Structured-light control and spatial breathing of Elegant Hermite High-Order Cosine-hyperbolic Gaussian beams in strongly nonlocal nonlinear media

Abstract

Abstract The propagation behavior of an Elegant Hermite High-Order Cosine-hyperbolic Gaussian beam (EHHOChGB) in a strongly nonlocal nonlinear medium (SNNM) is theoretically investigated. Using the Snyder–Mitchell model in combination with the Huygens–Fresnel integral (Collins form), analytical expressions for the complex amplitude, rms beam width, and curvature radius are derived, enabling a detailed description of beam evolution in both weakly and strongly decentered configurations. The moment method is employed to evaluate the rms beam width and determine the critical power for self-trapping, providing quantitative insight into the transition between diffraction-dominated, solitonic, and over-focusing regimes. Numerical simulations reveal that the EHHOChGB exhibits periodic self-reconstruction governed by the interplay between diffraction and nonlocal self-focusing. At the critical input power, the beam maintains an invariant rms width and curvature, forming a stationary spatial soliton; below and above this threshold, it displays bounded breathing oscillations. The results show that the beam order parity, modulation index, and decentering parameter jointly control the propagation regime: odd Hermite orders generate dark-centered solitons, while even orders form bright breathing modes. Increasing the decentering parameter enhances internal interference and modulation sensitivity, demonstrating that the nonlocal medium acts as a tunable optical potential for structured beams. This study extends the Snyder–Mitchell framework to complex Hermite–cosh–Gaussian families and provides new insights into nonlocal self-trapping, power-dependent beam shaping, and nonlinear optical confinement.