Fresnel and Fraunhofer diffraction of a Gaussian laser beam by fork-shaped gratings

Abstract

Expressions describing the vortex beams that are generated by the process of Fresnel diffraction of a Gaussian beam incident out of waist on fork-shaped gratings of arbitrary integer charge p, and vortex spots in the case of Fraunhofer diffraction by these gratings, are deduced. The common general transmission function of the gratings is defined and specialized for the cases of amplitude holograms, binary amplitude gratings, and their phase versions. Optical vortex beams, or carriers of phase singularity with charges mp and -mp, are the higher negative and positive diffraction-order beams. The radial part of their wave amplitudes is described by the product of the mpth-order Gauss-doughnut function and a Kummer function, or by the first-order Gauss-doughnut function and the difference of two modified Bessel functions whose orders do not match the singularity charge value. The wave amplitude and the intensity distributions are discussed for the near and far fields in the focal plane of a convergent lens, as well as the specialization of the results when the grating charge p=0; i.e., the grating turns from forked into rectilinear. The analytical expressions for the vortex radii are also discussed.

Fraunhofer diffraction of a Laguerre–Gaussian laser beam by fork-shaped grating

“…D is period of the rectilinear grating and plays the same role for the FSG far from its pole. While, the specifications of the transmission coefficients depend on the type of the grating, and we define them as in [17] The performed integration over the azimuthal variable ϕ leads to the following expression…”

Effects of aberrations in vortex-beams generated with amplitude diffraction gratings

“…Traditional description of OVs was based on Laguerre-Gauss polynomials, 25, 26 but they do not completely describe OVs generated from diffraction gratings. In recent years, the study of OVs created from diffraction gratings has turn its direction into this more accurate description based on Kummer functions 19,20,27 instead of traditional Laguerre-Gauss polynomials.…”

Design of a compact setup to generate and test optical vortex beams

“…A possible way for obtaining optical vortices is based in the use of fork like holograms, as has been widely studied [7][8][9][10][11][12][13][14]. In order to test the vortices an interferometric set-up is required.…”