We demonstrate that very high performance, extremely compact, scalar corrugated feedhorns can be designed and constructed by optimizing the excitation and phasing of the HE11, HE12, and HE13 modes near the throat of the horn while limiting excitation of higher order modes. We present the design and measurement of two families of dual-profiled horn, both with a directivity of 20 dBi that couple with very high efficiency to a fundamental Gaussian mode. The first was optimized for sidelobe performance and features sidelobes approaching -60 dB for a horn length of only 15.6 lambda. The second was designed to minimize horn length and to achieve sidelobe levels below -35 dB for a horn that is only 4.8. long. The horns exhibit excellent coupling to the fundamental free-space Gaussian mode, with LG(00) power coupling of 99.92% and 99.75%, respectively. We demonstrate excellent agreement between simulation and experiment at 94 GHz and simulate the performance over a 20% bandwidth. High-performance compact scalar horns are of interest because they reduce manufacturing risk at high frequencies, and reduce size and weight at lower frequencies, which can be important in horn arrays and space applications, where horn arrays often have serious weight and size restrictions.