Ring Interferometry 9783110277920, 9783110277241

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Grigorii B. Malykin, Vera I. Pozdnyakova Ring Interferometry

De Gruyter Studies in Mathematical Physics

Editors Michael Efroimsky, Bethesda, Maryland, USA Leonard Gamberg, Reading, Pennsylvania, USA Dmitry Gitman, São Paulo, Brazil Alexander Lazarian, Madison, Wisconsin, USA Boris Smirnov, Moscow, Russia

Volume 13

Grigorii B. Malykin, Vera I. Pozdnyakova

Ring Interferometry Translated by Alexei Zhurov

Physics and Astronomy Classification Scheme 2010 02.20.-a, 02.60.Cb, 02.70.Uu, 03.30.+p, 03.75.Dg, 05.10.Ln, 05.40.-a, 05.40.Ca, 07.60.Ly, 07.60.Vg, 42.15.-i, 42.25.Dd, 42.25.Hz, 42.25.Ja, 42.25.Kb, 42.25.Lc, 42.50.Wk, 42.65.Hw Authors Dr. Grigorii B. Malykin Russian Academy of Sciences Institute of Applied Physics Ul’yanov Street 46 603950 Nizhny Novgorod Russian Federation [email protected] Dr. Vera I. Pozdnyakova Russian Academy of Sciences Institute for Physics of Microstructures GSP-105 603950 Nizhny Novgorod Russian Federation [email protected]

ISBN 978-3-11-027724-1 e-ISBN 978-3-11-027792-0 Set-ISBN 978-3-11-027793-7 ISSN 2194-3532

Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. © 2013 Walter de Gruyter GmbH, Berlin/Boston Typesetting: le-tex publishing services GmbH, Leipzig Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen Printed on acid-free paper Printed in Germany www.degruyter.com

Contents List of abbreviations List of notations 1

Introduction

xi xiii 1

2 Fiber ring interferometry 8 8 2.1 Sagnac effect. Correct and incorrect explanations 8 2.1.1 Correct explanations of the Sagnac effect 2.1.1.1 Sagnac effect in special relativity 8 9 2.1.1.2 Sagnac effect in general relativity 2.1.1.3 Methods for calculating the Sagnac phase shift in anisotropic media 9 10 2.1.2 Conditionally correct explanations of the Sagnac effect 2.1.2.1 Sagnac effect due to the difference between the non-relativistic gravitational scalar potentials of centrifugal forces in reference frames moving with counterpropagating waves 10 2.1.2.2 Sagnac effect due to the sign difference between the non-relativistic gravitational scalar potentials of Coriolis forces in reference frames moving with counterpropagating waves 10 2.1.2.3 Quantum mechanical Sagnac effect due to the influence of the Coriolis force vector potential on the wave function phases of counterpropagating waves in rotating reference frames 11 2.1.3 Attempts to explain the Sagnac effect by analogy with other effects 11 11 2.1.3.1 Analogy between the Sagnac and Aharonov–Bohm effects 2.1.3.2 Sagnac effect as a manifestation of the Berry phase 12 12 2.1.4 Incorrect explanations of the Sagnac effect 12 2.1.4.1 Sagnac effect in the theory of a quiescent luminiferous ether 2.1.4.2 Sagnac effect from the viewpoint of classical kinematics 13 2.1.4.3 Sagnac effect as a manifestation of the classical Doppler effect from 14 a moving splitter 2.1.4.4 Sagnac effect as a manifestation of the Fresnel–Fizeau dragging effect 15 15 2.1.4.5 Sagnac effect and Coriolis forces 2.1.4.6 Sagnac effect as a consequence of the difference between the orbital angular momenta of photons in counterpropagating waves 16 2.1.4.7 Sagnac effect as a manifestation of the inertial properties of an electromagnetic field 16

vi

2.1.4.8 2.1.4.9 2.2 2.2.1 2.2.2 2.2.2.1 2.2.2.2 2.2.2.3 2.2.2.4

2.2.2.5 2.2.2.6 2.2.2.7 2.2.3 2.2.4 2.2.5 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.4.1 2.3.4.2 2.3.4.3 2.3.4.4 2.3.5 2.3.6 2.3.7 2.4 2.5

Contents

Sagnac effect in incorrect theories of gravitation 16 Other incorrect explanations of the Sagnac effect 17 Physical problems of the fiber ring interferometry 17 Milestones of the creation and development of optical ring interferometry and gyroscopy based on the Sagnac effect 17 Sources for additional nonreciprocity of fiber ring 20 interferometers General characterization of sources for additional nonreciprocity of fiber ring interferometers 20 21 Nonreciprocity as a consequence of the light source coherence Polarization nonreciprocity: causes and solutions 21 Nonreciprocity caused by local variations in the gyro fiber-loop parameters due to variable acoustic, mechanical, and temperature actions 23 Nonreciprocity due to the Faraday effect in external magnetic 23 field Nonreciprocal effects caused by nonlinear interaction between counterpropagating waves (optical Kerr effect) 23 Nonreciprocity caused by relativistic effects in fiber ring interferometers 24 Fluctuations and ultimate sensitivity of fiber ring 24 interferometers Methods for achieving the maximum sensitivity to rotation and processing the output signal 25 Applications of fiber optic gyroscopes and fiber ring interferometers 26 Physical mechanisms of random coupling between polarization 28 modes Milestones of the development of the theory of polarization mode linking in single-mode optical fibers 28 30 Phenomenological models of polarization mode coupling Physical models of polarization mode coupling 31 32 Inhomogeneities arising as a fiber is drawn Torsional vibration 32 33 Longitudinal vibration 33 Transverse vibration Transverse stresses 34 34 Inhomogeneities arising in applying protective coatings 34 Inhomogeneities arising in the course of winding Rayleigh scattering: the fundamental cause of polarization mode coupling 35 35 Application of the Poincaré sphere method. . . Thomas precession. Interpretation and observation issues 36

Contents

3 3.1 3.1.1 3.1.2

3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.4

vii

Development of the theory of interaction between polarization modes 38 Phenomenological estimates of the random coupling 38 38 Small perturbation method Expanding the scope of the small perturbation method by partitioning the fiber into segments whose length is equal to the depolarization length 40 A physical model of the polarization mode coupling 41 A model of random inhomogeneities in SMFs with random twists of 41 the anisotropy axes Connection between the polarization holding parameter and statistics of random inhomogeneities 42 Polarization holding parameter in the case of random and regular twisting 45 Statistical properties of the polarization modes for fibers with random 47 inhomogeneities Evolution of the degree of polarization of nonmonochromatic light 55 55 Small perturbation method A method for modeling random twists 57 A mathematical method for modeling random twists in the presence of 63 a regular twist Analytical calculation of the limiting degree of polarization of nonmonochromatic light 68 Increasing of the correlation length of nonmonochromatic light traveling through a single-mode fiber with random inhomogeneities 69 72 Anholonomy of the evolution of light polarization

4.5 4.6 4.7

76 Experimental study of random coupling between polarization modes A rapid method for measuring the output polarization state 76 Method for measuring the polarization beat length and 79 ellipticity Experimental comparison of the accuracy of different methods 86 Influence of winding of single-mode fibers on the amount of 89 the polarization holding parameter Experimental study of the polarization degree evolution of light 92 93 Method of fabricating ribbon single-mode fibers 95 Method for removing the effect of photodetector dichroism

5 5.1 5.2

Fiber ring interferometers of minimum configuration 98 98 Polarization nonreciprocity of fiber ring interferometers 107 Fiber ring interferometers with a single-mode fiber circuit. . .

4 4.1 4.2 4.3 4.4

viii

5.3 5.3.1 5.3.2 5.3.3 5.3.4

5.3.5 5.3.6 5.3.7 5.3.8 5.3.8.1 5.3.8.2 5.3.9 5.4

6 6.1 6.2 6.2.1 6.2.2 6.2.3 6.3 6.3.1 6.3.2 6.3.3 6.3.4 6.3.5

7 7.1 7.1.1

Contents

Zero shift, deviation, and drift of fiber ring interferometers 110 Applicability conditions for the ergodic hypothesis 110 Influence of the amount of random twist of the fiber 131 131 Influence of the location of the random inhomogeneity Influence of the mutual coherence of nonmonochromatic light in the main and orthogonal polarization modes at the point of inhomogeneity 132 Approximate calculation of the temperature zero drift 132 Calculation of the zero shift deviation of the FRI by the small 136 perturbation method Calculation of the zero shift deviation with the extended small perturbation method 139 Calculation of the zero shift deviation by the method of mathematical modeling of random inhomogeneities 139 140 Zero shift deviation of an FRI with a high-birefringence fiber 142 Zero shift deviation of an FRI with a low-birefringence fiber Calculation of the zero shift deviation of FRIs 144 Domains of application of the different methods for calculating 146 PN Fiber ring interferometers of nonstandard configuration 148 148 New type of nonmonochromatic light depolarizer for FRIs 156 Zero drift and output signal fading in an FRI with a polarizer Small perturbation method. The quasi-axis model 156 157 Extended small perturbation method Method of mathematical modeling of random inhomogeneities in fibers 158 163 Fiber ring interferometers without a polarizer 164 FRIs with circularly polarized input light Modulation method for removing the zero shift in a fiber ring interferometer without a polarizer 167 Fiber ring interferometer with a depolarizer of nonmonochromatic light 169 Fiber ring interferometer with a circuit made from a uniformly twisted 170 fiber Zero shift deviation in FRIs without a polarizer and with a circuit made from a high-birefringence fiber in a limited temperature range 171 172 Geometric phases in optics. The Poincaré sphere method Application of the Poincaré sphere method 172 Analysis of the properties of the Pancharatnam phases. The Poincaré 172 sphere

Contents

7.1.1.1 7.1.1.2 7.1.2 7.1.2.1 7.1.2.2 7.1.2.3 7.1.3 7.1.3.1 7.1.3.2 7.1.3.3 7.2 7.3 7.4 7.5 7.5.1 7.5.2

8 8.1 8.1.1 8.1.2 8.1.3 8.2 8.3 8.4

9 9.1 9.1.1

ix

Type I Pancharatnam phase 172 Type II Pancharatnam phase 173 Birefringence in SMFs due to mechanical deformations 175 175 Kinematic phase in SMFs Bending induced linear birefringence of SMFs 176 Twisting-induced circular birefringence of SMFs. The spiral polarization 176 modes Rytov effect and the Rytov–Vladimirskii phase in SMFs and FRIs in the case of noncoplanar winding 177 177 Rytov effect in the FRI circuit fiber Rytov–Vladimirskii phase and PP2 in SMFs with noncoplanar winding 179 180 Rytov phase detection in FRIs Polarization nonreciprocity in FRIs. Nonreciprocal geometric phase 182 Determination of a polarization state ensuring the absence of NPDCM 189 Criticism of unsubstantiated hypotheses relating to geometric 191 phases Opto-mechanical analogies relating to light propagation in SMFs 195 The analogy between the Rytov effect polarization optics and Ishlinskii effect in classical mechanics 195 An opto-mechanical analogy of an SMF with twisting of the linear 198 birefringence axes Time-dependent, nonlinear, and magnetic effects 201 Influence of the second harmonic of the phase modulation 201 frequency In-phase and quadrature components of the parasitic phase modulation 201 203 Numerical estimates of the incidental phase modulation Optimal harmonic of the phase modulation frequency 206 Experimental investigation of the piezo transducer’s 207 nonlinearity Methods for removing the influence of the nonlinear Kerr effect Influence of random inhomogeneities on the Faraday zero shift 215 deviation

209

Relativistic effects in optical and non-optical ring interferometers 220 220 Sagnac effect for waves of any nature in special relativity 220 Sagnac effect in the laboratory frame of reference

x

9.1.2 9.2 9.2.1 9.2.1.1 9.2.1.2 9.2.1.3 9.2.2 9.3 9.3.1 9.3.1.1 9.3.1.2 9.3.1.3 9.3.1.4 9.3.2 9.4 9.4.1 9.4.2

10 Index

Contents

Sagnac effect in a rotating frame of reference. Zeno’s relativistic paradox 223 Non-optical Sagnac sensors of angular velocity 226 A ring interferometer based on slow acoustic or magnetic waves 226 226 Advantages of using slow waves in ring interferometers Choosing an optimal frequency of the slow waves in ring interferometers 227 A method for detecting the phase difference between 229 counterpropagating waves in slow-wave ring interferometers A ring interferometer based on de Broglie waves of pions 232 236 Influence of Thomas precession on the zero shift Thomas precession as a corollary of Ishlinskii’s solid angle theorem applied to the angle of relativistic aberration 236 236 Thomas precession Ishlinskii’s theorem as a classical analogue of Thomas precession 237 Observed rotation of an object rapidly moving in a circular path and 238 Thomas precession Physical meanings of the Thomas precession and Ishlinskii angle 241 Influence of Thomas precession on the zero shift of ring interferometers based on de Broglie waves of matter particles with spin 241 243 Potential usage of FRIs for detecting fundamental effects Verification of the basic postulates of special and general relativity using FRIs 243 Analysis of the possibility of detecting nonreciprocal effects with 246 FRIs Conclusion 299

250

List of abbreviations For the reader’s convenience, we give a list of abbreviations frequently used throughout the book: CTE CVC DP EMONLB FOG FRI GP GR GRL GRT KP NGPCM NPDCM PFRI PMD PN PP PSR RA RVP SD SFLS SL SMF SPM SR STR TP

coefficient of thermal expansion current–voltage characteristic dynamic phase electromagnetic optical nonreciprocity linear birefringence fiber optic gyroscope fiber ring interferometer geometric phase general relativity gas ring laser general theory of relativity (same as general relativity) kinematic phase nonreciprocity geometrical phase of counterpropagating modes nonreciprocity phase difference of counterpropagating modes polarization fiber ring interferometer polarization mode dispersion polarization nonreciprocity Pancharatnam phase polarization state of radiation Rytov angle Rytov–Vladimirskii phase superluminescent diode superfluorescent fiber light source semiconductor laser single-mode optical fiber spiral polarization modes special relativity special theory of relativity (same as special relativity) Thomas precession

List of notations Below is a list of main notations used throughout the monograph: a Ax,y ABC α α α

αk αRyt 2α b β β0 βc βc βe βEM βH βind βk βK βP c ◦

C

d D D± dΦ DΦ δ δn± δt ΔF

semi-major axis of the polarization ellipse amplitudes of the normalized Jones vector components Ex,y spherical triangle on the Poincaré sphere rotation angle of the SMF axes (azimuth of the major axis of a natural polarization mode); Chapters 3 to 7 detection ratio; Chapter 9 angle by which a body rotates when it completes one revolution around a circle due to Thomas precession (α = χ , where χ is the solid angle); Chapter 9 rotation angle of the SMF axes at the output of the kth segment total Rytov angle for N coils of a noncoplanar SMF longitude on the Poincaré sphere, twice the azimuth of the major axis of a natural polarization mode; Chapter 7 semi-minor axis of the polarization ellipse linear birefringence of an SMF unperturbed (intrinsic) linear birefringence of an SMF circular birefringence of an SMF effective circular birefringence due to the Rytov effect elliptical birefringence of an SMF electromagnetic nonreciprocal linear birefringence (EMNLB) nonreciprocal circular birefringence due to the Faraday effect winding-induced linear birefringence of an SMF birefringence of the kth segment of an SMF linear birefringence due to the Kerr effect linear birefringence due to the Pockels effect speed of light in vacuum degree Celcius outer diameter of an SMF winding diameter of an SMF Jones matrix of depolarizer for opposing directions of light propagation eikonal increment eikonal increment for a wave propagating in a single direction along a closed path phase difference between the light components that has traveled along the slow and fast axes of an SMF nonlinear correction to the unsaturated refractive index of the SMF core due to the nonlinear optical Kerr effect temperature quasiperiod radio range resonance width

xiv ΔH Δθ Δλ Δn Δnc Δnd ΔnH ΔP Δt Δt  Δt ± ΔϕH 0 Δϕn

ΔΩ e Ex,y ε f F g γ γ γ γ Γ h ˜ h

ˆ h

List of notations

half-width of ferromagnetic resonance aberration angle spectral linewidth of nonmonochromatic light at half maximum refractive index difference between the slow and fast axes of an SMF refractive index difference between the slow and fast axes of the FRI circuit SMF refractive index difference between the slow and fast axes of the depolarizer SMF refractive index difference between circular modes in magnetic field optical power difference between counterpropagating waves in the FRI circuit travel time difference for counterpropagating waves in a rotating ring interferometer in the laboratory frame of reference travel time difference for counterpropagating waves in a rotating ring interferometer time changes for counterpropagating light waves around the ring in the laboratory frame of reference nonreciprocal phase difference between counterpropagating waves in magnetic field constant output phase difference between counterpropagating waves due polarization nonreciprocity zero shift (bias) of an FRI expressed in ◦/h unit-length vector electric field components of a light wave extinction ration of a polarizer effective amplitude of phase modulation radio range frequency photoelasticity coefficient intensity of radiation transferred from one orthogonal mode to the other due to imperfections in the SMF; Chapters 3 and 5 Lorentz factor; Chapter 9 Rytov angle for a single coil of a noncoplanar SMF; Chapter 7 magnetomechanical ratio; Chapter 9 rotation angle of the linear birefringence axes in one coil of a noncoplanar SMF polarization holding parameter, spectral density of random perturbations of linear birefringence on the polarization beat length in an SMF polarization holding parameter determined through the energy exchange process between natural polarization modes as they propagate through an SMF spectral power density of the spatial component of random circular birefringence (random twists) whose period corresponds to the coil length

List of notations

 h2 h3 h4 H H ϑ Θ Θ0 Θk Θmax ΘABC I Iinterf Itotal J Jk k k k

K K K  Knonin K± κ

l l± lcoh ldep lk l L Lb Lw

xv

reduced Planck constant decrement of exponential change in the intensities of orthogonal elliptical polarization modes in regularly twisted SMF decrement of decaying oscillations of the intensities of orthogonal elliptical polarization modes in regularly twisted SMF parameter characterizing the spatial oscillation period of the intensities of orthogonal elliptical polarization modes in regularly twisted SMF magnetic field strength pitch of SMF winding angle between the segments of a two-segment depolarizer or an arc on the Poincaré sphere (determined by the context) twist of an SMF regular twist of an SMF twist of the kth segment of an SMF maximum twist of an SMF solid angle subtended by spherical triangle ABC on the Poincaré sphere light intensity at the output of an FRI interference signal intensity at the FRI output total light intensity at the FRI output light coherence matrix at the SMF output Bessel function of the first kind of order k wave number imaginary part of the wave number of a slow wave wave vector, k = (kx , ky , kz ) kelvin inertial frame of reference inertial frame of reference inertial frame of reference that instantaneous ly accompanies the noninertial frame Knonin noninertial rotating frame of reference Jones matrix of the FRI circuit for opposing directions of light propagation intensity of radiation transferred from one orthogonal mode to the other due to imperfections in the SMF length of a segment of an SMF path length of counterpropagating waves in a rotating ring interferometer relative to the laboratory frame of reference coherence length of nonmonochromatic light in an SMF depolarization length of nonmonochromatic light in an SMF length of the kth segment of an SMF mean segment length of an SMF total length of the SMF in the FRI circuit polarization beat length of an SMF coil length of an SMF

xvi

λ λ0 λm m m0 Mk n ˜± n N Nm No ν ξk p p P Π r R R

ρ s SABC Sk Sm Sϕ (t −1 ) σp t t± T ± (α) τ U  (U ± )Kin

List of notations

light wavelength in vacuum mean wavelength of nonmonochromatic light de Broglie wavelength relativistic mass of matter particle rest mass of matter particle Jones matrix of the kth segment of an SMF refractive index saturated refractive index of the SMF core for opposing waves due to the nonlinear optical Kerr effect integer number of pions entering the pion counter per unit time number of photons entering the photodetector at the FRI output per unit time light source frequency modulation phase of the kth harmonic of frequency F degree of polarization of nonmonochromatic light mean value of polarization degree degree of linear polarization of nonmonochromatic light Jones matrix of polarizer radius vector radius of a ring interferometer; Chapters 2, 8, and 9 parameter characterizing the light intensity transferred from one mode to the other due to imperfections in the single-mode fiber; Chapters 3, 4, and 9 coefficient proportional to the winding-induced linear birefringence βind of an SMF area of the FRI fiber circuit coil projection onto the plane perpendicular to the angular velocity area of spherical triangle ABC on the Poincaré sphere normalized Stokes vector components area of de Broglie wave interferometer spectral density of the phase difference between counterpropagating waves at the FRI output, a function of the temperature frequency root-mean-square deviation of polarization degree from the mean time or temperature (determined by the context) times in which counterpropagating waves travel around the ring in the laboratory frame of reference Jones matrix of rotation by the angle α for opposing directions of propagation of light twist of a noncoplanar SMF per unit length gravitational potential in the inertial frame of reference  gravitational potential in the noninertial frame Knonin

List of notations

Υ v vm vs vφ ± vφ V

V(Lk ) W± ϕ 0 ϕn  ϕn  ϕn

ϕNGPCW f ϕNGPCW s ϕNGPCW (x,y)

ϕnon Φ ΦS χ χ χ3 2χ ψ z ω ω Ω Ω Ω ΩT

xvii

interferometric visibility speed speed of matter particles speed of sound phase velocity of waves in a ring interferometer, no rotation phase velocities of counterpropagating waves in a rotating ring interferometer relative to the laboratory frame of reference Verdet constant interferometric visibility function at the FRI output due to the kth imperfection in the circuit total Jones matrix of an FRI for opposing directions of light propagation phase difference between counterpropagating waves at the FRI output time-varying zeroth-order output phase difference between counterpropagating waves due polarization nonreciprocity time-varying first-order output phase difference between counterpropagating waves due polarization nonreciprocity time-varying second-order output phase difference between counterpropagating waves due polarization nonreciprocity nonreciprocal phase difference between counterpropagating waves in the FRI circuit nonreciprocal phase difference between counterpropagating waves in the FRI circuit for the fast birefringence axis nonreciprocal phase difference between counterpropagating waves in the FRI circuit for the slow birefringence axis nonreciprocal phase difference between counterpropagating waves for orthogonally polarized modes of the FRI circuit eikonal phase difference due to the Sagnac effect ellipticity of a natural polarization mode of an SMF; Chapters 3 to 6 solid angle; Chapter 9 real part of a medium’s nonlinear third-order susceptibility latitude on the Poincaré sphere, twice the ellipticity of a natural polarization mode of an SMF; Chapter 7 phase difference between light rays in the X and Y axes coordinate along an SMF angular frequency of light source angular frequency of orbital motion angular velocity magnitude of a fiber optical gyroscope; Chapters 2, 5, and 9 angular velocity magnitude, Ω = 2π F ; Chapter 8 angular velocity vector of a fiber optical gyroscope; Chapter 7 angular frequency of Thomas precession

1 Introduction The Sagnac effect manifests itself in a rotating ring interferometer, where two waves traveling in opposite directions acquire a relative phase difference directly proportional to the angular speed of the interferometer, the area covered by the interferometer, and the wave frequency. This is a kinematic effect of special relativity (SR), which follows from the relativistic addition of two velocities, the phase velocity of a wave and linear rotational velocity of the interferometer. However, it is noteworthy that some authors still treat the Sagnac effect ambiguously despite its physical simplicity, which includes attempts to reduce it to a known classical effect and attempts to negate, directly or indirectly, the validity of special relativity. The Sagnac effect does not only apply to electromagnetic waves but can also apply to de Broglie and particle waves as well as acoustic, magnetostatic, and other waves. Angular velocity sensors whose operation relies on the Sagnac effect are widely used in gyroscopy and navigation as well as to address a number of other fundamental and applied problems. Currently, Sagnac rotation sensors operating on electromagnetic waves of optical (and near-infrared) range have found wide practical application. These include gas ring lasers (GRL) and fiber ring interferometers (FRI) based on single-mode optical fibers (SMF). When a GRL rotates, counterpropagating waves acquire a frequency difference proportional to the angular velocity. The phenomenon of mutual capture of counterpropagating waves, which is caused by light scattering on various optical elements, is the main factor that restricts the ultimate sensitivity of GRLs. First GRLs appeared about 50 years ago; by now, they have been studied quite well and will not be discussed in what follows. FRIs, which appeared about 30 years ago, have several advantages over GRLs, including the absence of mutual capture of counterpropagating waves, the possibility of determining the rotational direction, a significantly decreased weight, a shorter warm-up time, simplicity of manufacturing and operation, lower production costs, and greater acceleration and vibration resistance. Currently, FRIs are used not only for traditional purposes of gyroscopy and navigation but also in precision optical systems developed for the detection of gravitational waves, verification of basic postulates of special relativity, and discovery of new effects in general relativity. In addition, FRIs are employed in the following fundamental and applied areas: – in geophysics, detecting seismic rotations, seismic waves, and effects caused by gravitational waves, measuring Earth’s rotation period fluctuations, and revealing the effect of the Sun and Moon on the Earth’s rotation; – finding new nonreciprocal optical effects; – measuring polarization mode dispersion in SMFs; – measuring chromatic dispersion and determining the dependence of SMF refractive index on light intensity;

2

– – – – – – – – – –

Introduction

measuring the Lorentz dispersion term in the Fresnel drag coefficient; creating optical filters; non-contact measurements of fluid flow rates and acoustic emissions from a heated surface; creating pressure sensors; monitoring optical surface profiles; creating electric current and magnetic field sensors; developing optical switches for optical fiber distribution networks for extra large arrays of digital data; creating safe (protected) communication systems; developing distributed systems from individual sensors; creating wavelength dependent multiplexers, etc.

Due to the above, optimization of FRI optical systems in order to increase their sensitivity limits to the angular velocity of rotation is an important task. It is also topical to study FRI schemes that are easier to manufacture and, at the same time, provide a sufficiently high accuracy of measurement in some practically important cases. When the FRI was created, the researchers immediately faced a new phenomenon–polarization nonreciprocity (PN). Even in the absence of true nonreciprocal effects, such as the Sagnac effect (no rotation), Faraday effect, Fresnel–Fizeau drag effect, and others, when the conditions of the reciprocity theorem are certainly satisfied, this phenomenon leads to a phase difference between counterpropagating waves at the FRI output. Polarization nonreciprocity is due to the fact that the input radiation, with a given polarization, will generally excite different polarizations in the counterpropagating waves in the FRI circuit. Polarization nonreciprocity is, in a sense, a geometric effect, since the phase difference it causes between counterpropagating waves depends on how the anisotropy axes of the SMF at both ends of the FRI circuit are oriented. The phenomenon restricts the ultimate sensitivity of the FRI. There are a number of other factors that limit the maximum sensitivity of the FRI. These include optical shot noise and thermal noise at the input to the signal processing unit, the nonlinear optical Kerr effect, external magnetic field effects, transient effects associated with nonsymmetric changes in the single-mode fiber optic length of the FRI circuit, non-ideal operation of the phase modulator, Rayleigh scattering effects, and some others. However, the factor that affects the maximum sensitivity of the FRI mostly of all is linear interaction (coupling) between polarization modes occurring at random inhomogeneities of the SMF. The linear polarization-mode coupling causes the natural modes of the SMF to be randomly elliptic; the ellipticity changes its sign in time by a random law due to temperature variations in the fiber and, in addition, the magnitude and sign of the ellipticity on the light wavelength. All of these factors extremely complicate the analytical analysis of the light polarization state, especially in the case of non-monochromatic sources. As applied to the FRI, the linear polarization-mode coupling causes the phase difference of counterpropa-

Introduction

3

gating waves, induced by polarization nonreciprocity, to vary with the temperature of the SMFs in the FRI circuit rather than remain constant. Thus, apart from a zero shift, unrelated to rotation, the FRI acquires a thermal zero drift. Over the years, we carried out theoretical and experimental studies of the linear polarization-mode coupling in single-mode optical fibers as well as its effect on the ultimate sensitivity of fiber optic gyroscopes (FOGs), angular velocity sensors. These studies have been published in [38–40, 48–50, 59, 157, 227–231, 272, 276, 480– 580, 611–613, 615, 617, 618, 902, 931]. The current monograph summarizes this activity. It has been nearly 40 years since the creation of quartz SMFs. During this time, great progress has been made in fiber optic technology: the theoretical limit of optical losses has been reached and optical fibers have been created with virtually zero chromatic dispersion in the operating wavelength range. SMFs have found wide application in optical communication and manufacturing sensors for various physical parameters. Two types of fiber optic sensors, homodyne and interferometric, for measuring different physical parameters are known. Interferometric sensors are most advanced; these include fiber ring interferometer and also Michelson and Mach–Zehnder fiber optic interferometers. This kind of sensor converts the quantity that is measured into a phase change of an optical signal. Despite the great progress in fiber optic technology, as noted above, there is a serious problem that limits the maximum speed of information transmission in fiber optic communication lines as well as the sensitivity of fiber interferometers, in particular, the FRI. In SMFs, there are two mutually orthogonal polarization modes propagating with different speeds and exchanging energy on fiber inhomogeneities. Even if there is only polarization mode excited at the fiber input, both polarization modes will begin to propagate at a certain distance from the entry point, which is mainly determined by the amount of linear birefringence of the fiber. Since the distribution of inhomogeneities along the SMF is random, the amplitude relationships between the orthogonal polarization modes as well as the phase difference are also random. Because the polarization mode coupling has a random character, ultrashort pulses become wider as they propagate through communication lines. Consequently, in fiber optic interferometers (including FRIs), the phase change between the interfering counter waves is random at the output. This random phase change is referred to as a zero drift. In addition, random changes in the state of polarization at the output lead to random variations in the visibility of the interference, which also affects the accuracy of measurements. The random inhomogeneities can arise in SMFs in the process of drawing fibers from a preform, which is entirely dependent on the SMF technology, and in the process of winding the fiber on the sensor coil, which depends on the method of winding. Consequently, it is important to develop methods for monitoring changes in the polarization state and changes in the polarization invariance parameter during the winding. No less important is the task of creating the types of SMF in which the polarization mode coupling practically does not increase for any method of winding.

4

Introduction

The degree of polarization is an important characteristic of non-monochromatic radiation in an optical fiber is. Indeed, the standard deviation of the zero shift in FRIs from the average, which is due to random polarization-mode coupling, is associated with both the degree of polarization at the input of the FRI circuit and the changes in the degree of polarization in the circuit. For example, if a depolarizer of non-monochromatic radiation with an insufficiently long optical length is set up at the input of an FRI with a weakly anisotropic SMF circuit, the degree of polarization can recover inside the circuit, thus leading to a significant increase in the deviation of the FRI zero shift. For this reason, it is important to develop a computational method and carry out experimental studies to analyze the evolution of the degree of polarization as non-monochromatic radiation propagates through an optical fiber. In order to calculate the zero shift deviation for an FRI with an arbitrarily birefringent SMF circuit and assess the evolution of the degree of polarization of non-monochromatic radiation in single-mode fibers, one needs to have an adequate description of the linear polarization-mode coupling in the fibers. This is a very complex mathematical problem, especially for non-monochromatic sources, which suggests that the statistical characteristics of random inhomogeneities in the fibers must be known. This issue has been addressed in a large number of studies, which will be discussed in the literature review; however, all these studies used the method of small perturbations, which only allows one to obtain reasonable results for fibers with strong linear birefringence on quite limited lengths. For single-mode fibers with weak linear birefringence, the method of small perturbations is, as a rule, inapplicable. The development of a rigorous theory of linear polarization-mode coupling required us to construct an adequate model of random inhomogeneities in singlemode fibers that would reflect both the physical nature of the inhomogeneities and their statistical characteristics. In many studies addressing this issue, no physical model of random inhomogeneities is considered at all; instead, a phenomenological approach is employed where the so-called polarization-holding parameter is introduced. The value of this parameter is inversely proportional to the fiber length on which the light intensities of the input polarization modes, excited and unexcited, are approximately equal. Other studies consider a model of random inhomogeneities where a single-mode fiber is represented as a set of randomly oriented phase plates with linear birefringence. This model is clearly incorrect, since the orientation of the birefringence axes can undergo a discontinuity along the fiber. Some studies use the assumption that there are randomly distributed coupling centers of polarization modes along the fiber length; however, no distribution statistics or physical nature of the coupling is specified. It is known from a number of theoretical and experimental studies that if fibers have a regular twist such that the induced circular birefringence significantly exceeds the unperturbed linear birefringence the polarization-mode coupling at random inhomogeneities decreases. However, the dependence of the polarization-holding parameter on the fiber twist per unit length has not been previously obtained.

Introduction

5

Widely used in polarization optics is the Poincaré sphere method. In some cases, this method allows one to calculate, without using the complex Mueller and Jones matrix methods, the polarization state of light at the output of complex optical systems as well as the phase increment in a light beam or phase difference between two light beams. Moreover, the Poincaré sphere method allows one to calculate geometric optical (topological) phases in a fairly simple manner; these phases, associated with the light propagation topology (in an ordinary or parametric space), accumulate in addition to the main phase (per unit length) as light propagates in a certain path. The well-known Rytov phase is an example of a geometric optical phase; it is associated with the propagation of light along a curved line. Another example is the Pancharatnam phase, which is associated with the evolution of the polarization state along the light beam. It is noteworthy that the Poincaré sphere method has not previously been used to calculate the output phase difference caused by conditional polarization nonreciprocity of the FRI circuit. We have developed mathematical techniques to do so. In recent years, major progress has been made in the implementation of Sagnac sensors based on de Broglie waves of material particles. In particular, the sensitivity of recent interferometers based on de Broglie waves of sodium and cesium atoms is already as high as that of the best fiber ring interferometers. Despite the fact that these studies are currently still in the laboratory stage, the sensitivity of de Broglie wave interferometers will significantly exceed that of fiber ring interferometers. This is due to the fact that the de Broglie wavelengths are many orders of magnitude shorter than the optical wavelengths. However, there are a number of obstacles, both technical and fundamental, that limit the ultimate sensitivity of de Broglie wave interferometers. One of the obstacles is the evolution of the quantum mechanical spin state of material particles in counterpropagating beams. This evolution is caused by both the interaction between the electric and magnetic fields (or, respectively, their scalar and vector potentials) and the Thomas precession, the special relativity effect that results in a change of the spin state as a particle moves in a curved line. This required researchers to suggest non-optical Sagnac schemes that would be free from the effects associated with the change of the polarization state of counterpropagating waves. The main objectives of this monograph are to give a comprehensive analysis of the influence of the linear polarization-mode coupling as well as other polarization and phase effects on the ultimate accuracy of recording the Sagnac effect (i. e., the angular velocity of rotation) and to assess the possibility of studying fundamental effects using fiber ring interferometers and some other non-optical Sagnac sensors. To this end, we have done the following: – constructed an adequate physical model of random inhomogeneities in singlemode optical fibers; – determined the dependence of the polarization-holding parameter on the amount of intrinsic linear birefringence of a single-mode fiber, coefficient of photoelasticity of the fiber material, statistical parameters of random inhomogeneities in the fiber, and amount of regular twist of the fiber (if any);

6

–

– –

–

– –

–

– – – – – – –

–

–

Introduction

analyzed the evolution of the polarization degree of non-monochromatic light, based on the model of random inhomogeneities, as the light beam travels through a fiber, including the case of regular twisting of the fiber; determined the asymptotic value of the polarization degree as the fiber length increases indefinitely; evaluated the statistic characteristics of natural polarization modes of singlemode optical fibers with inhomogeneities; investigated different methods for measuring the polarization-holding parameter in fibers, compared the accuracies of the methods, and established the potential area of application of each method; analyzed the influence of fiber winding parameters on the polarization-holding parameter and investigated methods for evaluating the ellipticity of natural polarization modes of the fiber; assessed the possibility of creating single-mode fibers for which the winding would not increase the polarization-mode coupling; computed the parameters of different schemes of fiber ring interferometers with a single-mode fiber circuit with arbitrary unperturbed linear birefringence in the presence of random inhomogeneities, including the case of regular twisting of the fiber circuit; formulated conditions of applicability of the ergodic hypothesis for fiber ring interferometers; for some special cases, derived analytical expressions, by the Jones matrix method, of the zero shift deviation in a fiber ring interferometer due to polarization-mode coupling in the fiber circuit; studied new schemes of simplified fiber ring interferometers and simple methods for removing the zero shift and drift; assessed new, more efficient types of depolarizers of non-monochromatic light for fiber ring interferometers; investigated the phenomenon of polarization nonreciprocity for fiber ring interferometers; suggested using the Poincaré sphere method for calculating the zero shift due to polarization nonreciprocity; analyzed nonlinear and unsteady processes affecting the zero drift; proposed methods for removing or significantly reducing their influence; investigated the physical nature of the Sagnac effect; assessed the possibility of using fiber ring interferometers to detect a number of new fundamental effects, including relativistic ones; worked out requirements for the parameters of the interferometers ensuring a sufficient accuracy of detection; assessed the possibility of creating non-optical Sagnac rotation sensors free from polarization nonreciprocity and analyzed the influence of some effects, including relativistic ones, on the operation of these sensors; found an adequate relation between the orbital speed of a material particle and the Thomas precession frequency.

Introduction

7

We refer the reader who wishes to further explore some of the issues discussed in this book to our papers [38–40, 48–50, 59, 157, 227–231, 272, 276, 480–580, 611–613, 615, 617, 618, 902, 931]. However, before we proceed to discuss the aims and objectives of this monograph, we give a review of the major publications that address the issues related to the topic of the monograph. This allows us to state more clearly the main directions of our research and define its place amongst the studies in this area. Research papers dedicated to experimental methods for measuring various parameters of single-mode fibers and light polarization will be discussed in relevant sections of Chapter 4. We are grateful to our colleagues and coauthors I. A. Andronova, Z. E. Arutyunyan, D. E. Burlankov, A. V. Bychkov, E. M. Dianov, I. S. Emelyanova, V. M. Gelikonov, G. V. Gelikonov, A. B. Grudinin, A. N. Guryanov, D. D. Gusovskii, S. V. Ignatyev, S. V. Kofanov, M. M. Kucheva, S. A. Kharlamov, V. P. Khrulev, D. V. Kutyrev, E. G. Malykin, N. D. Milovskii, I. M. Nefedov, Yu. I. Neimark, G. V. Permitin, E. L. Pozdnyakov, A. E. Rozental, I. A. Shereshevskii, O. B. Smirnov, D. P. Stepanov, G. A. Vugalter, E. I. Yakubovich, and Yu. I. Zaitsev. Our joint work with them have have formed the basis of this book. We would also like to thank V. L. Ginzburg for valuable discussions of some issues in special relativity, in particular, the relativistic Sagnac effect, A. Yu. Ishlinskii for discussing some questions on specific features of topological effects in classical mechanics, S. Walter, V. V. Vladimirskii, Vl. V. Kocharovskii, Yu. A. Kravtsov, V. N. Listvin, V. N. Logozinskii, V. I. Ritus, A. I. Smirnov, Yu. M. Sorokin, F. R. Tangherlini, F. Hasselbach, and A. Chakrabarti for useful discussions of a broad range of issues, and A. M. Sergeev and E. A. Khazanov for their attention to our work. We are thankful to K. V. Rozenberg for measuring the dependence of the scaling factor of a fiber optic gyroscope on the rotation time, S. E. Mozharovskaya and D. V. Shabanov for their help in work, G. V. Kolesnikova, T. N. Fedotkina, and E. Yu. Shnyrova for their help in finding some publications, and V. V. Vorontsova, Z. G. Malykina, S. N. Novikova, T. I. Sokolova, L. F. Chaplygina, M. N. Sharov, and E. N. Yalymova for their assistance in making figures. We also wish to thank the reviewers who helped improve this monograph, M. Efroimsky for encouraging discussions, A. I. Zhurov for translating the book into English and useful remarks, and the DeGruyter Publishing House for the speedy and efficient production. The research presented in this monograph was partially supported by the Russian Foundation for Basic Research (project Nos. 94-02-03916, 96-02-18568, 96-15-96742, 99-02-16265, 00-15-96732, and 00-02-17344) and Russian Federation President’s Council for Support of Leading Scientific Schools (project Nos. NSh1622-2003-2, NSh7738-2006-2, NSh1931-2008-2, NSh3800-2010-2, and NSh-5430.2012.2).

2 Fiber ring interferometry based on the Sagnac effect (literature review) 2.1 Sagnac effect. Correct and incorrect explanations The Sagnac effect [741–743] (see also the reviews [32, 252, 314, 498, 519, 695, 888]) manifests itself in a rotating ring interferometer, where two waves of arbitrary nature traveling in opposite directions acquire a relative phase difference. This is a relativistic kinematic effect [474], which, as shown in Section 9.1 (see also our papers [508, 512, 902]), follows from the relativistic addition of velocities, the phase velocity of an electromagnetic wave and linear rotational velocity of the ring (the platform to which the interferometer is attached). Along with the Michelson–Morley experiments [603, 607], the Sagnac effect represents one of the basic experiments of special relativity [498, 519]. By now, the Sagnac effect has been detected, apart from the optical range waves, for radio waves [100], X-rays [904], and non-electromagnetic, de Broglie waves of matter particles such as electrons [314, 642], neutrons [215, 917], calcium atoms [718], sodium atoms [438], and cesium atoms [302]). It has been shown theoretically that the Sagnac effect must also exist for gamma rays [906] as well as surface acoustic waves, surface magnetostatic waves (the so-called slow waves) [643, 902], de Broglie waves of π -mesons [503], and the Bose gas wave function [198]. It is known that the Sagnac effect can be explained [32, 252, 314, 498, 508, 512, 695, 888], for both optical and non-electromagnetic waves, in completely different ways, including ones negating the theory of relativity. To some extent, this has led to confusion in this issue and even allowed some authors to call the Sagnac effect mysterious [938]. No strict and thorough consideration of the issue from the positions of special relativity has been conducted in the most general case.

2.1.1 Correct explanations of the Sagnac effect An explanation of the Sagnac effect will be said to be correct if it allows one to obtain an exact expression of the phase difference of counterpropagating wave in a rotating ring interferometer without any restrictions on the system parameters, which include the linear rotational velocity of the ring (the platform to which the interferometer is attached), phase velocity of waves (optical and de Broglie), etc. All correct explanations of the Sagnac effect rely of the theory of relativity.

2.1.1.1 Sagnac effect in special relativity Albert Einstein was the first to consider the propagation of waves in a rotating ring interferometer using special relativity in his paper [225], published in 1914, where he

Sagnac effect. Correct and incorrect explanations

9

analyzed the experiments by F. Harress [310]. The papers [23, 184, 209, 210, 234, 320, 426, 428, 431, 433, 435, 473–475, 599–602, 633, 695, 734, 798, 919, 925, 926] also treat the Sagnac effect from the viewpoint of special relativity. However, as noted above, none of these papers considered the most general case, where the interferometer has an optical medium and rotates at an arbitrary angular velocity. The possibility of treating the Sagnac effect within the framework of special relativity was also noted by Ginzburg [282]. The fact that special relativity theory remains valid in non-inertial reference frames was pointed out by Wien [918], Einstein (in communication with Shankland) [778], and Misner, Thorne, and Wheeler [620]. This issue will be discussed in detail in Section 9.1. It should be noted that the very existence of the Sagnac effect refutes a number of theories that are alternative to special relativity. In particular, as shown in our paper [522], the presence of the Sagnac effect refutes Ritz’s ballistic hypothesis [722, 723], according to which the speed of light is added to the velocity of the light source.

2.1.1.2 Sagnac effect in general relativity A classical analysis of the Sagnac effect using general relativity can be found in the course of theoretical physics by Landau and Lifshitz [421], where a metric tensor is invoked to calculate the propagation time difference between counterpropagating waves in a rotating reference frame. If there are no gravitational fields, the elements of the metric tensor are determined in the same manner as in special relativity (e. g., see [474]) using the invariance of the interval. Consequently, the discussion as to whether purely kinematic problems in rotating reference frames in the absence of gravitational fields should be treated using general or special relativity is a pure formality – this is a matter of definition, since all calculations within general relativity [421] and special relativity [474] are mathematically identical. If there are inhomogeneous gravitational fields or if the ring interferometer rotates with angular acceleration, the Sagnac effect must be analyzed using general relativity [86, 242, 396, 790, 895–897]. If there is an optical medium, especially if the dispersion of the refractive index must be considered, it is much more complicated to calculate the Sagnac effect using general relativity than special relativity; this complexity may become a source of errors (e. g., see [790, 791, 894, 899]). Langevin [425] and Silberstein [798] were the first to consider the Sagnac effect using general relativity in 1921. Subsequently, this approached was also used in the papers [27, 29, 65, 72, 86, 158, 176, 242, 243, 245, 317, 372, 421, 694, 696, 790, 813, 817, 855, 894, 897, 899].

2.1.1.3 Methods for calculating the Sagnac phase shift in anisotropic media Maxwell’s equations are often used to evaluate the Sagnac phase shift for electromagnetic waves [31, 54, 176, 242, 317, 372, 396, 443, 623, 695, 698, 755, 764, 768, 784, 789, 790, 859, 884, 894–899, 934]. In each specific case, the calculations are based

10

Fiber ring interferometry

on either special relativity [695, 698, 755, 789, 859, 884] or general relativity [31, 176, 242, 317, 372, 396, 443, 623, 755, 764, 768, 784, 790, 859, 884, 894–899, 934]. In [764], the usage of Maxwell’s equations is combined with an incorrect approach to calculating the Sagnac phase shift. It is noteworthy that if the optical medium used in the ring interferometer is anisotropic, this fact is much easier to take into account with the Jones matrix method [794], which is derived from Maxwell’s equations.

2.1.2 Conditionally correct explanations of the Sagnac effect An explanation of the Sagnac effect will be said to be conditionally correct if it allows one to obtain an approximate expression of the phase difference of counterpropagating wave in a rotating ring interferometer provided there are some restrictions on the system parameters (linear rotational velocity of the ring, phase velocity of waves, etc.). All conditionally correct explanations of the Sagnac effect rely on analyzing how the Newtonian non-relativistic scalar and vector potentials of the equivalent gravitational field generated by the inertia (centrifugal and Coriolis) forces affect the time dilation in the rotating reference frame or the phase change of a matter particle’s wave function.

2.1.2.1 Sagnac effect due to the difference between the non-relativistic gravitational scalar potentials of centrifugal forces in reference frames moving with counterpropagating waves As shown in our paper [508], rotating (non-inertial) frames of reference moving with the points of a fixed phase (phase fronts) of counterpropagating waves can be replaced, in accordance with the equivalence principle [421, 855], with equivalent inertial frames in which fictitious gravitational fields arise that create apparent gravitational forces coinciding in magnitude and direction with the centrifugal forces in the non-inertial frames. Since the velocity magnitudes of counterpropagating waves are different, the fictitious gravitational fields cause different time dilations for these waves [222, 656, 855]. The calculation method based on this approach is valid only if the ratio of the non-relativistic gravitational scalar potential of centrifugal forces to the speed of light squared is much smaller than unity [421].

2.1.2.2 Sagnac effect due to the sign difference between the non-relativistic gravitational scalar potentials of Coriolis forces in reference frames moving with counterpropagating waves With the scalar gravitational potential, one can also calculate the Sagnac phase difference in an inertial frame of reference equivalent to a rotating non-inertial frame. In the former, the phase velocities of counterpropagating waves are equal in magnitude

Sagnac effect. Correct and incorrect explanations

11

and, apart from centrifugal accelerations, the waves experience Coriolis accelerations equal in magnitude. The Coriolis acceleration is codirectional with the centrifugal acceleration if the wave front moves in the direction of rotation and is contradirectional otherwise. Note that the Coriolis force in non-potential, just like the Lorentz force. These forces are gyroscopic [841] or solenoidal [838], which means that they do not perform work, since they are always perpendicular to the body velocity. Nevertheless, it is possible to introduce a scalar gravitational potential of the Coriolis force, which is mathematically analogous to the Lorentz force [179, 659, 771] inside a solenoid. With this potential, one can evaluate the Sagnac phase shift up to terms of the order of v 2/c 2 [508].

2.1.2.3 Quantum mechanical Sagnac effect due to the influence of the Coriolis force vector potential on the wave function phases of counterpropagating waves in rotating reference frames Unlike the scalar potential, the vector potential of the Coriolis force can be introduced in a perfectly correct way. In accordance with the quantum mechanical laws, a vector potential affects the phase of a wave function. Calculations of the phase difference between counterpropagating de Broglie waves can be found in [27–29, 314, 321, 752, 917]. The calculations are based on solutions of appropriate Schrödinger, Dirac, and Klein–Gordon equations [752]. In addition, the calculations are often performed in the Wentzel–Kramers–Brillouin (WKB) approximation [28, 314, 917]. The resulting expression of the phase difference for de Broglie waves coincides, up to small relativistic corrections, with that obtained using special relativity for waves of arbitrary nature.

2.1.3 Attempts to explain the Sagnac effect by analogy with other effects Drawing an analogy between different effects does not mean attempting to reduce one effect to another. The purpose of an analogy is to clarify the physical meaning of one effect by comparing it with another, simpler one or an effect that is better known and easier to understand. Consequently, an explanation by analogy cannot be considered incorrect.

2.1.3.1 Analogy between the Sagnac and Aharonov–Bohm effects There are a number of papers [28, 65, 210, 244, 311, 312, 321, 451, 752, 817, 929] that draw an analogy between the Sagnac effect and Aharonov–Bohm effect [5, 657, 803]. However, this analogy is quite formal and superficial; the similarity between the two effects is that in either case, the vector potential of non-potential gyroscopic forces has an impact on the phase of the wave function. There is a significant difference be-

12

Fiber ring interferometry

tween these effects; specifically, the Aharonov–Bohm effect can only exist for quantum objects and vanishes for macroscopic bodies, whereas the Sagnac effect is observed for both quantum and macroscopic objects.

2.1.3.2 Sagnac effect as a manifestation of the Berry phase The papers [32, 65, 209, 314, 387] treat the Sagnac effect as a manifestation of the Berry phase [98], which is a topological phase (see also the reviews [6, 30, 97, 130, 387, 891]). In order to understand the difference between the ordinary phase and Berry phase, let us have a look at this simple example. The characteristic feature of the Berry phase for waves of arbitrary nature is that the visibility of the interference pattern is not impaired no matter how much the phase increases [244]. At the same time, as the ordinary phase difference increases the visibility of the interference pattern deteriorates, because the actual line width is always finite. In fiber ring interferometers, as reported in [150], the interference pattern deteriorates as the angular velocity of rotation increases, which indicates that the Sagnac effect cannot be topological. Thus, although there are surely some analogies between the Sagnac and Aharonov–Bohm effects and Berry phase, the similarity is quite formal and does not reflect the physical meaning of these effects.

2.1.4 Incorrect explanations of the Sagnac effect The vast majority of incorrect explanations of the Sagnac effect stem from ignoring or a direct denial of the theory of relativity as well as from trying to reduce this kinematic effect of special relativity to another, well-known effect of classical physics. Some of the incorrect explanations result from a misunderstanding of the theory of relativity or mistakes in the analysis.

2.1.4.1 Sagnac effect in the theory of a quiescent luminiferous ether Historically, the explanation based on the assumption that there exists a quiescent light-bearing ether was the first [498]. The resulting time difference between counterpropagating waves is accurate up to small relativistic corrections. This approach to calculate the Sagnac phase difference was employed by Lodge [465, 466], Michelson [604–606], and Sagnac [741–743], who were strong supporters of the theory of luminiferous (light-bearing) ether, as well as Silberstein [798]. Approaches based on considering the Sagnac effect from the viewpoint of luminiferous ether have not completely become part of the history of physics – such studies are still being published in journals today. For example, see the fairly recent studies [411, 412, 586, 924]. It is essentially this approach that was used in some review papers [56, 96] and textbooks [155, 647].

Sagnac effect. Correct and incorrect explanations

13

So the question arises as to why the obviously false assumption of the existence of luminiferous ether can lead to a correct, up to small relativistic corrections, result. The question can be answered as follows: (i) the assumption of an ether that is not carried along by a rotating interferometer leads to the conclusion that the ether is quiescent in the non-rotating (laboratory) frame of reference, thus suggesting that the speed of light is constant in any direction relative to this frame, which does not contradict the special theory of relativity (a non-rotating frame of reference is inertial); (ii) if the interferometer involves an optical medium, one uses Fresnel’s dragging coefficient [255], which can be obtained from the relativistic velocity addition law as a first approximation [224, 225]. This explanation, however, contains an inherent contradiction. For the Sagnac effect to exist, one has to assume that the ether is nether carried along by the rotation of the interferometer, nor by the Earth’s rotation [605, 606]. At the same time, the negative results of the Michelson–Morley experiments [603, 607, 608] and subsequently repeated experiments (see the review papers [252, 888]) can be explain within the concept of a luminiferous ether only if the ether is completely carried along by the Earth’s motion. Thus, as noted by Vavilov [888] back in 1956, the concept of a luminiferous ether leads to conflicting requirements: the ether must be entirely carried along by the Earth’s translational motion and absolutely not carried along by its rotation. Some of the advocates of the luminiferous ether theory assume that the ether is gradually dragged into rotation by a ring interferometer. If this assumption were correct, the phase difference between counterpropagating waves in a rotating ring interferometer would gradually reduce with time and vanish in the long run. Equivalently, this means that the so-called scaling factor of a fiber optic gyroscope, which relates the output signal with the angular speed, should slowly decrease with time as the gyroscope rotates. At the request of one of the authors (G. B. Malykin), this assumption was verified experimentally by the Joint-Stock Company “Fizoptika” (Moscow, Russia) using a commercially available fiber optic gyroscope VG910F. The measurement results, reported in [515], showed the failure of the assumption that the ether is dragged into motion by a rotating ring interferometer.

2.1.4.2 Sagnac effect from the viewpoint of classical kinematics In order to evaluate the amount of the Sagnac effect for waves of arbitrary nature, some researches use the Galilean velocity addition law to add the velocity of either of the counterpropagating waves to the linear rotational velocity of the device where the waves are separated and combined together. If there is an optical medium with refractive index n that rotates with the interferometer, the resulting expression of the travel time difference between counterpropagating waves coincides, up to small relativistic corrections, with that obtained from special relativity. This approach to evaluating the amount of the Sagnac effect is employed in a number of studies [54,

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Fiber ring interferometry

55, 158, 245, 252, 278, 337, 427, 431, 433, 695, 781] as well as the last two editions of the Russian physical encyclopedia [820, 821]. In case the Sagnac effect is considered for waves propagating through a material medium, such as acoustic and magnetostatic waves (which are called slow waves) [643, 902], the classical kinematic method leads to serious errors. For example, Newburgh et al. [643], utilized the Galilean law of velocity addition to arrive at the wrong conclusion that there is no Sagnac effect for slow waves. It should also be noted that results of [643] contradict the results obtained in Section 9.2.1 within special relativity. The analysis of the Sagnac effect for de Broglie waves of matter particles using the classical kinematic approach also leads to the null result [210]. The errors arising from using the classical kinematic approach to analyze the Sagnac effect are the result of choosing the Galilean rather than relativistic law of velocity addition. In optics, using Fresnel’s dragging coefficient corresponds, in the first approximation, to the relativistic velocity addition law; the resulting expression is valid up to small relativistic corrections, as noted above. In all other cases, the classical kinematic approach leads to incorrect results.

2.1.4.3 Sagnac effect as a manifestation of the classical Doppler effect from a moving splitter This explanation is based on treating the rotating beam-splitting mirror (splitter) of a ring interferometer in a fixed frame of reference as a moving light source that emits a shorter wave in the rotation direction and a shorter wave in the opposite direction as compared with the case where there is no rotation. The advocates of this approach believe that there are different numbers of wavelengths per unit length of the ring in counterpropagating waves, which creates a phase shift in the output interference pattern. This approach is fundamentally incorrect, since the source and receiver must move relative to each other for the Doppler effect to arise. In this case, however, the splitting mirror is simultaneously the source and receiver and so cannot move relative to itself. In [508], we demonstrated the incorrectness of using the Doppler effect approach to analyze the Sagnac effect. To this end, we considered an optical case and applied this approach to evaluate the amount of the Sagnac effect. In optics, the amount of the Doppler effect is independent of whether the medium filling the space between the light source and receiver moves or not [252, 424]. The resulting phase difference between counterpropagating waves was found to be by a factor of 2n2 larger (n the refractive index) than that obtained using special relativity. Actually, the Sagnac phase difference is independent of the refractive index of the medium filling the interferometer, which follows from rigorous analyses relying on special relativity [225, 508, 695, 902] and also from the experimental studies by Pogany [680–682] and Bershtein [100] (see also the reviews [498, 888]). However, this fact is not trivial and, in the days of first fiber ring interferometers, the influence of the refractive index of the interferom-

Sagnac effect. Correct and incorrect explanations

15

eter sensitivity was widely discussed [243, 301, 430, 791, 883]. Detailed analyses and discussions of the Sagnac effect can be found in [86, 96, 158, 209, 215, 217, 245, 314, 431, 433, 700, 755, 781, 813] as well as [353] (a course of wave optics), [820, 821] (two editions of the Russian physical encyclopedia), [686] (encyclopedia of quantum electronics), and [822] (a course of quantum electronics). It was shown in our paper [508] that using the Doppler effect approach for waves propagating through a material medium in a rotating ring interferometer (e. g., ordinary acoustic waves) leads to an exaggerated Sagnac phase difference by a factor of 2c/vs (where 2vs is the sound speed and c is the light speed in vacuum) as compared to the true value [902]. Likewise, for de Broglie waves of matter particles, the Doppler effect approach exaggerates the true value by a factor of 2c/vm (where vm is the speed of matter particles).

2.1.4.4 Sagnac effect as a manifestation of the Fresnel–Fizeau dragging effect This explanation was suggested by Harress [310], who was first to conduct measurements of the phase difference between counterpropagating waves in a rotating ring interferometer (see the papers [313, 388] and reviews [32, 252, 314, 498, 695, 888, 948]). Harress believed that if there in no medium, there will be no phase difference. Accordingly, he made mistakes in the processing of the experimental results; these mistakes were corrected by Einstein [225]. In [428], apart from using special relativity, Laue attempted to reduce the Sagnac effect to the Fresnel–Fizeau dragging effect, while treating Harress’s experiments [310] as analogues of Zeeman’s experiments [946, 947] to measure the dragging coefficient. However, the Sagnac effect is by no means a corollary of the Fresnel–Fizeau effect, since the former is observed even if there is no optical medium in the interferometer. This issue was discussed quite widely in the days of first fiber ring interferometers [430, 883] and also fairly recently [440, 755, 909, 910].

2.1.4.5 Sagnac effect and Coriolis forces As shown above, the Sagnac effect can be treated as a consequence of different time dilations or phase changes of de Broglie waves in counterpropagating waves due to the non-relativistic scalar or, respectively, vector potential of Coriolis forces in a rotating frame of reference. Silberstein [798, 799] suggested an explanation of the Sagnac effect based on the direct consideration of the effect of the Coriolis forces on the counterpropagating waves. According to his reasoning, in a three-mirror ring interferometer, the optical path of the wave codirectional with the rotation represents a triangle with slightly convex sides, while that of the contradirectional wave, a triangle with slightly concave sides. The areas of the triangles are different. However, shortly after, Lunn [475] showed that although the shapes of the triangles are not exactly the same, their areas are nevertheless equal, since the area change due to the Coriolis deviation

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Fiber ring interferometry

of either of the counterpropagating beams is fully compensated by the respective area change due to the change in the angle of incidence on the next mirror. There are recent studies by Bashkov [74, 75, 78] where the erroneous statement of Silberstein [798, 799] is repeated.

2.1.4.6 Sagnac effect as a consequence of the difference between the orbital angular momenta of photons in counterpropagating waves The method of analysis of the Sagnac effect employed by Pomerantsev [689] (see also Molchanov [631]) is based on considering the change of the orbital angular momenta and energy of photons generated in a rotating ring laser. This explanation of the Sagnac effect is close to that based on the Doppler effect – in both cases, one uses the assumption that the counterpropagating waves have different frequencies, and the adequate expression of the frequency difference is a mere coincidence, because the phenomena in question are first-order effects in v/c . However, as shown by Einstein [225], the light frequency in a rotating frame of reference remains unchanged in the order of v/c .

2.1.4.7 Sagnac effect as a manifestation of the inertial properties of an electromagnetic field In the study [764], Schulz–DuBois derives an expression of the Sagnac frequency difference between counterpropagating waves in a ring laser. The derivation is based on the inertial properties of the electromagnetic field in a fiber ring resonator. The approach by Schulz–DuBois is somewhat similar to that based on the Doppler effect. It is noteworthy that the inertial properties of waves (more precisely, wave packets) used in gyroscopic instruments such as solid state wave gyroscopes [957] as well as quantum gyroscopes, which are based on macroscopic quantum properties of superfluid helium [67, 663, 766]. These instruments, along with the Foucault pendulum and mechanical gyroscopes [339, 341, 765], are angular position sensors, whereas Sagnac effect based devices are angular velocity sensors. This is the fundamental difference between the devices, those employing the properties of physical bodies or wave packets to maintain their orientation in space and those based on using the Sagnac effect.

2.1.4.8 Sagnac effect in incorrect theories of gravitation Currently, among the numerous theories of gravitation, there are so-called incorrect ones, which contradict the results of classical experiments conducted within the Solar System to test general relativity [920–922]. Yilmaz’s scalar theory [935–938] is one of them. In [938], Yilmaz considers the Sagnac effect from the viewpoint of his theory. It follows from the results of [938] that, in particular, the refractive index of vacu-

Physical problems of the fiber ring interferometry

17

um in a rotating frame of reference is not the same for counterpropagating waves: 1/n± = 1 ± ΩR/c . In Yilmaz’s opinion, this is what causes the Sagnac effect. However, as will be shown in Section 9.1.1, the amount of the Sagnac effect is independent of the refractive index of the medium.

2.1.4.9 Other incorrect explanations of the Sagnac effect There are a number of other incorrect explanations of the Sagnac effect, which are, however, much less frequent than the above. These include: the Sagnac effect as a consequence of the time inversion violation for counterpropagating waves in a rotating ring interferometer [334, 335, 816], an explanation from the viewpoint of an extended understanding of the locality hypothesis [593]; a consequence of the Fermat principle in a rotating frame of reference [415], a manifestation of an adiabatic invariant in a rotating frame [251], a consequence of Schwarzschild’s solution [767] in a rotating frame [644, 729], an explanation based on the assumption that rotation is relative rather than absolute [143], and a consequence of the twin paradox in a rotating frame [209]. Analyses of the specific errors of the studies [143, 209, 251, 334, 335, 415, 593, 644, 729, 816] can be found in our paper [508].

2.2 Physical problems of the fiber ring interferometry based on the Sagnac effect Physical problems of fiber ring interferometry call for further consideration and systematization. There are a number of review papers related to these issues, which include [54, 71, 96, 120, 134, 144, 159, 190, 208, 234, 414, 431, 433, 434, 437, 449, 655, 671, 758, 781, 796]. These papers are 15–30 years old and so do not reflect the most recent results in this area. Recent reviews (e. g., see [638]) mainly deal with technological aspects of production and design of fiber optic gyroscope.

2.2.1 Milestones of the creation and development of optical ring interferometry and gyroscopy based on the Sagnac effect This section lists the milestones of the creation and development of optical ring interferometry and gyroscopy based on the Sagnac effect. These include: 1893–1897 The statement of the problem of detecting rotation of a reference frame by means of a ring interferometer belongs to Lodge [465, 466] (1893–1897). 1909–1911 The first measurements of the phase difference between counterpropagating waves in a rotating ring interferometer filled with glass were performed by Harress [310]; he was wrong to believe that he was studying the Fresnel–Fizeau dragging effect.

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1913–1914 The first experimental studies specially dedicated to this phenomenon were carried out by Sagnac [741, 742] (1913), who also suggested in 1914 that this effect should be used for gyroscopy and navigation purposes [743]. 1914 Einstein showed [225] that in a rotating ring interferometer, the dragging coefficient coincides with the classical Fresnel dragging coefficient. It follows that the amount of the Sagnac effect is independent of the refractive index and dispersion of the optical medium filling the interferometer and moving with it; this was experimentally confirmed by Pogany [680–682] (1926–1928). 1921 Silberstein [798] suggested that a ring interferometer should be used for detecting the Lense–Thirring general relativity effect [439, 846–848]. A similar suggestion was made by Scully et al. [769] (1981). 1925 Michelson was the first to detect Earth’s rotation with a 630 × 340 m ring interferometer [604]. 1950 The first multi-circuit ring interferometer was built by Bershtein using a radiofrequency cable to measure the Sagnac phase difference for radio waves [100]. Also he was the first to apply the phase modulation method (the so-called triangle method) [99, 479] and an electronic scheme for processing the output signal. 1952–1955 Gorelik [138, 290] and Bershtein [101–104] suggested and implemented the method of modulation interferometry, where the optical phase difference was varied harmonically. 1958 Wallace suggested, in his patent, a scheme of a fiber optic interferometer for the first time [906]. 1962 Rosenthal suggested measuring the rotational angular velocity with a passive ring resonator [730] (implemented by Ezekiel and Balsamo [235] in 1978). He also proposed the idea of laser ring gyroscopy (implemented by Macek and Davis [476] in 1963). 1965 McLaughlin [596] suggested the method of asymmetric phase modulation for a ring interferometer based on discrete optical elements. 1976 The first fiber ring interferometer using a single-mode optical fiber with a monochromatic light source, a 0.63 μm He-Ne laser, was made by Vali and Shorthill [881, 882]. 1979 Almost simultaneously, three research teams – Schiffner et al. [761], Ulrich and Johnson [873], and Logozinskii et al. [289, 470] – studied the phenomenon of polarization nonreciprocity (PN) in fiber ring interferometers, which causes a phase difference in counterpropagating waves even if there is no rotation (known as the zero shift), and suggested methods for reducing polarization nonreciprocity. 1979–1980 Alekseev, Bazarov et al. [21] and Ulrich [871] applied asymmetric harmonic phase modulation in fiber ring interferometers. In addition, Ulrich suggested the so-called minimum scheme of an FRI [871], which proved to be most successful and has found wide application.

Physical problems of the fiber ring interferometry

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1980 Shupe [792] demonstrated that random nonsymmetric thermal fluctuations in the optical path length of the single-mode optical fiber cause a zero drift of the interferometer. 1980 Dianov, Prokhorov et al. [141] applied a compensatory method for suppressing excessive fluctuations of the light source in processing the output signal of the photodetector. 1981 Bohm et al. [119] utilized an non-monochromatic light source (a superluminescent diode) in an FRI. In 1983, the same team applied to the FRI a single-mode fiber Lyot depolarizer [126]. 1981 Kintner [384] demonstrated that the zero shift due to polarization nonreciprocity in the minimal scheme of a ring interferometer is proportional to the extinction coefficient of the polarizer. In 1982, Pavlath and Shaw [670] analyzed the relationship between the relative rotation of the anisotropy axes of the single-mode optical fiber at the FRI circuit input and the amount of the zero shift. 1981–1982 Kaplan and Meystre [359] and Birman and Logozinskii [112] showed that if the intensities of counterpropagating waves in a fiber ring resonator [359] or a fiber ring interferometer [112] are unequal, a phase nonreciprocity arises due to the nonlinear optical Kerr effect. In 1982, Bohm et al. [125] studied the influence of the external magnetic field on the FRI zero shift due to the Faraday effect. 1982 Kingsley [383] suggested that large FRIs should be used to detect gravitational waves. 1984–1986 Burns and Moeller [149] and Kozel, Listvin, et al. [401] studied some special cases of the relationship between the parameters of the SMF loop, light source, and deviation of the FRI’s zero. To sum up, there are three main variants, one active and two passive, of optical gyroscopy based on the Sagnac effect. The active variant is implemented as a ring laser [56, 72, 158, 176, 245, 414, 919], in which rotation results in a frequency difference between counterpropagating waves. The passive variants are implemented as a fiber ring interferometer [414, 431, 433, 781], where rotation results in a phase difference between counterpropagating waves, and a passive fiber ring resonator [235, 431, 730], where rotation causes a natural frequency difference and a phase difference between counterpropagating waves in a resonance region. A fiber optic gyroscope (FOG) can be constructed on the basis of a fiber ring interferometer or a fiber ring resonator. At the same time, FRIs can be used to make not only FOGs but also instruments for measuring other physical effects. This suggests that the concepts of a FOG and an FRI are not exactly the same. In this monograph, we consider FOGs based on FRIs (IFOGs, interferometric fiber optic gyros).

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2.2.2 Sources for additional nonreciprocity of fiber ring interferometers 2.2.2.1 General characterization of sources for additional nonreciprocity of fiber ring interferometers Experimental studies of fiber ring interferometers revealed a number a number of additional signals arising at the output, which are identical to rotation but are not related to it. The sources of the additional signals can be divided into the following six groups: (i) effects related to light scattering and reflection in the fiber tract, (ii) the phenomenon of polarization nonreciprocity, (iii) effects related to local reciprocal unsteady changes in fiber parameters as they are excited nonsymmetrically with respect to the middle of the fiber loop, (iv) proper nonreciprocal phenomena such as the Faraday, Fresnel–Fizeau, and other effects, (v) phenomena caused by the nonlinear optic Kerr effect, and (vi) special relativity and general relativity effects. The errors in FRI readings can be split into a constant and a time-variant part, the zero shift and zero drift. Since there are different definitions of the zero drift in the literature, this issue should be looked at more carefully. A zero drift of a fiber ring interferometer is a random change of the phase difference between counterpropagating waves; this change can be due to different phenomena such as nonsymmetric (with respect to the loop middle) changes in the fiber temperature (the so-called Shupe effect [792]), temperature fluctuations in the optical path length and fiber birefringence (for a nonmonochromatic light source, only if there is a random coupling between polarization modes in the fiber), changes in the magnitude and/or direction of the external magnetic field (Faraday effect), photocurrent shot noise, and others. Since the main objective of the monograph is to study the influence of linear coupling between polarization modes on the operational qualities of the FRI, the phenomenon of polarization nonreciprocity will be given special attention. Since FRIs (FOGs) detect angular velocity of rotation, the amount of the zero drift must have the units of ◦/(hour K). Thus, the zero drift is a derivative quantity of the relationship between the counterpropagating wave phase difference at the FRI output (zero shift) and the SMF temperature. At the same time, in some studies, the zero drift is understood as the root mean square deviation, or essentially the confidence interval, of the temperature evolution of the zero shift. In what follows, for definiteness, the zero shift of an FRI (units of ◦/hour) will be understood as its mean (mathematical expectation), the zero shift deviation (also measured in ◦/hour), as the root mean square deviation, and the zero drift (measured in ◦/(hour K)) will be understood as the maximum value of the first derivative of the zero shift with respect to the fiber temperature. Whenever temperature zero drift is mentioned as a phenomenon rather than quantity, it will be understood as the fact that the zero shift is dependent of the temperature. If an FRI is used as angular velocity sensor, the primary source of measurement error is the shot noise of the photocurrent. If it is used as a navigation device, i. e., an angular position sensor, in which the output signal is integrated over a long time, the main source of error is the temperature zero drift [459].

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2.2.2.2 Nonreciprocity as a consequence of the light source coherence The first fiber optical gyroscopes [301, 871, 881, 882] used a helium-neon laser as the source of monochromatic light with a wavelength of 0.63 μm. For this reason, the effect that were first to be discovered were related to the coherence of the laser sources employed and could not be detected in Sagnac’s classical experiments [741– 743]. When a monochromatic source of radiation is used, significant phase nonreciprocity arises which is caused by dual-linking, due to slow and fast birefringence axes of the SMF loop; the nonreciprocity depends on the loop length [35], which is subject to temperature variations. Back reflections and scattering [279, 343], including Raleigh scattering [123, 148, 195, 654], in the optical tract of an FRI produce an additional, rotation-unrelated signal with an arbitrary phase; once coherently added to the useful signal, it results in an additional phase difference between counterpropagating waves and causes a response of the laser distorting the output signal. A number of different ways for reducing the laser response were considered [36, 199, 699]; however, what was found to be a drastic measure to remove the effects of the back reflections and scattering on both the source and the signal measured was the use of broadband semiconductor radiation sources such as superluminescent diodes (SD) with high spectral intensity and short coherence length, 4–20 μm [119, 581, 927]. Superfluorescent fiber light sources (SFLS) [13, 14, 16] also find application in FOG schemes.

2.2.2.3 Polarization nonreciprocity: causes and solutions The Jones matrix method [19, 68, 133, 277, 291, 332, 350, 351, 794, 845, 933] provides the most convenient way of calculating phase and polarization characteristics of light in optical media, including the FRI zero shift caused by polarization nonreciprocity. If there are no nonreciprocity effects (rotation, magnetic field, etc.), the fiber ring interferometer must satisfy the conditions of the traditional reciprocity theorem [281, 423]. In this cases, the Jones matrices of the SMF loop for counterpropagating waves are the transposes of each other [649]. Strictly speaking, Jones matrix calculations are only valid for monochromatic light. Nevertheless, this method can also be applied in conjunction with the coherence matrix [401]. The polarization nonreciprocity phenomenon is caused by the polarization of counterpropagating waves at the FRI output and arises despite the fact that the waves have the same polarization at the input, because of the different sequences of anisotropic elements the waves meet as they travel in the loop. The polarization nonreciprocity is partially due to the relative rotation of the anisotropy axes at the ends of the SMF loop. Methods to control the alignment of the axes are discussed in [43]. The polarization nonreciprocity arises even if the characteristics of the FRI satisfy the traditional reciprocity theorem [281, 423, 649]. In a minimal scheme of a fiber ring interferometer [871], which will be discussed in Chapter 5, the suppression of the zero shift caused by polarization nonreciprocity

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Fiber ring interferometry

is proportional to the proportional to the extinction coefficient of the polarizer [384]. Note that presently there are a number of different types of fiber polarizer available [92, 111, 212, 223, 463, 468, 653, 863, 887, 905, 907]. As shown in [401], the amount of deviation of the zero shift is inversely proportional to the square root of the light source bandwidth, which suggests that sources with a maximum possible bandwidth are more advantageous. Since the zero shift caused by polarization nonreciprocity as well as its deviation are proportional to the degree of polarization of the radiation at the FRI output [401], some studies consider using a depolarizer of nonmonochromatic light instead of or along with a polarizer [10, 117, 119, 126, 146, 256, 349, 458, 733, 827, 829]. As a simple depolarizer, one can use a phase plate with linear birefringence or an anisotropic fiber segment [458] with a length exceeding the depolarization length of the nonmonochromatic light, such that the light traveled along the orthogonal axes of anisotropy becomes mutually incoherent. The Lyot depolarizer [467] consists of two phase plates or two anisotropic fiber segments [126, 182, 189, 431, 624, 655, 908] with a length ratio of 1 : 2 and anisotropy axes rotated at an angle of 45◦ . There is a solid fiber optic Lyot depolarizer [11]. The degree of suppression of polarization nonreciprocity achieved through using depolarizers depends on their number, position, and quality [10, 117, 119, 126, 146, 256, 349, 458, 733, 827, 829]. Chapter 6 will discuss the characteristics of fiber ring interferometers with different kinds of depolarizer located in different parts of the FRI scheme. The polarization nonreciprocity can also be due to anisotropy and losses in the input beam splitter; this phenomenon even occurs is a minimal FRI scheme with two splitters. This issue is now quite well understood [44, 164, 238] and will not be discussed in what follows. It is noteworthy that until recently there was one theoretical study [401] that allowed the calculation of the zero shift deviation caused by polarization nonreciprocity, but only provided that the small perturbation method is applicable. The method of [401] fails to calculate the effect for FRIs with cheap, weakly anisotropic SMF loops. A rigorous solution to this problem, which has yet not been obtained so far to the best of our knowledge, will be given Chapters 5 and 6 in the most general case. Another important issue stems from the fact that, according to the method of [401], the zero shift deviation is calculated by averaging over the ensemble of independent realizations of random inhomogeneities in the fiber, whereas in a real SMF loop there is only one realization of random inhomogeneities. Moreover, it is assumed that the loop satisfies the ergodicity conditions [401], but no substantiation of this assumption is given. The issue of validity and applicability of the ergodicity hypothesis will be resolved in Chapter 5.

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23

2.2.2.4 Nonreciprocity caused by local variations in the gyro fiber-loop parameters due to variable acoustic, mechanical, and temperature actions The third group of rotation-unrelated additional input signals includes acoustic, mechanical, and temperature actions that are locally reciprocal but nonsymmetrically excited about the middle of the fiber loop. These actions lead to variations in the amplitude, polarization, and phase of one wave delayed with respect to the counterpropagating wave [84, 792]. The influence of unsteady effects vanishes whenever the product of the perturbation frequency of the fiber’s optical length by the light propagation time through the loop equals π N , where N is an integer [45, 93]. The influence of nonsymmetric thermal perturbations can be significantly reduce through using different types of winding of the FRI circuit [196]. The effects related to the operation of the phase modulator, which is always nonsymmetric with respect to the middle of the fiber loop, form a separate subgroup. These effects lead to an additional nonreciprocal signal in the second harmonic of the modulation frequency. Along with phase modulation, amplitude modulation can also arise as well as light polarization [17, 34, 46, 93, 270, 347, 826]. The papers [12, 18] suggested a compensation method for polarization modulation based on using a Faraday element, which reverses the polarization [52, 274, 591]. The study [375] sowed that the presence of even harmonics (especially the second harmonic) in the phase modulation leads to an additional signal in the first harmonic, which can be in phase or in antiphase with the useful signal. Methods for reducing the effect of the second harmonic will be discussed in Chapter 8.

2.2.2.5 Nonreciprocity due to the Faraday effect in external magnetic field Some of the factors causing the fourth-group nonreciprocity are effects arising when an external constant magnetic field is applied [124, 125, 262, 326, 327, 431, 744, 828]. Such effects as well as their instability when in service impair the accuracy characteristics of fiber optic gyroscopes and result in the need for magnetic shielding. This issue will also be discussed in Chapter 8.

2.2.2.6 Nonreciprocal effects caused by nonlinear interaction between counterpropagating waves (optical Kerr effect) The fifth group consists of nonreciprocal nonlinear effects caused by the intensity difference between counterpropagating waves. These effects are due to the dependence of the fiber refractive index, caused by the nonlinear optical Kerr effect, on the light intensity [818]. The amount of the effect is determined by the quadratic nonlinearity of the refractive index of the light-guiding core material and is related to the high density of the optical power in the single-mode fiber due to the small core diameter (4–8 μm). These effects were studied in [94, 112, 237, 237, 358, 359]. Counterpropagating waves with collinear polarizations but different intensities form a standing wave

24

Fiber ring interferometry

structure, which fixes in the medium, due to the nonlinear Kerr effect, a multilayer virtual mirror where the counterpropagating waves get reflected [431, 561, 610, 614, 616]. In this case, the nonreciprocal nonlinear output phase difference is affected by the reflections at the nonlinear virtual mirror and determined by the intensity difference between the counterpropagating waves as well as the nonlinear polarization. This effect is significantly weakened if broadband light sources are used. For such light sources, the standing wave structure of the refractive index, essential for nonreciprocity to arise, is only the same for different wavelengths at the coherence length of the nonmonochromatic light near the middle of the fiber circuit [91, 188, 258]. (For FRIs with superluminescent diodes the coherence length is from several to several hundred μm.) Using a polarizer does not reduce the phase nonreciprocity caused by the Kerr effect. In case the counterpropagating waves have different polarizations, additional nonlinear nonreciprocal polarization effects can arise, since either wave changes its polarization due to the interaction with the other, noncollinear wave [113, 561, 610, 612–618, 756, 914, 930, 944, 945]. The contribution of these effects to the phase nonreciprocity of FRIs is significantly less than that of the nonlinear Kerr effect caused by collinear counterpropagating waves. These issues will be discussed in Chapter 8.

2.2.2.7 Nonreciprocity caused by relativistic effects in fiber ring interferometers It should be reminded that the Sagnac effect is relativistic [421, 474, 508, 902]. It was suggested in [769] that large FRIs should be used to detect the Lense–Thirring effect (general relativity) [439, 846–848], which represents a gravitational analogue of the phenomenon of electromagnetic induction and is that a rotating mass affects differently the phase of corotational and counterrotational waves. Using large FRIs was also suggested for precision tests of the basic postulate of special relativity that the speed of light is isotropic. Gravitational waves can also cause a phase difference between counterpropagating waves in FRIs. In 1992, the LIGO (Laser Interferometer Gravitational-Wave Observatory) project was initiated in the USA to detect gravitational waves using a 4 km Michelson interferometer. As an alternative to the LIGO project, the possibility of using a fiber ring interferometer of a special design is being discussed in the literature [383] to carry out fundamental experiments for detecting gravitational waves.

2.2.3 Fluctuations and ultimate sensitivity of fiber ring interferometers The influence of noise on the ultimate sensitivity of the ring interferometer was thoroughly studied over 50 years ago by Bershtein [100]; he considered a ring interferometer operating in the radio frequency range. In the optical range, however, there are a number of specific features. The sensitivity affected by noise of different ori-

Physical problems of the fiber ring interferometry

25

gin at the photodetector output was studied by Andronova and Bershtein [36] as well as a number of other authors [20, 47, 85, 116, 151, 153, 389, 448, 471, 472, 627, 628, 630, 673, 687]. The noise comes from several sources [639]: (i) quantum (shot) noise, relating to discreteness of photons and photoelectrons, (ii) natural noise of the light source, due to the beats of its spectral components [20, 36, 153] arising in the quadratic detection of the optical spectrum, (iii) equilibrium thermal fluctuations of the refractive index of the SMF [471, 628, 630], (iv) fluctuations due to light scattering, (v) flicker noise, arising when the supply current passes through a semiconductor light source. The amounts of these contributions depend on whether or not there is phase modulation [116, 627]. The noise spectral density of the output photocurrent due to shot noise is inversely proportional to the photocurrent, while that due to the natural noise of the light source is inversely proportional the spectral width of the light source [47, 376]. The output spectral density of phase fluctuations due to equilibrium fluctuations of the refractive index is directly proportional to the fiber length and temperature squared and inversely proportional to the light wavelength and mode diameter [911]. It is only low-frequency perturbations that get into the reception bandwidth at the modulation frequency; the frequency of these perturbations does not exceed the reception bandwidth [151, 628]. The light scattered in the fiber circuit results in an additional interference noise signal [775]. The shot noise and superluminescent diode noise make the greatest contribution to the threshold sensitivity in measuring the angular velocity of a fiber ring interferometer based on the first harmonic of the modulation frequency and optimal amplitude of the phase modulation [36]. The ultimate sensitivity can be increased through increasing the output power while the modulation depth of the shot noise exceeds that of the source noise; then the sensitivity becomes independent of the photocurrent, since the source noise at the FRI output begins to prevail. These kinds of noise can be significantly reduced by compensating them using part of the original source power [141, 151, 628, 703]. As shown in our paper [520], the temperature zero drift in an actual fiber optic gyro, which is caused by polarization nonreciprocity in the interferometer, usually exceeds the zero drift due to shot noise. For this reason, the issues as to how the source light fluctuations? affect the ultimate sensitivity of the FRI will not be discussed in what follows; these issues are now sufficiently well understood.

2.2.4 Methods for achieving the maximum sensitivity to rotation and processing the output signal The intensity of the interference signal at the FRI output is proportional to the cosine of the phase difference between counterpropagating waves; as a result, the FRI is insensitive to the rotation direction and, at low angular velocities, weakly sensitive to

26

Fiber ring interferometry

rotation. Hence, an obvious method to achieve the maximum sensitivity is to change the working point to a segment where the dependence of the interference signal on the phase difference is sufficiently steep. In the literature, there have been a number of suggestions as to how to change the working point to a steep segment. In particular, these suggestions included using: (i) a nonreciprocal phase element based on the Faraday effect [200], (ii) frequency splitting between counterpropagating waves [199, 699] with the aid of discrete [398] or fiber [652] optoacoustic modulators, (iii) the nonlinear optical Kerr effect with different intensities of counterpropagating wave [358, 359], (iv) a 3 × 3 input splitter that creates an initial phase difference [154, 684, 780], (v) a mismatching between the wavefronts of counterpropagating waves [141], (vi) losses in the input beam splitter [328], and (vii) frequency modulation of the light source [191, 952, 953]. However, none of the methods listed has found wide application. As noted above, the method of nonsymmetric phase modulation has proved to be most effective [21, 871]. In this case, the photocurrent contains the sum of harmonics of the phase modulation frequency [93]. If there is no rotation, the output photocurrent contains only even harmonics of the modulation frequency. In case there is rotation or any other nonreciprocal phase difference, there are even as well as odd harmonics. If the rotation velocity changes, one only uses, as a rule, the first harmonic of the modulation frequency. In case the input signal amplitude is not constant, special compensatory methods can be used [105, 376]. Methods for constructing linear dependences between the angular velocity of rotation and output signal in a wide range of angular velocities were studied in [121, 122, 174, 191, 213, 220, 221, 257, 295, 325, 344, 365, 369, 370, 373–375, 381, 382, 436, 674, 857, 870]. Methods that use the whole set of harmonics of the modulation frequency were studied in [121, 174, 220, 257, 295, 857]. The possibilities of applying the heterodyne methods were analyzed in [191, 370, 674]. In some cases, these methods were found to be capable of increasing the sensitivity [105]. It is noteworthy that a change in the fluctuation level can be a sign of rotation [142].

2.2.5 Applications of fiber optic gyroscopes and fiber ring interferometers Interferometric FOGs are used for measuring the angular velocity and rotation angle, whereas FRIs find application in a variety of different areas unrelated to object rotation. By now, fiber optic gyroscopes have entered the production stage and taken a fairly extensive niche of moderate-precision gyroscopic instruments [677, 678]. The sensitivity range of FOGs is quite wide, from 100◦/hour to 0.005◦/hour. Most of commercial instruments are based on the minimum FRI scheme [871]. Although far from being complete, the list of companies manufacturing fiber optic gyroscopes includes: Honeywell, Litton Corp., KVH Inc., Fibersense Technology Corp., Andrew Kintec Corp. (USA); Photonetics, SFIM, Litef, IMAR (Europe); CSIR (South Africa); Com-

Physical problems of the fiber ring interferometry

27

munications Research Laboratory, Tamagawa (Japan); CJSC Fizoptika, Production Association Korpus, OJSC Perm Scientific-Industrial Instrument Company (Russia). FOGs have a fairly wide range of practical applications for gyroscopy, orientation, and navigation purposes [145, 219, 414, 669, 751, 754, 839]. FOGs with a sensitivity of the order of 10◦/hour are mounted on ground vehicles that must move in accordance with a preset algorithm; these include motor cars, electric cars, robots, and various agricultural machines [145]. In addition, FOGs are used for rail-track laying and well drilling. According to some forecasts, in the near future, FOGs will begin to take over the niche of navigational gyroscopes, previously occupied by the laser and mechanical gyroscopes. Fiber ring interferometers can also be utilized as ordinary phase detectors of variable actions [51]. The paper [176] investigates the possibility of applying FRIs for solving some geophysical problems. FRIs are used for determining fluid flow velocity [450, 725, 864], Lorentz correction to the Fresnel–Fizeau dragging coefficient [203], and parameters of single-mode optical fibers such as birefringence [377], chromatic dispersion [3, 87], polarization mode dispersion [286, 658, 801], and the dependence of the nonlinear refractive index on the light intensity [57] and length of polarization beats [417]. FRIs are also used for contactless monitoring of optical surface profile [162, 785], as current sensors [115, 127, 357, 447, 726, 787, 788], electric field sensors [726], magnetometers [95], fiber strain sensors [952], for measuring the extinction coefficient of fiber polarizers [33], as optical frequency filters [110, 178, 316, 379, 380, 390, 636, 889], temperature sensors [814, 840, 932], optical attenuators [932], optical switches [329, 330, 348, 416, 462, 717, 779, 928], optical microphones [868] or hydrophones [293, 403], curvature sensors [253], and for measuring object velocity based on the Doppler effect [595]. FRIs find a number of other applications [156, 165, 177, 181, 315, 323, 324, 378, 418, 445, 478, 598, 668, 774, 869, 880]. The papers [118, 303, 464, 732] deal with military applications of interferometric FOGs and the paper [352] addresses their commercial applications. The polarization fiber ring interferometer (PFRI) suggested in [648, 650] finds application in measuring magnetic field [53, 588] and determining fluid flow velocity [725]. It can also be used for detecting nonreciprocal linear birefringence in crossed electric and magnetic fields in optical fibers, which is a new fundamental optical effect studied previously in [651] for fluids. This issue will be discussed in Chapter 9.

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Fiber ring interferometry

2.3 Physical mechanisms of random coupling between polarization modes in single-mode optical fibers and mathematical models for describing these mechanisms The coupling between orthogonal polarization modes arising in single-mode optical fibers is conventionally characterized by the h-parameter [354, 583, 710]. This coupling causes (i) changes in the degree of polarization of non-monochromatic light as it travels through the fiber [152, 941], (ii) output signal fading and a temperature drift, due to polarization nonreciprocity, of the zero of the interference pattern [149, 401] in Mach–Zehnder [371] Michelson [912] fiber interferometers, and (iii) a number of other unwanted phenomena, in particular, short-pulse widening in long communication lines based on SMFs [283, 460, 511, 567, 690–692, 697, 858, 866]. Below we will consider different physical mechanisms of the origination of this coupling and discuss their hierarchy based on the polarization mode contribution. We will also consider different mathematical models that describe polarization mode coupling. These models can be categorized into two groups: phenomenological, dealing with some abstract inhomogeneities in SMFs if at all, and physical, taking into account specific perturbations in SMFs. Inhomogeneities arise as the fiber is drawn from a preform and also as a coating is applied and the fiber is wound on a coil or laid in a communication line. Apart from technical factors causing polarization mode linking, there is a fundamental cause, Raleigh scattering.

2.3.1 Milestones of the development of the theory of polarization mode linking in single-mode optical fibers 1944 Ginzburg [280] showed that in an optical medium with linear birefringence and twisting, the natural polarization modes are spiral elliptic modes, or spiral polarization modes, which are elliptic modes in the spiral frame of reference that follows the twisting. 1952 Budden [140] considered the evolution of a state of polarization as radio waves travel through a one-dimensional plane-layered medium with linear and/or circular birefringence. 1968 Kravtsov [407] found, by the method of quasi-optical approximation in geometric optics, a link between the state of polarization of an electromagnetic wave traveling through a gyrotropic (due the Faraday effect) and anisotropic plasma in a non-planar curvilinear path and the length dependence of the magnetic field. This link is determined by the Budden–Kravtsov equations, which, as shown later by Naida [637] (1970), can be reduced to a Riccati equation. In 1972, Azzam and Bashara [69] obtained a similar equation for an optical medium in the special case of a planar path.

Physical mechanisms of random coupling between polarization modes

29

1972 Kapron, Borrelli, and Keck [360] discovered that single-mode optical fibers, which should basically be polarization-isotropic, actually have some linear and circular birefringence randomly varying along the fiber length. 1973 Suvorov [825] determined normal mode parameters in the spiral frame of reference for an electromagnetic wave traveling through an anisotropic plasma in an external magnetic field. 1974–1975 Marcuse introduced the notion of a polarization holding parameter (hparameter) in counterpropagating waves [583] and copropagating waves with orthogonal polarization [584]. However, subsequent studies showed that unlike microwave waveguides, the polarization mode coupling between counterpropagating waves in single-mode optical fibers is many orders of magnitude lower than between copropagating waves, since the characteristic length of inhomogeneities in UHF waveguides is commensurable with the electromagnetic wave length, whereas that in SMFs is approximately four orders of magnitude longer than the light wave length. 1978 Kaminov et al. [706] made an SMF with strong linear birefringence. 1980 Monerie and Jeunhomme [634] suggested a physical model of random inhomogeneities in SMFs where a fiber is treated as a collection of randomly oriented linear phase plates. They also showed that the phenomena due to the orthogonal polarization mode coupling cannot be strictly described in the language of the intensity coupling between two given (unperturbed) polarization modes, that is, in terms of the h-parameter. 1983 Zeleznyakov, Kocharovskii, and Kocharovskii [951] studied the propagation of spiral elliptic modes in SMFs and obtained, for some specific dependences of the linear birefringence angle of twist on the fiber length, exact or approximate solutions for the polarization state evolution. 1983 Moeller, Burns, and Chen considered the cases of strong [626] and moderate [152] linear birefringence and found that if both equal-weight polarization modes are excited by nonmonochromatic light at the input of a fiber segment whose length considerably exceeds the length of depolarization, some residual polarization is observed at the output of the segment; this residual polarization is due coupling between the polarization modes at random inhomogeneities. 1983 Bohm et al. [126] established a link between the residual polarization of nonmonochromatic light at the input of a finite-length fiber segment with strong linear birefringence and the h-parameter as well as a link between the linear birefringence and the light source bandwidth. A more rigorous solution was obtained by Zalogin, Kozel, and Listvin [941] (1986). 1985–1989 The studies [160, 456, 672, 831, 867] showed that the h-parameter is a Lorentz function of the amount of liner birefringence. It follows that the random-inhomogeneity spatial spectrum power is also a Lorentz function of the space frequency. The significant difference in the values of h between copropagating and counterpropagating waves is explained by the fact that the former

30

Fiber ring interferometry

is equal to the random-inhomogeneity spectrum power over the space period of polarization beats in the fiber, while the latter is that over the space period equal to half the light wavelength in the fiber [831] (1986). 1987 Grudinin and Sulimov [297] showed that if the fiber has segments whose length exceeds the depolarization length of nonmonochromatic light and the anisotropy axes have constant orientation along these segments, the degree of polarization of the light traveling through the fiber will effectively decrease regardless of the weight the polarization modes had at the fiber input. 1987–1988 Marrone et al. [590] (1987) and Alexandrov et al. [8] (1988) demonstrated experimentally that single-mode optical fibers have random twisting of birefringence axes.

2.3.2 Phenomenological models of polarization mode coupling A characteristic feature of polarization mode coupling models [152, 354, 397, 583, 710, 745, 777, 941] is that they are described phenomenologically, while real physical mechanisms of polarization mode coupling are considered. Actually, the calculations are performed using the small perturbation method, which is applicable when the total light intensity transferred from one polarization mode to the other on some fiber length is small compared with the total light intensity in the entire fiber. Moreover, multiple intensity transfers from one mode to the other usually cannot be taken into account. This approach has the advantage that the calculations can be performed analytically and, in particular, one can obtain simple relations for the degree of polarization of nonmonochromatic light in SMFs [152, 941] as well as zero shift and zero shift deviation in FRIs [149, 401]. The studies [152, 777] use a model of point coupling between polarization modes without considering the real physical mechanism of the coupling. It should be noted that these studies omit some small terms in the entries of the Jones matrix [794] that characterizes the SMF and so the determinant of the matrix is not equal to unity despite there are no light losses. This suggests that the results of [152, 777] may not be quite correct. In [745], the main parameters characterizing the polarization model coupling in SMFs are chosen to be the mean square of the number of random inhomogeneities per unit length and autocorrelation function of random inhomogeneities along the length as well as the bandwidth and spectral shape of the light source. These parameters are inconvenient for calculations, since the first two of them are difficult to measure experimentally. The method of [397] for calculating the polarization characteristics of SMFs relies on using the Müller matrix [794], which does not allow one to take into account the light phase. In particular, this method is unsuitable for calculating the zero shift and zero shift deviation in FRIs.

Physical mechanisms of random coupling between polarization modes

31

In [126, 941], the main parameter used to characterize the total coupling between orthogonally polarized modes along a length ldep is the product hldep , provided that hldep  1 and hL  1, where ldep is the depolarization length of nonmonochromatic light in the SMF and L is the total length of the SMF. The small perturbation method [126, 941] is the most adequate, as compared to the other phenomenological models, in describing the polarization mode coupling at inhomogeneities. However, in weakly anisotropic SMFs, the condition hldep  1 does not hold, and hence the calculation method of [126, 941] is only suitable for strongly anisotropic SMFs of finite length. Furthermore, just as [152], the studies [126, 941] omit some small terms in the entries of the Jones matrix. The model of [941] was employed in the papers [452– 454, 456]. The study [852] used an abstract Gaussian random process to describe the polarization mode coupling. The results of [852] cannot be utilized for calculating the evolution of the nonmonochromatic light polarization in SMFs or zero shift deviation in FRIs. Thus, in the most general case, phenomenological models cannot provide an adequate description of the polarization mode coupling (which was shown by Monerie and Jeunhomme [634] back in 1980), since they are not based on real physical processes causing the coupling. Phenomenological models can only be used when the conditions for the small perturbation method hold.

2.3.3 Physical models of polarization mode coupling In some studies (e. g., see [84, 634]), real single-mode optical fibers are assumed to be equivalent to a set of ideal linear phase plates and the plate lengths and azimuths of the birefringence axes are randomly distributed. This model has a number of flaws. First of all, the mechanism of how jump discontinuities in the azimuths of birefringence axes arise is not clear. Secondly, since the azimuths are discontinuous functions of the fiber length, the fiber must undergo a discontinuity where an azimuth has a jump. Nevertheless, this model is still in use (e. g., see [885]), especially for calculating the polarization mode dispersion in SMFs (e. g., see [185, 192, 193, 460, 691, 858]), which widens short pulses in fiber optic communication lines [205, 206, 299] and, hence, limits the maximum information transfer density. Experimental studies show that the model of randomly oriented linear phase plates does not correspond to actual perturbations of the linear birefringence axes in SMFs. For example, Alexandrov et al. [8] investigated the orientation of the linear birefringence axes in strongly anisotropic SMFs by directly breaking a fiber and observing under a microscope the orientation of the elliptic intermediate straining shell. The results revealed two types of twisting of the linear birefringence axes, monotonic adiabatic and nonmonotonic. The monotonic twisting was of the order of 1 rad/m; it

32

Fiber ring interferometry

does not cause coupling of orthogonally polarized modes. The nonuniform twisting was of the order of 2 rad/m (with peaks up to 8 rad/m) and has a scale of about 2 cm; it causes coupling of orthogonally polarized modes and, as will be shown in Chapter 3, determined the value of the h-parameter. No saltatory changes in the azimuths of the birefringence axes were observed. Similar results were obtained in [2, 590]. When the linear birefringence axes are twisted, the natural modes of SMFs become elliptical [73, 84, 280, 297, 634, 876, 933, 951]. Ginzburg [280] was first to show that in optical media with intrinsic linear birefringence and twisting, so-called spiral polarization modes occur, which remain elliptical in the local frame of reference that follows the twisting. Using the spiral polarization modes is principally equivalent to the regular method of Jones matrices in a linear basis and can be advantageous in some cases. To this end, one has to change to the elliptical basis corresponding to the mutually orthogonal spiral polarization modes [68], where the Jones matrix for the fiber segment of interest will be diagonal, which significantly facilitates the calculations. It is noteworthy that about 30 years ago, Sakai et al. [746, 747] investigated the fundamental possibility for a random twist to occur. However, determining the effect of a random twist on the polarization mode coupling was not amongst the objectives. Thus, the situation required developing a physical model of random inhomogeneities in SMFs that would take into account random twists of the linear birefringence axes. A key feature of the underlying theory is an adequate prediction of statistical parameters of the random twists. This model will be considered in Chapter 3. It will be shown below that random twisting of the linear birefringence axes is not the only but the main cause of polarization mode coupling in SMFs. To this end, we will discuss how inhomogeneities can arise in SMFs.

2.3.4 Inhomogeneities arising as a fiber is drawn As a fiber is drawn from a preform, three kinds of vibration can arise, transverse, longitudinal, and torsional. In addition, internal stresses may appear in the fiber. To begin with, we consider torsional vibration, since, as will be shown below, it is this kind of vibration that makes the main contribution to the polarization mode coupling.

2.3.4.1 Torsional vibration Figuring out the cause of torsional vibration of fibers is one of the objective of this monograph. This issue will be considered in detail in Chapter 3. Here we note that, according to the measurements of [2, 8, 590], the correlation length of random twists is about ∼ 2 cm.

Physical mechanisms of random coupling between polarization modes

33

2.3.4.2 Longitudinal vibration As a single-mode optical fiber arising as it is drawn from a preform and solidifies, the arising longitudinal vibration causes fluctuations in the fiber diameter and, consequently, fluctuations in the core diameter, which lead to changes in the speed at which light travels through the fiber due to changes in the refractive index. In case there is only linear birefringence, a change in the core diameter along the fiber will lead to a change in the amount of the birefringence alone without affecting the directions of the birefringence axes. If, in addition, there is no regular or random twisting of the axes, then varying linear birefringence cannot cause polarization mode coupling. If there is regular twisting, then fluctuations in the linear birefringence should theoretically lead to coupling between elliptical polarization modes in the frame of reference rotating with the twisting [951]. Fluctuations in the SMF diameter were theoretically investigated by Krawaric and Watkins [409]. They showed that the correlation length is approximately 90 cm. The standard deviation of the diameter was found to be about 2% of the diameter. Since the correlation length of core diameter fluctuations significantly exceeds that of random twists (∼ 2 cm), it can be assumed that the amount of linear birefringence varies adiabatically along the fiber length. Thus, fluctuations in the core diameter cannot lead to an actual increase in the polarization mode coupling. In order to estimate the effect of core diameter fluctuations on the h-parameter in case there are random twists of the birefringence axes, we performed some calculations showing that the value of the h-parameter does not change in weakly anisotropic SMFs and increases, though insignificantly (by no more than 1–2%), in strongly anisotropic SMFs.

2.3.4.3 Transverse vibration In the case of transverse vibration arising as a fiber is drawn from a preform, one observes linear birefringence induced by bending [239, 355, 715, 875]. If the plane of bending vibration coincides with the direction of the fast or slow axis of the SMF, this vibration can only result in an increase or decrease of the linear birefringence. In general, when this is not the case, the axis of total linear birefringence will not coincide with those of intrinsic birefringence; consequently, the SMF will be equivalent to a collection of linear phase plates corresponding to random segments of the SMF. The correlation function of random inhomogeneities in SMFs caused by transverse vibration (random bending) has the same form as that of inhomogeneities caused by random torsional vibration [872, 877]. For a telecommunication SMF made by Fujikura Ltd. (with the refractive index difference between the axes Δn = 10−8 ) wound into a hank 70 cm in diameter, the method of polarization reflectometry showed random variations in Δn to be about 5 × 10−9 with the correlation length of these variations to be about 16 m [460], which is much longer than those for

34

Fiber ring interferometry

torsional and longitudinal vibrations. Hence, the effect of transverse vibration of a fiber, arising as it is drawn from a preform, on the polarization model coupling is much weaker than the effect of torsional and longitudinal vibrations.

2.3.4.4 Transverse stresses If radially symmetric fibers did not acquire internal stresses in the process of drawing from a preform, they would not have linear birefringence, which means that their two natural polarization modes would be degenerate. In reality, such stress always arise and lead to slight linear birefringence (Δn = 10−6 –10−8 ). There is no data available from the literature on the amount of random linear birefringence and correlation length of these disturbances. Apparently, this is due to the fact that the correlation length of random stresses is quite large and the corresponding birefringence is seen to be regular in experiments.

2.3.5 Inhomogeneities arising in applying protective coatings The inhomogeneities arising in the course of applying a protective coating to SMFs are essentially indistinguishable from those relating to transverse stresses, discussed above; as the coating solidifies, it causes similar stresses. However, the transverse stresses produced in the coating can be significantly higher (depending on the coating material) than those in the SMF itself. In addition, the effect of the coating stresses can be revealed experimentally by removing the coating. Such experiments will be discussed in Chapter 4. Sometimes, the fiber is required to be glued to a surface such as that of a piezoceramic material. Significant transverse stresses can arise in such cases as well [275, 413].

2.3.6 Inhomogeneities arising in the course of winding The inhomogeneities arising in the course of winding an optical fiber on a drum or laying a communication cable are largely the same as those arising in the process of drawing; these include twisting and bending. In this case, fiber diameter fluctuations are practically not revealed, since this would require applying considerable variable tensile forces to the fiber; expressions of the birefringence induced by winding and tension simultaneously can be found in [355, 716]). Just as in the case of drawing, here the main contribution to the polarization mode coupling is made by random twists of the fiber. In winding fibers on large drums, the value of the h-parameter remains the same for all diameters greater than a certain threshold value. For small drum diameters,

Application of the Poincaré sphere method. . .

35

the h-parameter begins to increase rapidly, approximately inversely proportionally to the diameter [713]. Light scattering then increases as well, going into both its own and orthogonal polarization mode. The amount of scattering depends on the winding diameter, outer fiber diameter, and refractive index difference at the interface between the core and reflective shell [585]. The value of the h-parameter considerably increases if the drum has bulges [874], in which case the fiber experiences very steep radial bendings; an analysis of microbending losses in single-mode fibers was performed in [58], and it was shown in [446] that microbendings cause an effective coupling between core and cladding. Furthermore, the h-parameter significantly increases when the winding is performed randomly rather than tightly and evenly; in the former cases, an upper coil can cross several lower coils so that practically each coil rests on a multiple budges [691]. This issue will be discussed in detail in Chapter 4.

2.3.7 Rayleigh scattering: the fundamental cause of polarization mode coupling Polarization mode coupling arises even if no disturbances are introduced in drawing, coating, and winding. There is a fundamental factor that causes it: Rayleigh scattering of light in SMFs. Rayleigh scattering is the elastic scattering of light by the atomic structure of the fiber core material. Rayleigh scattering adds a correction to the h-parameter, decreasing with the light wavelength as λ−4 [131, 655]. The value of the h-parameter due to Rayleigh scattering in quartz SMFs ranges from 10−8 m−1 (for λ = 1.3 μm) [831] to 3–8 × 10−8 m−1 (for λ = 0.63 μm) [457]. It is noteworthy that Rayleigh scattering in SMFs is characterized by some polarization anisotropy, which makes about 5% [457, 461].

2.4 Application of the Poincaré sphere method for polarization calculations in fiber optic gyros. The geometric-optical and topological phases The milestones of the development of the Poincaré sphere method include: 1891–1892 Poincaré [683] suggested a method for projecting polarized light onto a sphere; subsequently, it was called the Poincaré sphere method. A detailed description of the method can be found in a number of monographs, for example, [68, 131, 291, 705, 793, 794]. A Russian translation of the portion of [683] relating to the issue in question can be found in the appendix of the paper [499]. 1938–1941 Rytov [737, 738] and Vladimirskii [892] showed that as a light beam travels in a nonplanar curvilinear path, its plane of polarization acquires some addition-

36

Fiber ring interferometry

al rotation, which is characterized by an additional geometric phase, known as the Rytov–Vladimirskii phase. 1956 Pancharatnam [664, 665] showed that even if light travels in a straight line but its state of polarization undergoes changes, there arises an additional geometric, called Pancharatnam phase, which can be conveniently displayed on the Poincaré sphere. Lately, the Poincaré sphere method has become widespread due to the necessity for calculating the geometric (topological) phase, often referred to as the Berry phase [98], in polarization optics, quantum mechanics, and classical mechanics (e. g., see the surveys [30, 97, 130, 387, 499, 891]). The method allows one to calculate the geometric phase easily and graphically through the solid angle determined by a closed curve on the Poincaré sphere, where the curve corresponds to a change in the state of polarization of light or a change in the spin of a subatomic particle. However, so far, no method has been developed for calculating the phase difference due to polarization nonreciprocity in fiber ring interferometers using the Poincaré sphere. Such a method will be considered in Chapter 7.

2.5 Thomas precession. Interpretation and observation issues The fact that the phase difference between counterpropagating waves of arbitrary nature due to the Sagnac effect is inversely proportional to the wavelength (see relation (9.6)) makes it tempting to use de Broglie waves of material particles with nonzero mass (such as electrons, neutrons, atoms, etc.). The relativistic kinematic effect known as the Thomas precession is observed for particles with spin as well as macroscopic objects [521, 540, 720, 849, 850]. It manifests itself as rotation (precession) of the local frame of reference traveling with the object in a curvilinear path [540, 633]. In particular, the precession of the axis of a gyro moving in a circular orbit is also the Thomas precession [540]. If material particles (e. g., electrons) had the same orientation of their spins (same polarization) at the input of a ring interferometer, then the spin orientations of the particles in the counterpropagating waves would be different at the output. When two counterpropagating waves interfere with each other, this will result in a phase difference unrelated to rotation [29, 593]; the order of magnitude of this phase difference is determined by the ratio v 2/c 2 , where v is the speed of the particle and c is the speed of light. Thus, the Thomas precession is an effect that fundamentally restricts the accuracy of measuring the angular velocity of rotation in Sagnac interferometers based on de Broglie waves. It has been over 80 years since the publication of Thomas’s articles [849, 850] and over 100 years since the publication of Sommerfeld’s articles [810, 811], which considered, in the most general case, the relativistic precession of a rigid body as it moves in

Thomas precession. Interpretation and observation issues

37

a curvilinear path. Despite this, as shown in [521] (see also [720, 721]), different studies obtain different relations between the angular velocity ΩT of the Thomas precession of a gyro’s rotation axis and the orbital angular velocity ω of the gyro. Indeed, it follows from the results of [70, 89, 194, 240, 345, 621, 633, 850] that as the linear orbital velocity tends to the speed of light, the angular velocity of the Thomas precession ΩT tends to infinity, whereas from [81, 166–168, 304, 305, 504, 506, 523, 524, 540, 719– 721, 806–808, 823] it follows that ΩT tends to ω. About ten more versions of the relation between ΩT and ω can be retrieved from the literature [197, 241, 246, 247, 254, 261, 285, 336, 391, 404, 444, 477, 667, 759, 760, 812, 913, 939, 943]. So, there are a number of completely different results on the same issue available in the literature. This requires a thorough and adequate derivation of the relation between ΩT and ω, which will be performed in Section 9.3. The above survey of the most important publications relating to the questions discussed in the present monograph indicates that some of these questions have not been studied previously, while others, although considered, have been solved either incorrectly or for some special cases only. There is no clarity on a number of questions, as the results of different studies are contradictory. Thus, the survey helps us helps to define the main directions of our research.

3 Development of the theory of linear interaction (random coupling) between polarization modes in single-mode optical fibers 3.1 Phenomenological estimates of the random coupling between polarization modes 3.1.1 Small perturbation method As noted in Chapter 2, the most adequate descriptions of the small perturbation method, compared to the other phenomenological studies, as applied to analyzing the polarization mode coupling at inhomogeneities of single-mode optical fibers (SMF) were given in [126, 941]. However, the studies [126, 941] omitted some terms of the second order of smallness in the entries of the Jones matrix characterizing the SMF. As a result, the determinant of the Jones matrix was not equal to unity, which formally indicates the presence of dichroism, although there are no light losses. Such a matrix is no longer the Jones matrix of an elliptical phase plate as it must be in the presence of unperturbed linear polarization modes [73, 84, 280, 296, 297, 634, 876, 933, 951]. From now on, we use the following rigorous expression of the Jones matrix of an elliptical phase plate given in the monograph by Shurkliff [794] to be employed in conjunction with the small perturbation method: 

i sin 2R sin δ2 e−iγ cos2 R eiδ/2 + sin2 R e−iδ/2 , B= (3.1) δ i sin 2R sin 2 eiγ cos2 R e−iδ/2 + sin2 R eiδ/2 where δ = βL is the phase difference between the light beams that traveled along the slow and fast axes of a fiber segment of length L, β = 2π Δn/λ is the linear birefringence of the fiber, λ is the light wavelength in vacuum, and Δn is the difference between the refractive indices along the slow and fast axes. As will be shown in Section 3.2.4, κ ≡ sin2 (R) = 12 [1 − exp(−2hL)] is the intensity of the light that transferred from one orthogonal mode to the other at inhomogeneities in the fiber segment of length L, γ is the phase of this light, and h is the polarization holding parameter [354, 583, 710]. The small perturbation method is applicable provided that hL  1 .

(3.2)

This condition ensures smallness of the off-diagonal entries of the Jones matrix. Note that κ and γ are integral parameters, which characterize the total intensity and phase of the light transferred from one polarization mode to the other at a large number of random inhomogeneities in a fiber segment of length L. As will be shown in Section 3.2.4, if condition (3.2) holds for an ensemble of independent implementations

Phenomenological estimates of the random coupling

39

of random inhomogeneities, we have κ const and γ is uniformly distributed on the interval [0, 2π ]. If condition (3.2) holds, we can expand κ in a series in the small parameters hL and confine ourselves to the linear approximation: κ = 12 [1 − exp(−2hL)] ≈ hL. By setting a ≡ hL, we get sin2 (R) ≈ a and cos2 (R) ≈ 1 − a. Then expression (3.1) can be rewritten as 

δ (1 − a)eiδ/2 + a e−iδ/2 2i (1 − a)a sin 2 e−iγ ˜= . (3.3) B δ 2i (1 − a)a sin 2 eiγ a e−iδ/2 + (1 − a)eiδ/2 Note that, despite the fact that all entries of the Jones matrix (3.3) are approximate, we ˜ = 1 unlike [126, 152, 941]. Thus, the calculation results obtained using (3.3) have det B will not lead to errors due to the fictitious dichroism arising when the determinant of the Jones matrix is not equal to unity. If one considers the propagation of nonmonochromatic light through an SMF using the small perturbation method, then one should keep in mind that the polarization mode coupling on the fiber length ldep = λ20 /(Δλ Δn), where λ0 is the mean wavelength of the light source and Δλ is its spectral width measured at the half height. This was shown in [483, 941] for some practically important cases, in particular, when one needs to calculate the evolution of the polarization degree of nonmonochromatic light in SMFs and zero shift deviation in FRIs. Then, when studying the transfer of nonmonochromatic light form one polarization mode to the other due to random coupling between the polarization modes on a fiber segment of length ldep , one should use the Jones matrix (3.1) or (3.3) with the total segment length L substituted for by the depolarization length ldep : Bdep = B(L = ldep ) ,

(3.4)

˜ = ldep ) . ˜dep = B(L B

(3.5)

If the calculations are to be carried out for the entire segment length L, one must impose the condition hldep  1 (3.6) in addition to (3.2) in order to ensure that the small perturbation method is applicable. It should be noted that ldep  L in many practically important cases; however, the opposite relation ldep  L is also possible. One usually assumes that conditions (3.2) and (3.6) suffice to ensure the applicability of the for nonmonochromatic light. However, one more condition, Δλ  λ0 , (3.7) must be imposed. The physical meaning of condition (3.2) is smallness of the relative intensity of the light transferred, on the fiber length L, from one polarization mode to the other. Condition (3.6) suggests smallness of the normalized electric field of the light transferred, on the fiber length ldep from one polarization mode to the other. Condition (3.7) ensures a relatively narrow spectrum of the nonmonochromatic

40

Development of the theory of interaction between polarization modes

light. The point is that the amount of the linear birefringence β = 2π Δn/λ depends on the light wavelength, and hence the phase difference between the light that traveled along the slow and fast axes of the fiber segment, δ = βL, is also a function of the wavelength. Furthermore, as will be shown in Section 3.2.2, the value of the hparameter is also a function of the wavelength. If condition (3.7) does not hold for different wavelengths in the nonmonochromatic light spectrum, we have κ ≠ const and δ ≠ const in the Jones matrix (3.1), which significantly complicates the procedure of averaging over the spectrum.

3.1.2 Expanding the scope of the small perturbation method by partitioning the fiber into segments whose length is equal to the depolarization length The number of necessary conditions for the applicability of the small perturbation method can be reduce to two by partitioning the single-mode fiber into segments of equal length ldep . As shown in our papers [483, 534], each of these segments has its own value of γ , statistically independent of the other segments, while the parameter a is the same for all segments (the entire length of the fiber). Then the Jones matrix of the entire fiber is expressed as BΣ =

N 

Bk ,

(3.8)

˜k , B

(3.9)

k=1

or ˜Σ = B

N  k=1

˜k ) is the Jones matrix of the kth segment of length ldep , given by (3.4) where Bk (or B (resp., (3.5)), and N = L/ldep . In this case, the applicability of the small perturbation method does not require setting condition (3.2). However, the calculations become more complicated, as it is required to perform averaging over all N segments, which is only possible by numerical simulation on a computer. Note that for traditional nonmonochromatic light sources such as superluminescent diodes and active superfluorescent fibers, condition (3.6) only holds for strongly anisotropic SMFs. Thus, even partitioning of the entire fiber length into segment of length ldep does not make the small perturbation method universally applicable; it just removes the restriction ˜k = 1, we on the overall length of the fiber. It should also be noted that since det B ˜Σ = 1 and so there is no error accumulated as the product in (3.9) is comhave det B puted, despite the fact that the entries of the matrices Bk are replaced by their linear approximations. The main conclusion of Section 3.1 is that phenomenological models of the random coupling between polarization modes can be used to describe mode interaction processes only if the applicability conditions of the small perturbation method are satisfied, since these models deal with integral parameters.

A physical model of the polarization mode coupling

41

3.2 A physical model of the polarization mode coupling 3.2.1 A model of random inhomogeneities in SMFs with random twists of the anisotropy axes Consider the mechanism for the appearance of random twists in SMFs. As it is drawn from a preform, the fiber moves from top to bottom while gradually solidifying and then passes through a roller having a groove to hold the fiber. The roll applies to the fiber a friction force that has not only a longitudinal component (along the fiber) but also a centrifugal component (sometimes called the rolling force), which drags the fiber to one or the other sloping wall of the roller’s groove causing twisting. After that, the tension force takes the fiber back to the groove center and then the process repeats. It is clear that the thus excited torsional oscillations are random, with the current angular velocity jump being independent of the previous one. The torsional oscillations are transmitted to the entire fiber length, including the top part where the material has not fully solidified yet. As a result, once the fiber has solidified, its birefringence axes “memorize” the random twists. Note that sometimes, after having been drawn from the preform, the fiber is passed through a pair of mutually perpendicular freely rotating shafts, which, just as the roller, serve to fix the fiber position in space. Torsional oscillations arise in the fiber in this case, too. The physical basis for the model in question is considered here because of the random twists of the birefringence axes experimentally observed in SMFs [2, 8, 590]; these twists are discussed in detail in Chapter 2. Suppose a single-mode fiber is partitioned into a number of segments of random length lk each of which having uniformly changing (twisting) azimuths of the linear birefringence axes, Θk . For each segment, its neighbors have different twisting, but the azimuth α is assumed to be a continuous function of the fiber length coordinate l and its derivative, Θ = dα/dl is piecewise constant. The amount of random twist Θk in each segment is assumed to be uniformly distributed on the interval [−Θmax , Θmax ], with the twists in all segments being independent from one another. The random segment lengths lk are also assumed to be independent and distributed exponentially with density function ρl (z) = e−z/l /l and mean l. As noted above, if the linear birefringence axes have twisting, the natural modes of the SMF become elliptical in the frame of reference that follows the twisting [73, 84, 280, 297, 634, 876, 933, 951]. Since, according to the model adopted, each fiber segment has a constant twisting, the Jones matrix of the kth segment in the laboratory frame of reference is expressed as [933]:

where αk =

k j=1

(Bk )tw = T (−αk )Mk T (αk−1 ) , Θj lj , T (α) is the Jones matrix of turn through the angle α,

 cos α sin α T (α) = , − sin α cos α

(3.10)

(3.11)

42

Development of the theory of interaction between polarization modes

Mk is the Jones matrix of the kth fiber section in the local spiral reference frame, ⎛ ⎞ iβ 2(1 − g)Θk βk lk βk lk βk lk + sin sin ⎜cos ⎟ 2 βk 2 βk 2 ⎟ ⎜ ⎜ ⎟, Mk = ⎜ (3.12) ⎟ βk lk iβ βk lk βk lk ⎠ ⎝ 2(1 − g)Θk cos − − sin sin βk 2 2 βk 2 β is the natural (unperturbed) linear birefringence of the fiber, βk = 2 2 β + [2(1 − g)Θk ] is the elliptical birefringence of the kth normal mode in the spiral frame, and g is the photoelasticity coefficient of the fiber material. The theoretical value of g quartz is 0.08 [715]; the experimental values reported in [876] and [73] are 0.065 ± 0.005 and 0.073, respectively. Note that the matrix Mk characterizes the kth spiral polarization mode whose rotation through the angle αk − αk−1 is taken into account by the matrices T (−αk ) and T (αk−1 ). It should be mentioned

that the representation (3.12) is used in a number of studies [73, 876, 933, 951]. Since, as shown in Chapter 2, the main cause for the appearance of polarization mode coupling in single-mode fibers are random twists, which arise as the fiber is drawn from the preform (i. e., then the fiber is fabricated), the natural polarization modes are spiral polarization modes. In each segment, where the amount of random twist is constant, the polarization mode has a constant magnitude and elliptical sign.

3.2.2 Connection between the polarization holding parameter and statistics of random inhomogeneities In order to determine the model parameters, maximum fluctuation magnitude Θmax and the mean segment length l and to show that the model does not contradict the experimental data, let us find the expression of the h-parameter (holding parameter) within the framework of the model of random inhomogeneities suggested. The value of the h-parameter, determined as the spectrum of random perturbations of the linear birefringence, is related to the amount of random inhomogeneities, c(z), as a function of the fiber fiber length coordinate z as follows [8, 941]: 1 L→∞ L

h = lim

2   L      c(z) eiβz dz ,  

(3.13)

0

where c(z) = (1−g)Θ(z), L is the fiber length, and the angle brackets denote the averaging over the ensemble of fibers. By the Wiener–Khintchin theorem [287, 582, 739, 740], the h-parameter is related to the correlation function of the process c(z) as follows: ∞ 2

h = 2c (z)

ϕ(u) cos(βu)du , 0

(3.14)

A physical model of the polarization mode coupling

43

where c 2 (z) is the variance and ϕ(u) is the correlation function of c(z). The suggested model of the random function c(z) satisfies the conditions of Doob’s theorem [214] (see also [609]) and so is a stationary normal Markov process. Hence, ϕ(u) = e−u/l ,

c 2 (z) =

2 cmax , 3

(3.15)

which can also be obtained by the direct calculation of c(z1 )c(z2 ). Note that, for the exponential law of distribution of the random segment lengths, the correlation length of the random process c(z) equals lc = l [739, 740]. Substituting (3.15) into (3.14) yields the expression of the h-parameter for the suggested model of random twisting of the birefringence axes h=

2 [(1 − g)Θmax ]2 l . 3 1 + β2 l2

(3.16)

Note that a very similar dependence of the h-parameter on Θmax and l can be found in [672]. In the papers [160, 453, 831, 867], it is also shown that h = h0 /[1 + β2 l2 ]. Figure 3.1 displays the dependence of h on Δn defined by formula (3.16) with Θmax = 1.9 rad/m (cmax = 1.75 rad/m), l = 2 cm, g = 0.08, and λ0 = 800 nm. These value of Θmax and l correspond to those measured in [2, 8, 590]. Also shown in Fig. 3.1 are experimental values of the h-parameter obtained by different authors [60, 152, 364, 483, 484, 490, 712, 802, 874] (some data can also be found in the review [645]) for different kinds of single-mode fiber (with strong, moderate, and weak birefringence) fabricated by different technologies and operating in the wavelength range 0.63–1.1 μm. It is apparent from Fig. 3.1 that the experimental data fit the graph h(Δn), based on formula (3.16), quite well. This validates the suggested model of random twisting of the linear birefringence axes. There is another approach to determining the values of the h-parameter based on considering the process energy exchange between natural polarization modes as they travel through a single-mode fiber [354, 583, 710]. This approach relies on the formulas ˜

I1 (z) =

1 + e−2hz , 2

˜

I2 (z) =

1 − e−2hz , 2

(3.17)

where I1 and I2 are the intensities of the natural polarization modes as functions of the fiber length coordinate z in case there is only the first mode excited at the SMF input. It was shown in our paper [572] that for a sufficiently long fiber with a large number of random segments N such that N ≈ z/l  1, the polarization holding ˜ in (3.17) is expressed as parameter h ⎛ ⎞ 1 + β2 l2 2(1 − g)Θ 1 l max ⎠. ˜=− ln ⎝ arctan h (3.18) 2l 2(1 − g)Θmax l 1 + β2 l2

44

Development of the theory of interaction between polarization modes

10−1

10−2

10−3

h, m−1

10−4

10−5

10−6

10−7

10−8 10−8

10−7

10–6

10−5 Δn

10−4

10−3

10−2

Fig. 3.1: Theoretical dependence h(Δn) (solid curve) and experimental values of h obtained by different studies: + [152],  [483], • [60], ⊗ [484], ∗ [490],  [874], × [364],  [802], ♦ [712]).

Formulas (3.16) and (3.18) are significantly different. However, as shown in our paper [572], expression (3.18) is asymptotically equivalent to (3.16) for ξ  1, where ξ=

2(1 − g)Θmax l . 1 + β2 l2

The condition ξ  1 suggests that the circular birefringence induced by the maximum possible random twisting is small as compared with the intrinsic linear birefringence of the single-mode fiber, or as compared with the reciprocal correlation length ˜ and h against β for of random inhomogeneities. Figure 3.2 compares the graphs of h l = 2 cm and three different values of the maximum random twist of the fiber Θmax . In the practically important case of Θmax = 1.9 rad/m, which corresponds to actual random twists arising in the process of drawing of a single-mode fiber from the pre˜ and h coincide with a rather high accuracy for all values of β. For set, the values of h ˜ and h only coincide for sufficiently large linear birefrinlarger Θmax , the values of h gence β. Since the two definitions of the polarization mode coupling parameter (3.13) ˜ , are equivaand (3.17), as well as the resulting expressions (3.16) for h and (3.18) for h

A physical model of the polarization mode coupling

45

102

100

h (m−1)

3

10–2 2

10–4 1

10–6

10–8

100

102 β (rad/m)

104

˜ (formula (3.18), solid line) and h (formula (3.16), dotted line) on Fig. 3.2: Dependences of h the amount of linear birefringence β for l = 2 cm and three different values of Θmax : 1, Θmax = 1.9 rad/m; 2, Θmax = 19 rad/m; and 3, Θmax = 190 rad/m.

lent for actual single-mode fibers, we will only use h to denote the polarization mode coupling parameter in what follows. ˜ It is worthwhile to explain the physical meaning of the difference between h and h. At small perturbations, the contribution to the random coupling between polarization modes is only made by the first spectral component of the spatial power spectrum of random perturbations on the polarization beat length. However, at large perturbations, the higher harmonics of the spatial power spectrum are dominant; the weights with which the harmonics are summed up depend on the value of the parameter ξ . Hence, expression (3.18) is more general. In case the fiber is would on a reel without considerable additional perturbations and so the random twists are mainly due to the drawing process, expressions (3.18) and (3.16) practically the same result and so one would prefer the simpler formula (3.16). However, if the fiber is wound on the reel with significant perturbations (e. g., see [484, 493, 691, 874]), the value of the h-parameter considerably increases and so one should use formula (3.18).

3.2.3 Polarization holding parameter in the case of random and regular twisting The experimental studies [748, 749] of the polarization mode coupling in a 4 m long regularly twisted single-mode fiber (with Δn = 1.2 × 10−5 and λ = 1.15 μm) showed that the coupling becomes weaker as the twisting increases. It follows from the mea-

46

Development of the theory of interaction between polarization modes

surement results that if, in the absence of regular twisting, the value of the effective hparameter for linearly polarized natural modes is 6 × 10−4 m−1 , then, with a regular twist Θ0 = 100 rad/m, when the natural modes become practically circular, it decreases to 2.5 × 10−5 m−1 . One of the hypothetical mechanisms explaining the weakening of the natural polarization mode coupling in a twisted single-mode fiber suggests that the twisting causes a strong circular dichroism [363]. However, it should be noted that no observable circular dichroism in twisted SMFs was found in the studies [748, 749]. Another mechanism is based on the fact that the amount of elliptical birefringence increases with increasing the regular twist Θ0 and, hence, decreasing the polarization beat length. This should lead, in the presence of random twists and in accordance with (3.16) and (3.18), to a decrease in value of the h-parameter. It should be noted that twisting of fibers suppresses the unperturbed linear birefringence [73, 688, 876] and causes a decrease in the polarization mode dispersion [264–266, 441, 442, 750, 763]. In our study [572], it was shown that for fibers with practically achievable parameters and regular twisting, the intensities of both polarization modes, provided that N ≈ z/l  1 and the first mode is only excited at the fiber input, as expressed as I1,2 (z) =

1 ± r2 e−2h2 z ± |r3 |e−2h3 z cos(r4 + h4 z) , 2

(3.19)

where the rj are some coefficients. It is apparent that we have three rather than one hparameter here. The most important parameter out of the three is the one that make the greatest contribution to I1 and, correspondingly, has the greatest influence on the polarization mode coupling. It is clear that this h-parameter has the least magnitude and appears with the largest coefficient in (3.19). Let us discuss the physical meanings of these polarization holding parameters in a uniformly twisted single-mode fiber. The parameter denoted h2 has the same phys˜ -parameter above if there is no twisting, and hence is the expoical meaning as the h nential decrement of the intensities of orthogonal elliptical polarization modes. It is apparent from (3.19) that apart from the exponential decay, the intensities of the polarization mode have oscillations. The parameter h3 is the decay decrement of these oscillations, while h4 defines their spatial period. The reason why these oscillations arise is that, as shown in [572] for an SMF with constant and random twisting as well as linear birefringence, the natural elliptical polarization mode supplied to the fiber input does not coincide with the averaged (over the random twists) elliptical polarization mode. For realistic linear birefringence parameters as well as regular and random twisting, the oscillation amplitudes are quite small and, moreover, the oscillations decay very rapidly (after about half the period of polarization beats), thus practically not affecting the polarization mode coupling process, which is determined by the parameter h2 [572].

A physical model of the polarization mode coupling

47

10−2

h2 (m−1)

10−4 10−6 10−8 β=3.927rad/m 10−10 β=0.3927rad/m 10−12

β=0.03927rad/m 0

200

400

600 Θ0 (rad/m)

800

1000

Fig. 3.3: Dependence of the main polarization-holding parameter h2 , which characterizes the mode coupling in twisted single-mode fibers, on the amount of regular twist Θ0 for l = 1 cm, Θmax = 1.75 rad/m, and three different values of the linear birefringence β.

In the general case, one cannot express h2 analytically. However, in the special case where the circular birefringence induced by the regular twisting is significantly greater than the intrinsic linear birefringence of the fiber, the parameter h2 can be expressed as follows [572]: h2 ≈

2 β2 Θmax . 24 (1 − g)2 lΘ04

(3.20)

Thus, h2 decreases inversely proportionally to Θ04 as the regular twists increase. Figure 3.3 displays h2 versus the regular twist Θ0 for three different values of the linear birefringence. It is noteworthy that single-mode fibers break as large twists (Θ0  200 rad/m) [757]; therefore, efficient suppression of the parameter h2 can only be realized in practice for fibers with weak linear birefringence.

3.2.4 Statistical properties of the polarization modes for fibers with random inhomogeneities This section aims to show that a single-mode fiber whose length significantly exceeds the correlation length of random inhomogeneities can be characterized with a Jones matrix whose one or more parameters have a prescribed statistical distributions corresponding to an ensemble of independent implementations of the random inhomo-

48

Development of the theory of interaction between polarization modes

geneities. In other words, the purpose of this section is to find, given a fixed light wavelength λ, integral statistical properties of the polarization modes of a singlemode fiber with random inhomogeneities and obtain the probability distributions of the parameters of its Jones matrix. Another purpose of this section is to determine the most rational form of such a representation and conditions for its validity as well as to relate the parameters of the integral Jones matrix, where it makes physical sense, to the value of the h-parameter. The calculations are based on the physical model of random inhomogeneities described in Section 3.2.1 and performed using a mathematical modeling method. It should be reminded that, as shown in [280, 825, 951], each fiber segment with unperturbed linear birefringence that has a random length and a random twist, which induces a circular birefringence, is equivalent to an elliptical phase plate. Furthermore, the Poincaré–Herpin theorem [19, 218, 291, 322, 332, 499] (see also [9, 704, 843]) states that a set of elliptical phase plates is equivalent to a single elliptical phase plate. Consider two possible forms of representation of the Jones matrix for an elliptical phase plate. One of them is presented above in Section 3.1.1 (see relation (3.1)). The parameters R and γ are related to the ellipticity of the natural modes χ = arctan(b/a), where a and b are the semi-major and semi-minor axes of the polarization ellipse, and the azimuth of the major axis α as follows [794]: α= χ=

1 2 1 2

arctan(tan 2R cos γ) ,

(3.21)

arcsin(sin 2R sin γ) .

(3.22)

It is apparent from (3.21) and (3.22) that for constant R and variable γ , the ellipticity χ and azimuth α change simultaneously in such a way that α = 0 at maximum |χ| and χ = 0 at maximum |α|. Let us write out the Jones matrix of a phase plate with ellipticity χ in the frame of reference defined by the axes of the polarization ellipse (e. g., see [804]): ⎞ ⎛ δ δ δ sin 2χ sin 2 cos 2 + i cos 2χ sin 2  ⎠. B =⎝ (3.23) − sin 2χ sin δ2 cos δ2 − i cos 2χ sin δ2 The matrices B and B  (see (3.1) and (3.23)) are related by B = T (−α)B  T (α) ,

(3.24)

where |T (α)| is the matrix of rotation through the angle α (see (3.11)). Formally, the description of a fiber segment with the matrix B given by (3.1) and that with the product of matrices (3.24) are completely equivalent. Choosing which form of representation is more preferable is reduced to choosing which description is the more rational and adequate to characterize the changes in the natural polarization modes for different independent realizations of the random inhomogeneities as the fiber temperature changes, provided that the ergodic hypothesis holds true.

49

A physical model of the polarization mode coupling

R = |a|

γ=0

γ = π/4

γ = π/2

γ = 3π/4

γ=π

γ = 5π/4

γ = 3π/2

γ = 7π/4

γ = 2π

γ = π/2

γ = 3π/4

γ=π

γ = 5π/4

γ = 3π/2

γ = 7π/4

γ =2π

R = −|a|

γ=0

γ = π/4

χ=0

χ = |χmax|/2

χ = |χmax|

χ = |χmax|/2

χ=0

χ = −|χmax|/2

χ = −|χmax|

χ = −|χmax|/2

χ=0

α=0

α = |αmax|/2

α = |αmax|

α = |αmax|/2

α=0

α = −|αmax|/2

α = −|αmax|

α = −|αmax|/2

α=0

α=0

0