Investigation of spline approximation approach of surface data for synthesis of high-efficient beam shapers

When specifying optical surfaces of complex shape, polynomial representation methods are often used. However, if the equation of the curve is not described by a standard polynomial, the approximation error at the edges of such functions increases sharply - this introduces huge oscillations. Freeform surfaces of a complex geometric representation are achieved often as a result of design of beam shapers. Spline methods allow minimizing the effect of oscillations after the surface approximation of the beam shapers. In this paper, a comparative analysis of methods for approximation of freeform surfaces is given using the example of Focal-piShaper synthesis and single collimating lens with one freeform surface. The approximation results for the Focal-piShaper prove the high efficiency of the spline approximation, while in the case of aberration-free collimating lens it is preferable to use polynomial regression.