The objective of this study is to investigate the propagation dynamics of finite-energy Airy beam (FEAB) at the interface between a linear dielectric and a nonlinear medium in the fractional Schr?dinger equation (FSE). This research aims to explore the influence of system parameters, such as the Lévy index, nonlinear diffusion coefficient, guiding parameter, incident angle, and initial beam amplitude, on the propagation characteristics of FEAB. The aim is to enhance the current understanding of light-matter interactions in composite media and provide guidance for optical device design.

A split-step Fourier method is employed to numerically solve the nonlinear fractional Schr?dinger equation (NLFSE) for FEAB propagating at the interface between linear and nonlinear media. The model incorporates the fractional diffraction effect by varying the Lévy index (1≤α≤2) and considers key parameters, including the nonlinear diffusion coefficient μ, guiding parameter p, incidence angle v, and initial beam amplitude A. The Airy beam is launched from the nonlinear medium (x<0) with a specific transverse position relative to the interface. The beam evolution is explored systematically with parameter variations under both standard (α=2) and fractional diffraction regimes.

The results reveal that the transverse oscillation of FEAB at the interface strongly depends on the initial incident position and Lévy index. For example, when the incident position d increases from negative to positive, the oscillation period first decreases and then increases (Fig. 2), showing a nonlinear dependence on d. Under fractional diffraction conditions, a critical Lévy index is observed at α=1.4?1.5. Below this threshold, the beam forms a linear propagating breathing soliton, while above this threshold, transverse oscillation of localized waves is maintained. The nonlinear diffusion coefficient μ significantly affects FEAB propagation in the standard diffraction regime (α=2). Increasing μ enhances transverse oscillation and reduces the oscillation period (Fig. 4). The waveguide parameter p induces transverse oscillation and slightly shortens the oscillation period. The incidence angle v controls oscillation amplitude and period, where larger absolute values of v lead to higher oscillation amplitudes and longer periods. Lastly, the initial beam amplitude A directly influences the beam’s focusing effect, increasing localized beam intensity, reducing beam width, and shortening the transverse oscillation period.

This study provides a comprehensive analysis of the propagation dynamics of FEAB at the interface between linear and nonlinear media under both standard and fractional diffraction regimes. The results demonstrate the critical role of the Lévy index in determining the beam evolution mode: below a threshold value (α=1.4?1.5), the beam forms a linear propagating breathing soliton, while above the threshold, transverse oscillation is maintained. System parameters such as the nonlinear diffusion coefficient, guiding parameter, incident angle, and initial beam amplitude provide effective means to control the transverse dynamics of FEAB. These findings offer valuable theoretical insights into the active control of beam propagation at complex interfaces and hold potential applications in optical device design and optical switching.