Conformal Mapping and Complex Coordinates in Cassegrainian and Gregorian Reflector Antennas

In a Gregorian or Cassegrainian reflector antenna, the complex coordinate u' of an output ray is related to the corresponding input coordinate u by a bilinear transformation, u' = (au + b)/(cu + d). We discuss the properties of this transformation, derive its coefficients a, b, c, d, and give explicitly the conditions that must be satisfied in order that symmetry be preserved. The conditions are expressed directly in terms of the parameters that specify the path of the principal ray, which is the ray corresponding to the feed axis. The results are directly related to well-known properties of stereographic projections, and they are shown to be useful in the design of multi reflector antennas which minimize aberrations and cross-polarization.