A proposed dual-reflector feed for the spherical reflector antenna in Arecibo is presented. This is analyzed over a large frequency range: at the lower frequencies by physical optics (PO) integration, and at the higher ones by a geometrical optic (GO) ray tracing technique described in another work. The latter calculations are extended with the transition region theory (TRT) to include edge diffraction. The results clearly demonstrate the usefulness of the time efficient TRT method. However, they also show that PO integration is important, as this has detected an underillumination of the central region of the aperture. This effect is related to a similar problem with the line feeds, but can in the present case be reduced by moving the subreflectors away from the paraxial focus.