Freeform Lens Design for Scattering Data with General Radiant Fields

We show the existence of a lens, when its lower face is given, such that it refracts radiation emanating from a planar source, with a given field of directions, into the far field that preserves a given distribution of energies. Conditions are shown under which the lens obtained is physically realizable. It is shown that the upper face of the lens satisfies a pde of Monge-Ampère type.