Theoretical study on the diffraction-based generation of a 2D orthogonal lattice of optical beams: physical bases and application for a vortex beam multiplication

Abstract

A comprehensive theoretical study on the generation of a 2D orthogonal lattice of optical beams based on the near-field diffraction and Talbot effect is presented. First we investigate the near-field diffraction of an optical beam with a finite lateral extension from an infinite 2D orthogonal grating. It is shown that the resulting diffraction patterns over the Talbot planes depend on the following parameters: the period and opening ratio (OR) of the grating, wavelength and spatial spectral bandwidth of the incident beam, and the propagation distance. In terms of these parameters, we find multiplication conditions: the certain conditions under which a 2D orthogonal lattice of the Fourier transform of the incident beam is generated on the Talbot planes. Therefore, if the incident beam is Fourier-invariant and all the established multiplication conditions are fulfilled, the intensity profile of each of the individual Talbot images resembles the intensity profile of the incident beam. We consider the Laguerre–Gaussian beams having zero radial index as an important class of the vortex beams. We explicitly show that these beams are Fourier-invariant and we calculate their spatial spectral bandwidth. As a result, in the illumination of a 2D orthogonal binary grating with this kind of vortex beam, a 2D orthogonal lattice of the incident optical vortex is generated at the Talbot planes. Considering the obtained multiplication conditions, for the first time, to our knowledge, we determine a multiplication interval. This interval covers the propagation distances at which the vortex beam multiplication occurs. Moreover, we obtain the maximum possible value of the grating’s OR for the realizations of the vortex multiplication. It is shown that both the multiplication interval and the maximum value of the OR depend on the topological charge (TC) of the incident beam. With the aid of some practical examples and defining a multiplication quality factor, the mentioned results are verified quantitatively. In addition to the vortex beam multiplication effect, we consider another interesting phenomenon that results from the interference of the grating’s first diffraction orders. We call this phenomenon the first diffraction orders interference (FDOI) effect. We show that both the multiplication and the FDOI effects occur simultaneously but at different propagation distances. It is also shown that the multiplication and FDOI intervals separate and distance from each other by increasing the TC of the incident beam.