Conformal Mapping and Complex Coordinates on Cassegrainian and Gregorian Reflector Antennas

Gregorian and Cassegrainian reflector arrangements are needed for ground station and satellite antennas, and terrestrial radio relay systems. 18 In these antennas, a paraboloid of large aperture is combined with a smaller subreflector (an hyperboloid or an ellipsoid). The feed is placed at the antenna focal point, and it illuminates the subreflector with a spherical wave, which is then transformed by the two reflectors into a plane wave. Each input ray from the feed is thus transformed into an output ray parallel to the paraboloid axis. This transformation can be represented by a stereographic projection.9"11 Therefore, it is a conformal mapping--it transforms circles into circles, and it is described by the bilinear transformation 2397 where u is the "complex coordinate" of an input ray and u' the corresponding output coordinate. In this article, we discuss the properties of the bilinear transformation, derive its coefficients a, b, c, d, and give explicitly the conditions that must be satisfied to obtain circular symmetry, in which case b = c = 0, d = 1. (2) The results are related to well-known properties of stereographic projections, and they generalize previous results in Refs. 12 to 18. We first consider, in Section II, an ellipsoid illuminated by a spherical wave front S. We assume that S originates from one of the two foci of the ellipsoid, and determine the properties of a reflected wave front S', assuming geometric optics. We determine for each point P' of S' the corresponding point P of the incident wave front S and show that the correspondence P--* P' is everywhere a conformal mapping.