In the smooth-surface limit, the angular distribution of the light scattered from a surface maps the power spectral density of its residual surface roughness. This result is essentially independent of the scattering theory used and the statistical properties of the surface roughness. The power spectral densities of engineering surfaces are generally broad and increase with increasing spatial wavelength. As a result, practical surface finish parameters are not intrinsic properties of the surface, but depend, with varying degrees of sensitivity, on the bandwidth limits inherent in their measurement or dictated by their application. These issues are discussed with reference to two classes of finish parameters: those related to the central moments of the scattering spectrum, and those related to the coefficients in the expansion of the shape of the spectrum in inverse powers of the scattering angle. The significance of "1/02" scattering in this context is emphasized. A shot model of surface roughness is then introduced to gain further insight into the relationship between scattering and surface features. In this model inverse power terms are related to "edge" scattering effects from critical points in various types of elemental microdefects. The relationship between this view and electronic noise is pointed out; in particular, the correspondence between "1/62" scattering and "1 /f" or flicker -noise phenomena.

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Surface finish measurements are usually fitted to models of the finish correlation function which are parametrized in terms of root-mean-square roughnesses, delta, and correlation lengths, l. Highly finished optical surfaces, however, are frequently better described by fractal models, which involve inverse power-law spectra and are parametrized by spectral strengths, K(n), and spectral indices, n. Analyzing measurements of fractal surfaces in terms of sigma and l gives results which are not intrinsic surface parameters but which depend on the bandwidth parameters of the measurement process used. This paper derives expressions for these pseudoparameters and discusses the errors involved in using them for the characterization and specification of surface finish.

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We describe methods of predicting the degradation of the performance of a simple imaging system in terms of the statistics of the shape errors of the focusing element and, conversely, of specifying those statistics in terms of requirements on image quality. Results are illustrated for normal-incidence, x-ray mirrors with figure errors plus conventional and/or fractal finish errors. It is emphasized that the imaging properties of a surface with fractal errors are well behaved even though fractal-power spectra diverge at low spatial frequencies.

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