Expression for the reflectance of randomly rough surfaces derived with the Fresnel approximation

In this theoretical paper the formulas expressing the reflectance of randomly rough surfaces are derived within the framework of the scalar theory of diffraction. The Fresnel approximation has been used in the mathematical procedure since geometrical conditions of some real reflectometers do not correspond to the conditions of the Fraunhofer diffraction of light. The formulas found are generally different from the formulas derived within the framework of the Fraunhofer approximation. By means of the numerical analysis of these formulas the differences between both the Fresnel and the Fraunhofer approximations are shown. It is also found that the formulas corresponding to the Fresnel approximation give the same results as the formulas derived by means of the Fraunhofer approximation if certain geometrical conditions are valid. These geometrical conditions are determined in this paper. The formulas corresponding to thin films with randomly rough boundaries are also introduced.