Modeling and analysis of the mid-spatial- frequency error characteristics and generation mechanism in sub-aperture optical polishing

In the field of ultra-precision manufacturing, the mid-spatial-frequency (MSF) error can severely affect the performance of the optical elements, but it is rather difficult to quantitatively predict the MSF error distribution. In this paper, the piecewise-path convolution (PPC) analysis is established to investigate the characteristic and the mechanism of the MSF error. The path type, tool influence function (TIF), feed rate, movement type, etc. are all considered mathematically in the analysis. This method can quantitatively predict the MSF error distribution. The coupling relationship among the path type, TIF and the MSF error are proved through the filtering theory. Besides, the analysis reveals the mathematical relationship between the tool movement type (orbital motion, radial runout) and the MSF error; the results show that the tool motion can also introduce non-negligible MSF error. Based on the research above, two selection formulae of path type, TIF and polishing parameters are provided for low MSF error polishing, which gives the theoretical guidance for the parameter selection in deterministic polishing. Practical experiments demonstrate the validity of the analysis results and conclusions.