Based on the Huygens–Fresnel integral and ABCD matrix, the propagation equation for the Lorentz–Gauss vortex beam (LGVB) in a gradient-index medium (GRIN) is rederived. The evolution of the intensity and phase distributions of an LGVB through a GRIN medium are numerically calculated as a function of the gradient-index parameter with changes in the incident beam parameters. The results showed that the propagation path and intensity distributions changed periodically with increasing propagation distance. In contrast, phase distributions change at multiples of or , depending on whether the values are odd or even, respectively. At the same time, the parameters of the gradient index determine the periodic values of the Lorentz–Gauss vortex beams during propagation, and as increased, the period of evolution decreased. The Lorentz–Gauss vortex beam propagating through the gradient index will develop from a square beam to a Gaussian vortex beam more quickly with an increase of . In addition, the topological charge affects the size of the dark spot at the center of the beam and the size of the beam, causing the phase distributions to change periodically in the medium. This study is beneficial for laser optics and optical communications.

中文翻译：

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洛伦兹-高斯涡旋光束在梯度折射率介质中的传播特性

基于惠更斯-菲涅尔积分和ABCD矩阵，重新推导了洛伦兹-高斯涡旋光束(LGVB)在梯度折射率介质(GRIN)中的传播方程。通过 GRIN 介质，LGVB 的强度和相位分布的演变可作为梯度折射率参数随入射光束参数变化的函数进行数值计算。结果表明，随着传播距离的增加，传播路径和强度分布发生周期性变化。相反，相位分布以倍数变化或者，分别取决于值是奇数还是偶数。同时，梯度指数参数决定了洛伦兹-高斯涡旋光束在传播过程中的周期值，随着梯度指数的增大，演化周期减小。随着梯度折射率的增加，洛伦兹-高斯涡旋光束将更快地从方形光束发展为高斯涡旋光束。。此外，拓扑电荷影响光束中心暗点的大小和光束的大小，导致介质中的相位分​​布周期性变化。这项研究对于激光光学和光通信是有益的。