Diffraction-free beams in fractional Schr\"odinger equation

TL;DR

This paper explores how Gaussian beams propagate without diffraction in the fractional Schrödinger equation, revealing splitting, conical diffraction, and chirp-dependent deflections, and introduces a Talbot effect for these beams.

Contribution

It provides analytical and numerical analysis of diffractionless Gaussian beams in FSE, uncovering new propagation behaviors and the Talbot effect for such beams.

Findings

1D Gaussian beams split into nondiffracting beams during propagation.

2D Gaussian beams undergo conical diffraction.

Chirp influences beam deflection trajectories.

Abstract

We investigate the propagation of one-dimensional and two-dimensional (1D, 2D) Gaussian beams in the fractional Schr\"odinger equation (FSE) without a potential, analytically and numerically. Without chirp, a 1D Gaussian beam splits into two nondiffracting Gaussian beams during propagation, while a 2D Gaussian beam undergoes conical diffraction. When a Gaussian beam carries linear chirp, the 1D beam deflects along the trajectories $z=\pm2(x-x_0)$, which are independent of the chirp. In the case of 2D Gaussian beam, the propagation is also deflected, but the trajectories align along the diffraction cone $z=2\sqrt{x^2+y^2}$ and the direction is determined by the chirp. Both 1D and 2D Gaussian beams are diffractionless and display uniform propagation. The nondiffracting property discovered in this model applies to other beams as well. Based on the nondiffracting and splitting properties, we introduce the Talbot effect of diffractionless beams in FSE.