List of abbreviations -- List of notations -- 1 Introduction -- 2 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 -- 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 -- 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 -- 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 -- 2.1.3 Attempts to explain the Sagnac effect by analogy with other effects -- 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 -- 2.1.4 Incorrect explanations of the Sagnac effect -- 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 -- 2.1.4.3 Sagnac effect as a manifestation of the classical Doppler effect from a moving splitter -- 2.1.4.4 Sagnac effect as a manifestation of the Fresnel-Fizeau dragging effect -- 2.1.4.5 Sagnac effect and Coriolis forces.

2.1.4.6 Sagnac effect as a consequence of the difference between the orbital angularmomenta of photons in counterpropagating waves -- 2.1.4.7 Sagnac effect as a manifestation of the inertial properties of an electromagnetic field -- 2.1.4.8 Sagnac effect in incorrect theories of gravitation -- 2.1.4.9 Other incorrect explanations of the Sagnac effect -- 2.2 Physical problems of the fiber ring interferometry -- 2.2.1 Milestones of the creation and development of optical ring interferometry and gyroscopy based on the Sagnac effect -- 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 -- 2.2.2.2 Nonreciprocity as a consequence of the light source coherence -- 2.2.2.3 Polarization nonreciprocity: causes and solutions -- 2.2.2.4 Nonreciprocity caused by local variations in the gyro fiber-loop parameters due to variable acoustic, mechanical, and temperature actions -- 2.2.2.5 Nonreciprocity due to the Faraday effect in external magnetic field -- 2.2.2.6 Nonreciprocal effects caused by nonlinear interaction between counterpropagating waves (optical Kerr effect) -- 2.2.2.7 Nonreciprocity caused by relativistic effects in fiber ring interferometers -- 2.2.3 Fluctuations and ultimate sensitivity of fiber ring interferometers -- 2.2.4 Methods for achieving the maximum sensitivity to rotation and processing the output signal -- 2.2.5 Applications of fiber optic gyroscopes and fiber ring interferometers -- 2.3 Physical mechanisms of random coupling between polarization modes -- 2.3.1 Milestones of the development of the theory of polarization mode linking in single-mode optical fibers -- 2.3.2 Phenomenological models of polarization mode coupling.

2.3.3 Physical models of polarization mode coupling -- 2.3.4 Inhomogeneities arising as a fiber is drawn -- 2.3.4.1 Torsional vibration -- 2.3.4.2 Longitudinal vibration -- 2.3.4.3 Transverse vibration -- 2.3.4.4 Transverse stresses -- 2.3.5 Inhomogeneities arising in applying protective coatings -- 2.3.6 Inhomogeneities arising in the course of winding -- 2.3.7 Rayleigh scattering: the fundamental cause of polarizationmode coupling -- 2.4 Application of the Poincaré sphere method -- 2.5 Thomas precession. Interpretation and observation issues -- 3 Development of the theory of interaction between polarization modes -- 3.1 Phenomenological estimates of the random coupling -- 3.1.1 Small perturbation method -- 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 -- 3.2 A physical model of the polarizationmode coupling -- 3.2.1 A model of random inhomogeneities in SMFs with random twists of the anisotropy axes -- 3.2.2 Connection between the polarization holding parameter and statistics of random inhomogeneities -- 3.2.3 Polarization holding parameter in the case of random and regular twisting -- 3.2.4 Statistical properties of the polarization modes for fibers with random inhomogeneities -- 3.3 Evolution of the degree of polarization of nonmonochromatic light -- 3.3.1 Small perturbation method -- 3.3.2 A method for modeling random twists -- 3.3.3 A mathematical method for modeling random twists in the presence of a regular twist -- 3.3.4 Analytical calculation of the limiting degree of polarization of nonmonochromatic light.

3.3.5 Increasing of the correlation length of nonmonochromatic light traveling through a single-mode fiber with random inhomogeneities -- 3.4 Anholonomy of the evolution of light polarization -- 4 Experimental study of random coupling between polarization modes -- 4.1 A rapid method for measuring the output polarization state -- 4.2 Method for measuring the polarization beat length and ellipticity -- 4.3 Experimental comparison of the accuracy of different methods -- 4.4 Influence of winding of single-mode fibers on the amount of the polarization holding parameter -- 4.5 Experimental study of the polarization degree evolution of light -- 4.6 Method of fabricating ribbon single-mode fibers -- 4.7 Method for removing the effect of photodetector dichroism -- 5 Fiber ring interferometers of minimum configuration -- 5.1 Polarization nonreciprocity of fiber ring interferometers -- 5.2 Fiber ring interferometers with a single-mode fiber circuit -- 5.3 Zero shift, deviation, and drift of fiber ring interferometers -- 5.3.1 Applicability conditions for the ergodic hypothesis -- 5.3.2 Influence of the amount of random twist of the fiber -- 5.3.3 Influence of the location of the random inhomogeneity -- 5.3.4 Influence of the mutual coherence of nonmonochromatic light in the main and orthogonal polarization modes at the point of inhomogeneity -- 5.3.5 Approximate calculation of the temperature zero drift -- 5.3.6 Calculation of the zero shift deviation of the FRI by the small perturbation method -- 5.3.7 Calculation of the zero shift deviation with the extended small perturbation method -- 5.3.8 Calculation of the zero shift deviation by the method of mathematical modeling of random inhomogeneities -- 5.3.8.1 Zero shift deviation of an FRI with a high-birefringence fiber.

5.3.8.2 Zero shift deviation of an FRI with a low-birefringence fiber -- 5.3.9 Calculation of the zero shift deviation of FRIs -- 5.4 Domains of application of the different methods for calculating PN -- 6 Fiber ring interferometers of nonstandard configuration -- 6.1 New type of nonmonochromatic light depolarizer for FRIs -- 6.2 Zero drift and output signal fading in an FRI with a polarizer -- 6.2.1 Small perturbation method. The quasi-axis model -- 6.2.2 Extended small perturbation method -- 6.2.3 Method of mathematical modeling of random inhomogeneities in fibers -- 6.3 Fiber ring interferometers without a polarizer -- 6.3.1 FRIs with circularly polarized input light -- 6.3.2 Modulation method for removing the zero shift in a fiber ring interferometer without a polarizer -- 6.3.3 Fiber ring interferometer with a depolarizer of nonmonochromatic light -- 6.3.4 Fiber ring interferometer with a circuitmade from a uniformly twisted fiber -- 6.3.5 Zero shift deviation in FRIs without a polarizer and with a circuitmade from a high-birefringence fiber in a limited temperature range -- 7 Geometric phases in optics. The Poincaré sphere method -- 7.1 Application of the Poincaré sphere method -- 7.1.1 Analysis of the properties of the Pancharatnamphases. The Poincaré sphere -- 7.1.1.1 Type I Pancharatnamphase -- 7.1.1.2 Type II Pancharatnamphase -- 7.1.2 Birefringence in SMFs due to mechanical deformations -- 7.1.2.1 Kinematic phase in SMFs -- 7.1.2.2 Bending induced linear birefringence of SMFs -- 7.1.2.3 Twisting-induced circular birefringence of SMFs. The spiral polarization modes -- 7.1.3 Rytov effect and the Rytov-Vladimirskii phase in SMFs and FRIs in the case of noncoplanar winding -- 7.1.3.1 Rytov effect in the FRI circuit fiber.

7.1.3.2 Rytov-Vladimirskii phase and PP2 in SMFs with noncoplanar winding.