Statistical theory of rough boundaries for electromagnetic waves. II. Statistical Green functions and scattering cross sections

Based on the deterministic equations obtained in Part I [ J. Opt. Soc. Am. A2, 2244 ( 1985)], basic equations of the first- and second-order statistical Green functions are first obtained in the form of the Dyson equation and the Bethe–Salpeter (B–S) equation, respectively, by following precisely the same procedure as employed for scalar waves in a previous paper. The solution of the B–S equation is written in terms of an incoherent scattering matrix, which also enables various equations associated with the incoherent waves, including a relation that eventually leads to the optical relation, to be written exactly. Scattering cross sections of the boundary are obtained together with the precise optical relations for both the reflected and the transmitted waves. As typical examples, the cross sections of single scattering are obtained specifically for a slightly random boundary and a large-scale rough boundary, with varieties of scattering angle and polarization. For the reflected waves, the previous result for a slightly random boundary is reproduced; whereas, for the large-scale rough boundary, the cross sections are obtained consistently with power conservation, in contrast to those obtained by the conventional methods. An exact version of the shadowing function is also shown. Finally, a composite random boundary is investigated by applying the addition theorem of scattering matrices, which was previously introduced for a combined system of a random medium and random boundaries.