Mathematical modeling of random coupling between polarization modes in single-mode optical fibers: Integrated statistical properties of polarization modes of single-mode optical fibers in the presence of random inhomogeneities

Using mathematical modeling, an explicit form is found for the Jones matrix of a segment of a single-mode optical fiber with random inhomogeneities, whose length is considerably greater than the correlation length of random inhomogeneities. It is shown that parameters of this matrix are of a statistical nature. A rational representation of the matrix is proposed. It is shown that, under certain conditions, one of the matrix parameters may be treated as constant, whereas the other parameter is assumed to be continuously distributed on the interval [0, 2π]. As the latter parameter is changed, which corresponds to a change from one random realization of inhomogeneities in a single-mode fiber to another, the ellipticity and the azimuth of the major axis of the polarization ellipse of natural polarization modes of a single-mode fiber simultaneously change.     

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.     

Path-integral analysis of propagation in a waveguide with random inhomogeneities

The method of path integration is applied to the analysis of wave propagation in both a graded-index optical waveguide and in an otherwise homogeneous infinite medium whose refractive indices have random statistical inhomogeneities superposed upon a regular variation of refractive index with suitable averaged properties. The authors use techniques originally employed in the context of electron propagation in a set of random scatterers to calculate the averaged Green function describing paraxial wave propagation in a medium whose refractive index has statistical inhomogeneities. The concept of an averaged density of modes is introduced, and the paper presents detailed calculations of this quality for two limiting case. In the first, the correlation length associated with the distribution of inhomogeneities is zero along the direction of propagation. In the second, the Feynman variational technique is used to describe the propagator in a medium whose statistical inhomogeneities have an infinite correlation length along the direction of propagation. Comments are made about the intermediate case which is of greater relevance to real waveguides.     

Parabolic equation for nonlinear acoustic wave propagation in inhomogeneous moving media

A new parabolic equation is derived to describe the propagation of nonlinear sound waves in inhomogeneous moving media. The equation accounts for diffraction, nonlinearity, absorption, scalar inhomogeneities (density and sound speed), and vectorial inhomogeneities (flow). A numerical algorithm employed earlier to solve the KZK equation is adapted to this more general case. A two-dimensional version of the algorithm is used to investigate the propagation of nonlinear periodic waves in media with random inhomogeneities. For the case of scalar inhomogeneities, including the case of a flow parallel to the wave propagation direction, a complex acoustic field structure with multiple caustics is obtained. Inclusion of the transverse component of vectorial random inhomogeneities has little effect on the acoustic field. However, when a uniform transverse flow is present, the field structure is shifted without changing its morphology. The impact of nonlinearity is twofold: it produces strong shock waves in focal regions, while, outside the caustics, it produces higher harmonics without any shocks. When the intensity is averaged across the beam propagating through a random medium, it evolves similarly to the intensity of a plane nonlinear wave, indicating that the transverse redistribution of acoustic energy gives no considerable contribution to nonlinear absorption. Published in Russian in Akusticheskiĭ Zhurnal, 2006, Vol. 52, No. 6, pp. 725–735. This article was translated by the authors.     

Sound scattering by random fractal inhomogeneities in the ocean

Sound scattering by random volume inhomogeneities (fluctuations of the refraction index in a medium) with an arbitrary anisotropy is considered using the small perturbation method (Born’s approximation). Surfaces (boundaries) of the inhomogeneities are deemed to be fractal ones: the energy spectra of the refraction index fluctuations follow the power law with a nonintegral exponent. Formulas are obtained for the volume scattering coefficient. Frequency and angular dependences of the scattering coefficient and their relations to the fractal dimension of inhomogeneities with different kinds of anisotropy and different sizes (on the sound wavelength scale) are presented. The fractal dimension of the inhomogeneities is estimated.     

Investigation of statistical characteristics of waves in layered media with regular and random inhomogeneities based on the invariant imbedding method

We consider a model of a stationary problem of wave propagation in a layered half-space with regular and random inhomogeneities. The choice of regular perturbation corresponds to a linear profile of the wave velocity. Random inhomogeneities are simulated in the framework of the white noise model. We analyse the influence of inhomogeneities on the probability distribution of the reflection-coefficient phase and the imaginary value of the average wavefield at the boundary of the half-space.     

Boundary diffraction waves and the effective size of the inhomogeneities of the scattering object

A method for measuring the angular size of the halo from boundary diffraction waves without using two-position schemes is proposed. The relationship between the angular distortions of the beam and the size distribution of inhomogeneities is derived for a random transmitting screen. The effective size of inhomogeneities is expressed in terms of the ratio between the fourth-order and second-order moments of the distribution function. A method for estimating the correspondence between the effective size of the inhomogeneities and the parameters of the real screen is substantiated. The key relation for the estimation is the equality of the angular sizes of the halo for the simulated and real scattering objects.     

Mean field of a low-frequency sound wave propagating in a fluctuating ocean

Statistical characteristics of low-frequency sound waves propagating over long distances in a fluctuating ocean are important for many practical problems. In this paper, using the theory of multiple scattering, the mean field of a low-frequency sound wave was analytically calculated. In these calculations, the ratio of the sound wavelength and the scale of random inhomogeneities can be arbitrary. Furthermore, the correlation function of inhomogeneities is expressed in terms of a modal spectrum (e.g., internal waves modes). The obtained mean sound field is expressed as a sum of normal modes that attenuate exponentially. It is shown that the extinction coefficients of the modes are linearly related to the spectrum of random inhomogeneities in the ocean. Measurements of the extinction coefficients can therefore be used for retrieving this spectrum. The mean sound field is calculated for both 3D and 2D geometries of sound propagation. The results obtained can be used to study the range of applicability of the 2D propagation model.     

The angular dependence of the coefficient of sound reflection from anisotropic volume inhomogeneities of the ocean

The small perturbation method is used to consider the scattering of sound by random volume inhomogeneities of the ocean. The inhomogeneities are assumed to be horizontally anisotropic with their horizontal dimensions far exceeding the vertical ones. Expressions are obtained for the angular dependence of the scattering coefficient. Vertical and horizontal sections of the scattering pattern are presented, and the angular widths of its maxima are analytically estimated.     

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.     

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.     

Resonant transition radiation in plamsa with magnetic inhomogeneities

We calculate the spectrum of the resonant transition radiation generated by a fast charged particle moving in plasma with small-scale random magnetic inhomogeneities. We determine the conditions under which this type of transition radiation dominates over the transition radiation on plasma density inhomogeneities. We discuss possible applications of the resonant transition radiation in astrophysical and geophysical conditions.     

The influence of inhomogeneous properties of a system on the percolation process in two-dimensional space

The percolation process in a two-dimensional inhomogeneous lattice is studied by the Monte Carlo method. The inhomogeneous lattice is simulated by a random distribution of inhomogeneities differing in size and number. The influence of inhomogeneities on the parameters (critical concentration, average number of sites in finite clusters, percolation probability, critical exponents, and fractal dimension of an infinite cluster) characterizing the percolation in the system is analyzed. It is demonstrated that all these parameters essentially depend on the linear size of inhomogeneities and their relative area.     

Radiative Transport in a Periodic Structure

We derive radiative transport equations for solutions of a Schrödinger equation in a periodic structure with small random inhomogeneities. We use systematically the Wigner transform and the Bloch wave expansion. The streaming part of the radiative transport equations is determined entirely by the Bloch spectrum, and the scattering part by the random fluctuations.     

Effects of impurities on heat capacities at paramagnetic to ferromagnetic transitions

The influence of controlled impurities upon the detailed shape of the heat capacity of Gd near the Curie temperature is studied experimentally. The experimental results can be interpreted in terms of Watson's theoretical analyses which show that small scale, random inhomogeneities do not alter the shape of the heat capacity curve, whereas large scale inhomogeneities are found to reduce the sharpness of the critical singularities.     

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.     

Pairs of opposite decaying evanescent waves in random media and tunneling radiative transfer

A tunneling mechanism of radiative transfer through a dielectric random medium is revealed applying technique of Dyson and Bethe-Salpeter equations for electromagnetic wave multiple scattering by medium inhomogeneities (scatterers) with near fields effects in scattered fields. The mechanism consists in existing inside of a random inhomogeneity a pair of virtually opposite decaying evanescent waves whose interference results in energy flux.     

Acoustic diffraction tomography of the ocean

Tomographic monitoring of medium-scale inhomogeneities of the ocean using acoustic waves is considered. The basic integral equations are analyzed. A more detailed study is suggested for two patterns of diffraction tomography: pulsed translucent tomography of random inhomogeneities and hydroacoustic imaging. Some results of reconstruction of inhomogeneity images recovered based on laboratory-experiment data are presented.Institute of Applied Physics, Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 36, No. 8, pp. 738–751 August, 1993.     

Hydrodynamic Dispersion for Fluid Filtration Through a Porous Medium with Random Macroscopic Inhomogeneities

Radiophysics and Quantum Electronics - We study the convective diffusion of passive admixtures in the course of forced fluid filtration through a porous medium with frozen random inhomogeneities of...     

Influence of Inhomogeneity on Critical Behavior of Earthquake Model on Random Graph

We consider the earthquake model on a random graph. A detailed analysis of the probability distribution of the size of the avalanches will be given. The model with different inhomogeneities is studied in order to compare the critical behavior of different systems. The results indicate that with the increase of the inhomogeneities, the avalanche exponents reduce, i.e., the different numbers of defects cause different critical behaviors of the system. This is virtually ascribed to the dynamical perturbation.