Numerical integration of an aspheric surface profile

Abstract

This paper deals with the design of aspherics on surfaces with steep curves in systems with very large numerical apertures, and small angular fields of view, with a goal to achieve rigorous axial stigmatism for specified finite axial conjugate points. In lens design, where numerical apertures are moderate, aspheric surface descriptions based on Feder's equation of a conic section with several polynomial terms give satisfactory results to a certain point. For larger numerical apertures plot of aberrations start showing ripples; increasing the number of polynomial terms is no remedy. This problem is similar to Runge's phenomenon and it is probably related to it, but yet not well understood. Alternative surface descriptions have been proposed with apparently some degree of success. Here we propose a method for designing aspheric surfaces in systems with large numerical apertures. In this method the profile of the aspheric is presented as a table of coordinates of key points, the points of intersection of meridian key rays with the surface. Such rays are rigorously corrected for spherical aberration for specified finite conjugates. For tracing other rays through the gap between key points, interpolation with a spline of degree 3 is proposed. Correction with this method has proved satisfactory for certain high aperture examples designed with a small number of key rays. For systems with still higher apertures the number of key rays may have to be increased and the interval between key points reduced accordingly but the degree of the spline is maintained to avoid ripples similar to Runge's phenomenon.