On the theory of the synthesis of offset dual-shaped reflectors-case examples

In an earlier paper by V. Galindo-Israel et al. (see ibid., vol.AP-35, p.887-96, August 1987) the geometrical optics (GO) principles, constraints, and requirements of the dual- and single-offset-shaped reflector synthesis problem were collected and developed into a set of nonlinear first-order partial differential equations (PDEs). Methods of solving these PDEs numerically were illustrated, as were certain problems that may arise. An extension of the methods by which solutions to the PDEs can be obtained is presented, together with several case examples. These examples are independently analyzed by GO and physical optics diffraction methods. The starting point for the integration over each reflector can be taken on the outer rim, at the center, or at an intermediate point-the intermediate starting point being the more general case. The utility of the speed of this synthesis method is demonstrated. This makes practical the incorporation of the synthesis into a search algorithm that can optimize one or more parameters of the reflector system. As an example, the optimization of the mapping equations for low cross polarization is discussed