分数衍射效应下余弦高斯光束传输特性的研究

Research on Propagation Characteristics of Cosine-Gaussian Beam under Fractional Diffraction Effect

基于变系数的分数阶导数薛定谔方程,采用分步傅里叶算法研究了余弦高斯光束的传输和相互作用。在线性情况下,单束无啁啾余弦高斯光束在传输过程中分成两束相同的高斯光束并形成Y形光束结构,而啁啾余弦高斯光束在传输过程中产生两束不对称的光束。当莱维指数为1时,两束啁啾余弦高斯光束相互作用之后形状保持不变。此外,余弦高斯光束在周期函数、线性函数和幂函数形式调制的分数衍射作用下,其传输结果分别展现出周期性的结构、抛物轨迹传输以及形状不变的传输。在非线性情况下,饱和非线性效应破坏余弦高斯光束的周期性结构,同时抑制余弦高斯光束的传输展宽。

Based on the fractional derivative Schr?dinger equation with variable coefficients,the propagation and interaction of cosine-Gaussian beams are studied by means of the split-step Fourier algorithm.In the linear case,a single non-chirped cosineGaussian beam splits into two identical Gaussian beams during transmission and forms a Y-shaped beam structure,while a chirped cosine-Gaussian beam generates two asymmetric beams during transmission.The value of chirped parameter affects the bifurcation of the cosine-Gaussian beams.The two branches of the cosine-Gaussian beams are asymmetrically distributed as the chirp parameter increases.One branch of the cosine-Gaussian beams disappears completely when the value of the chirp parameter is up to 1.5.When the Lévy index is 1,the shape remains unchanged after the interaction of the two chirped cosine-Gaussian beams.In addition,the cosine-Gaussian beam under the action of fractional diffraction modulated by periodic function,linear function,and power function,the transmission results show periodic structure,parabolic trajectory transmission,and shape-invariant transmission,respectively.In the nonlinear case,the saturating nonlinear effect destroys the periodic structure of the cosine-Gaussian beam while suppressing the propagation broadening of the cosine-Gaussian beam.Especially,the cosine-Gaussian beam can be trapped to form localized modes similar to an optical soliton,which retains its profile during propagations under the action of the saturating nonlinear effect.These results of the research may be used to control the cosine-Gaussian beam trajectory.