Matrix based algorithm for ion-beam figuring of optical elements

Abstract

An algorithm for solving the problem of local shape errors of optical surfaces correcting by a small-sized ion beam, based on the matrix representation is presented. The algorithm involves searching for points that rise relative to the average height on the surface, where exposure to an ion beam of a given shape will lead to a decrease in the RMS deviation of the surface shape from the calculated one. It is shown that the new approach makes it possible to significantly expand the range of spatial frequencies, in which the height of relief can be reduced using the given ion beam size, and provides better results than the method of minimizing the functional of convolution of the ion beam and the surface map. Moreover, the new approach does not lead to appearance of shape errors (concentric structure) defined by the ion beam size and scanning step. An experiment for minimizing of local surface shape errors based on an etching map calculated using new approach was done. Matrix algorithm allowed in one procedure to reduce surface shape errors for a concave spherical optical element made of fused silica with a diameter D = 100 mm and a radius of curvature R = −137.5 mm in the RMS by more than 3 times. The initial RMS was 4.5 nm, RMS after treatment is 1.36 nm.