Freeform Reflector Design With Extended Sources (2010)

Reflector design stemmed from the need to shape the light emitted by candles or lamps. Over 2,000 years ago people realized that a mirror shaped as a parabola can concentrate light, and thus significantly boosts its intensity, to the point where objects can be set afire. Nowadays many applications require an accurate control of light, such as automotive headlights, streetlights, projection displays, and medical illuminators. In all cases light emitted from a light source can be shaped into a desired target distribution with a reflective surface. Design methods for systems with rotational and translational symmetry were devised in the 1930s. However, the freeform reflector shapes required to illuminate targets with no such symmetries proved to be much more challenging to design. Even when the source is assumed to be a point, the reflector shape is governed by a set of second-order partial non-linear differential equations that cannot be solved with standard numerical integration techniques. An iterative approach to solve the problem for a discrete target, known as the method of supporting ellipsoids, was recently proposed by Oliker. In this research we report several efficient implementations of the method of supporting ellipsoids, based on the point source approximation, and we propose new reflector design techniques that take into account the extent of the source. More specifically, this work has led to three major achievements. First, a thorough analysis of the method of supporting ellipsoids was performed that resulted in two alternative implementations of the algorithm, which enable a fast generation of freeform reflector shapes within the point source approximation. We tailored the algorithm in order to provide control over the parameters of interest to the designers, such as the reflector scale and geometry. Second, the shape generation algorithm was used to analyze how source flux can be mapped onto the target. We derived the condition under which a given source-target mapping can be achieved with a smooth continuous surface, referred as the integrability condition. We proposed a method to derive mappings that satisfy the integrability condition. We then use these mappings to quickly generate reflector shapes that create continuous target distributions as opposed to reflectors generated with the method of supporting ellipsoids that create discrete sets of points on the target. We also show how mappings that do not satisfy the integrability condition can be achieved by introducing step discontinuities in the reflector surface. Third, we investigated two methods to design reflectors with extended sources. The first method uses a compensation approach where the prescribed target distribution is adjusted iteratively. This method is effective for compact sources and systems with rotational or translational symmetry. The second method tiles the source images created by a reflector designed with the method of supporting ellipsoids and then blends the source images together using scattering in order to obtain a continuous target distribution. This latter method is effective for freeform reflectors and target distributions with no sharp variations. Finally, several case studies illustrate how these methods can be successfully applied to design reflectors for general illumination applications such as street lighting or luminaires. We show that the proposed design methods can ease the design of freeform reflectors and provide efficient, cost-effective solutions that avoid unnecessary energy consumption and light pollution.