Light scattering from rough surfaces

We have reconsidered the theory of the scattering and absorption of light incident on a medium of dielectric constant $\epsilon$ that is bounded by a rough surface, when the roughness can be treated as a perturbation of the perfectly flat surface. We obtain explicit formulas for perpendicularly incident light by several different methods, both classical and quantum mechanical and compare our results with those obtained by previous authors. In particular, we consider the case of a metal of dielectric constant $1-\frac{{\omega }_p^2}{{\omega }^2}$ and discuss the excitation of surface plasmons. We show why the results obtained by Elson and Ritchie by a quantum-mechanical method based on a coordinate transformation do not agree in some cases with the results of the classical theory and with the results of a straightforward quantum treatment that is applicable only to normal incidence. We introduce a new coordinate transformation that does not suffer from the limitations of that used by Elson and Ritchie and thereby allows their general method to be applied, with correct results, for general polarization and incidence angle.