A puzzle in reflector shaping

The aerospace community has been using the commercially available software POS, produced by TICRA, to design shaped reflector antennas for many years. The program uses physical optics (PO) reflector analysis with an optimization subroutine developed by mathematicians. There are two convergence issues in this approach - one in the optimization subroutine and the other in the reflector surface. The convergence of the reflector surface is still an art form because the cost function involving the system trade-off is often not well defined. Even if the cost function is well defined, the antenna designer must trade off between the performance and CPU time. Thus, the reflector shaping solution can only be classified as (i) an engineering acceptable solution, and (ii) a cost effective (no further iteration "seems" necessary) solution. Many shaped reflectors have been designed and fabricated, and they work quite well, indeed. In spite of this fact, many puzzles remain. The paper looks at some of these questions and provides some answers.