Joint moments of the wave beam amplitude and inverse power in an absorbing random medium with lens properties

This paper presents a derivation of a system of closed equations for joint moments of the amplitude and inverse power of a wave beam propagating in a regularly inhomogeneous dissipative random medium. The radiation transfer in the medium is characterized by non-conservation of the total radiation energy flux and by the existence of power fluctuations. The statistics of the wave beam power fluctuations have been studied. Information on the power statistical characteristics is applied to close the system of equations for joint moments. For task parameters which are not very strict (an effective radius of the wave beam should be considerably less than the outer scale of the turbulence) a system of independent equations for arbitrary joint moments has been obtained. The equations for the first two lower joint moments of the beam intensity and inverse power have been solved analytically. With the solutions obtained the effective wave beam parameters were calculated, i.e. the beam mean displacement, effective broadening and tremble variance (the beam wandering variance) for the propagation of radiation in the refractive channel of an absorbing turbulent medium. Radically new characteristics of the behaviour of the effective parameters in random absorbing and transparent media have been revealed.     

Frequency dependence of the photonic noise spectrum in an absorbing or amplifying diffusive medium

A theory is presented for the frequency dependence of the power spectrum of photon current fluctuations originating from a disordered medium. Both the cases of an absorbing medium (“grey body”) and of an amplifying medium (“random laser”) are considered in a waveguide geometry. The semiclassical approach (based on a Boltzmann-Langevin equation) is shown to be in complete agreement with a fully quantum mechanical theory, provided that the effects of wave localization can be neglected. The width of the peak in the power spectrum around zero frequency is much smaller than the inverse coherence time, characteristic for black-body radiation. Simple expressions for the shape of this peak are obtained, in the absorbing case, for waveguide lengths large compared to the absorption length, and, in the amplifying case, close to the laser threshold. Received 8 August 2000     

Backscattering of stationary radiation in time-dependent random media: average intensity and intensity fluctuations

We consider backscattering of stationary radiation in a random medium whose wavespeed fluctuations depend on time and on space. We modify a previous derivation of the equations that govern the range-evolution of the spectra of the ensemble-averaged forward-and back-propagating components of the field and their second-order statistics, and extend the approach to treat the fourth-order statistics. The latter are governed by integro-difference equations that account for the broadening of the signal spectra due to the time-dependence of the random fluctuations. In the quasi-monochromatic regime, where spectra owing to a monochromatic excitation remain confined to a narrow band over extensive ranges, the integro-difference equations transform into ordinary differential equations that govern the time-dependence of the quantities of interest. We use this simplification to track the power fluxes and their fluctuations (scintillation) in a one-dimensionally stratified slab, where the wave-speed fluctuations depend on the range-coordinate normal to the planes of stratification, and also to treat modal propagation in a duct, where the wave-speed fluctuations depend on all three spatial dimensions. The results suggest that a Gaussian equilibrium is approached at large ranges, on a suitably defined backscattering scale that depends on the medium parameters and the geometry.     

Scattering losses from planar waveguides with material inhomogeneity

Abstract

We consider the propagation of guided waves in a planar waveguide that has a finite-length segment of inhomogeneous media. Except for the inhomogeneous segment, which varies in length from 0 to 160λ, the waveguide is piecewise homogeneous everywhere. The inhomogeneity is modelled by two-dimensional random permittivity fluctuations that are numerically generated from an assumed Gaussian correlation function.

In 2D, the Maxwell equations are solved in the frequency domain for both TE and TM polarization by using modal expansion methods, perfectly matched absorbing boundary layers and the R -matrix transfer matrix algorithm. The guided waves are excited by a Gaussian beam incident on the waveguide aperture. For various waveguide design parameters, numerical results are given for waveguide power loss per unit length of waveguide inhomogeneity. The power loss curves are calculated as the average from numerous realizations of the random permittivity and the coefficient of variation is also given.     

Statistical modelling of the backscattered field in a one-dimensional non-stationary stochastic problem

Abstract

Within the framework of an exact wave approach in the spatial-time domain, the one-dimensional stochastic problem of sound pulse scattering by a layered random medium is considered. On the basis of a unification of methods which has been developed by the authors, previously applied to the investigation of non-stationary deterministic wave problems and stochastic stationary wave problems, an analytical-numerical simulation of the behaviour of the backscattered field stochastic characteristics was carried out. Several forms of incident pulses and signals are analysed. We assume that random fluctuations of a medium are described by virtue of the Gaussian Markov process with an exponential correlation function. The most important parameters appearing in the problem are discussed; namely, the time scales of diffusion, pulse durations, the medium layer thickness or the largest observation time scale in comparison with the time scale of one correlation length for the case of a half-space. An exact pattern of the pulse backscattering processes is obtained. It is illustrated by the behaviour of the backscattered field statistical moments for all observation times which are of interest. It is shown that during the time interval when the main part of the pulse energy leaves the medium, the backscattered field is a substantially non-stationary process, having a non-zero mean value and an average intensity that decays according to a power law. There are various power indices for the different duration incident pulses, however, they are not the same as those of previous papers, which were obtained on the basis of an approximate and asymptotic analysis. We have also verified that the Gaussian law is valid for the probability density function of the backscattered field in the case of any incident pulse duration.     

Wave beam tremble within an absorbing random medium with lens properties

The present paper investigates the joint influence of fluctuations of the complex dielectric constant, ε, of a medium and its lens properties on random displacements of a radiation beam. Based on a non-traditional approach, an expression is obtained for the variance of the beam energy centre tremble, σ²_ρ, taking into account the contribution of fluctuations of the real and imaginary parts of ε, as well as that of their correlations. It was shown that the influence of fluctuations of the imaginary part of ε is particularly pronounced in the initial section of the path, as well as in paths of great length. Due to regular refraction, a considerable enhancement of the dependence of σ²_ρ upon the path length occurs, with the greatest contribution being made by fluctuations of the imaginary part of the dielectric constant. A positive correlation of random changes of the real and imaginary parts of ε was shown to result in a partial compensation of the beam tremble.     

Nonstationary two-flux model of radiation transport for optical tomography of scattering media

A nonstationary two-flux model is formulated for the transport of radiation in an inhomogeneous scattering medium and is applied to the situation where such a medium is irradiated by the narrow beam of a pulsed laser. It is shown that when the time distribution of the transmitted photons is measured it is possible simultaneously to reconstruct the two spatial functions (the coefficients of absorption coefficient and of scattering of the radiation by the medium) by means of an inverse Radon transformation and the solution of a system of nonlinear differential equations on the projection lines. An analytic solution is obtained in quadratures for these differential equations. The results constitute a method of solving problems in optical tomography in an inhomogeneous scattering medium Zh. Tekh. Fiz. 67, 61–65 (May 1997)     

1D waves in a random poroelastic medium with large fluctuations

We examine the problem of wave propagation in a random poroelastic medium. The porous medium is modelled as a Biot poroelastic solid whose constitutive parameters fluctuate substantially over finite distances. Our main results are asymptotic analytical expressions for the mean velocity-stress wave; this solution incorporates two distinct length scales. The effect of the fluctuations appears on the regular depth coordinate while the parameters of the effective medium arise on a shorter scale of distance. Thus the method that we apply, the theory of averaging, allows us to give a rigorous derivation of the effective medium parameters. It also provides the correction terms which are caused by the fluctuations in the random medium; we find that the relative effect of the latter increases in proportion to ω^1/2 where ω denotes the wave frequency. We also show that the fluctuations introduce significant attenuation of the fast Biot compressional wave and dispersion of the slow Biot wave. These results are illustrated by numerical examples using real oilfield data.     

Backscattering in stratified time-dependent random media—CW and narrow-band pulse propagation

Abstract

We consider backscattering in a random stratified medium where the wave-speed fluctuations depend on time and on the range coordinate, which is normal to the planes of stratification. For the limit where the correlation time is shorter than the mean scattering time, we first derive radiation-transport equations that govern the range evolution of the spectra of the ensemble-averaged forward-and back-propagating components of the field and their bichromatic coherence functions. The latter are governed by integro-differential equations that account for the broadening of the signal spectra due to the time dependence of the random fluctuations. When the correlation time is longer than both the period of the radiating signal and the travel time of a wavefront across a range correlation length, the spectrum of the radiation due to a monochromatic CW excitation remains confined to a narrow band over extensive ranges. This permits a quasi-monochromatic approximation, whereby the integro-differential equations produce ordinary differential equations that govern the quantities of interest. We use this approximation to track the power flux associated with the propagation of a narrow-band pulse.     

Polarization fluctuations in the radiation from cosmic sources due to scattering in a random inhomogeneous magnetoactive plasma

We use Feynman integrals along the trajectories to obtain expressions for the fourth-order statistical moments of polarized radiation propagating through a random inhomogeneous plasma. We write down the correlation functions and dispersion of the fluctuations in the Stokes parameters for the case of small fluctuations in the wave filed. We analyze the correlation functions of the Stokes parameters as a function of the polarization of the radiation from the source and the characteristics of the inhomogeneous plasma. We show that for each of the normal waves, the amplitudes and the phases experience different amounts of decorrelation in a random inhomogeneous magnetoactive plasma. As a result, fluctuations occur in the circular polarization.Radio Astronomical Institute, Academy of Sciences of the Ukrainian SSR. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 34, No. 7, pp. 738–747, July, 1991     

Experimental verification and refinement of the model of light-beam brightness fluctuations in a turbid medium with randomly nonuniform absorption

We study experimentally the brightness fluctuations of the light beam at the output of a layer of model turbid medium with strongly anisotropic scattering and randomly nonuniform distribution of an absorbing substance. The earlier developed theory of the light-field fluctuations is refined to broaden the limits of its applicability. It is shown that the refined model of fluctuations agrees well with the experiment and can be used for determining the parameters of absorbing irregularities of a turbid medium from the measured characteristics of brightness fluctuations of the light beam transmitted through the medium. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 51, No. 3, pp. 247–263, March 2008.     

慢波介质自由电子激光器

本文采用一维流体模型导出了慢波介质自由电子激光器的线性色散方程,建立了电子和波相互作用的同向回旋同步机理和反向回旋同步机理.     

Backscattering of stationary radiation in time-dependent random media: average intensity and intensity fluctuations

We consider backscattering of stationary radiation in a random medium whose wavespeed fluctuations depend on time and on space. We modify a previous derivation of the equations that govern the range-evolution of the spectra of the ensemble-averaged forward-and back-propagating components of the field and their second-order statistics, and extend the approach to treat the fourth-order statistics. The latter are governed by integro-difference equations that account for the broadening of the signal spectra due to the time-dependence of the random fluctuations. In the quasi-monochromatic regime, where spectra owing to a monochromatic excitation remain confined to a narrow band over extensive ranges, the integro-difference equations transform into ordinary differential equations that govern the time-dependence of the quantities of interest. We use this simplification to track the power fluxes and their fluctuations (scintillation) in a one-dimensionally stratified slab, where the wave-speed fluctuations depend on the range-coordinate normal to the planes of stratification, and also to treat modal propagation in a duct, where the wave-speed fluctuations depend on all three spatial dimensions. The results suggest that a Gaussian equilibrium is approached at large ranges, on a suitably defined backscattering scale that depends on the medium parameters and the geometry.     

Transformation of the Angular Power Spectrum of Scattered Radiation in a Turbulent Absorbing Magnetoactive Plasma

We consider the oblique incidence of a small-amplitude plane electromagnetic wave on a layer of turbulent absorbing plasma in a uniform external magnetic field. The equations for the first two statistical moments of the angular power spectrum of scattered radiation are derived in the geometrical-optics approximation. We show that two asymmetry factors of the problem, i.e., the oblique incidence and the influence of the anisotropic medium, compensate for each other along a certain direction and impede the appearance of the effect of displacement of the power-spectrum maximum and the effect of anomalous spectrum broadening. We find the condition at which such a case of wave propagation is realized. The dependence of the angular-spectrum shape on the distance from the plasma-layer boundary is studied numerically without using the small-angle approximation. The calculations confirm the results obtained using the geometrical-optics approximation.     

The random analogue of Epstein's transition layer

Abstract

For a quasi-homogeneous, random medium with variance, varying along the same direction as tanh, the mean Green function is obtained as an exact solution of Dyson's equation in a bilocal approximation. The coherent part of the field of a plane wave, falling on a bounded, randomly fluctuating medium with a non-sharp boundary, is studied in detail. In the case of small-scale fluctuations, a medium of this kind is shown to be a random analogue of a transient Epstein layer.     

Self-organization of a light-sustained discharge

A theoretical model is proposed to describe a light-sustained discharge sustained by the radiation from a neodymium laser. The model is based on the Navier-Stokes equations. A solution of these equations is found in the form of a quasisimple wave, and it is shown that the evolution of the system exhibits bifurcation, where the point of bifurcation determines the energy deposition into the discharge. A study is made of the transition of the system to the stable state with complicated space-time and functional structures and a high level of organization (the degree of ordering is estimated from the change in the information entropy). Within this model calculations are made of the macroscopic parameters of the discharge, the level of fluctuations and the required power, and they agree with experimental data. Zh. Tekh. Fiz. 67, 22–26 (June 1997)     

Statistical modelling of the backscattered field in a one-dimensional non-stationary stochastic problem

Within the framework of an exact wave approach in the spatial-time domain, the one-dimensional stochastic problem of sound pulse scattering by a layered random medium is considered. On the basis of a unification of methods which has been developed by the authors, previously applied to the investigation of non-stationary deterministic wave problems and stochastic stationary wave problems, an analytical-numerical simulation of the behaviour of the backscattered field stochastic characteristics was carried out. Several forms of incident pulses and signals are analysed. We assume that random fluctuations of a medium are described by virtue of the Gaussian Markov process with an exponential correlation function. The most important parameters appearing in the problem are discussed; namely, the time scales of diffusion, pulse durations, the medium layer thickness or the largest observation time scale in comparison with the time scale of one correlation length for the case of a half-space. An exact pattern of the pulse backscattering processes is obtained. It is illustrated by the behaviour of the backscattered field statistical moments for all observation times which are of interest. It is shown that during the time interval when the main part of the pulse energy leaves the medium, the backscattered field is a substantially non-stationary process, having a non-zero mean value and an average intensity that decays according to a power law. There are various power indices for the different duration incident pulses, however, they are not the same as those of previous papers, which were obtained on the basis of an approximate and asymptotic analysis. We have also verified that the Gaussian law is valid for the probability density function of the backscattered field in the case of any incident pulse duration.     

Wander of a Gaussian-Beam Wave Propagated Through a Non-Kolmogorov Turbulent Atmosphere

Both increasing experimental evidence and some results of theoretical investigation have shown that there exist two kinds of turbulence in the aerosphere, Kolmogorov and non-Kolmogorov turbulence. Thus, it is necessary to improve the theory of optical wave propagation through atmospheric turbulence, namely, study the laser-beam propagation in non-Kolmogorov turbulence, before analyzing the joint influence of the Kolmogorov turbulence and non-Kolmogorov one on satellite laser communication. The beam wander will lead to the performance degradation of satellite laser communication systems and exert an influence on the achievement and stability of its links. In this paper, we consider a theoretical power spectrum of refractive-index fluctuations with a generalized power law in order to derive the variance of Gaussian-beam wave wander in weak turbulence for a horizonal path and analyze the influence of spectral power-law variations on the beam wander. We show that the expression for the beam-wander variance is of concise closed form and independent of the optical wavelength.