Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles

Surface scatter effects from residual optical fabrication errors can severely degrade optical performance. The total integrated scatter (TIS) from a given mirror surface is determined by the ratio of the spatial frequency band-limited " relevant " root-mean-square surface roughness to the wavelength of light. For short-wavelength (extreme-ultraviolet/x-ray) applications, even state-of-the-art optical surfaces can scatter a significant fraction of the total reflected light. In this paper we first discuss how to calculate the band-limited relevant roughness from surface metrology data, then present parametric plots of the TIS for optical surfaces with arbitrary roughness, surface correlation widths, and incident angles. Surfaces with both Gaussian and ABC or K-correlation power spectral density functions have been modeled. These parametric TIS predictions provide insight that is useful in determining realistic optical fabrication tolerances necessary to satisfy specific optical performance requirements. 1 Introduction and Overview Surface scatter phenomena continue to be an important issue in diverse areas of science and engineering in the 21st century. In some applications, the total amount of scattered radiation is of primary concern. In other applications, knowing the angular distribution of the scattered light is crucial. Recall that the reflectance of a surface is defined as the ratio of the (total) reflected radiant power divided by the incident radiant power. However, for real surfaces (exhibiting some residual surface roughness) the total reflected radiant power consists of two components: one specularly reflected (obeying the law of reflection) and the other diffusely reflected (scattered). Following the work of Bennett and Porteus, 1 which built upon the earlier work of Davies, 2 the fraction of the total reflected radiant power remaining in the specular beam after reflection from a single moderately rough surface is given by