Fresnel and Fraunhofer diffraction of (l,n)th-mode Laguerre-Gaussian laser beam by a fork-shaped grating

We study theoretically the problem of diffraction of a Laguerre-Gaussian (LG) laser beam with radial mode number n and azimuthal mode number l, by a fork-shaped grating (FG) with integer topological charge (TC) p. The diffracted wave field amplitude and intensity are calculated at any distance behind the FG and in the back focal plane of a convergent lens. The zeroth diffraction order is obtained as an (l,n)th-mode LG beam. The higher, mth diffraction order beam is described in the radial direction through a product of the Gauss-doughnut function of order by the finite sum of hypergeometric Kummer functions. It can be a vortex beam with increased or reduced TC compared to that of the incident beam, or it can be a non-vortex beam. The obtained results are specialized for two particular cases: when the incident LG beam is with zeroth radial mode number and azimuthal mode number l, and when the incident beam is with zeroth azimuthal mode number and radial mode number n. The presented research results can find interest in optical trapping experiments, fibre-optic multiplexing and quantum information processing.