Offset Dual Reflector Shaping Based On the Pfaffian Integrability Condition

A rigorous theory for the synthesis of offset dual reflector antennas is presented. The synthesis problem is posed as a three dimensional vector field which becomes the surface normals to one of the reflectors. The ray trace which defines the vector field is calculated from the reflection law at one reflector and a constant path length requirement. Implicit in the definition of the vector field is the mapping from the input wavefront to the output wavefront. The Pfaffian integrability condition is applied to the vector field which leads to a partial differential equation (PDE) for the input to output mapping. Integrating this PDE coupled to the PDE for the shape of the first reflector yields the desired solution. The method allows for a wide choice of the input to output mapping and combined with a judicious choice of geometry produces practical designs which outperform traditional conic section designs. The use of an explicit existence theorem in the problem formulation is innovative and leads to simpler software which give excellent results. Unlike previous work, much of the algebra is done without reference to coordinate frames, simplifying the presentation and relegating algebraic details to symbolic software. A novel method for integrating the coupled PDE's is introduced.