A systematic perturbation theory is developed to study the interaction of electromagnetic waves at rough dielectric surfaces. The Ewald-Oseen extinction theorem is used as the basis of the perturbation theory. Explicit expressions for both first-order and second-order fields (in terms of the surface roughness parameter) at all points in space are presented. The incident field is treated very generally and thus it can be in any state of polarization. The surface is characterized by a structure function and hence the results are valid both for periodic and statistical surfaces. The first-order fields are used to calculate various types of scattering cross sections for arbitrary incident fields characterized by coherence matrices. The coherence matrix for scattered fields is given. The equivalent electric and magnetic surface currents on a flat surface which would lead to the same expressions for first-order fields as those obtained for the rough surface, are calculated using the extinction theorem. The extinction cross section is calculated using an appropriately formulated optical theorem. This leads in a straightforward manner to a change in the value of the reflectivity as compared to the value on a flat surface. The experiment of Teng and Stern is discussed briefly in the light of the expressions for extinction cross sections. Finally, first-order and second-order fields are used to discuss Smith-Purcell radiation, i.e., the radiation emitted by an electron moving parallel to a grating surface. This case corresponds to the conversion of evanescent waves into homogeneous waves due to surface roughness. The relation of some of our results to those of Crowell, Elson, Ritchie, Juranek, Lalor, Marvin et al., Maradudin, and Mills is also discussed.

• Received 18 February 1976

DOI:https://doi.org/10.1103/PhysRevB.15.2371