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.     

Localization of light energy on the nanometer scale in a silicon cone

The spatial structure of light fields in a metallized cone filled with a medium with complex dielectric function was studied on the basis of the exact solution of the eigenwave problem. It is suggested that silicon can be used as a core of optical probe in the visible spectral region. It is shown that the density of light energy at the output of optical probe can be drastically increased if silicon is used instead of glass fiber.     

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.     

Measuring temporal fluctuation on femtosecond scale in a random medium

We report an observation of femtosecond optical fluctuations of transmitted light when a coherent femtosecond pulse propagates through a random medium. They are a result of random interference among scattered waves coming from different trajectories in the time domain. Temporal fluctuations are measured by using cross-correlated frequency optical gating. It is shown that a femtosecond pulse will be broadened and distorted in pulse shape while it is propagating in random medium. The real and imaginary components of transmitted electric field are also distorted severely. The average of the fluctuated transmission pulses yields a smooth profile, probability functions show good agreement with Gaussian distribution.     

A solution of radiative transfer in isotropic plane-parallel atmosphere by integrating Milne equation

In this paper we solve the inversion problem of the radiative transfer process in the isotropic plane-parallel atmosphere by iterative integrations of the Milne integral equation. As a result, we obtain the scattering function in the form of a cubic polynomial in optical thickness. The author has already solved the same problem by iterative integrations of Chandrasekhar's integral equation. In the Milne integral equation, both the cosines of the viewing angles and the optical thickness are integral variables, while in Chandrasekhar's integral equation the cosines of the viewing angles are variables but the optical thickness is not. We derive several series of exponential-like functions as intermediate derivations. Their convergences are evaluated by the author's previous work in the solution of Chandrasekhar's integral equation. The truncated scattering function up to the third order in optical thickness thus obtained is identical to that obtained from Chandrasekhar's integral equation, though their apparent forms are different. Chandrasekhar pointed out that the solution of Chandrasekhar's integral equation does not have a uniqueness of solution. The Milne equation, in contrast, has been proven to have a unique solution. We discuss the uniqueness of the solution by these two methods.     

Transformation of the probability structures of the random fields of sources in the nonlinearly parametric mapping process

The formation of the functional operator procedure for solving the problem of the propagation of random fields generated by extraneous sources in a statistically inhomogeneous continuous medium is considered. The solution is represented in the form of a probability density functional or a characteristic functional as the basis for analyzing the likelihood ratio. The structure of operators introduced in the calculations correlates with the statistical properties of both the fields of random sources and the parameters of the propagation medium. In the framework of Gaussian models of sources and propagation parameters, the solution is constructed in a closed form. Representations for the mean fields and their correlation functions are derived.     

Time-dependent polarized radiation transfer in semi-infinite binary Markovian media

The problem of time-dependent radiation transfer in a semi-infinite plane-parallel random medium with Rayleigh scattering phase function including polarization is considered. The random medium is assumed to consist of two immiscible mixed materials with specular reflecting boundary. The mixing statistics of the two components of the medium is described by the two-state homogeneous Markovian statistics. A formalism, developed to treat radiative transfer in statistical mixtures, is used to obtain the ensemble-averaged solution. Two different weight functions are used to obtain the numerical results for the ensemble-average for reflectivity, radiant energy, and net flux of the medium at any time.     

Spectral and temporal speckle field measurements of a random medium

The zero-mean circular complex Gaussian field statistics of a random medium are experimentally demonstrated in the optical domain, thus verifying this key assumption of statistical optics. Using a frequency-tunable laser source in a fixed-path-length interferometer, we obtain optical field fluctuations in the time and frequency domains that clearly show that the ensemble-averaged temporal intensity converges to the photon transit time distribution, which for the samples used is in excellent agreement with a diffusion model.     

Majorization theory approach to the Gaussian channel minimum entropy conjecture

A long-standing open problem in quantum information theory is to find the classical capacity of an optical communication link, modeled as a Gaussian bosonic channel. It has been conjectured that this capacity is achieved by a random coding of coherent states using an isotropic Gaussian distribution in phase space. We show that proving a Gaussian minimum entropy conjecture for a quantum-limited amplifier is actually sufficient to confirm this capacity conjecture, and we provide a strong argument towards this proof by exploiting a connection between quantum entanglement and majorization theory.     

近场记录光盘的写入能量的研究

近场记录光盘是一种新的光盘记录技术,它有二个特点：（１）极高的记录密度;（２）很高的寻道速度。本文介绍了近场光盘技术的特性;提出了近场光盘记录技术中动态条件下读写中的写入能量地头盘间的能量传输问题进行了仿真计算。     

Multiple scattering effects on the radar cross section (RCS) of objects in a random medium including backscattering enhancement and shower curtain effects

This paper presents a theory of the radar cross section (RCS) of objects in multiple scattering random media. The general formulation includes the fourthorder moments including the correlation between the forward and the backward waves. The fourth moments are reduced to the second-order moments by using the circular complex Gaussian assumption. The stochastic Green's functions are expressed in parabolic approximation, and the objects are assumed to be large in terms of wavelength; therefore, Kirchhoff approximations are applicable. This theory includes the backscattering enhancement and the shower curtain effects, which are not normally considered in conventional theory. Numerical examples of a conducting object in a random medium characterized by the Gaussian and Henyey-Greenstein phase functions are shown to highlight the difference between the multiple scattering RCS and the conventional RCS in terms of optical depth, medium location and angular dependence. It shows the enhanced backscattering due to multiple scattering and the increased RCS if a random medium is closer to the transmitter.     

On the appearance of caustics for plane sound-wave propagation in moving random media

In this paper we derive expressions for the probability densities of the appearance of the first caustic for a plane sound wave propagating in moving random media. Our approach generalizes the previous work by White et al. and Klyatskin in the case of motionless media. It allows us to calculate analytically the probability density functions for two- and three-dimensional media and to express these functions in terms of the diffusion coefficient. Explicit equations are given for Gaussian and von Karman spectra of velocity fluctuations. If the random scalar or vectorial fluctuations of the medium have the same contribution to the refractive-index fluctuations, we demonstrate that in a moving medium caustics appear at shorter distances than in a non-moving one. The two-dimensional version of the theory is tested by numerical simulations in the case of velocity fluctuations with Gaussian spectra. Numerical results are in very good agreement with the theoretical predictions.     

Path integral and variation method in the band tail problem

A one-dimensional random system represented by a tridiagonal random matrix with diagonal elements distributed independently in Gaussian fashion as proposed by Bulatov and Birman [Phys. Rev. B 54 (1996) 16 305] for the calculation of the density of states in the band tails is formulated in terms of Feynman path integrals. It is shown that in the asymptotic limits both the prefactor and the exponent give the correct energy dependence for the density of states. A precise comparison of the random matrix theory and the path integral approach is given.     

Focal shift of focused truncated random electromagnetic beams

The focal shifts of focused truncated random electromagnetic beams are investigated. Based on the complex Gaussian expansion method for a hard-edged aperture function, the analytical propagation formula of cross-spectral density matrix for a random electromagnetic beam focused by an optical system with a thin lens and a circular aperture is derived. The Fresnel numbers related with the beam and system parameters are defined and used to examine focal shifts. The dependence of the focal shifts on the different Fresnel numbers and polarization distribution are discussed in detail with numerical examples.     

Calculation of coherent backscattering on the basis of Milne solution

In generalizing the Milne equation using the Wiener-Hopf method, the solution of the problem of multiple scattering by a medium with an isotropic single-scattering indicatrix for an arbitrary angle of incidence is found. The resulting solution is used as an initial approximation for calculating the intensity of coherent backscattering for an anisotropic indicatrix.     

Scattering of scalar light wave from a Gaussianben Schell model medium

The scattering of scalar light wave from a random medium with a correlation function of Gaussian–Schell model distribution is studied. It is shown that the properties of the scattered field, i.e., the spectral density and the spectral degree of coherence of the scattered field, are closely related to the properties of the scattering medium, including the scaled effective radius and the scaled correlation length of the correlation function.     

Localization of polymers in a finite medium with fixed random obstacles

In this paper we investigate the conformation statistics of a Gaussian chain embedded in a medium of finite size, in the presence of quenched random obstacles. The similarities and differences between the case of random obstacles and the case of a Gaussian random potential are elucidated. The connection with the density of states of electrons in a metal with random repulsive impurities of finite range is discussed. We also interpret the results obtained in some previous numerical simulations. Received 14 August 2001     

Partition function for an electron in a random potential

We compute the average partition function for an electron moving in a Gaussian random potential. A path integral formulation is used, with a trial action like that in Feynman's polaron theory. We compute the variational bound as well as the first correction in a systematic cumulant expansion. The results are checked against exact formulas for the onedimensional white noise problem. The density of states in the low-energy tail has the correct exponential energy dependence, and energy-dependent prefactor to within a few percent. In addition, the partition function goes over smoothly to the perturbation theory result at high temperatures.Work supported by the National Science Foundation.     

Multiple scattering effects on the radar cross section (RCS) of objects in a random medium including backscattering enhancement and shower curtain effects

Abstract

This paper presents a theory of the radar cross section (RCS) of objects in multiple scattering random media. The general formulation includes the fourthorder moments including the correlation between the forward and the backward waves. The fourth moments are reduced to the second–order moments by using the circular complex Gaussian assumption. The stochastic Green's functions are expressed in parabolic approximation, and the objects are assumed to be large in terms of wavelength; therefore, Kirchhoff approximations are applicable. This theory includes the backscattering enhancement and the shower curtain effects, which are not normally considered in conventional theory. Numerical examples of a conducting object in a random medium characterized by the Gaussian and Henyey–Greenstein phase functions are shown to highlight the difference between the multiple scattering RCS and the conventional RCS in terms of optical depth, medium location and angular dependence. It shows the enhanced backscattering due to multiple scattering and the increased RCS if a random medium is closer to the transmitter.     

Accurate reconstruction of the optical parameter distribution in participating medium based on the frequency-domain radiative transfer equation

Reconstructing the distribution of optical parameters in the participating medium based on the frequency-domain radiative transfer equation(FD-RTE) to probe the internal structure of the medium is investigated in the present work.The forward model of FD-RTE is solved via the finite volume method(FVM). The regularization term formatted by the generalized Gaussian Markov random field model is used in the objective function to overcome the ill-posed nature of the inverse problem. The multi-start conjugate gradient(MCG) method is employed to search the minimum of the objective function and increase the efficiency of convergence. A modified adjoint differentiation technique using the collimated radiative intensity is developed to calculate the gradient of the objective function with respect to the optical parameters. All simulation results show that the proposed reconstruction algorithm based on FD-RTE can obtain the accurate distributions of absorption and scattering coefficients. The reconstructed images of the scattering coefficient have less errors than those of the absorption coefficient, which indicates the former are more suitable to probing the inner structure.