Binary phase correctors cannot improve laser beam quality

Numerous publications have proposed using binary phase correctors to convert the near-field beams from higher-order-mode lasers or anti-phased laser arrays into far-field beams with a single strong on-axis lobe, and the use of phased arrays in general to obtain high power beams with good far-field properties. The counter-intuitive fact is, however, that within a broad range of reasonable assumptions binary phase correctors cannot improve the beam quality of laser beams at all-at least not the quality factor M² as usually defined based on second-order moments. Even from a "power in the bucket" viewpoint the output beam from an optimally phased or phase-corrected laser array may have substantially worse far-field beam properties than one would initially expect