The angular intensity correlation functions C (1) and C (10) for the scattering of light from randomly rough dielectric and metal surfaces

Abstract

We study the statistical properties of the scattering matrix S(q|k) for the problem of the scattering of light of frequency ω from a randomly rough one-dimensional surface, defined by the equation x 3=ζ(x 1), where the surface profile function ζ(x 1) constitutes a zero-mean, stationary, Gaussian random process. This is done by studying the effects of S(q|k) on the angular intensity correlation function C(q,k|q′,k′)=⟨I(q|k)I(q′|k′)⟩−⟨I(q|k)⟩⟨I(q′|k′)⟩, where the intensity I(q|k) is defined in terms of S(q|k) by I(q|k)=L −1 1(ω/c)|S(q|k)|2, with L 1 the length of the x 1 axis covered by the random surface. We focus our attention on the C (1) and C (10) correlation functions, which are the contributions to C(q,k|q′,k′) proportional to δ(q−k−q′+k′) and δ(q−k+q′−k′), respectively. The existence of both of these correlation functions is consistent with the amplitude of the scattered field obeying complex Gaussian statistics in the limit of a long surface and in the presence of weak surface roughness. We show that the deviation of the statistics of the scattering matrix from complex circular Gaussian statistics and the C (10) correlation function are determined by exactly the same statistical moment of S(q|k). As the random surface becomes rougher, the amplitude of the scattered field no longer obeys complex Gaussian statistics but obeys complex circular Gaussian statistics instead. In this case the C (10) correlation function should therefore vanish. This result is confirmed by numerical simulation calculations.