Tailored freeform surfaces for illumination with extended sources

Tailoring freeform surfaces for illumination with point sources is an established method to generate near-arbitrary illuminance distributions with high detail using a single freeform surface. Real, non-laser sources are not point sources, however. Replacing a point source in a tailored freeform system with an extended source blurs the point-source distribution. Accordingly, extended sources are often treated as a perturbation within a pointsource tailoring algorithm, which may work well for small sources. However, important applications like street lighting, wall-washing and automotive front lighting require some peak intensity in certain directions, which can only be achieved with a given source luminance when nearly the whole aperture contributes to that peak intensity. Then, the tenet of point source tailoring – a one-to-one relation of surface points to target points – breaks down. In this paper, we embrace extended sources instead of treating them as a perturbation: Illuminance at several target points is computed by integrating the luminance of the virtual source image over its finite projected solid angle in a noise-free, non-Monte-Carlo way. The freeform surface is parametrized; the shape of the distorted virtual source image, and thus the illuminance at the target points, becomes a function of the freeform surface parameters, leading to a system of nonlinear equations, which we solve iteratively. Using a solved-example problem, we also give first answers to whether solutions exist and are unique.