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The radiation characteristics of conical horns loaded with dielectric cones are investigated. The horn radiates low cross-polarisation over a wide band of frequencies with cones made of foam or solid dielectrics. They make high performance feeds for reflector antennas. A modal-matching technique is used to investigate the behaviour of the higher order modes excited along the horn that influence the radiated cross-polarisation. A parametric study is described to assess the influence of the aperture and cone diameters, the flare angle, and the permittivity of the dielectric.