Unveiling the link between fractional Schr\"odinger equation and light propagation in honeycomb lattice

TL;DR

This paper reveals a connection between the fractional Schr"odinger equation and light propagation in honeycomb lattices, demonstrating similar behaviors like conical diffraction, and discusses how potentials can disrupt this link.

Contribution

It establishes a novel link between the fractional Schr"odinger equation and light behavior in honeycomb lattices using the Dirac-Weyl equation, highlighting the effects of perturbations.

Findings

Gaussian beam propagation shows similar behavior in FSE and HCL

The connection breaks when an additional potential disrupts lattice periodicity

Conical diffraction is observed in both FSE and HCL simulations

Abstract

We establish a link between the fractional Schr\"odinger equation (FSE) and light propagation in the honeycomb lattice (HCL) - the Dirac-Weyl equation (DWE). The fractional Laplacian in FSE causes a modulation of the dispersion relation of the system, which in the limiting case becomes linear. In the HCL, the dispersion relation is already linear around the Dirac point, suggesting a possible connection with the FSE. Here, we demonstrate this connection by describing light propagation in both FSE and HCL, using DWE. Thus, we propagate Gaussian beams according to FSE, HCL around the Dirac point, and DWE, to discover very similar behavior - the conical diffraction. However, if an additional potential is brought into the system, the link between FSE and HCL is broken, because the added potential serves as a perturbation, which breaks the translational periodicity of HCL and destroys Dirac cones in the dispersion relation.