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We present a method to efficiently multiply or divide the orbital angularmomentum (OAM) of light beams using a sequence of two optical elements. Thekey-element is represented by an optical transformation mapping the azimuthalphase gradient of the input OAM beam onto a circular sector. By combiningmultiple circular-sector transformations into a single optical element, it ispossible to perform the multiplication of the value of the input OAM state bysplitting and mapping the phase onto complementary circular sectors.Conversely, by combining multiple inverse transformations, the division of theinitial OAM value is achievable, by mapping distinct complementary circularsectors of the input beam into an equal number of circular phase gradients. Theoptical elements have been fabricated in the form of phase-only diffractiveoptics with high-resolution electron-beam lithography. Optical tests confirmthe capability of the multiplier optics to perform integer multiplication ofthe input OAM, while the designed dividers are demonstrated to correctly splitup the input beam into a complementary set of OAM beams. These elements canfind applications for the multiplicative generation of higher-order OAM modes,optical information processing based on OAM-beams transmission, and opticalrouting/switching in telecom.

Multiplication and division of the orbital angular momentum of light with diffractive transformation optics