Oblique-incidence optical reflectivity difference from a rough film of crystalline material

Formation of a rough film of crystalline material on a smooth substrate resulting from kinetic roughening in epitaxy or erosion causes disproportionate changes in reflectivity for s- and p-polarized light. I present a mean-field theory of optical reflectivity difference defined as ${(r}_{p}\ensuremath{-}{r}_{p0}{)/r}_{p0}\ensuremath{-}{(r}_{s}\ensuremath{-}{r}_{s0}{)/r}_{s0}\ensuremath{\equiv}{\ensuremath{\Delta}}_{p}\ensuremath{-}{\ensuremath{\Delta}}_{s}$ from such a rough film, with ${r}_{p0}$ and ${r}_{s0}$ being the reflectivities of the bare substrate, and ${r}_{p}$ and ${r}_{s}$ being the reflectivities after the rough film forms on the substrate. In the limit that the average film thickness is less than the optical wavelength \ensuremath{\lambda}, I found that ${\ensuremath{\Delta}}_{p}\ensuremath{-}{\ensuremath{\Delta}}_{s}$ consists of a term that varies linearly with the average film thickness and a term that is proportional to the surface density of step edge atoms. I apply such a theory to the analysis of growth and ion erosion of a number of crystalline materials studied with the oblique-incidence optical reflectivity difference (OI-RD) technique.