Reference-wave solutions for high-frequency fields propagating in random media

Ray theory plays an important role in determining the propagation properties of high-frequency fields and their statistical measures in complicated random environments. According to the ray approach, the field at the observer can be synthesized from a variety of field species arriving along multiple ray trajectories resulting from refraction and scattering from boundaries and from scattering centers embedded in the random medium. For computations of the statistical measures, it is desirable therefore to possess a solution for the high-frequency field propagating along an isolated ray trajectory. For this reason, a new reference-wave method was developed to provide an analytic solution of the parabolic-wave equation.     

Modeling of high-frequency propagation in inhomogeneous background random media

When a high-frequency electromagnetic wave propagates in a complicated scattering environment, the contribution at the observer is usually composed of a number of field species arriving along different ray trajectories. In order to describe each contribution separately the parabolic extension along an isolated ray trajectory in an inhomogeneous background medium was performed. This leads to the parabolic wave equation along a deterministic ray trajectory in a randomly perturbed medium with the possibility of presenting the solution of the high-frequency field and the higher-order coherence functions in the functional path-integral form. It is shown that uncertainty considerations play an important role in relating the path-integral solutions to the approximate asymptotic solutions. The solutions for the high-frequency propagators derived in this work preserve the random information accumulated along the propagation path and therefore can be applied to the analysis of double-passage effects where the correlation between the forward-backward propagating fields has to be accounted for. This results in double-passage algorithms, which have been applied to analyze the resolution of two point scatterers. Under strong scattering conditions, the backscattering effects cannot be neglected and the ray trajectories cannot be treated separately. The final part is devoted to the generalized parabolic extension method applied to the scalar Helmholtz's equation, and possible approximations for obtaining numerically manageable solutions in the presence of random media.     

Approximating moments of acoustic fields in random media

Starting from a path integral representation of the fourth moment of an acoustic field, two types of approximations are considered in detail: a 'zero-kink' approximation and a 'single-kink' approximation. The expressions 'zero-kink' and 'single-kink' describe families of paths which are used to approximate path integrals over all continuous paths. The single-kink approximation reduces the propagation problem to a phase screen problem with fluctuations on the phase screen containing information about fluctuations along entire paths from source to receiver.     

Approximating moments of acoustic fields in random media

Abstract

Starting from a path integral representation of the fourth moment of an acoustic field, two types of approximations are considered in detail: a ‘zero-kink’ approximation and a ‘single-kink’ approximation. The expressions ‘zero-kink’ and ‘single-kink’ describe families of paths which are used to approximate path integrals over all continuous paths. The single-kink approximation reduces the propagation problem to a phase screen problem with fluctuations on the phase screen containing information about fluctuations along entire paths from source to receiver.     

Domain wall depinning in random media by ac fields

The viscous motion of an interface driven by an ac external field of frequency omega(0) in a random medium is considered here in the nonadiabatic regime. The velocity exhibits a smeared depinning transition showing a double hysteresis which is absent in the adiabatic case omega(0)-->0. Using scaling arguments and an approximate renormalization group calculation we explain the main characteristics of the hysteresis loop. In the low frequency limit these can be expressed in terms of the depinning threshold and the critical exponents of the adiabatic case.     

Radiation propagation in random media: From positive to negative correlations in high-frequency fluctuations

We survey research on radiation propagation or ballistic particle motion through media with randomly variable material density, and we investigate the topic with an emphasis on very high spatial frequencies. Our new results are based on a specific variability model consisting of a zero-mean Gaussian scaling noise riding on a constant value that is large enough with respect to the amplitude of the noise to yield overwhelmingly non-negative density. We first generalize known results about sub-exponential transmission from regular functions, which are almost everywhere continuous, to merely “measurable” ones, which are almost everywhere discontinuous (akin to statistically stationary noises), with positively correlated fluctuations. We then use the generalized measure-theoretic formulation to address negatively correlated stochastic media without leaving the framework of conventional (continuum-limit) transport theory. We thus resolve a controversy about recent claims that only discrete-point process approaches can accommodate negative correlations, i.e., anti-clustering of the material particles. We obtain in this case the predicted super-exponential behavior, but it is rather weak. Physically, and much like the alternative discrete-point process approach, the new model applies most naturally to scales commensurate with the inter-particle distance in the material, i.e., when the notion of particle density breaks down due to Poissonian—or maybe not-so-Poissonian—number-count fluctuations occur in the sample volume. At the same time, the noisy structure must prevail up to scales commensurate with the mean-free-path to be of practical significance. Possible applications are discussed.     

A hierarchical model for random walks in random media

We show that the random walk generated by a hierarchical Laplacian in ^d has standard diffusive behavior. Moreover, we show that this behavior is stable under a class of random perturbations that resemble an off-diagonal disordered lattice Laplacian. The density of states and its asymptotic behavior around zero energy are computed: singularities appear in one and two dimensions.     

McKean-Vlasov limit for interacting random processes in random media

We apply large-deviation theory to particle systems with a random mean-field interaction in the McKean-Vlasov limit. In particular, we describe large deviations and normal fluctuations around the McKean-Vlasov equation. Due to the randomness in the interaction, the McKean-Vlasov equation is a collection of coupled PDEs indexed by the state space of the single components in the medium. As a result, the study of its solution and of the finite-size fluctuation around this solution requires some new ingredient as compared to existing techniques for nonrandom interaction.     

Potentials for almost Markovian random fields

A positive almost Markovian random field is a probability measure on a lattice gas whose finite set conditional probabilities are continuous and positive. We show that each such random field has a potential and in the translation invariant case an absolutely convergent potential. We give a criterion for determining which random fields correspond to pair potentials, or in generaln-body potentials. We show that two translation invariant positive almost Markovian random fields have the same finite set conditional probabilities if and only if one minimizes the specific free energy of the other.     

Distribution functions for fluids in random media

A random medium is considered, composed of identifiable interactive sites or obstacles equilibrated at a high temperature and then quenched rapidly to form a rigid structure, statistically homogeneous on all but molecular length scales. The equilibrium statistical mechanics of a fluid contained inside this quenched medium is discussed. Various particle-particle and particle-obstacle correlation functions, which differ from the corresponding functions for a fully equilibrated binary mixture, are defined through an averaging process over the static ensemble of obstacle configurations and application of topological reduction techniques. The Ornstein-Zernike equations also differ from their equilibrium counterparts.     

Some limit theorems for random fields

We prove a central limit theorem with remainder and an iterated logarithm law for collections of mixing random variables indexed byZ ^d ,d≧1. These results are applicable to certain Gibbs random fields.     

Gaussian random fields

Two results on Gaussian random fields are presented. The first characterizes the unit Gaussian random field by a strong independence property and the second determines Gaussian random fields that are generated by stochastic processes.     

Lasing in random media

A random laser is a non-conventional laser whose feedback mechanism is based on disorder-induced light scattering. Depending on whether the feedback supplied by scattering is intensity feedback or amplitude feedback, random lasers are classified into two categories: random lasers with incoherent feedback and random lasers with coherent feedback. A brief survey of random lasers with incoherent feedback is presented. It is followed by a review of our recent experimental work on random lasers with coherent feedback, including measurement of the lasing threshold, lasing spectra, emission pattern, dynamical response, photon statistics, speckle pattern and the investigation of relevant length scales. Large disorder leads to spatial confinement of the lasing modes, that is the foundation for the micro random laser. Some theoretical models of random lasers with coherent feedback are briefly introduced. The study of random lasers improves our understanding of the interplay between light localization and coherent amplification.     

Lepton pair production in high-frequency laser fields

The production of electron-positron and muon-antimuon pairs in high-frequency laser fields via few-photon absorption is considered. It is assumed that an intense X-ray laser beam collides either with a relativistic ion beam or with a second, equally intense laser beam. We study the generation of free e ⁺ e ^? pairs, free μ⁺μ^? pairs, and bound-free e ⁺ e ^? pairs where in the latter case the electron is born in a low-lying atomic orbit of the projectile nucleus. Effects resulting from the finite nuclear size, the laser’s polarization state, and its magnetic field component are examined, which are testable experimentally by virtue of upcoming X-ray free-electron laser (XFEL) devices.