Reflection and Transmission by a Random Medium

TLDR

The mean‑field description of a bounded medium with a random refractive index satisfies an integrodifferential equation when the amplitude and correlation length of the fluctuations lie in suitable ranges. The authors solve this equation for plane‑wave incidence on semi‑infinite and slab random media with homogeneous background, derive explicit reflection, transmission, and coupling coefficients for exponential correlation, and provide Wiener‑Hopf factorization and curved‑boundary methods for other correlation functions. They find that modeling the mean wave in a bounded random medium requires a transition layer near the boundary, as the unbounded‑medium refractive index alone is insufficient.

Abstract

The mean field is considered for a bounded medium with a refractive index having a real random part. It has been shown previously that for suitable ranges of the amplitude and correlation length of the refractive index fluctuations, this field satisfies a certain integrodifferential equation. This equation is solved for a plane wave incident from either side on the plane boundary of a semi‐infinite random medium, for the Green's function of a semi‐infinite random medium, and for a plane wave incident on a slab of random medium, provided that the background refractive index is homogeneous throughout. The cases of both one‐ and three‐dimensional fluctuations are considered, and explicit expressions are given for the reflection,transmission, and coupling coefficients for a medium with an exponential correlation function.A Wiener‐Hopf factorization required for other correlation functions is described, as are methods for treating reflection and transmission at a curved boundary of a random medium. A Wiener‐Hopf factorization required for other correlation functions is described, as are methods for treating reflection and transmission at a curved boundary of a random medium. A principal finding is the inadequacy of treating the mean wave in a bounded random medium by using just the refractive index for an unbounded random medium, for in addition we must include a transition layer near the boundary.