Paraxial wave propagation in random media with long-range correlations

TL;DR

This paper analyzes how long-range correlated turbulence affects paraxial wave propagation, revealing multiscale randomization of wave travel time and field, with implications for imaging and communication in turbulent atmospheres.

Contribution

It provides a mathematical framework for wave propagation in media with non-integrable, long-range correlations, extending existing models to more realistic atmospheric turbulence.

Findings

Wave travel time is randomized by fractional Brownian motion.

Wave field is described by a Schrödinger-type equation driven by Brownian noise.

Scattering causes decorrelation and pulse deformation, impacting imaging and communication.

Abstract

We study the paraxial wave equation with a randomly perturbed index of refraction, which can model the propagation of a wave beam in a turbulent medium. The random perturbation is a stationary and isotropic process with a general form of the covariance that may be integrable or not. We focus attention mostly on the non-integrable case, which corresponds to a random perturbation with long-range correlations, that is relevant for propagation through a cloudy turbulent atmosphere. The analysis is carried out in a high-frequency regime where the forward scattering approximation holds. It reveals that the randomization of the wave field is multiscale: The travel time of the wave front is randomized at short distances of propagation and it can be described by a fractional Brownian motion. The wave field observed in the random travel time frame is affected by the random perturbations at long distances, and it is described by a Schroedinger-type equation driven by a standard Brownian field. We use these results to quantify how scattering leads to decorrelation of the spatial and spectral components of the wave field and to a deformation of the pulse emitted by the source. These are important questions for applications like imaging and free space communications with pulsed laser beams through a turbulent atmosphere. We also compare the results with those used in the optics literature, which are based on the Kolmogorov model of turbulence.