Mathematical modeling of random coupling between polarization modes in single-mode optical fibers: X. Verification of the ergodic hypothesis for fiber ring interferometers

The problem of the validity of the ergodic hypothesis as applied to a fiber ring interferometer (FRI) is considered on the basis of a comparison between magnitudes of the zero drift of an FRI calculated upon changing temperature of the fiber and upon random realizations of inhomogeneities in a single-mode optical fiber (SMF). The physical nature and statistical characteristics of random inhomogeneities in an SMF, types of polarization nonreciprocity in an FRI, and thermo-optical parameters of an SMF are analyzed. An algorithm for calculation of the zero drift of an FRI on changing temperature is proposed. The conditions under which the ergodic hypothesis is satisfied in an FRI are formulated. In particular, it is shown that many random inhomogeneities have to be placed on the depolarization length of polychromatic radiation in the SMF loop of an FRI; otherwise, the zero drift of the FRI calculated by the method of averaging over an ensemble of independent realizations may significantly exceed its actual value. Numerical estimations are made. It is shown that thermostabilization of an FRI with a polychromatic radiation source may significantly reduce its zero drift.     

Mathematical modeling of random coupling between polarization modes in single-mode optical fibers: XIV. Spectral characteristics of nonreciprocal phase difference of counterpropagating waves at the output of a fiber ring interferometer at a change in the fiber temperature

Distributions of the spectral density of the dependence of the phase difference of counterpropagating waves in a fiber ring interferometer (FRI) on the temperature of the single-mode optical fiber (SMOF) in the FRI loop (temperature spectra of the FRI zero drift) due to polarization nonreciprocity have been obtained by Fourier analysis for different spectral linewidths of the FRI radiation source and different linear birefringences of the SMOF in the FRI loop. It is shown that a change in the temperature of the SMOF in the FRI loop changes mainly the SMOF linear birefringence. This effect leads to a change in the phases of the radiation that is transferred from one polarization mode to another at those points in the SMOF where the random twisting of the axes of unperturbed linear birefringence of the SMOF changes. The effect of the magnitude of the jump under consideration, its location with respect to the nearest FRI loop end, and the magnitude of the cross-correlation (visibility) function of the radiation transmitted along the slow and fast SMOF axes from the loop end to the point where a jump in twisting of the SMOF axes occurs on the character of the temperature spectra of the FRI zero drift is determined. It is shown that in the case where the depolarization length of nonmonochromatic radiation in an SMOF is smaller than the average length of the SMOF segment on which random twisting is constant, the qualitative and quantitative characteristics of the temperature spectra of the FRI zero drift depend strongly on the polarization state of the radiation at the input of the FRI loop. The reason for this difference is explained.     

Mathematical modeling of random coupling between polarization modes in single-mode optical fibers: VII. Zero drift in a fiber ring interferometer with a circuit formed by a twisted optical fiber with weak natural linear birefringence

Zero drift and fading of the interference signal determined by polarization nonreciprocity at the exit from a fiber ring interferometer (FRI) of great length designed for recording a number of general relativity (GR) effects and made on the basis of a uniformly twisted single-mode optical fiber (SMF) with weak natural birefringence are considered. The calculations are carried out by numerical simulation using the model of random coupling between orthogonal polarization modes in an SMF, which was proposed in the first part of this work. The so-called minimum arrangement of an FRI with a nonmonochromatic radiation source, including a linear polarizer located between two beam splitters, is considered. Numerical estimates are made. It is shown that the FRI with a circuit on the basis of a twisted SMF has a significantly smaller zero drift and significantly smaller fading of the interference signal than the FRI with a circuit formed by an untwisted SMF. It is shown that the FRI produced on the basis of a twisted SMF provides the polarization sensitivity necessary for detecting the GR effects under study.     

Methods for calculating the temperature zero drift initiated in fiber ring interferometers by random inhomogeneities in single-mode optical fibers

The specific features of the existing methods used for calculating the temperature zero drift initiated in fiber ring interferometers (FRIs) by linear coupling between polarization eigenmodes at random inhomogeneities in single-mode optical fibers are analyzed. The ranges of applicability of each method are determined. It is demonstrated that numerical simulation with a variation in the temperature of the single-mode optical fiber of the FRI loop is the most complex and, at the same time, the most universal method.     

Mathematical modeling of random coupling between polarization modes in single-mode optical fibers: XII. Dependence of the zero drift in fiber ring interferometers in orthogonal polarization modes on the range of temperature variation in an optical fiber

Temperature dependences of the zero drift in a fiber ring interferometer (FRI) without a polarizer in the presence of random inhomogeneities in a single-mode optical fiber in the FRI loop are obtained by numerical simulation. The results are compared with known experimental data. It is shown that FRIs using a superluminescent diode as a source of nonmonochromatic radiation and a loop made of a single-mode optical fiber with weak linear birefringence can find application as low-accuracy fiber-optic gyroscopes and FRIs with a loop made of a single-mode optical fiber with strong linear birefringence can be used as medium-accuracy fiber-optic gyroscopes.     

Mathematical simulation of random coupling of polarization modes in single-mode optical fibers: IX. Zero drift in a fiber ring interferometer with a loop made of an optical fiber with strong linear birefringence

Numerical simulation of random inhomogeneities in a strongly anisotropic single-mode optical fiber (SMOF) forming a loop in a fiber ring interferometer (FRI) designed according to the minimal scheme is used to obtain the dependences of an interference signal at the FRI output and the zero shift and drift of an interference pattern of counterpropagating waves on the angles of orientation of the axes of linear birefringence in an SMOF at both inputs of the FRI loop. It is shown that the use of a superluminescent diode as a radiation source makes it possible to increase the sensitivity of fiber-optic gyroscopes fabricated on the basis of FRIs by 1–2 orders of magnitude and to obtain the values of real sensitivity required for navigation purposes.     

Mathematical modeling of random coupling between polarization modes in single-mode optical fibers: XIII. Conditions of applicability of the ergodic hypothesis for fiber ring interferometers

Some additional conditions of applicability of the ergodic hypothesis to fiber ring interferometers (FRIs) with a loop consisting of a single-mode optical fiber (SMOF) with random inhomogeneities are considered. It is shown by mathematical modeling that the change in the phase difference of counterpropagating waves at the FRI output with the SMOF temperature is not a stationary random process. However, in a fairly narrow temperature range, this dependence can be assumed to be locally stationary. The conditions determining this temperature range are formulated. It is shown for a fairly large ensemble of independent realizations of random inhomogeneities in an SMOF that, even when all conditions of ergodicity are satisfied with a large margin, there will always be at least one realization violating strict ergodicity. Thus, only conditional (approximate) ergodicity occurs in this case. Nevertheless, in calculation of the FRI zero drift in this situation, averaging over an ensemble of independent realizations of random inhomogeneities in the SMOF of an FRI loop can be performed with sufficient accuracy. As a result, calculations are simplified significantly. In the general case, when at least one of the conditions of ergodicity is not satisfied, averaging over temperature for each realization with subsequent averaging over the entire ensemble should be performed. It is shown also that, within this problem, we can speak only about quasi-ergodicity or emulation of ergodicity, since a change in the temperature of the SMOF of an FRI loop and successive enumeration of independent realizations of random inhomogeneities in the SMOF loop are radically different random processes. The parameters characterizing quasiperiodic temperature changes in the phase difference of counterpropagating waves at the FRI output are refined.     

Effect of polarization nonreciprocity in rotating fiber ring interferometers

The effect of polarization nonreciprocity in sensors of angular velocity of rotation—fiber ring interferometers (FRIs)—is considered for arbitrary polarization of eigenmodes of the single-mode optical fiber of an FRI loop and arbitrary polarization state of radiation at the FRI input. A new method for detecting the polarization nonreciprocity in an FRI is proposed. Numerical estimations are made.     

Mathematical modeling of random coupling between polarization modes in single-mode optical fibers: VIII. Zero drift in a fiber ring interferometer with a white light source

Different schemes of fiber ring interferometers (FRIs) with a broadband nonmonochromatic radiation source manufactured on the basis of air-silica microstructured single-mode optical fibers (SMOFs) are considered. This source is close in spectral characteristics to a white light source, because the width of its emission spectrum is comparable to the mean wavelength. It is shown that an increase in the width of the spectrum of the radiation source can lead to either a substantial decrease or an increase in the zero drift, depending on the radiation polarization at the FRI entrance. The latter fact has defied explanation within simple phenomenological models of random coupling between polarization modes in SMOFs of an FRI circuit. The observed increase in the zero drift of the FRI can be explained in terms of the dependence of the parameter of polarization conservation (the parameter h) on the light wavelength for highly anisotropic SMOFs. This dependence is weak for nonmonochromatic radiation sources with a relatively small spectral width, for example, superluminescent diodes, which are traditionally used in FRIs. In contrast, for substantially more broadband radiation sources (including air-silica SMOFs), the above dependence is well pronounced and can lead to a number of undesirable effects in FRIs. Different variants of the FRI design are analyzed. It is demonstrated that the zero drift can be noticeably decreased with an increase in the width of the spectrum of the radiation source for an arbitrary radiation polarization at the entrance of an FRI with a depolarizer of nonmonochromatic radiation and a circuit fabricated from a weakly anisotropic SMOF, for which the parameter h does not depend on the light wavelength. The numerical estimates are obtained.     

Influence of radiation coherence on the effects of polarization nonreciprocity of fiber ring interferometers

In this paper, we consider the effects of polarization nonreciprocity taking into account coherence or partial coherence of radiation at the output of a fiber ring interferometer (FRI) and discuss the possibility of occurrence of additional effects of polarization nonreciprocity at the output of an FRI with a broadband radiation source due to the coherent component of radiation resulting from the coupling of orthogonal modes at the output segments of the fiber. Allowing for the coherence, we estimate the temperature instability of the interferometer signal related to the polarization nonreciprocity. The features of the polarization nonreciprocity effects for a fiber ring interferometer with polarizer or dichroism are considered. It is shown that if the conditions of conventional reciprocity theorem (no rotation) are satisfied for the interferometer, then the polarization nonreciprocity effects become hidden and cannot be detected in the interference signal without a priori information on the character of birefringence and on the orientation of anisotropic elements of the fiber loop of the interferometer and polarizer.     

The possibility for a significant decrease in the zero drift of fiber ring interferometers with a circuit formed by a highly anisotropic optical waveguide and a broadband radiation source

It is shown that the zero drift of interference of counterpropagating waves at the output from a fiber ring interferometer can be completely eliminated in the first order of magnitude and significantly decreased in the second order of magnitude if a sufficiently broadband source of nonmonochromatic radiation is used in a fiber ring interferometer with a circuit formed by a highly anisotropic optical waveguide when the depolarization length of nonmonochromatic radiation in a single-mode fiber-optical waveguide becomes significantly shorter than the correlation length of random inhomogeneities. Numerical estimates are made.     

Effect of orthogonal-mode random coupling on polarization characteristics of single-mode fiber lightguides (SMFL) and SMFL-based ring interferometers Part II. Zero drift in fiber ring interferometers with nonmonochromatic radiation sources

A method is proposed for calculation of the zero drift and Fedding interference signal at the output of a fiber ring interferometer (FRI) based on a single-mode fiber lightguide (SMFL) with a nonmonochromatic radiation source. The fiber length is divided into sections that are equal to the depolarization length in the SMFL. It is shown that when the FRI is made of an anisotropic SMFL with weak orthogonal-mode coupling, the calculation results agree with those obtained earlier by other calculation methods. Results for an FRI made of a weakly anisotropic SMFL with strong orthogonal-mode coupling are obtained for the first time.Institute of Applied Physics, Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 37, No. 11, pp. 1473–1480, November, 1994.     

Mathematical modeling of random coupling between polarization modes in single-mode optical fibers: VI. Evolution in the degree of polarization of nonmonochromatic radiation traveling in a twisted optical waveguide

An analysis is made of the evolution of polarization of nonmonochromatic radiation travelling through single-mode optical fibers in the presence of random coupling between orthogonally polarized modes, which is caused by the random twist of axes of natural linear birefringence of an SMF, and in the presence of a regular axis twist. It is shown that the depolarization length of nonmonochromatic radiation in an SMF increases with increasing regular SMF twist, regardless of the presence of a random axis twist. Using the Monte Carlo method, the dependences of the mathematical expectation and the mean-square deviation of the degree of polarization of nonmonochromatic radiation on the fiber length for SMFs with different linear birefringences and regular axis twists are calculated. It is shown that the fiber length on which the degree of polarization of nonmonochromatic radiation reaches its limiting value increases and the limiting value itself decreases with an increasing regular twist. It is also shown that an induced SMF twist is able to improve parameters of a fiber interferometer, in particular, to decrease random phase changes and intensity fadings of the interference signal, which are caused by random coupling between polarization modes.     

Mathematical modeling of random coupling between polarization modes in single-mode optical fibers: XV. Dependence of integral statistical parameters of polarization modes in single-mode optical fibers with random inhomogeneities on the wavelength of light

Spectral dependences of integral statistical elements of the Jones matrix of an extended segment of a single-mode optical fiber (SMOF) with random inhomogeneities, whose length considerably exceeds the correlation length of random inhomogeneities, are found by the method of mathematical modeling. The boundaries of the ranges of optical wavelengths are determined, within the limits of which the parameters under consideration remain quasi-constant. This allows the derivation of analytical expressions for the degree of polarization of nonmonochromatic radiation in an SMOF and for the zero drift in fiber ring interferometers with the help of the approximate method of small perturbations. It is shown that the small-perturbation method leads to adequate results for SMOFs with strong and intermediate linear birefringence.     

Zero shift in fiber ring interferometer with wideband radiation source

It is shown that the polarization zero shift of a fiber ring interferometer (FRI) is chiefly determined by the Stokes components of the radiation, which are circularly and linearly polarized at an angle of 45° to the axis of the polarizer. A birefringent element between the polarizer and coupler in conjunction with a wideband source results in disappearance of the FRI zero shift, regardless of the mutual orientations of the polarizer, coupler, and birefringent element.Moscow Physics and Technology Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 34, No. 9, pp. 1001–1010, September, 1991.     

Mathematical modeling of random coupling between polarization modes in single-mode optical fibers: XVI. Depolarization and repolarization of polychromatic radiation in single-mode optical fibers with random inhomogeneities

The dependences of the degree of polarization of polychromatic radiation on the length of a single-mode optical fiber (SMF) with random inhomogeneities have been obtained by mathematical modeling. The case is considered where radiation having both polarization modes excited with equal weights of linear polarization is first introduced into a depolarizer of polychromatic radiation (a SMF segment with high linear birefringence) and arrives at an SMF with low linear birefringence. It is shown that the degree of polarization of radiation after transmission through the first segment becomes significantly suppressed and remains almost constant upon propagation through the second segment, after which it begins to sharply increase at some length; i.e., repolarization of radiation occurs. It is shown that repolarization of radiation depends weakly on the angle made by the axes of unperturbed linear birefringence of the first and second segments. The conditions for the length of the first segment (depolarizer) under which the degree of polarization remains minimum throughout the second segment are determined.     

Increase in the correlation length of nonmonochromatic radiation during propagation in a single-mode optical fiber containing random inhomogeneities and the influence of this increase on the operation of a fiber ring interferometer

An analysis is made of the conditions under which the coupling of orthogonal polarization modes at random inhomogeneities in single-mode optical fibers leads to an increase in the correlation length of a source of nonmonochromatic radiation. It is shown that when long-base fiber ring interferometers with a single-mode fiber ring system possessing weak linear birefringence are used, the correlation length of the nonmonochromatic radiation at the interferometer exit is increased, which means that the interference pattern can have satisfactory visibility even when there is an appreciable difference between the interferometer arms as a result of the Sagnac effect due to the Earth’s rotation. The calculations were made by mathematical modeling of random inhomogeneities in the fiber. Zh. Tekh. Fiz. 69, 140–143 (July 1999)     

Influence of the irregularity of twisting of the birefringence axes of single-mode optical fibers on the polarization nonreciprocity of fiber ring interferometers

A particular example of the so-called minimum-configuration fiber ring interferometer whose loop comprises two different-length segments of a single-mode optical fiber with the same linear birefringence but different twisting is used to show that even in the case where the birefringence axes at the loop input coincide with the polarizer transmission direction, the phenomenon of polarization nonreciprocity of the fiber ring interferometer can emerge. It is shown that polarization nonreciprocity can be eliminated by the proper adjustment of the axes of a single-mode optical fiber, but the above adjustment should be changed if the waveguide temperature is changed. It is also shown that polarization nonreciprocity is a function of the light wavelength and the polarization nonreciprocity value in a fiber ring interferometer varies quasiperiodically in response to the temperature change in an optical fiber with random irregularities.     

Methods for eliminating the influence of nonlinear Kerr effect on the zero drift of fiber ring interferometers

Nonlinear Kerr effect leads to the appearance of a periodic structure in the saturated refractive index of an optical fiber, which corresponds to a standing structure formed by counterpropagating waves in the circuit of a fiber ring interferometer (FRI). If the intensities of counterpropagating waves are slightly different, their reflection from this periodic structure leads to the appearance of a phase shift of interference of counterpropagating waves unrelated to rotation at the FRI output. If a nonmonochromatic radiation source is used in the FRI system, only radiation rereflected from the middle of the circuit makes a contribution to the phase shift. A method for eliminating the influence of the nonlinear Kerr effect on the zero shift of fiber ring interferometers is proposed. This consists in making the middle of the circuit discontinuous. Numerical estimates are made.     

Influence of the polarization mode dispersion on the propagation of ultrashort optical pulses in single-mode fiber lightguides with very weak linear birefringence and random inhomogeneities

We consider the influence of the polarization mode dispersion, which is stipulated by the presence of random inhomogeneities in single-mode fiber lightguides, on the propagation of ultrashort optical pulses in the fiber communication lines with very weak linear birefringence. Evolution of the envelope of ultrashort optical pulses and their spectra as functions of the length of a single-mode fiber lightguide with very weak linear birefringence and random inhomogeneities are obtained by the method of mathematical simulation. An increase in the pulse duration is shown to be proportional to the square root of the length of a single-mode fiber lightguide. The numerical-simulation results are compared with the results of experimental measurements of the polarization mode dispersion.