Scattering of electromagnetic waves at the corrugated interface between index-matched media

We consider the scattering of s- and p-polarized electromagnetic waves incident onto the periodically corrugated boundary between index-matched media. First we find the eigenmodes supported by this kind of interface by means of a perturbative solution to the corresponding homogeneous problem and show that these eigenmodes are intimately related to the occurrence of strong peaks in the curves of reflectivity versus angle of incidence. Second, we use the Rayleigh hypothesis to derive integral equations for the amplitudes of the reflected and transmitted waves and give a solution to these equations in the form of series of powers of the surface profile. This perturbative treatment is useful to follow the strong changes that a weak corrugation introduces in the reflectivity of index-matched interfaces.