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Some theoretical generalizations are given of two-reflector, rotationally symmetric microwave antennas fed by a plane wave across the feed aperture. So far as geometrical optics apply, the proposed designs offer 1) no reflection of energy back into the feed and 2) arbitrary illumination over the antenna aperture. The general solution is exhibited in terms of quadratures, and the reflector shapes for some simple cases are worked out in detail and plotted. Two families of antennas are found which image the feed aperture onto the secondary reflector. These antennas might be expected to have low spillover losses due to diffraction, but none of them is of practical proportions. It is shown in general that such imaging is incompatible with efficient illumination of the secondary aperture.