A spherical modal matching technique is developed to analyse and design wide flare angle corrugated conical horns. The theory is verified by experimental measurements. It is found that the junction at the throat of the horn controls the return loss and the higher order mode excitation. The optimum throat geometry depends on the flare angle. For narrow flare angle horns, the first slot should be half a wavelength deep. For horns with semi-flare angles between about 20 degrees and 40 degrees, a short smooth wall conical section gives lowest return loss. For horns of larger flare angle the slot depth should be constant.