Effects of surface roughness on the surface-polariton dispersion relation

The effects of surface roughness on the dispersion relation for surface polaritons at the rough interface between vacuum and a semi-infinite dielectric medium, characterized by an isotropic dielectric constant $\epsilon \left(\omega \right)$, have been determined. An integral equation is established for the Green's function for the matrix differential operator in Maxwell's equation for the macroscopic electric field in the presence of a rough interface, in terms of the corresponding Green's function in the presence of a plane interface. On the assumption that the Fourier coefficients of the surface roughness profile function are Gaussianly distributed random variables the integral equation can be solved. The position of the pole in the resulting solution corresponding to the frequency of the surface polariton has been determined through terms of $O\left({\delta }^2\right)$, where ${\delta }^2$ is the mean-square deviation of the surface from flatness. The resulting expressions for the real and imaginary parts of the surface-polariton dispersion relation are evaluated for two different choices of $\epsilon \left(\omega \right)$, the first corresponding to a diatomic polar crystal of cubic symmetry, the second corresponding to intraband transitions in a metal or semiconductor with free carriers. The effects of surface roughness are found to be significant only for wavelengths of the surface polariton comparable with or longer than the transverse correlation length characterizing the horizontal distribution of surface roughness.