A probabilistic theory of electromagnetic wave scattering from a slightly random surface 1. Horizontal polarization

The scattering of electromagnetic plane waves from a slightly random and perfectly conducting surface is analyzed by the probabilistic method developed in a previous paper. In view of the stochastic homogeneity of the infinite random surface the electromagnetic wave solution proves to be a homogeneous vector random field multiplied by an exponential phase factor. On the basis of the nonlinear theory of the Gaussian random field the electric field is represented in terms of orthogonal functionals associated with the Gaussian distributed surface with unknown coefficients of deterministic functions. Then the boundary condition on the random surface is transformed into a set of equations for such unknown functions, which is approximately solved for horizontal polarization to yield the stochastic solution involving multiple scattering. Unlike the solution by the perturbation method, the stochastic solution never diverges but has such a singularity similar to Wood's anomalies in the diffraction gratings that the partial wave scattered at the Rayleigh wavelength has a finite but large amplitude. The stochastic solution makes it possible to calculate many statistical properties of the scattering, such as the average and variance of the scattered electric field, the intensity of the incoherent scattering, the average power flow of the surface wave, and the scattering cross section per unit area. These quantities as well as the optical theorem are illustrated in the figures.