Ring interferometry

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

Dr. Vera I. Pozdnyakova
Russian Academy of Sciences
Institute for Physics of Microstructures
GSP-105
603950 Nizhny Novgorod
Russian Federation


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 .
© 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

8
2
Fiber ring interferometry
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
9

media
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
10
moving with counterpropagating waves
2.1.2.2
Sagnac effect due to the sign difference between the non-relativistic
gravitational scalar potentials of Coriolis forces in reference frames
10
moving with counterpropagating waves
2.1.2.3
Quantum mechanical Sagnac effect due to the influence of the Coriolis
force vector potential on the wave function phases of
11
counterpropagating waves in rotating reference frames
2.1.3
Attempts to explain the Sagnac effect by analogy with other
11
effects
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
15
effect
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
16
angular momenta of photons in counterpropagating waves
2.1.4.7
Sagnac effect as a manifestation of the inertial properties of
16
an electromagnetic field


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
17
Physical problems of the fiber ring interferometry
Milestones of the creation and development of optical ring
17
interferometry and gyroscopy based on the Sagnac effect
Sources for additional nonreciprocity of fiber ring
20
interferometers
General characterization of sources for additional nonreciprocity of
20
fiber ring interferometers
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
23
actions
Nonreciprocity due to the Faraday effect in external magnetic
23
field

Nonreciprocal effects caused by nonlinear interaction between
23
counterpropagating waves (optical Kerr effect)
Nonreciprocity caused by relativistic effects in fiber ring
24
interferometers
Fluctuations and ultimate sensitivity of fiber ring
24
interferometers
Methods for achieving the maximum sensitivity to rotation and
25
processing the output signal
Applications of fiber optic gyroscopes and fiber ring
26
interferometers
Physical mechanisms of random coupling between polarization
28
modes
Milestones of the development of the theory of polarization mode
28
linking in single-mode optical fibers
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
35
coupling
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
4
4.1
4.2

vii

Development of the theory of interaction between polarization
modes
38
38
Phenomenological estimates of the random coupling
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
40
length
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

42
of random inhomogeneities
Polarization holding parameter in the case of random and regular
45
twisting
Statistical properties of the polarization modes for fibers with random
47
inhomogeneities
Evolution of the degree of polarization of nonmonochromatic
55
light
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
68
nonmonochromatic light
Increasing of the correlation length of nonmonochromatic light traveling
69
through a single-mode fiber with random inhomogeneities
72
Anholonomy of the evolution of light polarization

4.5
4.6
4.7

Experimental study of random coupling between polarization modes
76
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

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

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
131
Influence of the amount of random twist of the fiber
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
132
inhomogeneity
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
139

perturbation method
Calculation of the zero shift deviation by the method of mathematical
139
modeling of random inhomogeneities
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
148
Fiber ring interferometers of nonstandard configuration
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
158
fibers
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
167
interferometer without a polarizer
Fiber ring interferometer with a depolarizer of nonmonochromatic
169
light
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
171
from a high-birefringence fiber in a limited temperature range
Geometric phases in optics. The Poincaré sphere method
172
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
175

Birefringence in SMFs due to mechanical deformations
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
177
the case of noncoplanar winding
177
Rytov effect in the FRI circuit fiber
Rytov–Vladimirskii phase and PP2 in SMFs with noncoplanar
179
winding
180
Rytov phase detection in FRIs
Polarization nonreciprocity in FRIs. Nonreciprocal geometric
182
phase
Determination of a polarization state ensuring the absence of
189
NPDCM
Criticism of unsubstantiated hypotheses relating to geometric
191
phases
Opto-mechanical analogies relating to light propagation in
195
SMFs

The analogy between the Rytov effect polarization optics and Ishlinskii
195
effect in classical mechanics
An opto-mechanical analogy of an SMF with twisting of the linear
198
birefringence axes
201
Time-dependent, nonlinear, and magnetic effects
Influence of the second harmonic of the phase modulation
201
frequency
In-phase and quadrature components of the parasitic phase
201
modulation
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

220
Relativistic effects in optical and non-optical ring interferometers

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
226
Non-optical Sagnac sensors of angular velocity
A ring interferometer based on slow acoustic or magnetic
226
waves
226
Advantages of using slow waves in ring interferometers
Choosing an optimal frequency of the slow waves in ring
227
interferometers
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
236
applied to the angle of relativistic aberration
236
Thomas precession
Ishlinskii’s theorem as a classical analogue of Thomas
237
precession
Observed rotation of an object rapidly moving in a circular path and

238
Thomas precession
Physical meanings of the Thomas precession and Ishlinskii
241
angle
Influence of Thomas precession on the zero shift of ring interferometers
241
based on de Broglie waves of matter particles with spin
243
Potential usage of FRIs for detecting fundamental effects
Verification of the basic postulates of special and general relativity
243
using FRIs
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 instantaneously 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;


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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);


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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.