We present a spectral representation for the effective dielectric
function of a sample that consists of two
homogeneous layers joined with a rough interface.
This spectral representation is closely related to the
Bergman-Milton spectral representation for bulk composites,
and is easily extended to multilayered materials. By comparing the
layered system to a reference layered system that has a flat
interface we form a surface spectral function that captures all the
effects of surface structure on the effective dielectric function of
the layered sample, and is independent of the dielectric functions of
the two layers. Because of the anisotropy of the layered system
there are two surface spectral functions, one for the case where the applied
field is parallel to the interface, and one for the case
where the applied field is perpendicular to the
interface. We discuss a reciprocity relationship between
these two spectral representations and present sum
rules that are directly related to the degree of surface roughness.
We present numerical calculations of the
surface spectral function for some model geometries,
including the Gaussian random surface that has been
extensively used to study light scattering from rough surfaces,
and show that the simulations verify the sum
rules and reciprocity relationships. We show how the surface
profile and interactions between layers of the
multilayered materials are related to the features of the
surface spectral function and we discuss the
possibility of determining the spectral function
directly from reflectivity measurements.
* Work done in collaboration with M. F. Thorpe (Michigan State University), A. R. McGurn (Western Michigan University) and D. J. Bergman (Tel Aviv University).