Peano-like paths for subaperture polishing of optical aspherical surfaces

Polishing can be more uniform if the polishing path provides uniform coverage of the surface. It is known that Peano paths can provide uniform coverage of planar surfaces. Peano paths also contain short path segments and turns: (1) all path segments have the same length, (2) path segments are mutually orthogonal at the turns, and (3) path segments and turns are uniformity distributed over the domain surface. These make Peano paths an attractive candidate among polishing tool paths because they enhance multidirectional approaches of the tool to each surface location. A method for constructing Peano paths for uniform coverage of aspherical surfaces is proposed in this paper. When mapped to the aspherical surface, the path also contains short path segments and turns, and the above attributes are approximately preserved. Attention is paid so that the path segments are still well distributed near the vertex of the surface. The proposed tool path was used in the polishing of a number of parabolic BK7 specimens using magnetorheological finishing (MRF) and pitch with cerium oxide. The results were rather good for optical lenses and confirm that a Peano-like path was useful for polishing, for MRF, and for pitch polishing. In the latter case, the surface roughness achieved was 0.91 nm according to WYKO measurement.