A superoscillation wave is a special superposition of propagating electromagnetic (EM) waves which varies with sub-diffraction resolution inside a fixed region. This special property allows superoscillation waves to carry sub-diffraction details of an object into the far-field, and makes it an attractive candidate technology for super-resolution devices. However, the Shannon limit seemingly requires that superoscillations must exist alongside high-energy sidebands, which can impede its widespread application. In this work we show that, contrary to prior understanding, one can selectively synthesize a portion of a superoscillation wave and thereby remove its high-energy region. Moreover, we show that by removing the high-energy region of a superoscillation wave-based imaging device, one can increase its power efficiency by two orders of magnitude. We describe the concept behind this development, elucidate conditions under which this phenomenon occurs, then report fullwave simulations which demonstrate the successful, power-efficient generation of sub-wavelength focal spots from propagating waves.