Statistical basis functions for modelling mid-spatial frequency errors in asphere manufacturing

Mid-spatial frequency manufacturing errors are often present in aspherical optics. These errors arise from the nature of the asphere manufacturing process, whereby many passes are made on multi-axis polishing machines with subaperture sized tools. The process results in mid-spatial frequency artifacts which can typically be characterised into 2 types of form error: rings and spokes. The standard tolerance specifications of form and slope error used in asphere manufacture does not capture the range of possible outcomes for an as manufactured part. The fact that the current tolerance standard does not adequately describe the range of outcomes for as manufactured aspheres has been known for some time. In this work, we present a set of basis functions which represent rings and spokes, combined with statistical form errors sampled from an appropriate power law statistical distribution in frequency space. We use real data to verify that our error representation is more efficient mathematically as compared with the standard Zernike decomposition.