Controllable inversion and focusing behaviors of Swallowtail-Gaussian beams in fractional Schrödinger equations
doi: 10.37188/CO.EN-2023-0018
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摘要:
在光场中引入一维燕尾突变函数，利用分步傅立叶方法研究了燕尾高斯（SG）光束在分数薛定谔方程（FSE）中的演化动力学，详细讨论了线性势、抛物线势、高斯势及无势的情况。在无势情况下，SG光束会因群延迟的变化而分裂成两个子光束，并且分裂轨迹会随着Lévy指数的增大出现弯曲。在线性势下，SG光束出现了周期性反转和聚焦行为，Lévy指数和线性势系数分别影响聚焦点峰值强度和反转及聚焦的演化周期，其反转和聚焦周期距离只受线性势影响而与Lévy指数无关。在抛物线势情况下，具有较大Lévy指数的SG光束的主瓣和旁瓣反转和聚焦从杂乱转变为周期性演化，其反转聚焦位置由抛物线势系数和Lévy指数共同决定。在高斯势中，光束的演化在势垒的约束下由于反射主瓣和旁瓣的干扰，窄势垒的周期性反转和聚焦出现杂乱混沌现象，而对于宽势垒，由于旁瓣减弱，周期性演化变得清晰。本文研究结果为利用高阶燕尾光波场实现光调制器和光开关提供了可能。
Abstract:By transferring a one-dimensional swallowtail catastrophe to an optical field, the evolution dynamics of the Swallowtail-Gaussian (SG) beams in fractional Schrödinger equations (FSE) with different potentials, which include the linear, parabolic, and Gaussian potential and non-potential cases, were investigated using the split-step Fourier method. In a non-potential case, the SG beams split into two sub-beams, and their splitting trajectories along straight lines can be curved with a larger Lévy index in FSE. In a linear potential case, periodic inversion and focusing behaviors are found, and a larger Lévy index can strengthen their peak intensities at focusing points and curve trajectories. However, the period distance of inversion and focusing is only affected by linear potentials rather than the Lévy index. In a parabolic potential case, the beams evolve from chaos interference into an apparent period in inversion and focusing of main and side lobes with a larger Lévy index, where the inversion and focusing position are combined and determined by parabolic potential and the Lévy index. In a Gaussian potential case, the evolution dynamics are evidently constrained within potential barriers. In a narrow barrier, the periodic inversion and focusing display chaotic behavior because of the interference of both the reflected main and side lobes. In contrast, the periodic evolution in a wider barrier becomes more prominent owing to the attenuation of the side lobes. The study of the SG beam in FSE offers the possibility of optical modulators and switches through the utilization of the higher-order swallowtail catastrophe wave fields.
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Key words:
- Swallowtail-Gaussian beam /
- Fractional Schrödinger equation /
- Lévy index
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Figure 1. Group delay and intensity split of one-dimensional Swallowtail-Gaussian beam without potentials for different Lévy indexes of α=1, 1.4, 1.8, and 2. White dotted lines: the evolutions of the main lobe
Figure 2. (Color online) Intensity and spectral distributions of SG beams with a linear potential for different Lévy indexes α and linear potential coefficients a.
Figure 3. Peak intensities and their positions of SG beams with a linear potential for different Lévy indexes α and linear potential coefficients a during propagation. (a) a = 8, (b) a = 4, (c) the peak intensity (Max) of the longitudinal coordinate in transmission varies with the change of the Lévy indexes α(a = −4, +4, −8, +8)
Figure 4. Intensity evolution of the SG beams with parabolic potential for different Lévy indexes and parabolic potential coefficients. White solid lines in (j)−(l): evolution trajectories of main lobes; Red and white dotted circles: inversion and focus positions of main lobes
Figure 5. (a) Peak intensities of SG beams for different parabolic potential coefficients during propagation; (b) peak intensity in reversal positions versus parabolic potentials
Figure 6. The evolution dynamics of SG beams for different Gaussian potentials and Lévy indexes. (a)−(c): α=1, w0=1; (d)−(f): α=1, x0=20; (g)−(i): x0=20, w0=1
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