Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media

TL;DR

This paper analyzes high-frequency wave propagation in long-range dependent random media, deriving a fractional Itô-Schrödinger equation to model laser beam propagation in turbulent atmospheres, considering paraxial and white-noise limits.

Contribution

It introduces a novel fractional white-noise limit and paraxial approximation framework for waves in long-range dependent random media, extending classical models to include fractional noise effects.

Findings

Derived fractional Itô-Schrödinger equation for wave propagation

Analyzed convergence using moment techniques due to long-range dependence

Provided insights into backscattering and mode coupling in random media

Abstract

This work is devoted to the asymptotic analysis of high frequency wave propagation in random media with long-range dependence. We are interested in two asymptotic regimes, that we investigate simultaneously: the paraxial approximation, where the wave is collimated and propagates along a privileged direction of propagation, and the white-noise limit, where random fluctuations in the background are well approximated in a statistical sense by a fractional white noise. The fractional nature of the fluctuations is reminiscent of the long-range correlations in the underlying random medium. A typical physical setting is laser beam propagation in turbulent atmosphere. Starting from the high frequency wave equation with fast non-Gaussian random oscillations in the velocity field, we derive the fractional It\^o-Schr\"odinger equation, that is a Schr\"odinger equation with potential equal to a fractional white noise. The proof involves a fine analysis of the backscattering and of the coupling between the propagating and evanescent modes. Because of the long-range dependence, classical diffusion-approximation theorems for equations with random coefficients do not apply, and we therefore use moment techniques to study the convergence.