Supporting quadric method in optical design of freeform lenses for illumination control of a collimated light

The illumination control (IC)   problem is an inverse problem concerned with determination of a refracting lens transforming an incident light beam into a beam illuminating a given target set with a prescribed radiation intensity distribution. It is solved here in the geometrical optics approximation for the case when the incident light beam is collimated and its total intensity is equal to the total intensity prescribed on the target. The normalized refraction index of the lens to be determined is a given constant n>1n>1. The input intensity is assumed only to be nonnegative and integrable and the output intensity distribution is assumed to be a positive Radon measure. The problem is formulated in weak form as an equation in measures. Under physically motivated assumptions it is shown that this equation has a solution which is a convex function defining a lens solving the IC problem. The solution is not unique. In contrast to the usual approaches in optical design, no a priori assumptions of any symmetry on the lens or data are made. The overall approach is based on the supporting quadric method [17] which allows us to obtain a solution to the IC problem as a limit of a sequence of specially constructed and physically meaningful discrete approximations.