In this paper, full wave solutions are derived for the like‐ and cross‐polarized electromagnetic fields diffusely scattered by two‐dimensional rough surfaces. These expressions for the diffuse scattered fields are used to obtain the random rough‐surface cross sections. The rough surface is characterized by a Gaussian joint probability density function for the surface heights and slopes at two points. These full wave results are compared with the associated small‐perturbation results and physical optics results. They are also compared with experimental and numerical results (based on Monte Carlo simulations). The earlier assumption that the surface height and slopes can be considered to be uncorrelated is examined. The impact of using the large‐radii curvature assumption and including self‐shadow is also considered.

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Full wave solutions are derived for vertically and horizontally polarized waves diffusely scattered across an interface that is two‐dimensionally rough separating two different propagating media. Since the normal to the rough surface is not restricted to the reference plane of incidence, the waves are depolarized upon scattering; and the single scattered radiation fields are expressed as integrals of a surface element transmission scattering matrix that also accounts for coupling between the vertically and horizontally polarized waves. The integrations are over the rough surface area as well as the complete two‐dimensional wave spectra of the radiation fields. The full wave solutions satisfy the duality and reciprocity relationships in electromagnetic theory, and the surface element scattering matrix is invariant to coordinate transformations. It is shown that in the high‐frequency limit the full wave solutions reduce to the physical optics solutions, while in the low‐frequency limit (for small mean square heights and slopes) the full wave solutions reduce to Rice's (1951) small perturbation solutions. Thus, the full wave solution accounts for specular point scattering as well as diffuse, Bragg‐type scattering in a unified, self‐consistent manner. It is therefore not necessary to use hybrid, perturbation and physical optics approaches (based on two‐scale models of composite surfaces with large and small roughness scales) to determine the like‐ and cross‐polarized fields scattered across the rough surface.

In this work full wave expressions are derived for the incoherent diffuse scatter cross sections (per unit area) for waves transmitted across rough surfaces. Both the like‐ and cross‐polarized cross sections are considered since the surfaces are rough in two dimensions. The full wave solutions are compared with the small perturbation solutions as well as with the physical/geometrical solutions. It is shown that when the heights and slopes are of the same order of smallness, the full wave solutions are in agreement with the small perturbation solution, while in the high‐frequency limit the full wave solution is in agreement with physical/geometrical optics solutions, provided that specular (stationary phase) points exist on the surface. Several illustrative examples are considered and the refractive index is assumed to be larger or smaller than unity. The critical angle for total internal reflection has a significant impact on diffuse scattering in the nonspecular direction. Since the full wave solutions account for specular point scattering as well as Bragg type scattering in a uniform, self‐consistent manner, it is not necessary to adopt the two‐scale model for the rough surfaces.