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

TL;DR

This paper derives analytical expressions for vortex beams generated by Fresnel and Fraunhofer diffraction of Gaussian beams through fork-shaped gratings, detailing their wave amplitudes, intensity distributions, and vortex radii.

Contribution

It provides new analytical formulas describing vortex beams from fork-shaped gratings, including special cases and their behavior in near and far fields.

Findings

Analytical expressions for vortex beam wave amplitudes and intensities.

Descriptions of vortex radii and diffraction patterns.

Specialization of results for different grating charges and configurations.

Abstract

Expressions describing the vortex beams, which are generated in a process of Fresnel diffraction of a Gaussian beam, incident out of waist on a 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 mp-th order Gauss-doughnut function and a Kummer function, or by the first order Gauss-doughnut function and a 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 field, 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.