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A 2 × 2 scattering matrix for rough surfaces that separate two different media is derived. To this end, explicit closed-form expressions for the nonspecular transmission scattering coefficients are derived for the rough-surface elements to complement previous derivations of the nonspecular reflection scattering coefficients. Both vertically and horizontally polarized electromagnetic excitations are considered here. Thus the elements of scattering matrices represent the polarization-dependent nonspecular transmission and reflection properties of each arbitrarily oriented and elevated element of the rough surface. The total nonspecularly transmitted and reflected fields are expressed as integrals (not integral equations). The full-wave solutions derived here are compared with the corresponding high-frequency physical-optics results, and the low-frequency perturbation results are also derived. They are shown to satisfy the reciprocity and duality relationships in electromagnetic theory, and they are invariant to coordinate transformations. These results may be applied to both deterministic and random rough surfaces.
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Bahar et al. (1988) studied this question.
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