We present a comprehensive study on designing a large-size horn antenna for radiating gigawatt-level high power microwaves (HPMs) in X-band. The horn taper angle and aperture dielectric window were optimized to achieve high power capacity, low reflection, and small mechanical deformation. It was particularly found that the HPM radiation patterns depend sensitively on the aperture dielectric plate thickness due to double reflections obeying Fresnel theorem. Theoretical analysis suggests that the optimal thickness should be chosen around integral times of half effective microwave wavelength in the dielectric.