the power coupling ratio
is equivalent to the square of the cosine of the angle θ between U and ai:
(B.41)
When U is perpendicular or orthogonal to ai, the coupling ratio C is zero.
Now, with the approximation of Gaussian modes, the power coupling ratio between a focused
Gaussian beam with a waist w0 equal to one of the modes or between two identical fundamental
modes may be calculated using the integral
(B.42)
Results are often given in decibels with an attenuation Γ defined by:
Γ = −10log[C] (in dB)
Strictly speaking, the attenuation is in decibels the proportion of light that is coupled, while the
loss is the complementary part which is uncoupled (an attenuation of −1 dB is 80%, and
corresponds to a loss of 20%) but very often the term loss is used for attenuation. There are
several kinds of misalignment that yield coupling attenuation or loss (Figure B.17):
Transverse misalignment d⊥ in the transverse (x, y) plane:
(B.43)
Longitudinal shift d// in the z propagation direction:
(B.44)
Angular misalignment θm:
(B.45)
It is important to notice that this depends on the mode diameter (MFD = 2w0) and not on the
core diameter.
Figure B.17 Misalignment-induced loss: (a) transverse; (b) longitudinal; and (c) angular.
The transverse and angular misalignments yield the same loss law with the ratio to the mode
radius w0 at 1/e2 and the ratio to the half-divergence angle θD/2 at 1/e2, respectively, because the
problem of angular misalignment can be considered a problem of transverse misalignment
between the virtual Gaussian far fields.
There is a −0.5 dB (90%) attenuation (i.e., a 10% loss) for:
d⊥ = MFD/6;
θm = θD/6;
d// = 2 MFD/3θD.
With a typical high-NA coil fiber (NA = 0.17) at 1,550 nm, which yields MFD = 2w0 = 0.84
λ/NA = 8 μm, θD = 1.5 NA = 0.25 rad, and then:
d⊥ = 1.3 μm;
θm = 0.04 rad = 2.5°;
d// = 20 μm.
These values are obtained with a dry connection (i.e., with a fiber interface in air), and the 4%
(−0.2 dB attenuation) Fresnel reflection loss must be added.
With an index-matched connection, there is no Fresnel reflection and the free-space divergence
θD of the mode has to be replaced in the formulae by the reduced divergence θD′ = θD/n in an
index-matched medium where the actual wavelength is reduced to λm = λ/n. This does not change
the transverse effect, but this same −0.5 dB attenuation is now obtained for an angular
misalignment θm = 2.5°/1.45 = 1.6°, or for a longitudinal shift d// = 1.45 × 20 μm = 30 μm. As
can be seen from this numerical example, the mechanical tolerances of the transverse alignment
are very difficult, but the longitudinal and angular alignments are less demanding in comparison.
B.4.3 Mode-Diameter Mismatch Loss of LP01 Mode
Another source of coupling loss is a mode diameter mismatch between two different fibers, or a
fiber and an integrated optic waveguide, or a fiber and a focused Gaussian beam. The result is
also deduced from an overlap integral which can be easily calculated with Gaussian modes. A
mismatch between two mode diameters 2w0 and 2w0′ at 1/e2 yields an attenuation:
(B.46)
and
(B.47)
An attenuation of −0.5 dB (i.e., a loss of 10%) is induced by a diameter ratio w0/w0′ = 1.4,
which shows that the diameter tolerance is not very critical for single-mode fibers.
In the case of elliptical Gaussian modes, encountered with elliptical core fibers but also
integrated-optic waveguide, the mode amplitude may be written as:
(B.48)
where w0x and w0y are the half-widths at 1/e2 along the minor and major axes instead of the
radius w0, and the attenuation due to a width mismatch is:
and
(B.49)
Note that when an elliptical mode has to be coupled to a circular mode, the lowest loss is
obtained when the radius w0 of the circular mode is equal to the geometrical mean value of the
half-widths of the elliptical mode:
(B.50)
For example, with an ellipticity as high as w0x′ = 4w0y′, then w0 is optimal when:
w0x′/2 = w0 = 2w0y′
and the loss is only 2 dB. With a ratio w0x′/w0y′ = 2, the optimal loss is only 0.5 dB, with:
This result for elliptical modes is useful for certain fibers that may have an elliptical core, but
also for coupling to integrated-optic circuit or semiconductor diodes, which usually have an
elliptical emission pattern.
This entire analysis of coupling to a single-mode fiber is assuming an input wave that is
spatially coherent; that is, the phases at all the points in a transverse plane are equal or at least
correlated. Spatially incoherent sources cannot be efficiently coupled in a single-mode fiber.
B.4.4 Mode Size Mismatch Loss of LP11 and LP21 Modes
This analysis of the coupling of the fundamental pseudo-Gaussian LP01 mode can be extended to
the case of the second-order pseudo-Gaussian-derivative LP11 mode. We have seen that an
important advantage of the Gaussian function is that it is invariant under Fourier transform (e–
πx2 and e–πσ2 form a pair of Fourier transforms), but the Gaussian derivative has similar
properties.
As a matter of fact, using the derivative theorem of the Fourier transform, it is found that the
Fourier transform of the derivative
This means that
when the LP11 mode is diffracted at the output of a fiber, it forms a free-space diverging beam
which keeps the same antisymmetric Gaussian-derivative profile (Figure B.18). Note that the
invariance is not complete, because the additional i term indicates that the Fourier transform of a
real odd function is a purely imaginary odd function. Optically, this yields an additional π/2
phase shift, called the Guoy effect, on the output wave as it is diffracted in free space.
Figure B.18 Far field of the LP 11 mode.
The pseudo-Gaussian-derivative beam that diverges in free space follows a law similar to the
Gaussian beam with a half-width at the maximum w′(z):
(B.51)
where w1 is the half-width at the maximum of the mode, and θD1 is the full divergence angle
between both extremes:
Figure B.19 Overlap with mode size mismatch: (a) symmetrical Gaussian axis and (b) antisymmetrical Gaussian-derivative
axis.
(B.52)
Finally, it is also possible to compute the coupling loss of the LP11 mode between two fibers. In
particular, a width mismatch yields an intensity coupling ratio:
(B.53)
Note that this formula is similar to those of the fundamental mode, but is at the fourth power
instead of the second.
This fourth power can be decomposed with a first power in the symmetrical Gaussian axis and
with a third power in the unsymmetrical Gaussian derivative axis.
For the four-lobe antisymmetrical LP21 mode that has an unsymmetrical Gaussian derivative
distribution of the mode in both orthogonal axes, it yields twice the third power, that is, the sixth
power:
(B.54)
B.5 Birefringence in a Single-Mode Fiber
B.5.1 Shape-Induced Linear Birefringence
So far, the problem of polarization has been obviated in the analysis by considering that the
polarization modes are degenerated in a perfect fiber. However, practical fibers have a residual
birefringence which modifies the state of polarization as the wave propagates. There are two
sources of birefringence of single-mode fibers:
Shape birefringence induced by a noncircular core;
Stress birefringence induced by an anisotropic stress through the elasto-optic effect.
The propagation constant difference ∆β induced by birefringence is defined by:
∆β = β2 − β1 (B.55)
where β2 and β1 are the propagation constants of the two eigen orthogonal polarization modes. It
may also be expressed with a birefringence index difference ∆nb between the equivalent indices
neq2 and neq1:
∆nb = neq2 − neq1 (B.56)
with
(B.57)
or a normalized birefringence B
where β is the mean value between β2 and β1 and neq is the mean equivalent index value between
neq2 and neqi.
The calculation of shape birefringence is difficult, but in the case of a small ellipticity of the
core, there is a linear birefringence which can be approximated by:
(B.58)
where f(V) is a term that depends on the normalized frequency V and that is equal to about 0.2 in
the practical range of use (1 < λ/λc < 1.5), and where
is the ellipticity of a
core that has a half-width ay along the major axis and ax along the minor axis (Figure B.20). Note
that the shape birefringence is proportional to the square of the normalized index step D of the
core, and that the fast principal axis is parallel to the minor axis and the slow principal axis is
parallel to the major axis.
Figure B.20 Elliptical core fiber.
B.5.2 Stress-Induced Linear and Circular Birefringence
Birefringence may also be due to an anisotropic normal stress Tn (Tn is positive for tensile stress
and negative for compressive stress), which destroys the isotropy of amorphous silica. Such
stress induces index variations and a uniaxial linear birefringence ∆nb with an extraordinary
index variation δne for the axis parallel to the stress and an ordinary index variation δno for the
two other orthogonal axes. Ignoring the effect of the dopant, their index variations are usually
expressed as:
(B.59)
(B.60)
(B.61)
where n = 1.45 is the index, E = 70 GPa is the Young modulus, p11 = 0.121 and p12 = 0.270 are
the elasto-optic coefficients, and v = 0.16 is the Poisson ratio for silica.
It yields
The n3/2 term, Poisson ratio v and Young modulus E in these formulae come from the fact that
the elasto-optic coefficients relate the change of dielectric impermeability ηi (the inverse of
dielectric permittivity εri) to the strain Si, with:
∆ηi = p11 Si (B.62)
for a parallel strain Si, and:
∆ηi = p12 Sj (j ≠ 1) (B.63)
for a perpendicular strain Sj. And there are:
(B.64)
and
(B.65)
(B.66)
A tensile (positive) stress decreases both extraordinary and ordinary indexes, and conversely a
compressive stress produces an increase, since index is related to the density of matter which
decreases under tension and increases under compression. It is interesting to notice that there is
mainly a change of the ordinary index which corresponds to a polarization perpendicular to the
stress direction. The change of the extraordinary index, parallel to the stress direction, is six
times smaller.
Figure B.21 Stress induced by bending.
When a fiber is bent, a transverse compressive stress Tnc is yielded parallel to the x-axis in the
plane of curvature, and is located in the center of the fiber where the light is guided (Figure B.21):
(B.67)
where Rclad is the radius of the fiber cladding and R is the radius of curvature.
This compressive stress creates a negative linear birefringence with an index difference:
(B.68)
The fast extraordinary axis is in the plane of curvature (Figure B.22).
Shear stress induced by twisting the fiber also creates birefringent effects. The analysis is more
complicated and requires consideration of the longitudinal field component of the fundamental
HE11 mode, which is usually ignored in the transverse LP mode approximation. It may be shown
that a twist rate tw (in rad/m) creates a circular birefringence:
∆βc = 0.14 tw (B.69)
With circular birefringence (see Section A.13.2), the eigen polarization modes are both circular
states of polarization, and a linear polarization is dragged by the fiber twist. The angle θp of
rotation of the linear polarization is proportional to the integrated twist angle θtw of the length L
of fiber:
Figure B.22 Bending-induced birefringence.
θp = ∆βc ⋅ L/2 = 0.07 θtw (B.70)
Note that the twist-induced circular birefringence ∆βc and the angle of rotation θp are
wavelength-independent to first order, while for bend-induced linear birefringence, it is the
birefringence index difference ∆nb, which is wavelength-independent.
B.5.3 Combination of Linear and Circular Birefringence Effects
When several birefringence effects are combined, the analysis of the resulting effect may be
tedious, but the use of a geometrical representation on the Poincaré sphere eases understanding.
On a Poincaré sphere the two circular states of polarization are placed respectively at each pole,
and the linear states of polarization are placed on the equatorial line (Figure B.23). The latitude
gives the degree of ellipticity of the state of polarization, and the longitude gives between 0° and
360°, the double of the angle of the direction of the major axis of the ellipse with respect to the
frame of reference. Two orthogonal states of polarization are opposite on the sphere. The change
of state of polarization due to birefringence is given by a rotation around a diameter of the
sphere. Circular birefringence is represented by a rotation around the polar diameter, and linear
birefringence by a rotation around an equatorial diameter corresponding to the position of the
principal axes. A quarter-wave plate yields a rotation of 90° and a half-wave plate yields a
rotation of 180°.
Figure B.23 Poincaré sphere: (a) top view and (b) side view.
The Poincaré sphere is used in bulk optics to explain the total effect of several birefringent
plates. However, its usefulness is limited because it is well known that the result of the
combination of several successive rotations around nonparallel axes is not straightforward. In
single-mode fiber optics, it is a much more powerful tool because it may explain simply the
effect of several sources of birefringence combined locally in the fiber. Geometrically, this does
not correspond to the combination of successive rotations anymore, but to the combination of
simultaneous rotations that can be simply added vectorially. In particular, the combination of
linear and circular birefringence is represented by an elliptical birefringence vector ∆βe, which is
the vectorial sum of the linear birefringence vector ∆βl and the circular birefringence vector ∆βc
(Figure B.24):
Figure B.24 Elliptical birefringence resulting from the vectorial combination of linear and circular birefringences.
∆βe = ∆βl + ∆βc (B.71)
and its magnitude is:
(B.72)
B.6 Polarization-Maintaining (PM) Fibers
B.6.1 Principle of Conservation of Polarization
As we have seen, a standard single-mode (SM) fiber has a residual birefringence due to shape
effects or spurious stresses. This modifies the state of polarization as the wave propagates, and
the output state is not stable over a long period of time. For an interferometric application like the
fiber gyro, it is very desirable to use a polarization-maintaining (PM) fiber, since interferences
require two waves in the same state of polarization to get high contrast (the terms “polarizationpreserving” or “polarization-holding” fibers were also used, but today it is clearly a PM fiber).
Such a conservation of polarization is obtained by creating a strong birefringence in the fiber.
When light is coupled in one eigenstate, it will remain in this state. For example, with a highlinear-birefringence fiber, light has to be coupled with a linear polarization that is parallel to one
of the two perpendicular principal axes of birefringence.
Polarization conservation could also be obtained with a strong circular birefringence, which
will then preserve a circular state of polarization. This technique would have the advantage of
eliminating the problem of alignment of principal axes, which is encountered with linear
birefringence. However, in practice this is very difficult to implement, and most PM fibers use
linear birefringence.
The phenomenon of polarization conservation in high-birefringence fiber is explained by the
effect of phase mismatch: when light is coupled in one eigen mode of polarization, a first defect
will couple some light in the crossed mode. However, the primary wave and the coupled wave
travel at different velocities because of birefringence, and the light that will be coupled by the
next defect will not be in phase with the coupled wave coming from the first defect. They do not
interfere constructively, and this limits the amount of power transferred in the crossed mode.
Practical polarization-maintaining fibers have a strong intrinsic linear birefringence. The first
possible solution is the use of a very elliptical core. However, getting a significant birefringence
requires a large index step of the core, since B is proportional to ∆2. This has two drawbacks: it
requires (1) a high level of dopant, which increases the loss, and (2) a very small core to remain
in the single spatial mode regime, which makes the mechanical tolerances of input coupling more
severe.
The second possible method, which is now widely generalized, is to use a stress-induced
birefringence with additional materials that have a thermal expansion coefficient larger than
silica (several 10–6/°C instead of 5 × 10–7/°C for pure silica). The fiber preform is fabricated
with two rods of highly doped silica (usually with boron, phosphorous, or aluminum) located on
each side of the core region. After pulling the fiber at high temperature, these highly doped rods
will tend to contract on cooling, but their thermal contraction is blocked by the surrounding
silica, which has a much lower thermal contraction. This puts the rods under tensile stress and by
reaction also induces stress in the core region where light propagates: there is a tensile stress Tn+
in the axis of the rods and a compressive stress Tn– in the orthogonal axis (Figure B.25).
Figure B.25 Stress-induced PM fiber showing the tensile and compressive stresses in the core region.
The rods are circular in the case of the panda fiber but the stressing region may have another
shape, particularly with bow-tie fiber or with an elliptical tiger-eye stressing structure (Figure
B.26).
Figure B.26 Stress structure of panda, bow-tie, and elliptical cladding (tiger-eye) PM fibers.
The birefringences induced by each stress are added and the total birefringence index
difference ∆nb is derived from:
(B.73)
(B.74)
and
(B.75)
(B.76)
The fast lower index axis is aligned along yy′, the axis perpendicular to the stressing structure,
and the slow higher index axis is aligned along xx′, the parallel axis. Such a fabrication technique
yields typical ⎪∆nb⎪ values of 3 to 7 × 10–4. The birefringence of the fiber is very often
expressed in terms of beat length Λ, which has already been defined as Λ = λ/∆nb. At λ = 1,550
nm, practical values of Λ are 2.5 to 3.5 mm. Such a value of the birefringence implies a very high
stress in the core region, since the Young modulus E = 7 GPa for silica:
(B.77)
It means strain of few tenths of percent.
Note that the birefringence index difference does not depend on the wavelength to first order,
and therefore the beat length is proportional to the wavelength for a given fiber.
B.6.2 Residual Polarization Crossed-Coupling
In practice, polarization-maintaining fibers are not perfect and yield a residual coupling in the
crossed-polarization mode. Before going through a detailed mathematical calculation, this
crossed-polarization coupling grows linearly with length. The ratio between the power P⊥
coupled in the crossed polarization mode and the input power P//in in the parallel polarization
mode of the PM fiber is given by the h-parameter, h standing for holding. This ratio is often
called polarization extinction ratio (PER) with:
This h-parameter is typically between 10–5 to 10–4 m–1, expressed also as a PER of −20 to −10
dB over 1 km.
Now, mathematically there are random coupling points along the fiber which can be described
with a stochastic process c(z): at a position z, there is a coupling of the amplitude Ap(z) of the
monochromatic primary wave in the crossed-polarization state, which yields a crossed amplitude
dAc(z):
dAc(z) = c(z) ⋅ Ap(z) ⋅dz (B.78)
Along a length L, the total crossed amplitude Ac(L) is the result of the sum integral of all the
dAc terms (Figure B.27). Taking into account the phase delay due to the propagation constant βp
of the primary mode and the propagation constant βc of the crossed mode:
Figure B.27 Distributed random coupling in the crossed-polarization mode.
Ap(z) = Ap(0)e–iβpz (B.79)
and the crossed-amplitude term dAc(z) yields at L:
dAcL(z) = dAc(z)e–iβc(L–z) (B.80)
Then
(B.81)
(B.82)
where ∆β = βp − βc is the difference of propagation constants due to the birefringence of the
fiber, with
Since c(z) is a stochastic process, Ac(L) is one also, which is the result of a stochastic integral.
Its statistical properties may be calculated simply, when c(z) is a stationary process (i.e., the
characteristics of the spurious random couplings are uniform along the fiber). It is possible to
define the autocorrelation Rc(z) of the stationary process c(z):
Rc(Z) = E{c(z) ⋅ c*(z − Z)} (B.83)
where E{} denotes the ensemble average. The power spectrum (or spectral density) is the Fourier
transform of the autocorrelation:
(B.84)
where σ is the spatial frequency. We have:
(B.85)
and
(B.86)
because of the fundamental result of linear transformation, which states that “the ensemble
average of the linear transformation L[x] of a stochastic process x is equal to the linear
transformation of the ensemble average of this stochastic process x”:
E{L[x]} = L[E{x}] (B.87)
Since AcAc* is the intensity Ic of the crossed-coupled wave, and ApAp* is the intensity Ip of the
input primary wave, the total intensity coupling ratio C has an ensemble average:
(B.88)
Applying the change of variable Z = z1 − z2 gives us:
(B.89)
Assuming that the length L of the fiber is much longer than the width of the autocorrelation
Rc(Z) of the coupling process, we have:
(B.90)
and since ∆β/2π = 1/Λ, where Λ is the birefringence beat length:
E{C} = L ⋅ Sc(1/Λ) (B.91)
The mean intensity coupling ratio E{C} in the crossed state of polarization is proportional to
the length L of the fiber and to the value of the power spectrum Sc of the stationary stochastic
coupling process c(z) for a spatial frequency equal to the inverse of the birefringence beat length
Λ. This term Sc(1/Λ) is often called the h parameter of the fiber (h standing for holding), with:
E{C} = h ⋅ L (B.92)
With stress-induced high-birefringence fibers, the typical values of the h parameter are 10–5 to
10–4 m–1, which corresponds to a polarization conservation of 20 to 10 dB over 1 km.
This mathematical calculation confirms the explanation of polarization conservation based on
phase mismatch. The random coupling c(z) has a power spectrum Sc(σ), and its spatial frequency
components do not usually yield crossed coupling, except for the frequency 1/Λ, which is the
only one to be phase-matched, because it is spatially synchronized with the propagation delay
induced by birefringence.
To carry out this calculation simply, we have assumed that the intensity of the coupled wave
remains very small compared to the intensity of the primary wave, and, therefore, that the light
which is recoupled in the primary mode may be ignored. When this is not the case, the mean
value of C tends to 1/2 because an equilibrium is reached with the same mean intensity in both
orthogonal polarizations, and (Figure B.28):
Figure B.28 Power crossed-coupling E{C} along a polarization-maintaining fiber.
(B.93)
The previous calculations were derived using ensemble averages on observations carried out
with a monochromatic source. It is well known with temporal stochastic processes that the
statistics of a process may be determined with temporal averaging on a single observation if the
process is ergodic:
E{x(t)} = < x(t) > (B.94)
and
R(τ) = E{x(t)x*(t − τ)} = Γ(τ) = < x(t)x*(t − τ) > (B.95)
where the brackets < > denote temporal averaging.
B.6.3 Depolarization of Crossed-Coupling with a Broadband Source
With the stochastic coupling process c(z), it is possible to define an equivalent to ergodicity with a
broad light spectrum that yields, on a single observation, a stable intensity coupling ratio Ct equal
to the ensemble average E{C}. The broad spectrum source has to be linearly polarized and
coupled along one principal axis of the high-birefringence fiber. At the output, most of the input
intensity is still in the input eigenstate of polarization, and there is some spurious intensity in the
crossed polarization that is not coherent with the primary wave. We have seen (Section A.12.3)
that the effect of a broad source in an interferometer may be explained by considering the
propagation of a wave train, which has a length equal to the decoherence length Ldc of the source.
When the path difference is larger than Ldc, there are two wave trains at the output of the
interferometer that do not overlap and cannot interfere.
Similarly, an input wave train coupled on both polarization modes of a high-birefringence
fiber propagates at different velocities on each mode, and there are at the output two wave trains
which do not overlap if their path difference is larger than Ldc (Figure B.29). The light then
becomes statistically depolarized; that is, the state of polarization varies randomly in time,
because the phases of both orthogonal mode are not correlated anymore. The length of fiber
required to get such a depolarization is called the depolarization length Ld. Since the optical path
of a high-birefringence fiber accumulated by one mode is n ⋅ Ld and the one accumulated by the
other is (n + ∆nb)Ld, the path difference must be:
(n + ∆nb)Ld − nLd = Ldc (B.96)
then
Ld = Ldc/∆nb (B.97)
or
Ld/Ldc = Λ/λ (B.98)
Taking a source with λ = 1,550 nm, and ∆λFWHM = 15 nm, which has a decoherence length
Ldc = λ2/∆λFWHM = 150 μm, a 5 × 10–4 birefringence index difference yields a depolarization
length Ld = 30 cm.
Figure B.29 Propagation of a wave train along both orthogonal polarization modes.
A fiber may be decomposed in segments with a length equal to Ld. When an input wave train is
coupled in one eigen polarization mode, it is possible to consider that a secondary wave train is
coupled in the crossed mode for each segment (Figure B.30). These secondary wave trains are not
coherent, and are therefore added in intensity. The total intensity coupling Ct is the sum of all the
random intensity couplings Ci along each segment:
Ct = ΣCi (B.99)
Figure B.30 Secondary wave trains coupled in the crossed-polarization mode (assuming it is the slow mode).
This sum is actually proportional to the ensemble average E{Ci}:
(B.100)
where N = L/Ld is the number of depolarization lengths Ld along the fiber length L. Then
(B.101)
and since E{Ci} = h ⋅ Ld, we find:
Ct = h ⋅ L = E{C} (B.102)
It is important to remember that the measurement of the intensity crossed-coupling ratio yields
a significant result only if the experiment is carried out with a broad spectrum, because with a
single experiment this actually yields an ensemble average of the couplings on all the
depolarization lengths along the fiber. With a monochromatic light, the result is a random
variable, and a single observation is not sufficient to deduce the statistical properties of the
process.
Note that the simple formula Ld/Ldc = 1/(∆nb) = Λ/λ that defined the depolarization length Ld is
valid only if the birefringence index difference ∆nb is wavelength-independent, which is the case
to first order with stress-induced high-birefringence fibers, which are the most commonly used
for getting polarization conservation.
With elliptical core fiber, the birefringence index difference has a significant wavelength
dependence, particularly with high core ellipticity, and this simple formula is not valid anymore.
The exact definition of the depolarization length Ld is the length of fiber required to have a
difference of group, not phase propagation time between both polarization modes equal to the
decoherence time τdc. For modes 1 and 2:
(B.103)
Since vg = dω/dβ and 1/vg = dβ/dω (see Appendix A),
(B.104)
with the birefringence ∆β = ω∆nb/c. If the birefringence index difference ∆nb is, as usual,
wavelength- (or frequency-) independent, then:
(B.105)
and the simple formula Ld/Ldc = 1/(∆nb) is retrieved. However, if it is wavelength-dependent,
then:
(B.106)
Note that the case of twist-induced circular birefringence yields a very peculiar result. In fact,
we have seen in Section B.5.2 that polarization is dragged angularly by the twist at a constant rate,
independently of the wavelength; that is, the birefringence ∆β is independent of wavelength (or
frequency). This implies d(∆β)/dω = 0; that is, mathematically, Ld tends to infinity. In practice,
because of second-order effects, the difference of group velocity between the two eigen circular
polarizations is not zero, but is much smaller than the difference of phase velocity. In this peculiar
case, the phenomenon of depolarization induced by the propagation of a broadband source in a
high-birefringence fiber is greatly reduced, since the depolarization length is much longer than
Ldc/∆nb.
B.6.4 Polarization Mode Dispersion (PMD)
In single-mode telecom fibers, spurious fiber birefringence was first seen as yielding unstable
state of polarization and requiring polarization-independent components with low polarizationdependent loss (PDL) and low polarization-dependent coupling (PDC).
However, as the data rate increased, the propagation delay due to birefringence started to yield
problems. It was called polarization mode dispersion, PMD, and more precisely coupled-mode
PMD (PMDc). It is a random process that grows as the square root of the propagation length. A
very low PMDc telecom fiber is below
The linear birefringence of a PM fiber yields also PMD. PMD is detrimental in telecom because
it broadens the input signal pulse, but it is useful in the fiber gyro because it separates crossedpolarization wave trains, equivalent to pulses, yielding depolarization.
However the linear birefringence of a PM fiber is not a random process, and its PMD effect is
called intrinsic PMD, PMDi. It grows linearly with length. Since the propagation time delay due
the birefringence index ∆nb is ∆nb ⋅ L/c, there is:
with a typical ∆nb = 5 × 10–4, it yields a PMDi = 1.7 ns/km.
Note that when a PM fiber is longer than 1/h (h being its holding parameter) we saw (Figure
B.23) that there is a statistic equilibrium with half the power in each polarization. In terms of
PMD, the PM fiber reaches the regime of coupled-mode PMD (PMDc) that grows as the squareroot of length.
However, since h = 10–4 to 10–5 m–1, 1/h is 10 to 100 km, which is longer than the length of
PM fiber used even in very-high-performance gyros.
B.6.5 Polarizing (PZ) Fiber
As we saw in Section B.8.1, a polarization-maintaining (PM) fiber has a strong linear
birefringence created by a stressing structure. The tensile stress Tn+ aligned along the structure
decreases mostly the index of its perpendicular polarization with little effect on its parallel
polarization. Conversely the compressive stress Tn–, orthogonal to the structure, increases
mostly the index of its perpendicular polarization with little effect on its parallel polarization.
However, this stressed region is localized around the core. In the axis orthogonal to the stress
structure, the index profiles for both polarizations converge to the index of the unstressed
cladding nclad (Figure B.31).
Figure B.31 Stress and index profiles of a PM fiber for the slow mode (dotted line) and the fast mode (solid line).
Operated far above the higher-mode cut-off wavelength (λ > 1.5λc), the mode profile widens
and its equivalent index decreases as seen in Section B.3.1. The equivalent index neq-slow of the
slow (high-index) polarization mode remains above nclad, and then guided, but the equivalent
index neq-fast of the fast (low-index) polarization mode gets below nclad which induces leakage,
and yields some polarization dependent loss (PDL) of the PM fiber.
Such a behavior can be optimized with specific index profile to get single-polarization
propagation for the slow mode, yielding a polarizing (PZ) fiber. However, as λ continues to
increase (λ > 1.7λc) the mode becomes so wide that bending loss increases even for this slow
mode.
It is possible to get a window of about 10% of the central wavelength where the PZ fiber can be
used (Figure B.32).
Figure B.32 Fast and slow mode attenuation of a PZ fiber.
B.7 All-Fiber Components
B.7.1 Evanescent-Field Coupler and Wavelength Multiplexer
When the phenomenon of interferences in bulk optics has been explained, we have made the
assumption of plane waves propagating in bulk interferometers that are mechanically aligned;
that is, the orientation of the mirrors and beam splitters is adequate for getting the same phase
difference for the whole light wavefront. In practice, these alignments are very delicate and there
may be interference fringes or rings. The difficult problems of fringe shape and localization, as
well as that of spatial coherence, have been obviated in the explanation, since single-mode fiber
optics, with which we are concerned, makes things much simpler: by principle, the fundamental
mode of a single-mode fiber is spatially coherent, and with an all-guided interferometer, there is
no spatial fringe modulation. Single-mode fiber optics is an ideal technology for making longpath interferometers because problems of mechanical alignments are limited to those of input
power coupling. System-wise, the effect of the transverse dimensions may be forgotten, and only
a device with a single curvilinear longitudinal coordinate has to be considered.
As we have already seen, the use of polarization-maintaining fibers is suitable for avoiding
problems of polarization control, but to take full advantage of this technology, it is desirable to
duplicate the various components of a bulk interferometer in a rugged all-guided form.
The main function in an interferometer is beam splitting to separate the input wave and
recombine the interfering waves. Such a function may be realized in an all-fiber form with
evanescent-wave coupling. Light may be coupled between two adjacent cores because the
evanescent tail of the bell-shaped fundamental mode extends into the cladding and can excite the
mode of the other fiber (Figure B.33).
Figure B.33 Principle of evanescent-wave coupling.
With two identical cores, the coupled power Pc follows a sine square law with respect to the
interaction length Lint, and the transmitted power Pt remains complementary:
Pc = Pin sin2(cs ⋅ Lint)
Pt = Pin cos2(cs ⋅ Lint) (B.107)
where cs is the coupling strength. The power is completely transferred into the second fiber for
the coupling length Lcp, with
cs ⋅ Lcp = π/2 (B.108)
and then starts to come back in the first fiber. To get a 50-50 or 3-dB coupler, the coupling
strength cs has to be adjusted to have the interaction length Lint equal to the half-coupling length:
(B.109)
The coupling strength is adjusted with the core-to-core distance dcc. A good order of
magnitude is dcc ≈ 5a (a being the core radius) and Lint ≈ 1 mm (i.e., a few thousands of
wavelengths in the fiber). Note that when the wavelength increases, the mode extends further into
the cladding, which improves the coupling strength. A good order of magnitude is a relative
change ∆cs/cs equal to three or four times the relative wavelength change ∆λ/λ. This wavelength
dependence is used to make all-fiber wavelength multiplexer-demultiplexer.
Figure B.34 Side-polished coupler: (a) half-coupler block; (b) grinding and side-polishing; and (c) coupler assembly.
Such evanescent-field fiber couplers were first fabricated with the side-polishing technique,
where the fiber is bonded in a curved groove sawed into a supporting silica block. The ensemble
is ground and polished laterally to remove the cladding and get access to the evanescent field.
Two identical blocks are then mated with an index-matching bonding to get power splitting
(Figure B.34). Side-polished couplers can be made very small (10 mm long) and have an
excellent ruggedness and a good thermal stability. Almost any polarization-preserving fibers may
be used, but this requires orienting the stressing-rod axis perpendicularly to the interface to
minimize crossed-polarization coupling (Figure B.35).
Figure B.35 Section view of a side-polished coupler with stress-induced birefringence fiber.
An alternative technique of fusion tapering has also been developed. Instead of removing the
cladding, two fibers are tapered by fusion and stretching, which reduces the distance between both
cores and also increases the diameter of the mode, which becomes loosely guided as the core
diameter decreases (Figure B.36). This technique is very advantageous with telecom single-mode
fibers, because the fabrication process may be automated and fusion provides an excellent
thermal stability. However, with polarization-maintaining fibers, this technique requires a specific
fiber structure to avoid loss induced by the highly doped stressing rods but this problem has been
overcome. The length of a fused coupler is usually larger (20 to 40 mm) than that of a sidepolished coupler, but fused-coupler technology is clearly the dominant approach today for
couplers and wavelength multiplexer.
Figure B.36 Fused-tapered coupler.
B.7.2 Piezoelectric Phase Modulator
Another important function to fulfill is phase modulation in order to use signal processing
techniques that improve signal-to-noise ratio. This can be simply implemented with an optical
fiber wound around a piezoelectric tube (Figure B.37). The driving voltage changes the tube
diameter and thus modifies the length L of the fiber. Ignoring mode dispersion effects, the phase
change is:
(B.110)
with
(B.111)
where, as we have already seen, p12 and p11 are the elasto-optic coefficients and v is the Poisson
ratio of silica. We have:
(B.112)
in relative value one gets:
(B.113)
and since nSiO2:
(B.114)
This 0.78 coefficient is also find in the variation of the relative wavelength change of a fiber
Bragg grating under a longitudinal strain ∆L/L.
Figure B.37 In-line phase modulator using fiber wound around a piezoelectric tube.
Such an all-fiber phase modulator is very simple to fabricate, but its efficiency is limited to the
sharp mechanical resonances of the piezoelectric tube. Three kinds of resonance may be used:
The very efficient loop resonance, which has a typical product frequency × diameter of about 50
kHz ⋅ cm (i.e., useful for a few tens of kilohertz);
The height resonance, which has a medium efficiency and a typical product frequency × height
of about 150 kHz ⋅ cm (i.e., useful for a few hundred kilohertz);
The thickness resonance, which has a low efficiency, but may work at a few megahertz, since the
typical product frequency × thickness is about 2 MHz × mm.
In a fiber gyro, piezo-electric phase modulators are then used for sine biasing phase
modulation (i.e., the open-loop configuration), but they do not work for wideband feedback
modulation as the phase ramp for the closed-loop configuration.
B.7.3 Polarization Controller
If PM fibers preserve the state of polarization, it is not the case with SM fibers which then require
polarization control. As seen in Section A.13.2, it can be performed with bulk-optic birefringent
plates, in particular half-wave (Λ/2) and quarter-wave (Λ/4) plates but it can be also done directly
in the SM fiber, by applying stress. For active control, they are devices that squeeze a fiber
inducing birefringence because of transverse stress.
It is also possible to use bending-induced birefringence detailed in Section B.5.2:
(B.115)
where Dclad is the diameter of the SM fiber cladding and D is the diameter of curvature.
For telecom SM fiber with Dclad = 125 μm a quarter-wave (Λ/4) delay is obtained at 1,550 nm
with one loop of 16.5 mm of diameter or two loops of 33 mm of diameter, and polarization is
adjusted by rotating these all-fiber phase plates or actually phase loops.
The general formula to obtain a delay of λ/m with N loops of diameter D is:
(B.116)
A polarization controller (Figure B.38) can be composed of a series of Λ/4, Λ/2, and Λ/4 loops
or only two Λ/4 loops.
Figure B.38 All fiber polarization controller (sometimes called Lefevre’s loops).
Sometimes called Lefevre’s loops, the polarization controllers were invented for the fiber gyro
in the early 1980s but have since found many applications in laboratory experiments, particularly
in telecoms.
B.7.4 Lyot Depolarizer
We already saw in Section B.6.3 that propagation in a PM fiber yields decoherence between the
orthogonal states because of birefringence produces a delay longer than the coherence of the
source entering light with a broad spectrum at 45° of the principal axes produces depolarized
light; however, if the input light is not at 45°, the power is not balanced for both orthogonal axes
and light is only partially depolarized.
Figure B.39 Front page of Bell Labs News journal, presenting the first 1 terabit/s transmission experiment in 1998. Note the
numerous all-fiber polarization controllers in the picture.
This problem is solved in bulk-optics with a Lyot depolarizer composed of a first birefringent
crystal followed by a second one with principal axes rotated by 45° and the double in length. This
can be duplicated in an all-fiber form with two pieces of PM fiber connected at 45° (Figure B.40).
B.7.5 Fiber Bragg Grating (FBG)
As we already saw in Section A.9.3, a fiber Bragg grating is a multiple-wave interferometer
working on a principle similar to the one of a multilayer dielectric mirror. A UV lateral grid
illumination of the core of the fiber induces a permanent periodic index increase that yields a
resonance effect and reflects the wavelength λR (see Appendix A):
λR = 2 ⋅ nSiO2 ⋅ ΛBragg (B.117)
where nSiO2 is the index of silica (≈ 1.45) and ΛBragg is the period of the grid illumination.
As we also saw in Section A.14, the relative width of the reflection bandwidth ∆λR/λR is at best
the inverse of the number N of periods ΛBragg :
(B.118)
Since ΛBragg is about 0.5 μm at 1,550 nm 104 periods (i.e., a length of 5 mm) are required to
get ∆λR = 0.15 nm.
In addition, as seen in Section B.1.5, fiber Bragg gratings follow the temperature dependence of
the index of silica and for a bare fiber:
(B.119)
However, with a coating, there is an additional thermal expansion and the relative variation is
about 10 to 11 ppm/°C instead of 6.35 ppm/°C.
Figure B.40 Principle of an all-fiber Lyot depolarizer with a length L1 of PM fiber connected to a length L2 = 2L1 at 45°.
Finally, as seen in Section B.7.2, a strain ∆L/L yields a relative Bragg reflection wavelength:
(B.120)
B.8 Pigtailed Bulk-Optic Components
B.8.1 General Principle
If fiber-optics makes use of all-fiber components and, as we shall see in Appendix C, integrated-
optic circuits, there are also components using fiber-pigtailed miniaturized bulk-optic bench.
The basic idea is that, as seen in Section A.11, a Gaussian beam keeps a constant diameter 2w0
with a quasi-planar phase front over a distance zR such that λzR/πw02 < 1. It can be approximated
considering that a beam waist 2w0 = Nλ remains about constant over a distance zR = N2λ, where
zR is known as the Rayleigh range (Figure B.41).
In addition, as seen in Section A.7.3, lens aberrations decrease with very short focal length.
Assuming a telecom fiber with NA = 0.12, and then a divergence of the output equal to 1.5NA, a
focal length f = 2 mm yields a collimated beam with a diameter 2w0 = 1.5 NA ⋅ f = 360 μm. At a
wavelength λ = 1.55 μm, the Rayleigh range zR is as high as 66 mm. Optical waves face
diffraction, but it is possible to manage to have a collimated beam with few tenths of millimeters
in diameter over a length of several tens of millimeters, where various bulk components can be
placed.
B.8.2 Optical Isolator
An isolator allows light to propagate in one direction but blocks it in the opposite direction. It is a
key component to protect lasers, amplifiers and ASE sources from spurious back-reflected light
that produces instabilities.
It is based on the nonreciprocal Faraday effect: a longitudinal magnetic field creates a circular
birefringence that rotates the orientation of the incoming polarization (see Section A.13.2).
Figure B.41 Rayleigh range z R of a Gaussian beam.
Light is first linearly polarized, sent in the Faraday rotator cell, which provides a 45° rotation
of the polarization axis, and goes through a second polarizer parallel to the polarization exiting
the cell. In the reverse direction, light is also polarized and rotated by 45° in the Faraday cell, but
since Faraday effect is nonreciprocal, this 45° rotation is in the opposite direction, and it is
exiting at 90° of the axis of the first polarizer that blocks it (Figure B.42).
Figure B.42 Principle of an optical isolator.
The principle of Faraday optical isolation is polarization dependent, but it is possible to get a
polarization-independent design by replacing the polarizers by polarizing splitters combiners:
they divide the input light in two orthogonal states of polarization that go separately through the
Faraday cell to experience isolation and that are recombined at the output.
Because of the development of EDFAs for telecommunications, isolators have become a very
common component in the 1,550-nm window, and they can be found in a compact and ruggedized
pigtailed package.
B.8.3 Optical Circulator
A circulator is actually an isolator where, instead of being blocked by a polarizer, the returning
light is sent into a third port by a polarization beam splitter (PBS) (Figure B.43).
Figure B.43 Principle of an optical circulator.
As for the isolator, a circulator comes in a fiber-pigtailed configuration as well as with a
polarization-independent design. It is also now a quite common component since it is used in
combination with fiber Bragg grating to get the reflected filtered wave without the loss of 6 dB
taking place with a 3-dB coupler (Figure B.44).
B.9 Rare-Earth-Doped Amplifying Fiber
Erbium-doped fiber amplifier (EDFA) was a huge revolution in optical fiber communications
during the 1990s, and it happened to be a quasi-ideal source for the fiber gyro.
Figure B.44 Comparison between the use of a 3-dB coupler and the one of a circulation with a Fiber Bragg grating (FBG)
reflector.
The basic scheme of an EDFA is a pigtailed pump laser diode at 980 nm (or 1,480 nm), which is
multiplexed with the signal in the 1,550-nm range that is amplified in the erbium doped fiber
(EDF) with a gain as high as 30 dB. There is an isolator at the output to avoid lasing on backscattering or back-reflection (Figure B.45).
This amplifying process is polarization-independent and an EDF is usually not preserving
polarization.
Figure B.45 Basic scheme of an erbium-doped fiber amplifier (EDFA).
As already discussed, an EDFA without an input signal emits a significant broadband optical
power through the process of amplified spontaneous emission (ASE). The basic broadband
spontaneous emission is partly guided in the aperture of the fiber and amplified by stimulated
emission, which preserves the broad spectrum of the seed spontaneous emission.
To obtain more power, an ASE source uses a double-path configuration. A mirror is placed at
the output to get spontaneous emission amplification also in the reverse direction. Such a doublepath ASE design requires however a high-rejection isolator to avoid lasing (Figure B.46). Typical
length of EDF for an EDFA or an ASE source is 5 to 20 meters.
If erbium at 1,550 nm is the privileged choice that takes advantage of the lowest attenuation of
silica fiber, note that other rare earths allow amplification and ASE in silica fiber: ytterbium and
neodymium in the 1,050-nm range and thulium in the 2-μm range.
Figure B.46 Basic scheme of an erbium ASE source.
B.10 Microstructured Optical Fiber (MOF)
So far, all the optical fibers we discussed were based on total internal reflection (TIR) between a
high-index core and a low-index cladding.
During the 1990s, appeared a drastically different concept. The vocabulary takes time to settle
down: we saw that PM (polarization-maintaining) fiber were called polarization-preserving or
polarization-holding, and ASE (amplified spontaneous emission) sources were super-radiant or
super-luminescent or super-fluorescent. These microstructured fibers are also called photoniccrystal fiber or photonic bandgap fibers. In any case these MOF are of two very different kinds:
solid core and hollow core.
Solid-core microstructured fibers have a silica core surrounded by air/vacuum channels that
lower the averaged index of the cladding (Figure B.47). They behave like ordinary fibers but with
interesting properties. In particular, the practical single-mode regime is wider in terms of
wavelength than the one of classical fibers. It is possible to have a large core without bending loss
when nonlinearity is a problem or, on the contrary, to have a very small core to increase
nonlinear effects. These fibers have also very specific dispersion properties.
For the fiber gyro, their advantage is not clear. One may outline that PM microstructured fibers
use shape birefringence and they seem to have less spurious twist than ordinary PM fibers, which
reduces the magnetic dependence of a fiber gyro. However, since μ-metal shielding is very
efficient, this advantage is not fundamental.
Figure B.47 Solid-core microstructured fiber.
Hollow-core microstructured fibers are working one a completely different principle. They are
composed of an air core surrounded by a honey-comb of air holes (Figure B.48).
Figure B.48 Hollow-core microstructured fiber (courtesy of Photonics Bretagne.).
Guidance in the low-index core is obtained through a resonance effect of the periodic partial
reflections at the air-glass interfaces. It should be viewed as a 2-D Bragg reflection.
Obviously such a matter-free fiber is of great interest for very-high performance gyroscope
since it drastically reduces matter-related spurious effects as the nonlinear Kerr effect or the
magneto-optic Faraday effect. It has also some interest for the Shupe effect since it suppresses the
thermal dependence of the index which is about 6 ppm/°C in relative value when the thermal
expansion of silica is only 0.5 ppm/°C but to take full advantage of this effect one has to reduce
the thermal expansion drag of the coating which can be on the order of 5 ppm/°C as we already
saw.
In any case, present attenuation of hollow-core fibers is still on the order of several dB/km, but
one can expect further improvement viewing the recent progresses.
B.11 Nonlinear Effects in Optical Fibers
So far, we have been concerned only by linear effects: the incoming frequency does not change
and wave velocity does not depend on power. However, even if the power involved in fiber-optics
is not very high, it is concentrated in a small core and the power density can become significant.
Potential nonlinearities are also integrated along a large length.
These nonlinear effects are very important in optical fiber communications but in a fiber gyro,
the only damaging nonlinear effect is the Kerr effect and we shall see that this problem is solved
with a dc broadband source in the interferometric fiber-gyro.
Of interest for the fiber gyro remain the stimulated Brillouin scattering (SBS) since it allows
distributed measurement of temperature and stress along a fiber, and they are useful to measure
the detailed temperature response of a fiber coil, with the method of Brillouin optical time
domain analysis (BOTDA).
Selected Bibliography
Agrawal, G.P., Nonlinear Fiber Optics, New York: Academic Press, 1995.
Dakin, J. C., and B. Culshaw, (eds.), Optical Fiber Sensor: Principles and Components, Norwood, MA: Artech House, 1988.
Daly, J. C., (ed.), Fiber Optics, Boca Raton, FL: CRC Press, 1984.
Desurvire, E., Erbium-Doped Fiber Amplifiers, New York: Wiley-Interscience, 2002.
Digonnet, M. J. F., “Broadband Rare-Earth Doped Fiber Laser Sources,” Ch. 9, Optical Fiber Rotation Sensing, New York:
Academic Press, 1994.
Erdogan, T., “Fiber Grating Spectra,” Journal of Lightwave Technology, Vol. 15, No. 8, 1997, pp. 1277–1294.
Jeunhomme, L. B., Single-Mode Fiber Optics, New York: Marcel Dekker, 1990.
Kashyap, R., Fiber Bragg Gratings, New York: Academic Press, 1999.
Marcuse, D., Theory of Dielectric Optical Waveguides, New York: Academic Press, 1974.
Miller, C. M., Optical Fiber Splices and Connectors, New York: Marcel Dekker, 1986.
Okoshi, T., Optical Fibers, New York: Academic Press, 1982.
Papoulis, A., Probability, Random Variables and Stochastic Processes, New York: McGraw-Hill, 1965.
Russel, P. S. J., “Photonic Crystal Fibers,” J.L.T., Vol. 24, No. 12, 2006, pp. 4729–4749.
Vassalo, C., Optical Waveguide Concepts, New York: Elsevier, 1991.
APPENDIX C
Fundamentals of Integrated Optics for the
Fibergyroscope
C.1 Principle and Basic Functions of LiNbO3 Integrated
Optics
C.1.1 Channel Waveguide
The concept of integrated optics is based on the use of microlithographic techniques to fabricate
optical components with waveguides on a planar substrate. The term planar lightwave circuit
(PLC) is by the way sometimes used for an integrated-optic circuit (IOC). Like integrated
electronics, it provides potential for integrating several functions on the same circuit which
improves compactness and reduces connections.
The basic element of an integrated-optic circuit is the strip or channel waveguide. It is
fabricated by increasing the index of refraction underneath the surface of a substrate with a
dopant in a narrow channel defined with microlithographic masking techniques. This substrate
acts as the equivalent of the surrounding cladding of an optical fiber (Figure C.1). In particular,
single-mode propagation, which is required in an interferometer, is obtained with a waveguide
width and depth of several micrometers and an index variation of a few tenths of a percent. These
values are very similar to the characteristics of the core of a single-mode fiber.
Figure C.1 Equivalence between (a) an integrated-optic waveguide and (b) an optical fiber.
However, because of the substrate surface, the cylindrical symmetry of a fiber is lost, which
makes the theory more complicated. In particular, the fundamental mode is not degenerated
anymore in terms of polarization. Instead of the hybrid HE11 fundamental mode of a fiber, the
fundamental mode is a transverse electric (TE) mode for the polarization parallel to the substrate
surface, and a transverse magnetic (TM) mode for the polarization perpendicular to the substrate
surface as seen for planar waveguide in Section A.6.2 (see Figure C.2). The longitudinal magnetic
component of the TE mode and the longitudinal electric component of the TM mode are usually
negligible, as is the case of the longitudinal electric and magnetic components of the HE11 mode
of a fiber, because the index step of the waveguide is also small. They should be called linearly
polarized (LP) modes as in a fiber but the habit is still to call them TE and TM.
Figure C.2 States of polarization of the HE11 modes of a fiber and the TE and TM modes of an integrated-optic waveguide.
C.1.2 Coupling Between an Optical Fiber and an Integrated-Optic Waveguide
To couple light into the waveguide, the sides of the substrate are polished with a sharp edge, and
fiber pigtails are butted against it with their core facing the waveguide (Figure C.3). Since the
fundamental modes of the fiber and of the waveguide have similar sizes, there is good coupling
efficiency (typically 80% to 90%, that is, a connection attenuation of 1 to 0.5 dB). Several
methods may be used to ruggedize these connections. In particular, the fiber may be held in a
small ferrule that is directly bonded to the substrate side (Figure C.4), the substrate typically being
1 mm in thickness, while the fiber diameter is on the order of one-tenth of a millimeter. Taking
into account a propagation attenuation of about 0.1 dB/cm, the fiber-to-fiber attenuation of a
pigtailed circuit is typically few to several dB, depending on the complexity of the circuit.
Figure C.3 End-fire coupling of a fiber with a waveguide.
Now LiNbO3 (lithium niobate) has indices of refraction around 2.2 when the one of silica fiber
is 1.45. It yields a 4% partial reflection at the connection, as seen in Section A.6.1. To avoid
guided back-reflection into the waveguide, the end-face is polished with a slant angle, as well as
the fiber. The angle pair is typically 10° in LiNbO3 and 15° for the fiber to respect the refraction
law and keep a good coupling (Figure C.5).
Figure C.4 Ruggedized coupling of a fiber with a waveguide.
C.1.3 Fundamental Mode Profile and Equivalence with an LP11 Fiber Mode
As already seen, an integrated-optic channel waveguide is quite similar to a fiber core, but with a
significant difference: the waveguide is buried just underneath the substrate top surface, yielding a
very high index step in the y direction perpendicular to this interface. Boundary conditions
impose a null field on this interface. It can be also considered that this interface act as a perfect
mirror working in total internal reflection under grazing incidence. As seen in Section A.6.1,
such reflection yields a π phase shift which leads to destructive interference between an incoming
wave and its reflection.
In the x direction parallel to the interface, the mode profile is similar to the one of a singlemode fiber, and follows the usual Gaussian curve seen in Section B.1.2. The waist wx is now the
mode half-width at 1/e in amplitude (or 1/e2 in power density) instead of the mode radius of the
LP01 mode of a fiber.
Figure C.5 Slant-angle faces to avoid backreflection.
In the perpendicular direction, the mode profile is actually very close to the normalized
derivative of a Gaussian function having a maximum for wy, as with the second order LP11 mode
of a fiber (see Section B.3.5) but with only one lobe (Figure C.6).
Figure C.6 Amplitude distribution of the fundamental mode of a channel waveguide: symmetrical Gaussian profile in the x
parallel direction and Gaussian-derivative profile in the y perpendicular direction.
The field (amplitude) profile can be decomposed as:
e(x, y) = e0 ⋅ e//(x)⋅ e⊥(y)
with a normalized Gaussian distribution e//(x) in the parallel direction:
and a normalized Gaussian-derivative distribution in the perpendicular direction:
The 1/e value of this normalized Gaussian derivative function is obtained for y ≈ 0.23wy and y
≈ 2.12wy, and then the full width at 1/e in amplitude and 1/e2 in intensity (power density) is about
2wy.
The similarity between the perpendicular distribution of the fundamental mode of a channel
waveguide and the distribution of the second-order LP11 mode of a fiber may be pushed
considering a virtual image with a π phase shift. The fundamental mode of a channel waveguide
has actually the same characteristics as the antisymetric second-order LP11 mode of an equivalent
fiber with a core that has the same width 2ax in the parallel direction and a width 2ay which is the
double of the height ay of the waveguide in the perpendicular direction (Figure C.8).
This similarity is also found in the equivalent index which follows the law of an LP11 fiber
mode (see Section B.3.5). In particular, if the fundamental mode of a fiber is unique for a
normalized frequency V following 0 < V < 2.4, and has theoretically not cut off, the fundamental
mode of a channel waveguide is unique for a normalized frequency V following 2.4 < V < 3.8 as
an LP11 fiber mode, and has a cutoff when V < 2.4, with V being calculated considering that the
core radius of the equivalent fiber is ay, the full height (in the perpendicular direction) of the
actual waveguide.
Figure C.7 Normalized mode profile (Gaussian-derivative) in the y perpendicular direction.
C.1.4 Mismatch Coupling Attenuation Between a Fiber and a Waveguide
The coupling attenuation between a waveguide and a fiber is calculated with an overlap integral
between the mode of the fiber and the mode of the waveguide as seen in Section B.4.
Figure C.8 Equivalence between (a) fundamental mode of a channel waveguide with a size 2ax × ay (b) L P 11 mode of a fiber
with a core size 2ax × 2ay .
Despite the Gaussian-derivative profile in the perpendicular direction, the coupling attenuation
due to mode mismatch can be simply calculated with a good approximation considering the
waveguide mode as having an elliptical Gaussian profile with a width 2wx in the parallel direction
and a width 2wy in the perpendicular direction, with 2w0 being the MFD of the fiber mode.
Following Appendix B, the coupling power attenuation is:
As seen in Section B.4.3, the lowest attenuation is obtained for a fiber mode diameter being the
geometrical mean of the widths of the elliptical waveguide mode.
A typical mode size of a LiNbO3 waveguide at 1,550 nm is:
2wx = 8 μm and 2wy = 5 μm
A 0.17-NA fiber having a mode diameter of 8 μm yields a coupling loss of 10% (i.e., a coupling
attenuation of 90% or 0.5 dB).
The lowest attenuation is obtained for a fiber mode diameter equal to the geometrical mean
, that is, an NA of 0.2 for the fiber. It is then 95% (i.e., 0.25 dB).
The total coupling attenuation is the sum of this mode mismatch attenuation and the attenuation
due to Fresnel partial reflection (4%, that is, an attenuation of 0.2 dB) as seen in Section C.1.2.
C.1.5 Low-Driving-Voltage Phase Modulator
Several materials, such as III-V semiconductors, silica over silicon, or glass waveguides, are
potential candidates for integrated-optic circuits, but for fiber gyro applications, the optimal
choice is lithium niobate (LiNbO3), which provides efficient phase modulation.
The fundamental function to fulfill is indeed phase modulation, and LiNbO3 has very good
electro-optic properties: by applying an electric E field with voltage-controlled electrodes, the
index of refraction seen by the optical wave is modified because of the electro-optic Pockels
effect, thus inducing a phase shift. This is actually used to make modulators in bulk form, but
integrated optics provides an additional advantage, since electrodes can be placed very close to
one another around the waveguide, while, in bulk form, space must be left to avoid diffraction of
the light beam. This shortens the length of the electric field line, compared to that of bulk
modulators, thus reducing the driving voltage for the required E field value in the material
(Figure C.9). The value of Vπ (i.e., the voltage required to produce a π radian phase shift) falls
into the range of a few volts instead of the hundreds of volts of the bulk form. This makes
LiNbO3 integrated-optics phase modulators compatible with low-voltage drive electronics. They
also have a quasi-flat frequency response from dc to several tens of megahertz.
C.1.6 Beam Splitting
The main advantage of LiNbO3 integrated optics is clearly low-driving-voltage wideband phase
modulation, but it also allows one to integrate several components on a unique multifunction
integrated-optic circuit (MIOC).
Figure C.9 Comparison of phase modulators: (a) integrated optics with a short driving-field line and (b) bulk optics with a longer
driving-field line.
Beam splitting with evanescent-field coupling can be performed as with fiber (see Section
B.7.1), bringing two waveguides close to each other. With a typical waveguide distance of few
channel-widths, an interaction length of several millimeters yields 50-50 (3 dB) splitting (Figure
C.10).
Figure C.10 The 50-50 evanescent-field coupler in integrated optics.
However, such an evanescent-field coupling is delicate to tune, and it is wavelength dependent.
The simplest way to perform 50-50 splitting in integrated optics is the Y-junction. Light from a
base waveguide is divided equally into two branch waveguides (Figure C.11). The 50-50 splitting
is ensured by symmetry and it is then wavelength independent. With an adequate design of the
branching region, the loss can be small (typically 5% to 10%, that is, an attenuation of 0.25 to 0.5
dB).
C.1.7 Polarization Rejection and Birefringence-Induced Depolarization
The last important function is polarization rejection. To integrate a polarizer on a multifunction
circuit (MIOC), as we shall see later, can be performed with an absorbing metallic overlay in
titanium-indiffused waveguide (C.2) or by single-polarization propagation in proton-exchanged
waveguide (C.3).
Figure C.11 The 50-50 splitting in a Y-junction.
Now, as we saw in Section B.6.3 with PM fiber, a birefringent waveguide yields depolarization
when a broad-spectrum source is used. This depolarization effect is very useful in a fiber-gyro
and it is very efficient in LiNbO3 circuit because of its strong birefringence. LiNbO3 is a uniaxial
crystal with a negative birefringence ∆nb, the difference between the extraordinary index ne and
the ordinary index n0. These index values are summarized in Table C.1 for the most usual fiber
wavelengths:
Table C.1 LiNbO3 Indices as a Function of Wavelength
LiNbO3
λ = 850 nm λ = 1,300 nm λ = 1,550 nm
Ordinary index n 0 = n x = n y (slow axes) 2.25
2.22
2.21
Extraordinary index n e = n z (fast axis) 2.17
2.15
2.14
∆n b = n e − n 0
−0.075
−0.073
−0.079
The ordinary index n0 has a temperature dependence:
while the extraordinary index ne has a temperature dependence more than 10 times higher:
It yields a relative thermal dependence of the birefringence of LiNbO3:
In terms of thermal expansion, there is for the x and y ordinary axes:
αxy LiNbO3 = 15 ppm/°C
and for the z extraordinary axis:
αz LiNbO3 = 7 ppm/°C
As with PM fibers, one can define a beat length ΛLiNbO3, the actual length that provides λ of
path difference between both polarization modes:
At 1,550 nm, there is:
ΛLiNbO3 (1550 nm) = 21 μm
Because of this strong birefringence, depolarization is very fast. The depolarization length Ld
defined in Section B.6.3 as Ld = Ldc/∆nb is much shorter than the circuit length. Taking again the
case of a source with λ = 1,550 nm, ∆λFWHM = 15 nm, and a related decoherence length Ldc =
λ2/∆λFWHM = 150 μm, the depolarization length Ld in lithium niobate is only 2 mm.
As it will be seen, the depolarization acts as a temporal filtering of the polarization in addition
of the actual rejection of the polarizer. Assuming an input wave train at 45° of the principal axes
of the LiNbO3 circuit, the attenuated component (in practice, polarized along the slow ordinary
axis in the TM mode) gets out of the circuit decorrelated with respect to the transmitted
component (in practice, polarized along the fast extraordinary axis in the TE mode) (Figure
C.12).
Figure C.12 Depolarization of the attenuated polarization TM mode in an LiNbO3 integrated optics.
C.2 Ti-Indiffused LiNbO3 Integrated Optics
C.2.1 Ti-Indiffused Channel Waveguide
The first fabrication technique of LiNbO3 integrated-optic waveguides was titanium (Ti)
indiffusion (Figure C.13). With photolithographic masking, very narrow strips of a thin film of
titanium are deposited on the substrate. By heating up the wafer to 900°C to 1,100°C for several
hours, titanium diffuses into the substrate and locally increases the index of refraction.
This provides single-mode guidance in narrow
Figure C.13 Process of Ti-indiffused waveguides in a LiNbO3 substrate: (a) deposition of Ti strips and (b) diffusion.
channels without degrading the low attenuation of the bulk material. Because this slow
indiffusion process takes place at high temperatures, Ti-indiffused LiNbO3 waveguides are very
stable over time.
C.2.2 Phase Modulation and Metallic-Overlay Polarizer with Ti-Indiffused
Waveguide
The electro-optic Pockels effect used for phase modulation is complicated, since LiNbO3 is a
uniaxial birefringent crystal, and the electro-optic efficiency depends on the respective
orientations of the driving electric field Ed and the optical electric field Eop.
The strongest electro-optic coefficient is the diagonal r33 term (r33 = 31 × 10–12 m/V); that is,
the most efficient phase modulation is obtained when both Ed and Eop fields are parallel to the
extraordinary z-axis (also called C-axis). In this case, the index change δnz is:
(C.1)
where Edz is the z-component of Ed. As with elasto-optic coefficients of silica (Section B.5.2), the
n3/2 term comes from the fact that r33 relates the change of dielectric impermeability η = 1/εr =
1/n2 to E and that ∆η = ∆(1/n2) = −2∆n/n3.
To get this optimal efficiency, an x-cut substrate (i.e., the x-axis is perpendicular to the substrate
surface) with a y-propagating waveguide (i.e., the waveguide is parallel to the y-axis) is needed.
Then the TE mode, which has a horizontal Eop field parallel to the z-axis, can be efficiently
modulated with planar metallic electrodes that are fabricated on both sides of the waveguide in a
second step of the photolithographic process. Under the electrodes, the driving Ed field is vertical
because of the electromagnetic boundary conditions on a metal; but the field lines bend under the
surface to connect both electrodes, and Ed is actually parallel to the horizontal z-axis in the
waveguiding region [Figure C.14(a)]. The TM mode is also modulated through the crossed r13
coefficient, since its optical Ed field is parallel to the x-axis [Figure C.14(b)], but the value of r13
is less than one-third of that of r33(r13 = 9 × 10–12 m/V).
Figure C.14 Phase modulations with an x-cut y-propagating waveguide: (a) TE mode and (b) TM mode.
With such a design, the TE mode has a typical Vπ value of 2 volts, and the TM mode has a
typical Vπ value of 7 volts for a 20-mm modulator length at a wavelength of 850 nm. Working at
a longer wavelength increases the Vπ value for the same modulator length. As a matter of fact,
the width of the waveguide and the electrode spacing have to be scaled up proportionally to the
wavelength ratio to keep the same optimal configuration. This increase of electrode spacing, and
thus of the field line length, yields an increase of the required driving voltage proportional to the
wavelength ratio. Furthermore, the phase change δϕ is inversely proportional to the wavelength
for a given index change δn since, δϕ = 2π δn ⋅ L/λ: this adds a second wavelength ratio
dependence. Then the Vπ value is increased proportionally to the square of the wavelength ratio.
For the same TE mode and 20-mm length, Vπ would be 6 volts at 1,550 nm instead of 2 volts at
850 nm. The thermal variation of Vπ is very significant; typically -800 ppm/°C.
Another important characteristic of integrated-optic phase modulators is their driving
bandwidth. They can work with a continuous voltage, even if some problems of long-term drift
may arise, and the upper frequency limit is dictated by the residual electrical capacitance of the
two electrodes, which are very close to one another and placed on a material that has a very high
dielectric permittivity at the usual modulation frequencies (∈LiNbO3 ≈ 30). A good order of
magnitude is a capacitance of 10 pF for a 10-mm length, which yields a bandwidth of few
hundreds of megahertz with a load resistor of 50Ω in parallel. This value is not limiting for fibergyro applications. Furthermore, the modulator response is very flat within this bandwidth, which
makes integrated optics the ideal technology to fulfill the important function of phase modulation
for fiber-gyro signal processing techniques.
For telecommunication applications requiring bandwidths of few tens of gigahertz, the
modulator design is much more sophisticated and needs to use traveling-wave electrodes with
matched impedance line.
This x-cut TE configuration is the most efficient and most typical design, but it may face the
problem of outdiffusion of Li2O at the surface of the substrate during the heating required for the
indiffusion of titanium. LiNbO3 is actually a compound of LiO2 and Nb2O5. The stoichiometric
composition is (Li2O)0·5(Nb2O5)0·5, but the material may withstand a slightly
nonstoichiometric composition (Li2O)x(Nb2O5)1–x, with x ranging from 0.48 to 0.5. In
particular, the highest material uniformity is obtained in the congruent composition (x = 0 ⋅ 486),
for which there is a composition equilibrium between the solid and liquid phases at the melting
point (1,243°C), where the crystal is grown. Outdiffusion of LiO2 decreases the value of x, which
does not modify the ordinary index no, but yields an increase of the extraordinary index ne. This
creates a parasitic planar waveguide for the extraordinary polarization (i.e., the TE mode with xcut), which may cause optical leakage or cross-talk between the useful indiffused channel
waveguides. Outdiffusion may be suppressed with techniques such as wetting the incoming gas
flow in the diffusion furnace or saturating the atmosphere with Li2O powder.
To avoid the effect of outdiffusion, it would be possible to work with the TM mode, which is
polarized along the ordinary x-axis with an x-cut substrate, but there is a drawback of lower
modulation efficiency. However, this is not desirable for fiber-gyro applications, because another
important component is the polarizer, and the efficient technique to fabricate such a component is
to cover the waveguide with a metallic layer that absorbs the TM mode while it transmits the TE
mode (Figure C.15).
Figure C.15 Attenuation of the TM mode with a metallic overlay polarizer.
However, having this transmitted TE mode polarized along an ordinary axis that is not affected
by outdiffusion, it is possible to use a z-cut substrate with a y-propagating waveguide. In this
configuration, one electrode must cover the waveguide to get a vertical drivingEd field parallel
to the z-axis in the waveguiding region and to modulate the phase of the TE mode through the r13
coefficient (Figure C.16). Even if it is not optimal, the modulation efficiency remains acceptable,
and, in addition, the covering electrode acts as a polarizer, which may be very useful.
For completeness, we may add that some specific components use alternative orientations and
other electro-optic coefficients. In particular, z-propagating waveguides are needed when the
effect of the birefringence of LiNbO3 has to be eliminated, but this is not advantageous for fiber
gyros, where, on the contrary, birefringence is very beneficial because it induces depolarization.
Figure C.16 Phase modulation of the TE mode with a z-cut y-propagating waveguide, one electrode acting as a polarizer that
attenuates the TM mode.
C.3 Proton-Exchanged LiNbO3 Integrated Optics
C.3.1 Single-Polarization Propagation
Ti-indiffused waveguides on a LiNbO3 substrate is a suitable technology for the fiber gyro, but,
ideally, a single-polarization waveguide would be preferable to ensure very good polarization
filtering. As we have already seen, outdiffusion of Li2O creates an increase of the extraordinary
index, which yields guidance only for a wave polarized along the z-axis. However, this technique
is difficult to control for channel waveguide, because the material used for masking must not be
diffused, and the induced lack of oxygen ions increases the attenuation.
A similar effect of single-polarization guidance is obtained with proton exchange, where the
LiNbO3 substrate is placed in a melted organic acid and H+ ions (i.e., protons) replace Li+ ions in
the crystal lattice. This technique does not degrade the attenuation and has the very attractive
property of increasing the value of the low extraordinary index while decreasing the value of the
high ordinary index. This yields guidance only for the z-polarized mode parallel to the
extraordinary axis. Such single-polarization waveguides provide a very high extinction ratio for
the nonguided crossed polarization.
Proton exchange is processed at a relatively low temperature (about 200°C to 300°C), which
avoids the problem of mask indiffusion, but the first experimental demonstrations suffered from
poor long-term stability. Annealing techniques have since solved the problem and annealed
proton-exchange circuits is today the optimal choice for fiber-gyro circuits, because high
polarization rejection is one of the important features required for high performance.
C.3.2 Phase Modulation in Proton-Exchanged Waveguide
Optimal modulation efficiency is obtained with the same x-cut, y-propagating orientation as for
Ti indiffusion. Electrodes, placed on both sides of the proton-exchanged waveguide, modulate the
transmitted TE mode with the strongest r33 electro-optic coefficient (Figure C.17).
Proton-exchanged waveguides show essentially no sensitivity to optical damage (i.e., index
drift under high-power illumination), which may be encountered with Ti indiffusion. However,
this advantage is not very important for fiber-gyro applications, where the optical power is not
very high.
Figure C.17 Proton-exchanged LiNbO3 phase modulator: x-cut, y-propagating waveguide with the transmitted TE mode and
the driving electrical field aligned on the z-axis (also called the C-axis).
Note that lithium tantalate LiTaO3 could be a possible alternative to LiNbO3 as the substrate
material. As a matter of fact, LiTaO3 has electro-optic properties very similar to those of
LiNbO3(r33LiTaO3 ≈ r33LiNbO3), and it is also used very often in bulk-optic phase modulators.
However, both crystals are also ferro-electric, and to get a stable electro-optic efficiency, a ferroelectric crystal has to be electrically poled to orient all the microscopic domains in the same
direction (as a ferro-magnetic material is magnetically poled to get a permanent magnet); but the
poling is lost when the temperature is raised above Curie temperature TC. For LiNbO3, TC =
1,150°C, and Ti indiffusion remains below TC; but for LiTaO3, TC = 610°C, which makes metal
indiffusion much more complicated because it requires repoling of each processed wafer. This
drawback of LiTaO3 disappears with proton exchange, since it is realized at only 200°C to 300°C.
However, LiTaO3 has a birefringence that is more than one order of magnitude lower than that
of LiNbO3 (∆nb is equal to 0.004 instead of 0.07). For fiber-gyro applications, this is a
disadvantage, since depolarization induced by birefringence is very beneficial, as we already saw.
C.3.3 Theoretical Polarization Rejection of a Proton-Exchanged LiNbO3 Circuit
The polarization rejection of a proton-exchanged LiNbO3 circuit is not due to an absorption
phenomenon: cross-polarized light is not guided and is diffracted in the substrate. Therefore,
some stray light may be partially coupled back into the output fiber.
Let us take an example of a 0.2-NA fiber that has a mode-field diameter 2w0 = 6.5 μm at 1,550
nm. Assuming that the nonguided crossed polarization is diffracted in a uniform medium, the full
divergence angle θD = 1.5 NA/n (see Section B.1.2) is 0.136 rad in LiNbO3 where n ≈ 2.2. After a
length L of 30 mm, the diffraction pattern diameter is 2w0′ = θD ⋅ L = 4 mm [Figure C.18(a)].
The recoupling ratio of the crossed polarization in the output fiber core (see Section B.4.3)
would then be, because of diameter mismatch:
(C.2)
and since w0′ >> w0:
that is only −50 dB.
Experimental results are better, but this may be explained simply with an interferometric
Lloyd’s mirror effect on the top interface of the substrate. As a matter of fact, the nonguided wave
is in total internal reflection on this interface: this yields a Lloyd’s mirror interferometer with
interferences between two sources, the input fiber mode and its virtual image [Figure C.18(b)].
Furthermore, total internal reflection under grazing incidence induces a π radian phase shift;
therefore, the central fringe located on the interface is a black fringe, which reduces drastically
the power density that gets to the output fiber located just below this interface.
Figure C.18 Recoupling of the nonguided polarization. (a) Assuming a uniform medium with an index n. (b) Lloyd’s mirror
effect with the π phase shift induced by total internal reflection (TIR) under grazing incidence, yielding a reduced overlap for the
parasitic coupling in the output core mode.
To evaluate this parasitic recoupling ratio more precisely, it is possible to consider that there is
actually diffraction of a second-order antisymmetric mode with a real lobe and a virtual image
lobe, which is recoupled in the second-order mode of an output waveguide composed of the
output fiber and its virtual image. As seen in Section B.4.4, this mode can be considered as a
pseudo-Gaussian derivative mode with a full divergence angle between both extrema:
(C.3)
where w1 is the half width at the maximum of the input fiber mode.
This half width w1 is about equal to w0, the half diameter at 1/e of the fundamental mode and:
θD1 ≈ θD/2
After a length L, the width 2w1′ at the maximum becomes:
2w1′ = θD1 ⋅ L
With the previous numerical example (λ = 1,550 nm, 2w1 = 2w0 = 6.5 μm, L = 30 mm):
θD1 ≈ 0.07 rad and 2w1′ = 2 mm
Because of the antisymmetry of the mode, the overlap is much less efficient [Figure C.18(b)]
and as seen in Section B.4.4, the coupling coefficient is:
That is −88 dB, close to the double of the previous case without the Lloyd’s mirror effect.
Now, the transmitted TE mode has some fiber-to-fiber attenuation. For a fiber-gyro circuit, it is
typically 6 dB (including the 3 dB splitting of the Y junction) and then the theoretical rejection is
reduced from 88 dB to 82 dB.
Finally, note that it is not surprising that the angle θD1 is about θD/2. The full width at 1/e in
amplitude (1/e2 in intensity/power density) of an antisymmetric Gaussian-derivative distribution
is about the double of the width between the maximum (see Figure C.18). The antisymmetric
Gaussian-derivative diffraction is the result of interference between two symmetric Gaussian
modes with a π phase difference. The full width of this interference pattern has to be about the full
width of the Gaussian diffraction of a single Gaussian mode.
C.3.4 Practical Polarization Rejection of Proton-Exchanged LiNbO3 Circuit
The theoretical analysis yielding a polarization rejection of about 80 dB assumes a thick
substrate. In practice there is total internal reflection on the bottom face that limits the actual
rejection. Stray light reflected in the middle of this bottom face reaches the output core, and this
light is not attenuated by the Lloyd mirror effect (Figure C.19).
Figure C.19 Recoupling of the beam reflected in the middle of the bottom face of the substrate.
To evaluate simply the coupling coefficient, one can consider the image of the input core
through the bottom face mirror sending the diffracted lobe toward the output fiber (Figure C.20).
One sees that the overlap between the diffracted lobe and the output core mode is in the regime of
symmetrical mode of Figure C.18(a) and not in the antisymmetrical regime of Figure C.18(b).
Considering the previous numerical example (λ = 1,550 nm, 2w0 = 6.5 μm, L = 30 mm)
yielding 2w1′ = 2 mm, and a substrate thickness b = 1 mm, the coupling coefficient is about 50
dB, and since the transmitted TE polarization has some attenuation (typically 6 dB as we saw for a
gyro MIOC), polarization rejection is actually about 40 to 45 dB only.
Figure C.20 Visualization of the overlap between the reflected diffracted lobe and the output core mode, using an imaged input
core.
C.3.5 Improved Polarization Rejection with Absorbing Grooves
To increase the polarization rejection of a proton-exchanged LiNbO3 circuit, one has to suppress
or at least reduce this reflection on the bottom face of the substrate. Antireflection (AR) coating
could be a solution but it is very difficult technologically because of the high index of LiNbO3,
and the grazing incidence.
Some reduction of the amount of reflected light may be also obtained by grinding the bottom
face, but because of the grazing incidence, it is not very efficient and the specular reflection
remains important.
The most common technique to improve the rejection is to fabricate absorbing grooves in the
bottom face to block the light reflected in the middle that is otherwise sent to the output fiber
(Figure C.21).
Figure C.21 Absorbing groove to block stray light.
This places the output core in the geometric shadow of the absorbing groove, and this can be
visualized easily with the imaged input core used earlier (Figure C.22).
Figure C.22 Geometric shadow of the absorbing groove using an imaged input core and an imaged groove.
Now, as seen in Section A.10.2, the light power is not perfectly attenuated in the geometric
shadow region because of knife-edge near-field Fresnel diffraction, and the semi-darkness decay
is not very fast. Figure C.23 displays this attenuation in log-log scale with respect to the
normalized abscissa X defined earlier as X =
It is interesting to note that it is a linear
function with a −2 slope.
Considering again the previous numerical example (λ = 1,550/2.2 = 700 nm in LiNbO3, D =
L/2 = 15 mm), the abscissa is normalized with
Assuming a groove depth d = 250 μm in a 1-mm-thick substrate, the output fiber core is placed
at 2d from the geometric shadow limits (i.e., 500 μm = 7X), which yields an attenuation of only 30
dB from one edge. To get only 10 dB of additional attenuation requires having a shadow three
times wider which is not possible in practice.
Figure C.23 Power density attenuation in the geometric shadow region due to knife-edge Fresnel diffraction as a function of
the normalized abscissa X =
(log-log scale).
In addition there are two edges (this is actually a diffraction by a strip), both Fresnel
diffractions interfere and, in the middle of the geometric shadow of the strip where the output
fiber is placed, there is constructive interference (similar to a 1-D Arago spot) increasing by
fourfold the power issued from a single edge, which reduces the attenuation by 6 dB, yielding
only 24 dB instead of 30 dB.
Figure C.24 Fraunhofer diffraction on the back of the groove.
Considering the 40 to 45 dB of rejection without groove, these additional 24 dB yield a
practical polarization extinction of 65 to 70 dB.
Some additional spurious light is also coming from the reflection on the back of the groove.
Since this groove is narrow and, in addition, receives the stray light with a grazing incidence, it
behaves like a Fraunhofer diffraction slit that disperses the light and reduces the power density on
the output core (Figure C.24). This diffraction yields an additional attenuation on the order of 35
to 40 dB.
To further increase rejection, it is possible to use additional vertical absorbing holes, when the
waveguide is not straight as it is the case with a Y junction. Their rejection is limited by Fresnel
diffraction in the geometric shadow region like with a groove (Figure C.25) and one typically get
an improvement of 15 to 25 dB.
Fresnel diffraction of a groove takes place in a plane perpendicular to the substrate surface,
while it takes place in a plane parallel to the surface with a hole. Both rejection effects are then
multiplied (or added in decibels) to yield a polarization rejection ratio on the order of 80 to 90
dB.
Figure C.25 Geometric shadow region created by a vertical hole placed between the branches of a Y junction.
C.3.6 Spurious Intensity Modulation
This analysis of polarization rejection of proton-exchanged LiNbO3 circuit applies also on the
problem of spurious intensity modulation of the phase modulator since they both involve
parasitic coupling of stray light in the output fiber.
Figure C.26 Comparison between recoupling of the unguided TM mode and recoupling of stray light of the guided TE mode.
For polarization rejection, the TM polarization is not guided, 100% is diffracted in the
substrate and 100% of the power density reaching the output core is coupled.
For the guided TE polarization, there is a mode mismatch at the input and typically 10% of the
input power is not coupled and diffracted into the substrate. It follows the same Lloyd’s mirror
effect but also the same problem of reflection on the bottom face as the crossed-polarized TM
wave. Since the output fiber receives 90% of the waveguide power, it has a potential of 10% to
couple back stray light. It can be calculated precisely but basically the spurious coupling of TE
stray light is about 20 dB below polarization rejection of the TM mode (Figure C.26).
Now we saw in Section C.1.7 that LiNbO3 birefringence yields depolarization between the
transmitted TE wave and the spurious cross-polarized TM wave, but this decoherence does not
take place between the transmitted TE wave and the spurious TE wave coming from stray light in
the substrate. Their optical path difference ΔLop is just related to the geometrical path difference
between the guided mode propagating along the length L of the circuit and the reflected stray light
propagating along the hypotenuse path
Going back to the previous example (a
circuit length L = 65 mm, and a circuit thickness b = 1 mm), the geometrical path difference is
only 30 μm (i.e., an optical path difference of 140 μm with ne = 2.14 for LiNbO3), which is
shorter than the decoherence length of Er-doped fiber ASE source.
Therefore, this spurious stray light interferes with the main wave, and as we saw several times,
the interference contrast is twice the amplitude ratio. A typical circuit with 60 dB of polarization
rejection has only 80 dB of spurious coupling of TE stray light, but it yields a power variation
(or intensity modulation) of ±2 × 10–4 when the phase difference between the guided mode and
the stray light is modulated. This is the case when there is a phase modulator, since it modulates
the phase of the guided wave but not the one of stray light.
Selected Bibliography
Hutcheson, L. D., (ed.), Integrated Optical Circuits and Components, New York: Marcel Dekker, 1987.
Marcuse, D., Theory of Dielectric Optical Waveguides, New York: Academic Press, 1974.
Vassalo, C., Optical Waveguide Concepts, New York: Elsevier, 1991.
Weiss, R. S., and T. K. Gaylord, “Lithium Niobate: Summary of Physical Properties and Crystal Structure,” Appl. Physics A, Vol.
37, 1985, pp. 191–203.
Wong, K. K., (ed.), “Integrated Optics and Optoelectronics II,” SPIE Proceedings, Vol. 1374, 1990.
APPENDIX D
Electromagnetic Theory of the Relativistic Sagnac
Effect
D.1 Special Relativity and Electromagnetism
Special relativity is based on the principle of equivalence of the inertial frames of reference for
all the laws of physics, particularly mechanics and electromagnetism, the main effect being that
light velocity c in a vacuum is the same in any inertial frame.
This yields the Lorentz transformation between the spatial coordinates and the time of two
reference frames moving with a constant translation velocity vt (along the x-axis) with respect to
each other:
(D.1)
with
Compared to the Galilean transformation of classical mechanics:
xG = x − vt t
yG = y (D.2)
zG = z
tG = t
There is a second-order (in vt/c) difference for the spatial coordinate x and a first-order
difference for the time t.
To simplify and condense equations, relativity laws are usually expressed with a fourdimensional notation using contravariant and covariant coordinates of four vectors and tensors. A
space-time four-vector 4x has contravariant coordinates defined by:
xμ = (x, y, z, ct) with μ = 1 to 4 (D.3)
and covariant coordinates defined by:
xμ = (−x, −y, −z, ct) with μ = 1 to 4 (D.4)
Contravariant coordinates are the usual coordinates, with:
(D.5)
where aμ are the basis eigenvectors. With the four-dimensional notation, the Σ term is omitted,
and the convention is to sum over any double index appearing once up and once down:
4x = xμa (D.6)
μ
The covariant coordinates are defined with the scalar product:
xμ = 4x ⋅ aμ (D.7)
With the orthonormal basis of a Euclidean frame of reference, contravariant and covariant
coordinates are equal. Some feeling of the difference between contravariant and covariant
coordinates may be obtained with a nonorthogonal basis of a two-dimensional plane (Figure D.1).
Contravariant coordinates are the usual coordinates defined by a parallelogram with side lengths
equal to x1 and x2 covariant coordinates defined by the scalar product x ⋅ a1 and x ⋅ a2
correspond to the perpendicular projection of x on each coordinate axis.
Figure D.1 Geometrical representation in a two-dimensional plane, of the contravariant coordinates x μ (the usual coordinates)
and the covariant coordinates x μ, defined with a scalar product.
The covariant coordinates are related to the usual contravariant coordinates with a second-rank
covariant tensor, the metric tensor gμv = aμ ⋅ av, which is a characteristic of the frame of
reference:
(D.8)
In an inertial frame, there are:
(D.9)
Now the Lorentz transformation may be written with:
where
(D.10)
Note that the length of a four vector and its square defined with a scalar product is invariant
under Lorentz transform:
(D.11)
or
(D.12)
where s2 is the space-time invariant.
Electromagnetism may also be described with this four-dimensional formalism. The vector
potential A and the scalar potential V form a four-vector 4A with contravariant components:
(D.13)
and covariant components:
(D.14)
This four-potential defined the electromagnetic field represented by a second-rank covariant
antisymmetric tensor Fμv. The usual definitions, E = −(∂A/∂t) − grad V and B = curl A, may be
written:
Fμv = ∂μAv − ∂vAμ (D.15)
where ∂μ is covariant and corresponds to the partial derivative with respect to the contravariant
space-time coordinate, ∂/∂xμ. It can be seen from this definition that Fμv is antisymmetric, since
Fμv = −Fvμ. Expressed in terms of the conventional E and B field components, the covariant
components of the field tensor are:
(D.16)
where μ is the column index and v is the line index. The contravariant components are given by:
Fμv = gμσ gvρFσρ (D.17)
where gμσ is the contravariant metric tensor related to the covariant metric tensor gμρ, with:
(D.18)
We have:
(D.19)
The first two Maxwell equations, M1 and M2, express that the field is derived from a potential
and may be written:
∂μFρσ + ∂σ Fμρ + ∂ρFσρ = 0 (with μ ≠ ρ ≠ σ ≠ μ) (D.20)
It is also possible to define a four-current vector 4J with:
Jμ = (jx, jy, jy, cρ)
Jμ = (−jx, −jy, −jy, cρ) (D.21)
The last two Maxwell equations, M3 and M4, may be written as:
∂μFμv = μ0Jv (D.22)
Note that the first two Maxwell equations use the covariant coordinates Fμv and that the last two
use the contravariant coordinates Fμv.
Finally, the propagation equation is:
∂ρ∂ρFμv = 0 (D.23)
where ∂ρ = ∂/∂xρ and ∂ρ = ∂/∂xρ, that is:
(D.24)
The important point is that Maxwell equations and their consequences, like the propagation
equation, are invariant under Lorentz transformation. In the new frame, the four vectors Aμ and
Jμ and the tensor Fμv are transformed accordingly to:
(D.25)
and these quantities
remains:
follow the same laws. In particular, the propagation equation
∂Lρ∂LρFLμv = 0 (D.26)
with ∂Lρ = ∂/∂xlρ and
This result is consistent with the fact that the velocity of light
in a vacuum remains equal to c in any inertial frame.
Note that the equation
is the basis of magnetism. With a motionless charge,
there is only an electric field E and the three magnetic components of
null:
in the rest frame are
To take into account the effect of a moving charge, we have to use the Lorentz transformation
between the frame where the charge is motionless and the frame where the charge is moving at an
opposite velocity −vt if vt is the velocity of the moving frame with respect to the rest frame of the
particle. Calculating
we obtain nonzero terms for
that is, it
appears a magnetic component B: even if it is sometimes obviated, the magnetic field is a pure
relativistic effect. Because the speed of the particles is usually much smaller than c, the pure effect
of B is usually much smaller than that of E. However, if this applies to one particle, in the case of
a conductor, then there is cancellation of the E effect, because there are as many positive particles
as negative ones, while the B effect is not cancelled, as the positive and negative particles do not
have the same speed. In this case, the global B effect becomes predominant despite the fact that the
elementary B effect for each particle is a lower effect. As will be seen, the problem of the Sagnac
effect has some similarities with that of the magnetic field: it is also a first-order effect that is
much lower than the zero-order effect, but the zero-order effect may also be nulled out because
of reciprocity, and in a ring interferometer the first-order effect may become predominant.
The main interest of these notations is that these results remain almost unchanged with any set
of coordinates, even if they are not Cartesian. In general, we still have:
∂μFρσ + ∂σ Fμρ + ∂ρFσρ = 0
Fμv = ∂μAv − ∂vAμ (D.27)
and the last two Maxwell equations are slightly modified:
(D.28)
where g is the determinant of the metric tensor. With an inertial frame and Cartesian coordinates,
we have gI = −1 and
The case of the propagation equation is much more complicated, as it uses the high-index
derivative ∂ρ in addition to the low-index derivative ∂ρ that is used in the other formulae, and
there is no simple generalization of the equation. To avoid this mathematical difficulty, it is
possible to define a propagation equation of the four-potential Aμ. As a matter of fact, the
equations:
(D.29)
yield a propagation equation:
(D.30)
So far, this analysis has been carried out with the assumption that there is a vacuum. If there is a
medium, a derived field tensor Gμv composed of the components D and H has to be used. The
first two Maxwell equations that show the relation with a potential are unchanged, but the two last
ones become:
(D.31)
where
is the free current four vector.
The derived field tensor Gμv is related to the field tensor Fσρ, with:
Gμv = χμvσρFσρ (D.32)
where χμvσρ is the constitutive tensor of the material. It depends on the relative permittivity ∈r
and the relative permeability μr, but also on the metric of the frame of reference, since Gμv is
used with its contravariant coordinates and Fσρ is used with its covariant coordinates.
In an inertial frame of reference, there are:
(D.33)
And the nonzero terms of the constitutive tensor χμvσρ of a motionless material are:
χ1212 = χ2121 = χ1313 = χ3131 = χ2323 = χ3232 = μ–1μ0–1 (D.34)
χ1414 = χ4141 = χ2424 = χ4242 = χ3434 = χ4343 = −∈μ0–1 (D.35)
In the most general case, the propagation equation of the four-potential is:
(D.36)
As an example of this analysis, we can consider the case of cylindrical coordinates with
contravariant (i.e., usual) values:
(D.37)
The metric tensor gcμv is:
(D.38)
Thus, the determinant gc = −r2 and
and the covariant coordinates
are:
xcμ = (−r, −r2θ, −z, ct) (D.39)
The electromagnetic four-potential 4Ac is defined by contravariant coordinates:
(D.40)
where Ar, Aθ and Az are the conventional coordinates of A, with orthogonal unit vectors ar, aθ
and az parallel to the equicoordinate lines (Figure D.2). The covariant coordinates are:
Figure D.2 Cylindrical coordinates.
(D.41)
For the covariant components of the field tensor, there are, similarly:
(D.42)
and for the contravariant components of the derived field tensor,
(D.43)
The nonzero terms of the constitutive tensor
are:
Note that, in the case of cylindrical coordinates, the general formulation that has been derived
is a way to retrieve the well-known expressions of the various vector operators (gradient,
divergence, curl, and Laplacian) with these non-Cartesian spatial coordinates. The constitutive
tensor
depends on μ and ∈ (i.e., the effect of the material), but also on r (i.e., the effect of
the non-Cartesian spatial coordinates).
D.2 Electromagnetism in a Rotating Frame
From a mathematical point of view, the tensor formalism derived in the previous section may be
applied to any kind of coordinates, but we have to define, from the physics point of view, what
coordinates can actually be used. Considering a given frame, the coordinates must be measurable
with internal experiments relative to this frame. For a translation, it is well known that the new
coordinates have to be derived with the Lorentz transformation. A Galilean transformation is
valid mathematically, but it yields new coordinates that cannot be measured with internal
experiments in the new frame.
In the case of a rotation of rate Ω, the new coordinates could be derived with an equivalent
Lorentz transformation, where the tangential speed rΩ replaces vt:
(D.44)
with:
Locally, this transformation is valid, and the potential
the field
and the derived field
defined with these coordinates correspond to the local values seen, particularly by a
medium at rest in this rotating frame. With such a motionless medium, the constitutive tensor
is equal to the constitutive tensor
defined for an inertial frame using cylindrical
spatial coordinates.
However, the angular coordinate is periodic, and, if the whole rotating frame is considered, this
periodicity will appear in the time coordinate. To analyze global experiments like the
measurement of a phase difference in a ring Sagnac interferometer, the time coordinate must be
univocal and thus independent of the periodic angular coordinate, and a Galilean transformation
may be used:
rRG = r
θRG = (θ − Ωt) (D.45)
zRG = z
tRG = t
In contrast to the case of translation, these new coordinates can be measured in the rotating
frame. A clock placed on the rotation axis has the same time as the time t in the inertial frame, and
it may synchronize the time in the whole rotating frame by sending cylindrical waves that
propagate perpendicularly to the displacement and are thus not modified. The angle θRG may be
defined as a constant portion of a full angle (i.e., 360°) which remains obviously constant.
Once it is admitted that a Galilean transformation is valid for a rotation, the general tensor
formalism may be applied. In particular, the propagation equation of a potential in a vacuum is,
with the four-dimensional formalism:
(D.46)
where
is the metric tensor corresponding to the contravariant coordinates
zRG, ctRG) and gRG is its determinant. We have:
= (rRG, θRG,
(D.47)
As can be seen, there are two nondiagonal terms
which yield a
difference of propagation velocities between co-rotating and counter-rotating waves. To the first
order in rΩ/c, the propagation equation in a vacuum becomes:
(D.48)
where the wave amplitude A is any component of the field or of the potential. Compared to the
familiar propagation equation in an inertial frame, there is an additional crossed term
which modifies the wave velocity proportionally to Ω.
With a co-rotating medium, the propagation equation is with four-dimensional formalism:
(D.49)
where the constitutive tensor
replaces
calculated with the constitutive tensor
This constitutive tensor
has to be
in an inertial frame using cylindrical
coordinates and the laws of transformation of coordinates between
order in rΩ/c, the propagation equation becomes:
and
To the first
(D.50)
It is very important to note that, if ∈r and μr appear at their usual place with ∂2A/∂t2, none of
them appear in the additional term
which remains identical to the case of a
vacuum. The change of wave velocity in a rotating frame, called the Sagnac effect, is independent
of the properties of the co-rotating medium, where light propagates.
Note that with this relativistic formalism, it becomes clear that H and D, involved in Maxwell
equations M3 and M4, are connected, as they compose the same contravariant tensor Gμv E and B
represent the basic field, derived from a potential and involved in the first two Maxwell equations,
M1 and M2, which are independent of matter. They form a covariant tensor Fμσ , and the derived
field tensor Gμv is connected to Fμσ with the constitutive tensor χμvρσ , which takes into account
the properties of the material, but also the properties of the frame of reference.
D.3 Case of a Rotating Toroidal Dielectric Waveguide
The case of the fiber gyro may be analyzed more precisely by considering a toroidal dielectric
waveguide, where R is the radius of curvature and a is the radius of the core (Figure D.3). As has
already been seen (Section B.3.1), an approximate solution of the fundamental mode of a straight
fiber is the pseudo-Gaussian mode of amplitude:
(D.51)
When the fiber is bent, the spatial propagation term βz may be replaced by βRθ, and there is a
centrifugal shift ∆Rc of the mode. The mode amplitude becomes:
(D.52)
with:
(D.53)
Using a perturbation method, the amplitude in a rotating frame with a co-rotating waveguide is
found to be:
Figure D.3 Toroidal waveguide.
(D.54)
There is an additional radial shift ∆RR due to rotation. It depends on the relative direction of
propagation and rotation through the sign of the product β ⋅ Ω:
(D.55)
but, in practice, it is completely negligible (with typical fiber parameters ∆RR ≈ 10–11 for Ω = 1
rad/s).
The perturbation ∆K on the propagation constant is:
(D.56)
Its relative value ∆K/βR is also very small, but, when the fiber is placed in a ring
interferometer, there is a measurement of the phase difference between the two opposite paths
with respect to 2π rad, and it accumulates with the length of the propagation path. After N turns of
counterpropagation with the respective propagation constants β + ∆K/R and −β + ∆K/R, the phase
difference becomes:
∆ϕ R = 4πN∆K (D.57)
(D.58)
(D.59)
This result, known as the Sagnac effect, depends only on the light frequency ω, the vacuum
light velocity c, and the total geometrical area enclosed by the coil A = NπR2. The indices of the
core and the cladding, the phase or group velocities of the mode, and the dispersion of the
medium or of the waveguide have no influence.
The Sagnac effect is a pure temporal delay ∆tR independent of matter, which is measured with a
clock, the frequency ω of the light source. Because ∆ϕ R = ω ⋅ ∆tR, there is:
(D.60)
However, as the wavelength λ in a vacuum is a more familiar quantity than the angular
frequency ω for optical waves, the formula is usually written as:
(D.61)
where D = 2R is the loop diameter, and L = N2πR is the total length of the waveguide coiled over
N turns.
Selected Bibliography
Arditty, H. J., and H. C. Lefèvre, “Theoretical Basis of Sagnac Effect in Fiber Gyroscope,” Springer-Verlag Series in Optical
Sciences, Vol. 32, 1982, pp. 44–51.
Arzelies, H., Cinématique Relativiste, (in French), Paris: Gauthier-Villars, 1955.
Landau, L. D., and E. M. Lipshitz, The Classical Theory of Fields, New York: Pergamon Press, 1975.
Lefèvre, H. C., and H. J. Arditty, “Electromagnétisme des milieux diélectriques linéaires en rotation et application à la propagation
d’ondes guidées,” (in French), Applied Optics, Vol. 21, 1982, pp. 1400–1409.
Post, E. J., “Sagnac Effect,” Review of Modern Physics, Vol. 39, 1967, pp. 475–493.
Post, E. J., “Interferometric Path-Length Changes Due to Motion,” Journal of the Optical Society of America, Vol. 62, 1972, pp.
234–239.
Tonnelat, M. A., Principles of Electromagnetic Theory and Relativity, London, U.K.: Gordon and Breach, 1966.
APPENDIX E
Basics of Inertial Navigation*
E.1 Introduction
Inertial navigation principle is simple: the craft acceleration is measured with respect to an
inertial frame with accelerometers. A computer integrates these acceleration measurements to
calculate velocity and then integrates the velocity to calculate position. Because the craft
orientation does not remain stable in the inertial frame, the orientation of the accelerometers is
maintained mechanically stable with respect to the inertial frame with a gimbaled system, or
acceleration measurements are projected by calculation in this inertial frame in the case of a
strap-down system. In both cases, gyroscopes that measure rotation rate are needed: in gimbaled
systems (Figure E.1), they are used for controlling torque motors that rotate the gimbals so that
gyroscope measurements remain zero and then keep a stable orientation of the accelerometers; in
strap-down systems (Figure E.2), they are used for computing the Direction Cosine Matrix, which
allows acceleration measurement projection into the inertial frame.
Therefore, in our 3-D world, an inertial system requires an inertial measurement unit (IMU)
composed of three gyroscopes and three accelerometers to provide full navigation function.
Some inertial systems may contain fewer sensors for price reduction if the function is limited.
If
inertial
navigation
principle
is
simple,
implementation is more
complex
for
multiple
reasons:
Accelerometers do not only
measure acceleration but
also the gravitational field.
Gyroscopes
measure
rotation rate with respect to
an inertial frame thatFigure E.1 Gimbaled navigation system.
includes Earth rotation
around its spinning axis and Earth rotation around the Sun.
In most applications, craft navigation is performed on the Earth or close to the Earth surface and
the gyroscopes sense also the rotational motion generated by the craft move on Earth surface
and the curvature of the Earth,
Earth is not a simple surface and it must be simplified into a model.
The inertial frame in which all inertial measurements (from gyroscopes and accelerometers) are
referenced to is not the most convenient one for representing craft position because it does not
rotate with the Earth.
Figure E.2 Strap-down navigation system.
Depending on the accuracy of its sensors, an inertial system provides different functions and
then has a different “jargon” name:
With gyroscopes having a bias stability of few degrees per hour, the system provides attitude and
heading, heading being initialized by an external mean (for example, magnetic sensor). It is an
Attitude & Heading Reference System (AHRS).
With a bias stability of 0.05° to 0.1° per hour, the system provides also attitude and heading, but
heading is autonomously computed without any external sensor. It is a gyrocompass.
With a bias stability better than 0.01° per hour, the system provides full autonomous navigation
capability: attitude and autonomous heading, but also autonomous calculation of velocity and
position. It is an Inertial Navigation System (INS).
Any of these systems can be enhanced by one or several measurements coming from other
noninertial sensors. Velocity measurements can be made available from Air Data Computer on
aircrafts, from odometer on land vehicles, from Doppler or electromagnetic log on vessels or
submarines. Position measurements can be made available from Global Navigation Satellite
Systems (GNSS) as GPS, Glonass, Galileo, or Beidou, from acoustic positioning systems, from
vision algorithms. Heading measurements can be made available from magnetic sensors, or from
dual antenna GNNS system.
Most of the time, the data fusion between inertial measurements and external noninertial
sensors is performed by using modern signal processing techniques, Kalman filtering being the
most popular one but not the only one.
E.2 Inertial Sensors
E.2.1 Accelerometers (Acceleration Sensors)
Almost all accelerometers are force sensors that measure the force that a proof mass needs to
follow the accelerometer case. From Newton’s law:
(E.1)
where m is the mass of the proof mass,
is the craft inertial acceleration,
is the gravitational
field (or gravity vector), and is the applied force on the proof mass that is measured.
Therefore, an accelerometer measures:
(E.2)
also called the specific force and there is no mean to separate
from
as it is well known.
From the accelerometer measurement, the only way to compute is to add
which means
that must be known (as a vector) at any location where the craft is navigating. This is done by
using a more or less complex model of gravity as a function of position on Earth. Obviously,
model complexity is related to requested navigation performance.
E.2.2 Gyroscopes (Rotation Rate Sensors)
Unlike accelerometers, gyroscopes that are used presently in inertial systems are based on
various principles. Historically, the first gyroscopes used the properties of the conservation of the
kinematic moment of a wheel spinning at high speed. This principle was used with several
different setups like gimbaled gyros, rate integrating gyros, flex or dry-tuned gyros, but different
principles are also used. Among them:
Vibrating gyroscopes based on the Coriolis effect: tuning forks, hemispherical resonating gyros
(HRG), and microelectromechanical system (MEMS) gyroscopes;
Nuclear magnetic resonance gyroscopes;
Optical gyroscopes with two different use of the same Sagnac effect: ring-laser gyros (RLG) and
fiber-optic gyros (FOG) (the subject of this entire book).
Whatever the technology, gyroscopes measure the angular rate with respect to the inertial
space, which means that, in addition to rotation with respect to the Earth, they also measure:
The angular rate of Earth around its spinning axis;
The angular rate of Earth around the sun;
The angular rate induced by the Earth’s curvature when the craft moves over the Earth surface
(known as the “craft rate”).
E.2.3 Classification of the Inertial Sensor Performance
Very often, inertial sensor performance is classified with respect to their application and as a
function of their bias stability.
The three main grades for gyros are the rate grade (>10°/h), the tactical grade (1 to 10°/h)
allowing attitude measurement, and the navigation grade (<10–2°/h) allowing pure (nonaided)
inertial navigation better than 1 nautical mile per hour.
The rate grade is usually split into an industrial grade (10 to 100°/h) and a consumer grade
(>100°/h).
There is also an intermediate grade (1 to 0.01°/h) between the tactical and inertial grades that is
used for gyrocompassing and GNSS-aided inertial navigation.
Finally, the high end of navigation grade is usually called the strategic grade (<0.001°/h) and
allows pure nonaided inertial navigation better than 1 nautical mile per day.
This is summarized in Table E.1, with in addition the required bias performance for the
accelerometers.
SI (International System) units are rarely used in inertial navigation. The correspondence is:
1°/h = 4.87 μrad/s and 1g = 9.81 m/s2.
Table E.1 Classification of Accelerometer and Gyroscope Performance
Accelerometers Gyroscopes
Consumer rate grade>50 mg
>100° /h
Industrial rate grade 10 mg
10 to 100° /h
Tactical grade
1 mg
1 to 10° /h
Intermediate grade
0.1 mg
0.01 to 1° /h
Navigation grade
0.05 mg
<0.01° /h
Strategic grade
0.005 mg
<0.001° /h
E.3 Navigation Computation
Nowadays, almost only strap-down systems are used, and we will concentrate the description on
this technology.
E.3.1 A Bit of Geodesy
Earth surface is not smooth and cannot be accurately represented by a simple geometrical surface.
This is why geodesists have defined models of the Earth that are easier to use.
The first one, the geoid, is the shape that the surface of the oceans (extended through the
continents) would take if it were submitted only to gravitation and rotation-induced centrifugal
force. All points of the geoid have the same potential energy (geoid is an equipotential surface),
and therefore at any point of the geoid, the gravity vector is perpendicular to the geoid local
surface. The geoid is a smooth surface, but it is an irregular surface, mainly due to
inhomogeneous density of Earth. It cannot be described by a simple mathematical model.
Nevertheless, it can be approximated by an ellipsoid, a rather simple mathematical surface
which is smooth, regular and that fits quite well the geoid since the maximum difference between
the geoid and the ellipsoid is less than 200m. The most commonly used reference ellipsoid
nowadays is known as the World Geodetic System (WGS84). This is with respect to this ellipsoid
that latitude, longitude, and elevation are defined.
E.3.2 Reference Frames
The inertial sensors measure rotation rates and specific forces with respect to inertial space.
Therefore, it would seem simpler to make all navigation computations in the inertial frame
defined as an Earth-centered frame that does not rotate with Earth and keeps a fixed orientation
with respect to inertial space. However, this frame is not convenient for representing the craft
position on the Earth, because this frame is not rotating with it.
A convenient frame for position on the Earth representation is the Earth frame: this frame is
Earth centered and rotates with Earth (Figure E.3).
Figure E.3 Inertial frame and Earth frame.
One of its axes is pointing to the North Pole, while a second axis pointing to the intersection of
the Greenwich meridian and the equatorial plane. In this frame, position on the Earth is
represented by the two angles of latitude and longitude.
If the craft position on Earth is conveniently represented in the Earth frame, this is not the case
for the orientation of the craft, as a common reference for attitude and heading is the local
vertical direction and North direction. This is why a third frame is used, the navigation frame
(Figure E.4). This frame moves with the craft at the surface of the Earth, or more exactly on the
reference ellipsoid surface. The vertical z-axis is orthogonal to the tangent plane to the ellipsoid
while the x-axis is pointing north and the y-axis is pointing east.
Figure E.4 Earth frame and navigation frame.
E.3.3 Orientation, Velocity, and Position Computation
Figure E.5 shows the principle of computation of orientation, velocity, and position. Because
orientation is computed in the navigation frame, angular rates are compensated for the Earth
rotation rate and craft rate and then integrated. Therefore, it is necessary to know or compute
initial orientation.
Specific forces measured by accelerometers are projected into the navigation frame. To
transform them into accelerations, they are compensated for gravity (gravity is computed using a
model stored inside the inertial system), as well as for Coriolis acceleration and centrifugal
acceleration due to Earth rotation. These accelerations are then integrated using initial velocity to
get instantaneous velocity. Finally, velocities are integrated using the initial position to compute
the present craft location (Figure E.5).
Figure E.5 Orientation, velocity, and position computation diagram.
E.3.4 Altitude Computation
Altitude is computed using the above algorithm. However, we must observe the following
phenomenon that makes altitude error grow exponentially, even with extremely performing
sensors. If, due to an error sensor, altitude is slightly wrong, the gravity model that takes into
account altitude for computing
will make an error on that will create an error on the craft
acceleration (remember that craft acceleration is not directly measured by accelerometer but only
with the specific force
) that will increase the altitude error. This loop will diverge rapidly.
Therefore, altitude computation must always be aided with an altitude external sensor like the
GNSS receiver, baro-altimeter (the pressure depends on altitude), or odometer.
E.4 Attitude and Heading Initialization
E.4.1 Attitude Initialization
We have seen above that accelerometer measurements provide only
What can be
considered as a difficulty for velocity and position computation will be useful for initial attitude
computation.
In applications where the inertial system is switched on while the craft is static with respect to
Earth, the acceleration is zero. Therefore, the accelerometers measure only the projection of
gravity on their sensitive axis. Knowing the value of gravity at the initialization location, it is
then possible to compute the two angles defining orientation of the craft with respect to the
direction of gravity, the vertical axis.
For applications where the inertial system is not static when switched on,
is not zero. We
integrate
(E.3)
is the velocity change during the integration time, while
is the velocity that an object
would reach in freefall during the integration time (300 m/s, i.e., Mach 1 after 30 seconds). So
after sufficient time,
tends toward
integration is equivalent to an averaging.
providing then the initial value of attitude, as an
E.4.2 Heading Initialization with Gyrocompassing
The definition of heading with respect to true North is the direction of the projection of Earth
rotation vector on the horizontal plane. The horizontal plane has been determined with previous
procedure. If craft were completely static with respect to Earth, then gyroscopes would only
measure Earth rotation rate and it would only be needed to project these measurements on
horizontal plane to find North direction. However, it is very scarce that a craft is completely static
in terms of rotation with respect to Earth: The Earth’s rotation rate is about 15° per hour. Any
small move of the craft (for example, somebody getting inside a land vehicle) creates an angular
velocity that may be hundreds or thousands of times this value, making the static assumption
completely wrong.
Because gyroscopes measure rotation with respect to inertial space, we use the gyroscope
measurements to stabilize the accelerometer measurements in inertial space frame. Assuming that
there is no craft linear acceleration (or that this acceleration is compensated with some velocity
aiding measurements), then we can see how gravity vector will vary over time with respect to
inertial space because of Earth rotation:
vector at time t,
vector at time t′ (Figure E.6).
Figure E.6 Gyrocompass principle.
If we differentiate these two vectors, we obtain a vector whose direction is West to East. This
way, we have found the East direction, making it easy to convert to the North direction. This
procedure for heading determination is known as gyrocompassing.
This procedure makes sense only if the gyroscope measurements are accurate enough so that
the gravity vector change in the inertial frame is only due to Earth rotation rate. If this is not the
case (gyroscope accuracy worse than one-hundredth of the Earth’s rotation rate to make it
simple), heading initialization accuracy using only inertial sensors will not be sufficient and other
means need to be used. Among them, the following aiding sensors for heading initialization can
be used: double-antenna GNSS receiver or magnetometer or flux gate.
This description provides a physical explanation of the way true North direction is found.
Obviously, equations in the inertial system do not follow completely this physical explanation.
E.5 Velocity and Position Initialization
Velocity and position cannot be initialized autonomously using only inertial measurements.
External assumptions or measurements have to be made.
For velocity, very often, the inertial system is switched on and initialized while the craft is not
moving with respect to Earth. Therefore, initial velocity can be automatically set to zero.
However, this assumption may not always be true. This is the case, for example, when initializing
the inertial system on a vessel at sea. In that case, velocity has to be initialized by measuring it
with an external sensor. Presently, the most popular one is the GNSS receiver, but other sensors
may be used, such as Doppler velocity log for vessels or odometer for land vehicles.
For position, initial craft location must be known to the inertial system. GNNS receiver
measurement is the most accurate and easy way to do so. When this is not possible, the initial
location must be known by using maps or referenced points.
E.6 Orders of Magnitude to Remember
The Earth’s rotation rate is precisely 15.04107° per hour; in one year, that is 365.2422 days, the
Earth rotates over 366.2422 turns, as it rotates one turn around the sun in addition to the daily
rotations.
The gyrocompassing procedure for 0.5° (i.e., 10–2 radians) heading accuracy requires
gyroscopes with an accuracy of 0.15° per hour (i.e., one-hundredth of the Earth’s rotation rate).
For achieving position accuracy of 1 nautical mile per hour (i.e., navigation grade
performance) using only inertial sensors, a gyroscope’s accuracy must be close to 0.01°/hour and
accelerometers accuracy must in the order of 5.10–4 m/s2 (i.e., 50 μg).
Gyroscopes in strap-down inertial systems must measure rotation rates reaching at least 100°
per second. The ratio between the required accuracy for 1 nautical mile per hour and the whole
dynamics is therefore 5 × 10–8. This explains why it is necessary to have extremely performing
gyroscopes to have a strap-down system with a good performance.
Even if static with respect to Earth, a craft at the equator will run about 40,000 km in a day, that
is, a velocity of about 1,700 km/h only due to the Earth’s rotation rate. At 45° of latitude, these
figures become 28,000 km and 1,200 km/h. These velocities are measured with gyros detecting
the Earth’s rotation. Inertial-grade performance is 1 nautical mile per hour, that is, about 2
kilometers per hour; therefore, at the equator, it requires an Earth rate measurement with a
relative stability of 2/1,700 of 15°h, that is, a bias stability of 0.015°/h.
Selected Bibliography
Bekir, E., Introduction to Modern Navigation Systems, New York: World Scientific, 2007.
Lawrence, A., Modern Inertial Technology: Navigation, Guidance, and Control, 2nd ed., New York: Springer, 1998.
Savage, G.P., Introduction to Strapdown Inertial Navigation Systems, Maple Plain, MN: Strapdown Associates, 1991.
Titterton, D. H., and J. L. Weston, “Strapdown Inertial Navigation Technology,” The American Institute of Astronautics and
Aeronautics & The Institution of Electrical Engineers, 2004.
* This appendix was written by Dr. Yves Paturel.
List of Abbreviations
A/D analog-to-digital
ADEV Allan deviation
AHRS Attitude and Heading Reference System
ARW angular random walk
ASE amplified spontaneous emission
ASIC application-specific integrated circuit
AVAR Allan variance
BS beam-splitter
CCW counterclockwise
CD compact disc
CW clockwise
D/A digital-to-analog
DVD digital video disc
EDF erbium-doped fiber
EDFA erbium-doped fiber amplifier
FBG fiber Bragg grating
FOG fiber-optic gyroscope
FPGA field-programmable gate array
FSR free spectral range
FWHM full width at half maximum
GNSS Global Navigation Satellite System
GPS Global Positioning System
HE hybrid electromagnetic
I-FOG interferometric FOG
IMU inertial measurement unit
INS Inertial Navigation System
IO integrated optics
IR infrared
ITU International Telecommunication Union
LD laser diode
LED light emitting diode
LH left handed
LP linearly polarized
LSB least sensitive bit or least significant bit
MEMS microelectromechanical system
MFD mode field diameter
MIOC multifunction integrated-optic circuit
MOF microstructured optical fiber
NA numerical aperture
OCDP optical coherence-domain polarimetry
OCDR optical coherence-domain reflectometry
OCT optical coherence tomography
OSA optical spectrum analyzer
OTDR optime time-domain reflectometry
PBS polarization beam splitter
PDL polarization dependent loss
PER polarization extinction ratio
PM polarization maintaining
PMD polarization mode dispersion
PSD power spectral density
PZ polarizing
PZT PbZx Ti1-x O3 (lead zirconatetitanate)
RLG ring laser gyroscope
R-FOG resonant FOG
RH right handed
RIN relative intensity noise
RRW rate random walk
SLD super-luminescent diode
SM single mode
TE transverse electric
TEM transverse electromagnetic
TIR total internal reflection
TM transverse magnetic
UV ultraviolet
WDM wavelength division multiplexing
List of Symbols
A(t) amplitude of an optical wave
Aμ or Aμ electromagnetic four-potential coordinates
A electromagnetic vector potential
A area or area vector
A area vector
a fiber core radius or acceleration
ae core radius of the equivalent step index fiber
ah size of a spatial filtering hole
a(f) frequency component of A(t)
B normalized birefringence
B magnetic field vector
b(V) normalized propagation constant
C power coupling ratio
c velocity of light in a vacuum (2.998 × 108 m ⋅ s–1)
c(z) random polarization coupling along a fiber
cs coupling strength in an evanescent-field coupler
curl curl operator
D diameter, distance, or diffusivity
D derived electric field vector
d distance or thickness
div divergence operator
E Young modulus
E{} ensemble average
E electric field vector
e exp (1) = 2.7183...
el ellipticity
Fμv or Fμv electromagnetic field tensor
F finesse of an optical cavity
f focal length or temporal frequency
mean temporal frequency
fm modulation frequency
fp proper or eigenfrequency of the coil
f force
g gravity
Guv or Guv derived field tensor
gμv or gμv metric tensor
grad gradient operator
H derived magnetic field vector
HE11 fundamental hybrid electromagnetic mode
h Planck constant (6.63 × 10–34 J ⋅ s)
h h-parameter of a polarization-maintaining fiber
I intensity of an optical wave
i intensity of an electrical current or pure imaginary unit value
Jn Bessel function of order n
Jμ or Jμ electrical four-current
j electrical current density vector
K thermal conductivity
k Boltzmann constant (1.38 × 10–25 J ⋅ K–1)
k0 wavenumber in a vacuum
k1 wavenumber in the core
k2 wavenumber in the cladding
km wavenumber in a medium
L length
Lbb common base-branch length
Lc coherence length
Lcp coupling length
Ld depolarization length
Ldc decoherence length
Lop optical path length
Lp length of a pulse
Lpc polarization correlation length
LP01 fundamental linearly polarized mode
LP11 second-order linearly polarized mode
LP21 third linearly polarized mode
LSB least significant bit
M magnetic polarization vector
M(t) modulation signal
N number of turns
Ṅ flow of uncorrelated particles
NA numerical aperture
n index of refraction
n1 index of refraction of the core
n2 index of refraction of the cladding
no ordinary index
ne extraordinary index
neq equivalent index
ng group index
P power or degree of polarization
P electric polarization vector
P perimeter
p pressure or parallel (polarization)
pij elasto-optic coefficients
ppm part per million
q electron charge
R radius or resistance
Rc autocorrelation (ensemble average)
Rf radius of the fiber cladding
R reflectivity
r radial coordinate
r radial vector
rij electro-optic coefficients
S recapture factor
s perpendicular (polarization)
T temperature
Ta absolute temperature
TC Curie temperature
Tm period of phase modulation
Tn normal stress
Tnc compressive stress
TE transverse electric mode
TM transverse magnetic mode
t time
T transmissivity
T(λ) transmission of a wavelength filter
V normalized frequency in a waveguide; electromagnetic scalar potential or voltage; Verdet
coefficient; visibility of interference fringes
v light velocity in a medium
v speed or velocity
w radius at 1/e2 in intensity of a Gaussian beam
w0 radius at 1/e2 in intensity of the waist of a Gaussian beam or radius at 1/e2 of the pseudoGaussian fundamental LP01 mode
w1 half width at the maximum of the LP11 mode
x, y, z Cartesian spatial coordinates
xj coordinate
Z impedance
Z0 impedance of a vacuum
ZR Rayleigh length
α attenuation per unit length of a fiber
α(f) modulus of a(f)
αF Fresnel-Fizeau drag coefficient
β propagation constant
Γ coupling loss
Γ(τ) autocorrelation function (time average)
γ (1 − vt2/c2)–1/2
γ(τ) normalized autocorrelation function (time average)
∆ normalized index difference or difference of
∆e normalized index difference of the equivalent step-index fiber
∆fFB feedback frequency difference
∆fFSR free spectral range of an interference
∆fFWHM full width at half maximum of the temporal frequency spectrum
∆fR rotation-induced frequency difference
∆fbw counting bandwidth
∆L difference of geometrical length
∆Lop difference of optical path length
∆nb birefringence index difference
∆β birefringence
∆ϕ difference of phase
∆ϕ e error of phase difference
∆ϕ F phase difference induced by Faraday effect
∆ϕ FB feedback phase difference
∆ϕ K phase difference induced by Kerr effect
∆ϕ m modulation of the phase difference
∆ϕ PR phase difference induced by phase ramp
∆ϕ R rotation-induced phase difference due to Sagnac effect
∆λFWHM full width at half maximum of the wavelength spectrum
∆σ FSR free spectral range of an inferometer
∆σ FWHM full width at half maximum of the spatial frequency spectrum
∆τg difference of group transit time through the coil
δ variation or shift of
δn index variation
δϕ phase shift
∈ small ratio
∈0 dielectric permittivity of a vacuum (8.854 × 10–12 F ⋅ m–1)
∈r relative permittivity of a medium
θ angle
θB Brewster angle
θC critical angle of total internal reflection
θD full divergence angle at 1/e2 of a Gaussian beam in a vacuum
θD′ full divergence angle at 1/e2 of a Gaussian beam in a medium
θinc angular increment of a gyroscope
Λ birefringence beat length
λ wavelength in a vacuum
mean wavelength
λc cutoff wavelength
λe emission wavelength
λm wavelength in a medium
μ0 magnetic permeability of a vacuum (4π × 10–7H ⋅ m–1)
μr relative permeability of a medium
v Poisson ratio
Π Poynting vector
π 3.14159...
ρ electrical charge density
ρcr cross-coupling polarization intensity
Σ summation
σ spatial frequency
mean spatial frequency
1σ root-mean-square value or standard deviation
τ temporal delay
τc coherence time, or characteristic time
τdc decoherence time
τg group delay
τϕ phase delay
ϕ phase
slope of analog phase ramp
ϕ b phase bias
ϕ m phase modulation
ϕ PR phase ramp
ϕ s phase step
χe dielectric susceptibility
χm magnetic susceptibility
χμvσρ constitutive tensor
Ω rotation rate
Ω rotation rate vector
Ωπ rotation rate inducing a π radian phase difference
Ωμ rotation rate inducing a 1-μrad phase difference
ω angular frequency
mean angular frequency
ωe emission angular frequency
< > temporal averaging
< ⎪ > generalized scalar product of complex functions
⎪ ⎪ absolute value
∇ vector differential operator
About the Author
Hervé C. Lefèvre is the chief scientific officer at iXBlue (www.ixblue.com) in France. He
graduated from the Ecole Normale Supérieure de Saint-Cloud in Physics and was awarded a
Doctorate in Optics-Photonics from the University of Paris-Orsay in 1979. His doctorate research
was performed at Thales (formerly Thomson-CSF) Central Research Laboratory and his thesis
subject was pioneering work on the fiber-optic gyroscope.
From 1980 to 1982, he was a postdoctoral research associate at Stanford University in
California, continuing R&D on the fiber-optic gyroscope. In 1982, he came back to the Thales
Central Research Laboratory and became head of the fiber-optic sensor team. In 1987, he joined
Photonetics, which was then a start-up, and became director of R&D. In addition to fiber-optic
sensors and gyroscopes, the company developed a very successful line of test instruments for
optical fiber communications. As the company grew, he moved to the position of chief operating
officer in 1999. At the end of 2000, Photonetics was acquired by the Danish group Nettest while
its fiber-gyro activity was spun out to create iXSea. Dr. Lefèvre remained in Nettest and managed
its Photonics division.
In 2004, he moved to iXCore, the parent company of iXSea, as the vice president for R&D with
a specific involvement in iXSea’s activity, and became the chief scientific officer of iXBlue at its
creation in 2010 as a merging of iXSea and several other subsidiaries of iXCore.
Dr. Lefèvre was also the president of the French Society of Optics (SFO) for the 2005–2007
period and the president of the European Optical Society (EOS) for the 2010–2012 period.
He has authored and coauthored more than 80 journal and conference publications about the
fiber-optic gyroscope and its related technologies and has been granted more than 50 patents.
He was awarded the Prize Fabry-de-Gramont in 1986 by the French Society of Optics (SFO)
and the Prize Esclangon in 1992 by the French Society of Physics (SFP).
Index
μ-metal shielding, 111, 159
Ωπ, 20–21
3x3 coupler, 165–166
A
Accelerometer, 1, 367–370, 373
Acoustic noise, 105
Acousto-optic modulator (AOM), 120–122
Aether, 7–8
Airy coefficient, 242
Airy pattern, 250
Allan
deviation, 27–30
variance, 27–30
All-digital processing method, 131–136, 153
All-fiber configuration, 53–54, 166–167, 198
Amplified spontaneous emission (ASE), 156–157, 207–208, 327
Angular increment, 16–17, 124, 130–131
Angular random walk (ARW), 25–26
Application specific integrated circuit (ASIC), 162
Attitude and heading reference system (AHRS), 368, 373–374
B
Backreflection, 57–58, 65–66, 333
Backscattering, 68–72, 190
Bandwidth, 31
Beat length, 76, 279, 339
Bending-induced birefringence, 303, 321
Bias drift, 26–28, 370
Bias noise, 25
Biasing modulation, 38–47
Birefringence, 267–272, 300–305
Bragg cell, 120–122
Bragg grating, 246, 324–325
Brewster incidence, 225–226
Brillouin optical time domain analysis (BOTDA), 329
Brillouin scattering, 329
C
Channeled spectrum, 145–147, 239–240, 256
Channel waveguide, 331–332
Chromatic dispersion, 218–219
Circular birefringence, 272
Circulator, 37, 153, 325–326
Closed-loop operation, 120–139, 152
Coherence, 254–263
length, 255, 261–262
function, 257, 260–261
Coherent detection, 89, 235
Combination of linear and circular birefringences, 109–110, 304–305
Contrast, 234–236
Coupler, 47–49, 318–321
Coupling loss, 293–300, 335–337
D
Deadband (or dead zone), 17, 139–140
Decibel (dB), 203–204
relative to 1 mW (dBm), 203
Decoherence length, 82, 255–257, 262–263
Degree of polarization, 85
Depolarization, 75–77
length, 76, 81, 273, 314
Dielectric waveguidance, 228–229
Diffraction, 248–254
grating, 246–248
Digital demodulation, 131–134, 153
Doppler effect, 12–13, 18
Dual phase ramp, 171
E
Eigen frequency, 41–47, 152, 281
Elasto-optic coefficients, 302
Electronic coupling, 139, 161–162
Electro-optic effect, 336, 340
Elliptical core fiber, 301–302
Equivalent index, 288–289
Erbium-doped fiber amplifier (EDFA), 326–327
Evanescent field coupler, 47–49, 316–319, 336–337
Excess RIN, 206–208
compensation, 157–158
F
Faraday effect, 107–111, 191, 324–
326
Far field, 250–251, 254
Few-mode fiber, 291–293
Fiber Bragg grating (FBG), 246, 323–324
Field-programmable gate array (FPGA), 162
Finesse, 19, 175, 243–244
Focal length, 230–231
Fourier ’s law of heat transfer, 101
Four-state modulation, 136–139
Fraunhofer diffraction, 248–250, 350
Free spectral range (FSR), 145–147, 239–240, 243–244, 256
Fresnel coefficients, 224–225
Fresnel diffraction, 250–252, 349–350
Fresnel-Fizeau drag effect, 7–8, 13–15, 17
Fringe visibility, 234, 260
Fundamental LP01 mode, 276–279, 285–290
Fused coupler, 318–319
Fused quartz, 275
G
Gaussian beam, 252–254
Geometrical optics, 229–232
Gimbaled navigation system, 367
Ground loop, 161–162
Group velocity, 218–221
dispersion, 46–47, 220–221
Gyrocompassing, 374–375
H
h-parameter, 75, 279, 308, 310–11
Half-wave plate, 269–270, 321
Heat
diffusion, 100–105
diffusivity, 101
Helium-neon (He-Ne) laser, 17–18, 155, 204
Hollow-core fiber, 198, 327–29
Huygens principle, 34–35
I
Impedance of a vacuum, 222
Index of refraction, 215–18
of silica, 218, 280, 290
of LiNbO3, 338–39
Inertial measurement unit (IMU), 367–69
Inertial navigation system (INS), 368, 370–73
Intensity modulation, 45–46, 351
Interference contrast, 234–36
Interferometer
Michelson, 89–90, 235–36
Mach-Zehnder, 50–51, 237
Fabry-Perot, 240–44
Sagnac, 9, 18–19, 34–35
Young double-slit, 238
Interferometric fiber-optic gyroscope (I-FOG), 20–22, 151–53
Isolator, 324, 326
J
Jones formalism, 182–85, 271
K
Kalman filtering, 369
Kerr effect, 112–16, 190
Knife-edge diffraction, 251, 349
L
Laplacian operator, 210
Lefevre’s loops, 321–22
Lens, 230–31
Lithium niobate (LiNbO3), 336–45
birefringence, 338
indices, 338–39
phase modulation, 336–37, 340–41, 344
thermal expansion, 339
Lithium tantalate (LiTaO3), 344
Lloyd’s mirror effect, 51, 345–47
Lock-in, 17, 139–40
Lyot depolarizer, 61, 87–88, 322–23
M
Magneto-optic effects, 107–12
Maxwell equations, 209, 216, 223, 356–57
Mean wavelength, 141–42, 156–57
Microstructured optical fiber (MOF), 198, 327–29
Minimum configuration, 46, 152
Mode
equivalent index, 288–89
field diameter (MFD), 277, 287
group index, 289–90
group velocity, 289–90
mismatch loss, 297–300
phase velocity, 288–89
Multifunction integrated-optic circuit (MIOC), 57, 153, 160
Multilayer dielectric mirror, 17, 245
Multiple-wave interference, 240–42
Multiplexer, 316–18, 326–27
N
Near field, 250–51, 254
Nonlinear effects, 112–16, 329
Numerical aperture (NA), 228, 276–77, 285–86
O
Open-loop operation, 119, 166–67, 198
Optical coherence-domain polarimetry (OCDP), 88–93, 152
Optical spectrum analyzer (OSA), 273–74
Optimal length, 23–24
Overlap integral, 294–96
P
Path-matched white light interferometry, 88–93, 152
Performance grades, 2, 196–97, 370
Phase front, 229
Phase ramp
analog, 122–26
digital, 126–31, 152
Phase velocity, 219, 288–89
Photon noise, 22–23, 160, 175, 205–06
Piezoelectric phase modulator, 53, 319–21
PIN diode, 160–61
Planar lightwave circuit (PLC), 331
Plane mirror, 229–30
Pockels effect, 336, 340
Poincaré sphere, 109–10, 304–05
Polarization controller, 53, 321–22
Polarization correlation, 81–84
Polarization crossed-coupling, 308–11
Polarization dependent loss (PDL), 61, 85, 317–18
Polarization extinction ratio (PER), 75, 308
Polarization-maintaining (PM) fiber, 60–61, 152, 306–08
Polarization mode dispersion (PMD), 314–15
Polarization nonreciprocities
amplitude-type, 59–60, 84–87
intensity-type, 59, 77–81
Polarization rejection of proton-exchanged LiNbO3 waveguide, 345–51
Polarizing (PZ) fiber, 315–316
Poynting vector, 222
Propagation equation, 210, 217
Proper frequency, 41–47, 152, 281
Protective coating, 279–81
Proton-exchanged LiNbO3 waveguide, 343–45
Q
Quarter-wave plate, 271–73, 321
R
Rare-earth doped fiber, 326–27
Rate ramp, 28–30
Rate random walk (RRW), 28–30
Rayleigh backscattering noise, 69V72, 190
Rayleigh scattering, 69–70, 190, 277
Rayleigh range, 324
Reciprocal
behavior of a beam splitter, 34–35
configuration, 36–38, 152
biasing modulation, 38–41, 152
Reciprocity of wave propagation, 33
Refraction, 223–27
Resonant fiber-optic gyroscope (R-FOG), 18–19, 175–93
Ring interferometer, 9, 18–19, 34–35
Ring-laser gyroscope (RLG), 15–18, 196–97
Ring resonant cavity, 19, 244
S
Sagnac effect
in a vacuum, 8–13
in a medium, 13–15, 151, 361–65
Sagnac interferometer, 9, 18–19, 34–35
Sagnac-Laue effect, 7–8
Sawtooth modulation, 121–22
Scale factor
accuracy, 25–27, 119, 140–47
linearity, 25–27, 119–40
Sensing coil, 158, 160
Serrodyne modulation, 122
Shape-induced birefringence, 300–01
Shupe effect, 3, 95–97
Silica
attenuation, 275–76
index, 218, 280, 290
group index, 290
thermal expansion, 280
Slant interface, 65–66
Snell law, 223–224
Spatial frequency, 201–02
Special relativity, 7, 12, 353–61
State of polarization, 212–15
Strap-down navigation system, 198, 369
Stress-induced birefringence, 301–04
Superluminescent diode (SLD), 154–56, 208
T
T-dot effect, 99–100
TE mode, 331–32
TE polarization, 228–29
Temperature dependence of
fiber Bragg grating, 323
LiNbO3 indices, 338
PM fiber birefringence, 281
proper frequency, 281
silica index, 280
Vπ of LiNbO3, 143
Temporal frequency, 201–202
Thermal detector noise, 160–61
Thermal expansion of
LiNbO3, 339
silica, 280
Titanium(Ti)-indiffused LiNbO3 waveguide, 340–43
TM mode, 331–32
TM polarization, 228–29
Total internal reflection (TIR), 226–28
Transverse magneto-optic effect, 111–12
Tunable laser, 35, 274
Two-wave interferometry, 232–35
U
Unpolarized source, 61–62, 156, 209
V
VLSB, 127
Vπ, 127, 336, 341
Vπ control loop, 138
Vector differential operator, 209–210
Verdet constant, 107–109
Vibration, 105
W
Waist, 252–54
Wavefront, 229–31
Wavelength, 201–02
control, 143–45
dependence of scale factor, 140–43
multiplexer, 316–19, 326–27
Wave train, 69, 75–77, 263
Wiener-Khinchin theorem, 259
Winding
dipolar, 98
quadrupolar, 98–99, 152
Y
Y-coupler configuration, 54–58, 160, 195
Y-junction, 49–53, 33