The optical phased array has been developed to realize solid-state optical beam steering following the advent of silicon photonics. Thus far, its feasibility has lacked either steering quality or optical efficiency, and its optimal design still remains unknown. Herein, we propose a scalable and wavelength-independent phased array design methodology that achieves steering quality and optical efficiency. This novel design methodology is based on a special aperiodicity from number theory of a complete residue system …