$$$Tailoring freeform lenses for illumination
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INTRODUCTION
Lens design is mostly associated with imaging optics. In imaging design all light rays originating in one point need to be transferred to one single point in the image plane. In contrast, illumination design “only” attempts to produce a predefined power distribution of light on a specified target surface from a light source of known intensity distribution. In illumination design it is mostly undesirable to form an image of the source on the target because this would lead to very high intensities. At present the most widely used method for illumination design is reverse engineering based on ray tracing. The idea is to minimize the difference between the desired distribution and the distribution of light predicted by Monte Carlo ray tracing by varying the reflector shape. This approach has its merits for relatively simple desired distributions. However, if the desired distribution is rich in detail reverse engineering suffers from two basic shortcomings: 1) the objective function to be minimized, i.e. the difference between desired and predicted distribution is blurred by stochastic error inherent in Monte Carlo procedures. 2) Optical performance tends to be a complicated function of the shape of the optical surfaces involved in so far as the observed power distribution depends very strongly and nonlinearly on the shape. Thus typically the objective function exhibits numerous local minima. It may be very difficult to find the global optimum from this multitude of local minima. In this contribution we present a constructive method based on solving a differential equation to tailor the shape of an optical surface such that it produces the desired distribution. Tailoring has been demonstrated in previous contributions for extended sources in a two-dimensional context [1, 2]. 2D tailoring of extended sources is based on the fact that there are precisely two edge rays in each reflector point. In the present contribution we extend the idea of tailoring to freeform surfaces, without the restriction of symmetry, for smalle general problem to find the shape of a refractive surface such as to produce a desired brightness distribution on a given target surface from a known point source leads to a boundary value problem with an elliptic partial differential equation of the Monge-Ampere type. This equation has been described and analyzed in the literature. The purpose of our contribution is to present a venue for a numerical solution as well as several solved examples. The essence of our algorithm for a numerical solution appears to be the explicit incorporation of the condition for the existence of a pseudopotential for a normal vector field.
Tailoring freeform lenses for illumination
Harald Riesa,b and Julius Muschawecka Optics & Energy Concepts, b Department of Physics, University of Marburg, Germany http://www.physik.uni-marburg.de
