An analytical Gaussian beam model for the radiation field of a corrugated horn antenna is presented. The model describes how the beam width and the location of the phase centre depend on the dimensions of the horn, and how they vary as a function of the distance from the horn aperture to the observation point. The model includes both the near-field and the far-field of the horn. These properties make the Gaussian feed model much better, for use in general work on reflector antennas, than the conventional cosnθ pattern. The Gaussian model is also used to define aperture-controlled and flareangle controlled horns, depending on whether the radiation patterns are frequency-sensitive or not.