Anjali Nair: Speckle formation of light in random media and the Gaussian conjecture

A well-known conjecture in physical literature states that high frequency waves propagating over long distances

through turbulence eventually become complex Gaussian distributed. The intensity of such wave fields then follows

an exponential law, consistent with speckle formation observed in physical experiments. Though fairly well-accepted

and intuitive, this conjecture is not entirely supported by any detailed mathematical derivation. In this talk, I

will discuss some recent results demonstrating the Gaussian conjecture in a weak-coupling regime of the paraxial

approximation.

The paraxial approximation is a high frequency approximation of the Helmholtz equation, where backscattering

is ignored. This takes the form of a Schrödinger equation with a random potential and is often used to model

laser propagation through turbulence. In particular, I will describe a diffusive scaling where the limiting probability

distribution of the wavefield is completely described by a second moment which follows an anomalous diffusion. The

proof relies on the asymptotic closeness of statistical moments of the wavefield under the paraxial approximation,

its white noise limit and the complex Gaussian distribution itself. An additional stochastic continuity/tightness

criterion allows to show the convergence of these distributions over spaces of Hölder-continuous functions. Numerical

simulations illustrate theoretical results.

This is joint work with Guillaume Bal.