Relation between the Height Distribution of a Rough Surface and the Reflectance at Normal Incidence

TLDR

A theory linking rms roughness of a plane surface to its specular reflectance at normal incidence has been reported for the case of small roughness relative to the wavelength. The present paper extends this theory, under certain restrictions, to shorter wavelengths. The authors show that rms roughness, rms slope, and autocovariance length are inadequate for the short‑wavelength regime, and they employ simple models of normal and abnormal surfaces to extract height distributions from normal‑incidence reflectance while accounting for phase changes due to roughness. For many normal surfaces, the height distribution can be readily determined from normal‑incidence reflectance measurements.

Abstract

A theory concerning the relation between the root-mean-square (rms) roughness of a plane surface and its specular reflectance at normal incidence has been reported previously for the case when the roughness is small compared to the wavelength of light. In the present paper the theory is extended with certain restrictions to shorter wavelengths. The inadequacy of parameters such as the rms roughness, the rms slope, and the autocovariance length for describing the reflectance in the shorter wavelength region is discussed. Particular attention is given to the problem of determining the distribution of heights of the surface irregularities from reflectance measurements at normal incidence. It is shown that for many surfaces, designated as normal surfaces, this distribution may be readily determined. Simple models for both normal and abnormal surfaces are used to illustrate the behavior of the reflectance in both cases and the consequent precautions necessary to obtain accurate height distributions. The role of the phase change due to roughness in determining the height distribution is also discussed.