Ray-Transformation Conditions and Shaped-Reflector Solutions for Three-Reflector Optics Without Cross Polarization

Reflector synthesis in geometric optics is extended to three-reflector systems producing field transformations that are spherical-to-planar in wavefront, of a prescribed form in power pattern and zero in cross polarization. Necessary and sufficient conditions for the existence of solutions are derived in terms of the input-to-output ray transformation and the surface of either the input or output reflector. A subset of the conditions, for a class of non-unique solutions, is interpreted geometrically in terms of the principal directions of the ray transformation and the lines of field polarization and wavefront curvature. These conditions are then used to derive integral solutions for offset-fed reflectors with any prescribed rotationally-symmetric power-pattern transformation. The reflector solutions are shown to exist for a continuous range of offset parameters, take the form of congruences of ellipses and include a special case where the main reflector is paraboloidal. Low-aberration solutions satisfying Abbe's sine condition are obtained in terms of standard functions. Illumination-enhancing solutions illustrate a possible application to compact-range illuminators for antenna measurements.