Dynamics of Hermite–Gaussian beams in the linear and nonlocal nonlinear fractional Schrödinger equations
Zhenkun Wu, Yagang Zhang, Jingmin Ru, Yuzong Gu
Abstract
We analytically and numerically investigated the dynamics of the Hermite–Gaussian (HG) beams in linear and nonlocal nonlinear fractional Schrödinger equations (SEs), in both one and two dimensions. In the linear regime, HG beams on the order of n>1 were observed to undergo an initial compression phase before they split into two sub-beams. The sub-beams were saddle-shaped; the separation between them linearly increased with the propagation distance. Due to their nonlinear propagation dynamics, HG beams quite unexpectedly followed a zigzag trajectory in the real space, which corresponds to a modulated anharmonic oscillation in the momentum space. Although the beam was well localized during propagation, it still broadened and may become unstable. In the two-dimensional case, a superposition of HG beams showed a linear evolution similar to that in the one-dimensional case; however, because of orbital angular momentum, the beams in two dimensions exhibited incomplete rotation during propagation. In the nonlinear regime, the superposed HG input beam first evolved in the real space into a filiform structure and then into an annular structure, with periodic inversion and variable rotation.