Autocovariance functions for polished optical surfaces

Autocovariance functions (ACF’s) G(x) for polished optical surfaces of CaF2, MgF2, and LiF are deduced from surface profiles determined by using microdensitometer analysis of micrographs of surface-shadowed carbon replicas. The different estimators allowing the ACF computation from its standard definition are reviewed, and their values are compared. ACF’s are also computed by using the fast-Fourier-transform algorithm. Results of both computations are in good agreement. It is shown that initial portions of ACF’s have a reasonable Gaussian form. The rms roughness height δ and the autocovariance length σ are deduced for each surface. The ACF’s of the surface slopes Gs′(x) are also computed, and it is shown that results obtained are consistent with results deduced from ACF’s for surface profiles. In particular, the standard relation between the second derivative of G(x) and Gs′(x) is reasonably verified. Finally, the exponential ACF model is discussed, and it is shown that this model would not be suitable to describe the initial portions of the ACF’s for the polished optical surfaces that we have studied.