Controllable autofocusing properties of Hermite-Gaussian vortex beams with quadratic phase modulation in fractional systems by two kinds of modulations

Abstract The autofocusing properties of Hermite-Gaussian vortex beams (HGVBs) with quadratic phase modulation (QPM) are studied using a split-step Fourier transform algorithm within fractional systems characterized by variable coefficients and potentials. Initially, the autofocusing behavior of HGVBs in free space driven by QPM is analyzed, revealing that the focusing characteristics are influenced by the Lévy index, topological charge, and QPM coefficient. The propagation dynamics are then examined under varying diffraction modulations. Cosine modulation induces phenomena such as oscillation, periodic evolution with single or dual focusing, with the focusing ability and period decreasing as modulation frequency rises. Under power function modulation, the beam stabilizes after focusing, with the focusing time and stabilization structure width governed by the modulation coefficient. The influence of external potentials is explored, where a parabolic potential induces self-focusing and defocusing, with the focusing effect being dependent on both the Lévy index and parabolic coefficient. The focusing characteristics of the beam in the frequency domain are opposite to those in the spatial domain. In an annular potential, the beam exhibits periodic reflection, ultimately evolving into chaos at higher Lévy indices. The focusing dynamics is also investigated when the Lévy index is less than 1. These findings highlight new opportunities for optical switching and manipulation.