Dispersion of surface plasmons and phonons in inhomogeneous media

The dispersion relation of surface plasmons or phonons is obtained, by solving Maxwell's equations, for the case that the dielectric function ε(ω) has a small exponential variation below the surface, perpendicular to the propagation direction. Only the usual two branches are found. The lower one exhibits the properties, at large propagation vector, that the limiting ω is determined by the surface value of ε and the shape depends on the spatial extent of the variation in ε.