Freeform lens design for scattering data with general radiant fields (Q1745398)

The authors prove the existence of a lens that refracts radiation which emanates from a planar source for a given set of directions. The surfaces constructed by the authors are free form and not necessarily rotationally symmetric. Precise conditions are given for which the lens is realizable. A general theory is developed to solve an illumination problem with two refracting surfaces, a planar source of radiation, and an arbitrary incident field. Aleksandrov solutions to this problem are also discussed. Finally, the upper face of the constructed lens is shown to satisfy a nonlinear partial differential equation of Monge-Ampère type.