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 malykin@ufp.appl.sci-nnov.ru Dr Vera I Pozdnyakova Russian Academy of Sciences Institute for Physics of Microstructures GSP-105 603950 Nizhny Novgorod Russian Federation vera@ipm.sci-nnov.ru 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 Introduction xi xiii Fiber ring interferometry 2.1 Sagnac effect Correct and incorrect explanations 2.1.1 Correct explanations of the Sagnac effect 2.1.1.1 Sagnac effect in special relativity 2.1.1.2 Sagnac effect in general relativity 2.1.1.3 Methods for calculating the Sagnac phase shift in anisotropic 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 3.1 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 4.1 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 5.1 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 6.1 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 7.1 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 8.1 8.1.1 8.1.2 8.1.3 8.2 8.3 8.4 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 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independent realizations of random inhomogeneities 50, 58, 68, 111, 115, 136, 146 – for temperature 111, 115, 136, 146 B beam splitter 26, 110, 148, 165 Birefringence – circular 47, 48, 64, 80, 175, 176, 179, 180, 189, 215, 248 – elliptical 42, 63 – linear 22, 27, 38, 42, 44, 46, 47, 57, 79, 80, 86, 111, 112, 175, 176 – twisting of axis 30, 31, 41, 43, 57, 131 Budden–Kravtsov equation 28, 81 C Chaplygin sleigh 198, 200 chromatic dispersion 1, 27, 112, 166 coefficient of thermal expansion 72, 111 coherence length 211 coherence matrix 55 coherent length 85 coning 196, 200 correlation length – of nonmonochromatic light 69, 70, 72 – of random inhomogeneities 32, 33, 43 Cotton–Mouton effect 80 coupling of polarization modes 2, 28, 30, 31, 46, 76, 151 current-voltage characteristic 230 D Darboux trihedron 179, 195 de Broglie waves 8, 14, 15, 36, 232–235, 241, 243 degree of polarization 4, 28, 30, 55 depolarizer of nonmonochromatic light 22, 148–150, 152, 154, 158, 163, 169, 244 dichroism 38, 46, 95, 167 Doppler effect 14, 27, 76, 222 dual linking 21, 99 E electromagnetic optical nonreciprocal linear birefringence 246–249 elliptical jacket 93 ergodic hypothesis 48, 111, 133 ergodicity conditions 22, 114, 122, 135 extinction ratio of polarizer 72, 96, 102 F Faraday effect 20, 76, 215, 247, 249 fiber optic gyroscope 17, 19, 21, 26, 98, 148, 153, 209, 215, 243 fiber ring interferometer 1–6, 14, 69, 102, 103, 148, 150, 163, 215, 243 – deviation of zero shift 4, 20, 100, 102, 110, 115, 136, 138, 139, 142, 144, 148, 158, 162, 167, 170, 189, 212, 217, 243, 244 – “minimal” diagram/minimum configuration 98, 100, 113, 148 – wanted signal fading 62, 139, 140, 145, 148, 149, 155, 156, 251 – zero drift 3, 20, 25, 99, 100, 111, 131, 133, 136, 146, 156, 244 – zero shift 3, 62, 95, 98, 100, 102–104, 114, 115, 128, 132, 137, 139, 148, 155, 159, 167, 189, 201, 204, 207, 209, 217, 232, 241 fiber ring resonator 16, 19 Fresnel–Fizeau effect 15 H Hänsch effect 211, 214 I Ishlinskii angle 197, 237, 241 Ishlinskii effect 197, 200 Ishlinskii theorem about solid angle 196, 236, 237 J Jones matrix – differential 73 – of elliptical phase plate 38, 48 – of fiber section 42 – of polarizer 108 – of turn 41 Jones vector 57, 96, 103 300 Index K Kerr effect 2, 24, 80, 168, 209, 213, 214, 247 L length – of depolarization 22, 39, 56, 64 – of polarization beats 79, 177, 183 Lense–Thirring effect 18, 24, 163, 213, 243 Lorentz transformations 221, 222 Lyot depolarizer 22 M Mach–Zehnder interferometer 28, 69, 182, 234, 235 Michelson interferometer 24, 28, 69, 182 modulation – incidental 201, 203, 206, 214, 249 – of amplitude 202 – of phase 18, 164, 201, 202, 208 Müller matrix 30, 81 N noncommutativity of finite rotations 196 nonreciprocal geometric phase of counterpropagating modes 184–190 nonreciprocity phase difference of counterpropagating modes 98, 106, 107, 110, 182, 188, 189, 191, 194, 210 O optical activity see birefringence, circular P Pancharatnam phase 172 – first type 172, 175, 187 – second type 172–175, 180, 181, 197, 200 parallel translation/transport 195–197, 200 Pfaffian equations 73 phase – dynamic 174, 179, 197 – geometric 172, 180, 181, 183 – kinematic 197 phase modulator 164, 204, 213 piezoelectric element 202, 204, 207, 213 Pockels effect 247, 249 Poincaré sphere method 35, 78, 172, 182, 186, 188 Poincaré–Herpin theorem 48, 157 polarization degree 4, 22, 28, 30, 76, 79, 82, 148 – evolution 39, 55–69, 92 polarization fiber ring interferometer 27, 246 polarization mode dispersion 1, 27, 46 polarization nonreciprocity 2, 21, 98–107, 110, 189 – first type 99, 164, 181 – second type 99, 180, 181 – third type 99 polarization state 28, 65, 76, 79, 81, 93, 107, 138, 165, 167, 168, 172, 189 – evolution 29, 72, 79, 82, 172, 183, 184, 189, 250 polarization-holding parameter 4, 38, 42, 43, 46, 89, 177 R reciprocity theorem 2, 21, 98, 209 ribbon single-mode optical fiber 93–95, 178, 247 Riccati equation 75, 81 Rytov angle 177, 178, 194 Rytov effect 177, 179, 181, 194, 195, 197, 200 Rytov–Vladimirskii phase 36, 179–181, 194, 197 S Sagnac effect 1, 8–17, 24, 107, 191, 192, 194, 220, 224, 225, 227, 232, 235 shot noise 20, 230, 233 Shupe effect 20, 121, 244, 245 single polarization single-mode optical fiber 98, 100, 103, 105, 185 slow waves 8, 14, 226–231, 235 small perturbation method 30, 31, 38–40, 55, 136, 139, 146, 156 spherical triangle 173, 185–188, 190 spiral frame of reference 28, 41, 42, 179 spiral polarization modes 28, 32, 176, 179 state of polarization 28 Stokes parameters see Stokes vector Stokes vector 55, 78, 110 stress cladding 112 stress jacket 95 T Thomas precession 36, 233, 236, 241 W Wiener–Khintchin theorem 42, 216 Y Yakubovich effect 211, 214 De Gruyter Studies in Mathematical Physics Volume 20 Valery A Slaev, Anna G Chunovkina, Leonid A Mironovsky Metrology and Theory of Measurement, 2013 ISBN 978-3-11-028473-7, e-ISBN 978-3-11-028482-9, Set-ISBN 978-3-11-028483-6 Volume 19 Edward A Bormashenko Wetting of Real Surfaces, 2013 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Sources for additional nonreciprocity of fiber ring. .. 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... 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