Exact solutions for lens and reflector shaping

We propose an exact solution technique for lens and reflector shaping, which determines the surface geometry of a reflector or a lens excited by a single feed element based on the specified far field information. Otherwise stated, given a desired far field pattern over a specified solid angle range, a single-reflector/single-feed antenna system can be obtained analytically via a closed form, exact solution. There is no need whatsoever to invoke optimization algorithms. Given a required two-dimensional far field pattern, the surface of the reflector or lens, illuminated by a single feed, is generated via a closed form solution, a solution which satisfies Maxwell's equations and all relevant boundary conditions. For the purpose of demonstration, three reflector antennas, and one lens, each with a single feed, have been constructed based on three different far field contour beam patterns. In each such case, the CPU required to generate a thousand surface points amounted to just a few minutes of real time. Solution verification was carried out by analytic examples, software simulations, and hardware measurement. Four anal3itic examples, including the standard planar, parabolic, elliptical, and hyperbolic reflectors, are presented herein. Also confirmed with commercial software was the performance of a shaped reflector which provides a specific far field contour beam. Additionally, a dish antenna measurement verified the perfect reconstruction of the reflector surface from its measured near field pattern. Excellent correlations were obtained in all demonstration cases.