Algebra of light: multiplication and division of orbital angular momentum

A novel type of diffractive optics is presented to perform the multiplication and division of the orbital angular momentum (OAM) of light beams in an efficient and compact way. The technique is based on the combination of circular-sector transformations, engineered to split and map conformally the azimuthal phase gradient of OAM beams onto complementary circular sectors. The optical elements have been fabricated with high-resolution electron-beam lithography and tested on an optical bench in the visible range in order to prove the designed optical operations on OAM beams.