Exploring the Potential of Cosine Modulations and Parabolic Potentials for Beam Shape‐Trajectory Control in the Fractional Schrödinger System

Abstract This study explores the transmission properties of off‐axis chirped Hermite Gaussian cross‐phase (HGCP) beams in fractional systems under cosine modulations and parabolic potentials. Cosine modulation induces periodic transitions in the light field, influenced by modulation frequency, Lévy index, cross‐phase (CP) coefficient, and Hermite Gaussian (HG) order, creating diverse patterns like rugby balls, water waves, and elliptical rings. The cross‐phase effect causes beam rotation, and diffraction enhances with increasing CP coefficient and Lévy index. Beam trajectories exhibit periodic oscillations, with amplitude growing with the Lévy index and chirp coefficients. Under parabolic potentials, the beam demonstrates autofocusing, defocusing, and periodic mode transitions, including HG to Laguerre‐Gaussian (LG) conversions. The beam propagation exhibits “spiral” and “straight line” oscillatory trajectories moving toward the origin, and periodic elliptical spiral trajectories depend on the Lévy index, off‐axis, and chirp parameters. These findings offer insights into beam dynamics in fractional Schrödinger equation (FSE) systems, with implications for optical communications and particle manipulation.