PT symmetry in a fractional Schrödinger equation

Abstract

We investigate the fractional Schrödinger equation with a periodic P T -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 onedimensional 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 P T -symmetric potential. This investigation may find applications in novel on-chip optical devices.

Soliton dynamics in a fractional complex Ginzburg-Landau model

“…The realization of the FSE theory in optical fields provides abundant possibilities for studies of the fractional-order beam-propagation dynamics. Subsequently, the propagation of beams in FSE with different external potentials and nonlinear terms was investigated [13][14][15][16][17]. In this vein, various soliton states based on FSE in Kerr nonlinear media and lattice potentials were reported recently [18][19][20][21][22][23].…”

One-dimensional gap solitons in quintic and cubic–quintic fractional nonlinear Schrödinger equations with a periodically modulated linear potential

“…Since then, interesting results on generating and manipulating linear and nonlinear propagation dynamics of laser beams in such fractional optical models were obtained. Some typical works include: Gaus- * zengjh@opt.ac.cn sian beams either evolved into diffraction-free beams [14] or undergone conical diffraction [15] during propagation without a potential, PT symmetry [16] and propagation dynamics of the super-Gaussian beams [17] , optical beams propagation with a harmonic potential [14,15,17] (which supports spatiotemporal accessible solitons too [18,19]) and periodic potentials [16,20], propagation management of light beams in a double-barrier potential [21], in the context of linear FSE regime; and in terms of nonlinear fractional Schrödinger equation (NLFSE) regime [22][23][24][25][26][27][28], including optical solitons (or solitary waves) without external potential [23,24], solitons supported by linear [25][26][27] and nonlinear [28] periodic potentials which refer, respectively, to optical lattice and nonlinear lattice as described below.…”

Fractional Schrödinger Equation in the Presence of the Linear Potential

“…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]. An optical realization of the space-fractional Schrödinger equation, based on transverse light dynamics in aspherical optical cavities, has been recently proposed in the study [13].…”