A Monge–Ampère problem with non-quadratic cost function to compute freeform lens surfaces

In this article, we present a least-squares method to compute freeform surfaces of a lens with parallel incoming and outgoing light rays, which is a transport problem corresponding to a non-quadratic cost function. The lens can transfer a given emittance of the source into a desired illuminance at the target. The freeform lens design problem can be formulated as a Monge–Ampère type differential equation with transport boundary condition, expressing conservation of energy combined with the law of refraction. Our least-squares algorithm is capable to handle a non-quadratic cost function, and provides two solutions corresponding to either convex or concave lens surfaces.