White-Noise Paraxial Approximation for a General Random Hyperbolic System

Abstract

In this paper we consider a general hyperbolic system subjected to random perturbations which models wave propagation in random media. We consider the paraxial white-noise regime, which is the regime in which the propagation distance is much larger than the diameter of the input beam or source, which is itself much larger than the typical wavelength, and in which the correlation length of the medium is of the same order as the diameter of the input beam or source. We introduce a general framework that allows to consider the general random hyperbolic system. Using invariant imbedding and asymptotic analysis we show how to derive the system of Itô-Schrödinger equations driven by Brownian fields that govern the wave propagation. The form of the diffraction operator and the covariance matrix of the Brownian fields are computed from the eigenvalues and eigenvectors of the unperturbed system and from the two-point statistics of the random fluctuations of the medium. Applications are given for acoustic, elastic, and electromagnetic waves.

Partially Coherent Electromagnetic Beam Propagation in Random Media

“…where ê1 and ê2 are the unit vectors in the transverse plane pointing in the x and y directions and the complex amplitude fields u j are the solution of the following statistically coupled Itô-Schrödinger equations [16,18,19]:…”

“…The derivation of ( 16) from the random three-dimensional scalar wave equation is presented in [16]. Its derivation for the Maxwell's equations (8)(9)(10)(11) is presented in [18,19]. In Eq.…”

Coupling of paraxial and white-noise approximations of the Helmholtz equation in randomly layered media

“…The paraxial and white-noise approximations are justified mathematically by taking the limits that correspond to the respective asymptotic scaling regimes described by these approximations. Because of the advantages of both of these approximations, the problem of simultaneously taking the respective limits that yield the paraxial and white-noise approximations has been studied for the Helmholtz equation [2] as well as more general wave propagation models [15,18]. In [2] the model corresponding to the simultaneous paraxial white-noise limit was derived in the case of randomly layered media for the high-frequency regime where the wavelength is small compared to the correlation length of the fluctuations in the medium.…”

Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium

“…The problem moreover decomposes into distinct mode problems parameterized by the transverse wavevector k [10]. In the general case with ν = ν(x, z) and µ = µ(x, z) the modes are coupled via a zero-mean coupling "matrix" which involves modes of all transverse wavevectors so that the coupling matrix is in fact a coupling operator [12,14] as shown below. By substituting the ansatz (7-8) into Eq.…”

“…corresponding to the boundary conditions (14)(15). The reflection operator evaluated at z = 0 carries all the relevant information about the random medium from the point of view of the reflected wave.…”