Progress of detecting orbital angular momentum states of optical vortices through diffraction gratings

Optical vortices are a new kind of laser beam and receiving more and more attention currently. The complex amplitudes of optical vortices comprise a helical term exp(ilφ), with l being the topological charge and φ the azimuthal angle. Each photon in optical vortices carries the orbital angular momentum (OAM) with a value of lħ, where ħ is the Planck's constant divided by 2Π. The topological charge l is the eigenvalue of optical vortices, and determines the helical wavefront distribution, thus also known as OAM state. Moreover, such an OAM state can be an infinite integer state. And vortices with various OAM states are orthogonal to each other, making it possible to be employed in high capacity data-transmission system. In addition, the above unique features contribute to their widely applications in lots of areas such as optical tweezers and spanners, rotation detection, quantum entanglement, etc. In these applications, detecting OAM states is basic, and greatly significant. Recently researchers have developed lots of approaches to detecting the OAM states, including the methods of interference, diffraction gratings, metasurface, etc. Of such approaches, the scheme of diffraction gratings is the simplest and most widely used, where one or more diffraction gratings are employed. When optical vortices propagate through such gratings, the OAM states are acquired immediately through capturing and analyzing the distinct OAM-related diffraction patterns. In this review, we focus on the techniques of detecting OAM states through diffraction gratings, which have been demonstrated by our group and other researchers. Some of the main detection gratings, including double-slit, triangle aperture and slit, angular slit, cylindrical lens, graduallychanging-period grating, annular grating, are introduced. In addition, schemes like composite fork grating, Dammann vortex grating and integrated Dammann grating, are presented to detect the OAM state for coaxial multiplexed vortices. Besides diagnosing OAM state, measuring the intensity proportion of each OAM channel, known as OAM spectrum, in multiplexed vortices is also necessary in some cases. Therefore we also introduce the ways to measure the OAM spectrum, e.g. the OAM mode sorter, the gray-scale algorithm.