$\mathcal{PT}$ symmetry in a fractional Schr\"odinger equation

TL;DR

This paper explores the effects of $\\mathcal{PT}$ symmetry in a fractional Schrödinger equation, revealing unique band structures and diffraction phenomena with potential applications in advanced optical devices.

Contribution

It introduces the analysis of $\\mathcal{PT}$-symmetric potentials in fractional Schrödinger equations, highlighting new diffraction behaviors and propagation characteristics.

Findings

Linear and symmetric band structure at critical points

Nondiffracting and conical diffraction of beams

Potential for high-power, narrow laser beam generation

Abstract

We investigate the fractional Schr\"odinger equation with a periodic $\mathcal{PT}$-symmetric potential. In the inverse space, the problem transfers into a first-order nonlocal frequency-delay partial differential equation. We show that at a critical point, the band structure becomes linear and symmetric in the one-dimensional case, which results in a nondiffracting propagation and conical diffraction of input beams. If only one channel in the periodic potential is excited, adjacent channels become uniformly excited along the propagation direction, which can be used to generate laser beams of high power and narrow width. In the two-dimensional case, there appears conical diffraction that depends on the competition between the fractional Laplacian operator and the $\mathcal{PT}$-symmetric potential. This investigation may find applications in novel on-chip optical devices.