Propagation dynamics of the Hermite–Gaussian beam in the fractional Schrödinger equation with different potentials

We have studied the propagation dynamics of the Hermite–Gaussian (HG) beam in the fractional Schrödinger system with linear, parabolic, and Gaussian potentials. The results show that the splitting of the beam without an external potential is influenced by the Lévy index. The splitting phenomenon disappears and a periodic evolution of the HG beam occurs when a linear potential is added to the equation. A shorter evolution period is shown with a larger linear potential coefficient, and its sign affects the laser beam’s deflection direction. The transverse amplitude of HG beams is proportional to the Lévy index. When taking into account a parabolic potential, the beam exhibits an autofocus effect during propagation. For a larger Lévy index, the focusing speed gets faster and the focal intensity is weakened. In addition, the transverse amplitude is smaller and the focusing speed is faster with a larger parabolic potential coefficient. In a Gaussian potential, the diffraction effect of the beam grows more pronounced as the Lévy index increases, which leads to a chaotic phenomenon in the beam. The propagation of HG beams is controlled by regulating the Gaussian potential height, potential width, and position of the potential. It is also found that the total reflectivity of the Gaussian potential barrier is stronger than the potential well for the same parameters. These features are significant for applications of optical communications, optical devices, and laser design.