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A design method for a horn-reflector antenna, which has a beam of arbitrary cross section, is developed. To show the effectiveness of the design theory, the measured data of an electrical model of Japan's domestic communications satellite antenna of this type are shown in comparison with calculated results. The design theory is based on geometrical optics, and is applicable not only to a horn-reflector antenna, but also to any reflector antenna. At the first step of the design theory, the shape of the wavefront near the aperture is determined to correspond to desired beam shape. The shape of the reflector is then determined by the law of the optical path.