The transformation of (l,n)th mode Laguerre-Gaussian (LG) laser beam, in the process of its diffraction by a curved fork-shaped grating with topological charge p, is theoretically studied. The analytical solutions for the diffracted wave field amplitudes are derived in the Fresnel regime and in the back focal plane of a convergent lens. The zeroth-diffraction order is found as (l,n)th mode LG beam. The higher, th diffraction-order beam is described in the radial direction through a product of the Gauss-doughnut function by the double sum of hypergeometric Kummer functions. Its topological charge can be increased or reduced compared to that of the incident beam, or it can be equal to zero. The radial intensity distributions are plotted and compared to the corresponding ones valid when a fork-shaped grating is used. The results are specialized for the cases of incident LG beams of modes (l, n = 0), (l = 0,n) and (l = 0,n = 0) i.e. Gaussian mode.

中文翻译：

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弯曲叉形光栅对(l,n)th模拉盖尔-高斯激光束的衍射

理论研究了第(l,n)模拉盖尔-高斯(LG)激光束在其被拓扑电荷为p的弯曲叉形光栅衍射过程中的变换。衍射波场振幅的解析解是在菲涅耳区域和会聚透镜的后焦平面中推导出来的。零次衍射级被发现为 (l,n)th 模式 LG 光束。通过高斯-甜甜圈函数与超几何库默函数的二重和的乘积，沿径向描述更高的衍射级光束。与入射光束的拓扑电荷相比，其拓扑电荷可以增加或减少，也可以为零。绘制径向强度分布并与使用叉形光栅时有效的相应分布进行比较。