Ray mapping approach for the efficient design of continuous freeform surfaces.

The efficient design of continuous freeform surfaces, which maps a given light source to an arbitrary target illumination pattern, remains a challenging problem and is considered here for collimated input beams. A common approach are ray-mapping methods, where first a ray mapping between the source and the irradiance distribution on the target plane is calculated and in a subsequent step the surface is constructed. The challenging aspect of this approach is to find an integrable mapping ensuring a continuous surface. Based on the law of reflection/refraction and an integrability condition, we derive a general condition for the surface and ray mapping for a collimated input beam. It is shown that in a small-angle approximation a proper mapping can be calculated via optimal mass transport - a mathematical framework for the calculation of a mapping between two positive density functions. We show that the surface can be constructed by solving a linear advection Eq. with appropriate boundary conditions. The results imply that the optimal mass transport mapping is approximately integrable over a wide range of distances between the freeform and the target plane and offer an efficient way to construct the surface by solving standard integrals. The efficiency is demonstrated by applying it to two challenging design examples, which shows the ability of the presented approach to handle target illumination patterns with steep irradiance gradients and numerous gray levels.