The electromagnetic field in a randomly inhomogeneous medium

Abstract
Maxwell's equations for a medium in which the "dielectric constant" is a slowly-varying random function of position are reduced to a scalar Helmholtz equation. The Helmholtz equation defines the electromagnetic field components as linear stochastic processes in the random refractive index. Spectral representations for the electromagnetic field components and the refractive index are then introduced, and the former is expressed in terms of a convolution of the latter. The relation between the spectral representations is found explicitly for the case when the average phase is constant. The case when a source is present is next considered. The inhomogeneous Helmholtz equation is expressed as a singular Fredholm integral equation whose kernel involves the Green's function for the homogeneous medium and the random refractive index. The conditions on the correlation function of the refractive index such that the Neumann series solution converges in the mean-square sense are investigated.