Offset dual reflector synthesis as a boundary-value problem

It is shown that the synthesis of offset dual reflectors under the assumptions of geometrical optics can be formulated using mathematically exact equations and numerical solutions in a number of important cases. The exact formulation is given, and the existence is demonstrated of numerical solutions to a certain problem. In this problem, rays within a symmetrical feed cone with a tapered power pattern are mapped through two reflections onto a circular output aperture over which the phase is uniform and the power is arbitrarily specified. The method outlined is shown to be capable of solving a number of problems hitherto thought to be insoluble except by approximate methods.