Surface discretization in optical aspheres production

Highly aspherical optical surfaces are commonplace in the optical industry today. Surface discretization, i.e., describing the surface via a point cloud or a grid, is crucial in several surface production processes such as machining, finishing, metrology, and characterization. The simplest way to discretize an aspherical surface is to generate a uniform x-y sampling in the vertex plane and orthogonally project that sampling onto the surface. However, such a method leads to excessively non-uniform samplings when the surface is highly aspherical, which is not rare in the current optical industry. To overcome this problem, we have studied the properties of three discretization methods applied to optical aspheres. The first essentially generates the involute of a circle embedded within the optical surface (CI). The other two methods are based on finding an intermediate mapping. Whereas one method finds this mapping by solving a Laplace-Beltrami partial derivatives system (LB), the other uses optimal mass transportation (OT). We compared the outcomes of these methods with the trivial projection method. Evidence shows that, when targeting uniformity, the CI and OT provide the best outcomes. Conversely, the CI and LB methods achieve better results if the quality target has grid cells close to equilateral.