Transfer function characterization of scattering surfaces revisited

Over twenty years ago a linear systems approach to modeling surface scatter phenomena was developed by considering it to be a scalar diffraction process resulting from random phase variations in the exit pupil of an optical system. This led to the derivation of a surface transfer function (STF) that relates the scattering behavior to the surface topography. The resulting model was analogous to, and an extension of, the highly successful application of linear systems theory to the understanding of image forming systems. Experimental angular scattering data was shown to be shift-invariant in direction cosine space with respect to incident angle (this led to a modest following among the radiometric community of BRDF curves plotted in the Harvey-Shack (beta) -(beta) o format). During the 1980s this STF was generalized to include: (1) the effects of small-angle scatter caused by \u27mid\u27 spatial frequency surface irregularities which span the gap between the traditional \u27figure\u27 and \u27finish\u27 errors, and (2) the extremely large incident angles inherent to grazing incidence Wolter Type I x-ray telescopes. Since no explicit smooth surface approximation is imposed, this STF can be utilized to predict the scattering behavior of rough surfaces not accurately modeled by vector perturbations techniques considered to be more rigorous by many investigators. Also, the scattering function is normalized by the total reflectance of the surface. Hence, the dominant polarization effects are included (in spite of the fact that this is basically a scalar treatment) by using the Fresnel reflectance coefficients for the desired polarization. In this paper it is emphasized that scattered radiance (not irradiance or intensity) is shift- invariant in direction cosine space paper to explain some non- intuitive scattering behavior reported in the literature. ©2004 Copyrigth SPIE - The International Society for Optical Engineering