Boundary conditions of the diffusion equation and applications

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Abstract
A particular expression of the mutual coherence function of a wave is first derived for a system of random layers with rough boundaries, starting from the unified Bethe-Salpeter equation that involves the random medium and boundaries on exactly the same footing. An effective scattering matrix of the medium is then introduced, which is the only medium-dependent quantity involved in the expression and is obtained as a boundary-value solution of the diffusion equation. The diffusion approximation is based on an eigenfunction expansion by using a set of eigenfunctions of the medium scattering cross section that can be even rotationally variant, and the first term is good enough in the diffusion region. Here each expansion coefficient is obtained as the solution of an integral equation with terms of the boundaries, and, for the first term, the equation can be converted to a diffusion equation with a source term, subjected to boundary conditions determined by the boundary scattering cross sections. Here the condition at each boundary is valid even when the averaged refractive indices of the two media differ from each other by a large amount, as contrasted to the condition of no reflection used so far when obtaining boundary-value solutions of diffusion equations. Specific expressions are obtained for both backscattered and transmitted waves through a random layer, along with numerical examples. The boundary condition is finally generalized to meet the case of a rough boundary between two random media of different kinds, consistent with power conservation.