Propagation Properties of Laguerre–Gaussian Beams with Three Variable Coefficient Modulations in the Fractional Schrödinger Equation

This article investigates the transmission characteristics of Laguerre–Gaussian (LG) beams under cosine modulation, power function modulation and linear modulation based on the variable coefficient fractional Schrödinger equation (FSE), respectively. In the absence of modulation, the LG beam undergoes diffraction-induced expansion as the transmission distance increases, with the degree of spreading increasing with a rising Lévy index. Under the cosine modulation, the evolution of the beam exhibits a periodic inversion, where the higher modulation frequency leads to a shorter oscillation period. The oscillation amplitude enlarges with a higher Lévy index and lower modulation frequency. When taking a power function modulation into account, the beam gradually evolves into a stable structure over propagation, with its width broadening with a growing Lévy index and modulation coefficient. In a linear modulation, the propagation of the LG beam forms a “trumpet-like” structure due to an accelerated diffraction effect. Notably, the transmission of the beam is not affected by the radial and azimuthal indices, but its ring number and phase singularity are changed correspondingly. The beam behaves in a similar evolutionary law under different modulations when the Lévy index is below 1. These findings offer valuable insights for applications in optical manipulation and communication.