Objective Diffraction can affect beam propagation. To reduce its influence, people have tried many means to search for a nondiffracted beam. Airy beam, featuring self-acceleration and self-healing, is a typical nondiffracted beam discovered in exploration. Unfortunately, it carries infinite energy. Later, an Airy beam with finite energy is obtained through truncation in practice. Airy-Gaussian beam is modulated by finite energy Airy beam, ranging from Airy beam to Gaussian beam by adjusting the distribution factor, which is convenient for research. Based on the nonlinear Schrödinger equation, researchers have studied the propagation of Airy-Gaussian beams in various nonlinear media such as Kerr medium, strongly nonlocal nonlinear media, photorefractive media, and obtained many intriguing phenomena.

Following the discovery of the nonlinear Schrödinger equation, the fractional Schrödinger equation, which is proposed in the category of quantum mechanics, is discovered. Longhi introduces it into optics, which sparks widespread interest and promptes a series of researches. The propagation and interaction characteristics of Airy and Gaussian beams have been extensively studied within the framework of the fractional Schrödinger equation. However, little research has been conducted on the Airy-Gaussian beam. Therefore, studying the propagation of the Airy-Gaussian beam modulated using the fractional Schrödinger equation is necessary.

The split-step Fourier method, considering that diffraction and nonlinearity act independently when the transmission distance is very small, is one of the most common methods to solve the nonlinear Schrödinger equation. Therefore, the transmission process is calculated in two steps, the influence of diffraction effect and nonlinearity effect is considered respectively, and the transmission result of the beam is obtained finally.

Methods The fractional Schrödinger equation model is used in this paper to study the interaction of dual Airy-Gaussian beams in the Gaussian potential and the effect of various parameters on the propagation process, including distribution factors, Lévy index, and barrier parameters, is thoroughly examined. The interaction process of dual Airy-Gaussian beams in the Gaussian potential is periodic.

Results and Discussions We first consider the effect of the Lévy index and the potential barrier’s position x₀ on the interaction process. With the increase of α, the diffraction effect becomes stronger, and the splitting phenomenon gradually disappears, accompanied by the larger angle between the two main lobes and the decreasing transmission period. When α takes a certain value, the energy exchange of the splitting sub-beams occurs after the collision. The position of the potential barrier will affect the propagation period, performing that when x₀ increases, the evolution period of the beam becomes larger (Fig. 1). When χ₀ is small, the interval parameter B has a certain influence on the period, and with its increment, the effect of the Airy-Gaussian beam on the period decreases, even disappears finally. And the number of peak points varies with the value of B at the same transmission distance (Fig. 2). In addition, when B is assigned different values, with the increase of the transmission distance, the number of peaks changes accordingly, whose number and intensity are always completely symmetric with x=0 in the transmission process (Fig. 3). In the case of in-phase and out-phase, the energy distributions of the beam are symmetric about the center axis. While energy transfer occurs in other phase conditions, the energy distributions are no longer symmetric (Fig. 4). When retaining other parameters unchanged, transmission/reflection ratio and period of beam interaction can be controlled by potential barrier depth p and potential barrier width d₀, both of which can decrease the transmission period. Simultaneously, the transmission of the beam in the wall of the potential barrier is weakened, whereas the reflection is enhanced (Fig. 5).

Conclusions The interaction of dual Airy-Gaussian beams in the Gaussian potential is studied using the split-step Fourier method in this paper, which is based on the fractional Schrödinger equation. The results show that the interaction between the two Airy-Gaussian beams in the Gaussian potential is periodic and the period can be changed by adjusting the potential barrier parameters. The increase of the potential barrier width and potential depth leads to the decrease of the period, whereas the change of the barrier position leads to the increase of the period. The depth and width of the potential barrier not only affect the periodic variation, but also affect the reflection and transmission of beams. With the increase of the depth and width of the potential barrier, the Gaussian potential’s reflection effect on the beam is enhanced, the transmission is weakened, and the beam reaches total reflection. The Lévy index mainly affects the splitting and diffraction of the beam. With the increase of the Lévy index, the splitting phenomenon gradually disappears, and the diffraction effect is strengthened. When the Lévy index increases to a certain value, chaos will appear after a propagation period. The distribution characteristics of the Airy-Gaussian beam can be adjusted by changing the distribution factor. When the distribution factor is small, the interval parameter can affect the beam period. In addition, the interaction between interval parameter and relative phase will affect the intensity distribution of beam in the transmission process. In the case of in-phase and out-phase, the energy of two beams remains symmetrical in the propagation process, but when a relative phase is π/2 or -π/2, the energy is no longer symmetrical and the energy transfer phenomenon appears. Changing the interval parameter also affects the direction of energy transfer. Our findings can be used to control the propagation direction of a light beam and the number of light beams generated. They have potential applications in optical switches, splitters, and other fields.