Propagation Dynamics of a Light Beam in a Fractional Schrödinger Equation

Abstract

The dynamics of wave packets in the fractional Schrödinger equation is still an open problem. The difficulty stems from the fact that the fractional Laplacian derivative is essentially a nonlocal operator. We investigate analytically and numerically the propagation of optical beams in the fractional Schrödinger equation with a harmonic potential. We find that the propagation of one- and two-dimensional input chirped Gaussian beams is not harmonic. In one dimension, the beam propagates along a zigzag trajectory in real space, which corresponds to a modulated anharmonic oscillation in momentum space. In two dimensions, the input Gaussian beam evolves into a breathing ring structure in both real and momentum spaces, which forms a filamented funnel-like aperiodic structure. The beams remain localized in propagation, but with increasing distance display an increasingly irregular behavior, unless both the linear chirp and the transverse displacement of the incident beam are zero.

Beam propagation management in a fractional Schrödinger equation

“…Dual-Airy states were predicted and the relevant optical implementation was suggested. This work opened a new area of beam dynamics in the FSE and inspired several intriguing studies on the beam propagation in both linear and nonlinear configurations 28–34 .…”

“…investigated the beam dynamics in the FSE with or without an external potential and found series of fascinating features, including the zigzag propagation of chirped Gaussian beam in a parabolic potential 28 , conical diffraction in -symmetry periodic lattices 29 , diffraction-free beams in uniform media 30 , and linear modes trapped in a harmonic-oscillator potential 31, 32 . Note that the experimental setting for the realization of beam propagation in the FSE was proposed very recently 30 .…”

“…Above p cr2 , an oblique input undergoes totally reflection and follows a zigzag trajectory, analogous to ref. 28. When p  ∈ [ p cr1 , p cr2 ], a beam can be partially transmitted and partially reflected.…”

Fractional Schrödinger Equation in the Presence of the Linear Potential

“…It should be noted that, in recent time, the case of α = 1 has attracted attention in the context of the propagation dynamics of wave packets under the use of the space-fractional Schrödinger equation [10,11,13]. In this study, we investigate the time evolution of the Cauchy initial wave packet…”

“…As an example, the fundamental solution to the Schrödinger equation for a free particle can only be written in closed form under consideration of the Fox H-function [9]. Very recently, the space-fractional Schrödinger equation has been employed for studying the propagation dynamics of wave packets in the presence of the harmonic potential [10] as well as of the free particle [11]. In addition, the fractional Schrödinger equation subject to a periodic P T -symmetric potential has been used to investigate the conical diffraction of a light beam [12].…”