Robust numerical solution to the Levi-Civita wavefront coupling problem via level set computation of the point characteristic function

Abstract

The Levi-Civita wavefront coupling problem consists of, given two prescribed wavefronts, obtaining a refractive or reflective surface coupling them. We propose a robust numerical method to solve Levi-Civita’s problem, whose rationale is to consider that Levi-Civita’s solutions are level surfaces of the point characteristic function established between points of incoming and outgoing wavefronts. The method obtains both surface data points and their normals, enabling a more robust surface reconstruction. We carry out a detailed error analysis of our method by means of comparing the surface data estimation with nominal surfaces obtained in reference tests offering analytical solutions to Levi–Civita’s problem. The method offers, in computer simulations, highly accurate results with moderate computational cost.

Extended-depth-of-focus wavefront design from pseudo-umbilical space curves

“…The method is purely within a geometrical optics framework, which is a reasonably good starting point. Once the incoming and outgoing wavefronts are prescribed, it is possible to provide their required coupling by means of, for instance, a refractive/reflective optical surface [9]. This kind of wavefront provides an extended depth-of-focus by means of a varying mean curvature through the spatial domain.…”