A beam-steering system having high positional resolution and fast switching speed is disclosed. Embodiments of the beam-steering system comprise a diffraction limited optical system that includes a reflective imager and two controllably rotatable MEMS elements. The optical system is characterized by a folded optical path, wherein light propagating on the path is incident on each MEMS element more than once. Each MEMS element imparts an optical effect, such as angular change, on the output beam. By virtue of the fact that the optical system is multi-bounce optical system, the optical effect at each MEMS element is multiplied by the number of times the light hits that MEMS element.