Design of pairs of reflectors

Given a 3D reflecting surface G, which is not necessarily rotationally symmetric, away from a point source, and a direction e, we design a reflecting surface F so that the system (G,F) reflects all rays into the direction e. We also solve the near-field problem when the direction e is replaced by a given point R. The surface F satisfies a system of first-order partial differential equations that can be explicitly solved in terms of G and e. We also apply the same ideas to designing refracting plates, of which the Schmidt corrector plate is an example.