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A step horn structure is presented that yields about 90% aperture efficiency with low cross-polar radiation. It is found that for obtaining high aperture efficiency, the horn aperture should consist of only the TEz type of modes in appropriate amplitudes and phases. The desired TEz modes are produced using multiple steps in the horn walls. The distances between the steps are critical to suppress the undesired TMz modes. Radiation characteristics of high efficiency circular and square horns are presented and potential applications of such high efficiency horns are discussed.