Optical design of freeform surfaces using radial basis functions with spatially variable shape parameters and local grid refinement

Free-form optics is transformative technology in contemporary optical engineering with the design toolset still under development. Recently, we introduced anisotropic Radial Basis Function (aRBF) expansion as a means for effective optical design and tolerancing. In our method, great emphasis is on choosing optimal shape parameters and grid type for placement of aRBFs. In this contribution, we study most general form of RBF based representation involving spatially variable shape parameters across arbitrary irregular domain. Major advantage offered by this approach is in reducing numerical ill-conditioning and increasing accuracy of approximation while using smallest number of RBFs. In addition, possible forms of local grid refinement are exemplified and studied. We tested our method on several use cases using both RBFs with and without compact support and compared the results obtained in multi-parameter optimization with the standard spline based local surface representations. Finally, we exemplify our method on a specific design case of freeform element originating in the design of chirped lens array and with the application for structured illumination in microlithography.