Optical response of microscopically rough surfaces

The effect of microscopic roughness on the complex normal-incidence reflectance r̃ at an interface between a transparent ambient and a microscopically rough substrate is treated analytically in the classical limit to lowest order in d/λ, where d is the effective thickness of the microscopically rough layer and λ is the wavelength of light. The ambient and substrate dielectric functions, ${\mathrm{\epsilon }}_{\mathit{a}}$ and ${\mathrm{\epsilon }}_{\mathit{s}}$, respectively, are assumed to be local, scalar, and isotropic. The mathematical definition of microscopic roughness is shown to include the distance to the observer as well as λ and the spatial Fourier components of the surface, which is characteristic of coherent (Fraunhofer) diffraction. The effect on r̃, expressed as a dielectric response of an equivalent thin film, reduces to standard effective-medium form, thereby providing a theoretical justification of the highly successful empirical approach. This expression further reduces to a surface integral of the local-field potential weighted by the component of the local normal surface vector parallel to the mean surface plane, a form that allows the microscopic-roughness-induced change Δr̃/r̃ in r̃ to be calculated directly if the local potential is known by symmetry, numerical analysis, conformal mapping, or other means. If ‖${\mathrm{\epsilon }}_{\mathit{s}}$‖ is large, as for semiconductors and metals, and if self-consistency effects can be ignored, then the problem becomes isomorphic to that of fluid flow and can be solved analytically for simple geometries by conformal mapping. In this limit, I obtain analytic solutions for the weak sinusoidal grating, the low ridge, and the isolated low step, all of which are incipiently rough surfaces that cannot be treated by effective-medium theory but which represent morphologies commonly encountered in crystal growth. The expressions for the ridge and step contain singularities that are logarithmic in the ratio of step height to separation, although these singularities are expected to be limited by self-consistency effects that remain to be established numerically. The present results also provide a theoretical framework for more general treatments.