Laguerre–Gaussian (LG) laser beams carrying orbital angular momentum are attractive vortex sources for a variety of photonic applications. In this work, we investigate the probing method based on a tilted biconvex lens for LG beams and pairs. We theoretically derive the formulation of the general astigmatic transformation of an LG vortex beam with non-zero radial index p and topological charge l after passing through a tilted biconvex lens. It is found that at a certain position after the focus of the lens, the field intensity of the LG beam is converted into a ( p+1) by ( p+|l|+1) slanted matrix of bright spots. The indices of p and l are obtained by examin-ing the converted intensity patterns. Theoretical calculations are in good agreement with experiments using LG beams with p up to 2 and l up to 34 emitted from a nonplanar ring vortex oscillator. Furthermore, we also found the method can also be used to estimate the mode comp-onents, the ratio of beam intensity and phase relationships of any two superposed LG beams.

携带轨道角动量的Laguerre-Gaussian（LG）激光束是各种光子应用中有吸引力的涡旋源。在这项工作中，我们研究了基于倾斜双凸透镜的LG光束和双光束的探测方法。我们从理论上推导了经过倾斜的双凸透镜后，具有非零径向指数p和拓扑电荷l的LG涡旋光束的一般像散变换的公式。发现在透镜聚焦之后的特定位置，LG光束的场强通过亮点的（p + | l | +1）倾斜矩阵转换为（  p +1）  。p和l的索引通过检查转换后的强度模式获得。理论计算与使用非平面环形涡旋振荡器发出的p高达2和l高达34的LG光束的实验吻合得很好。此外，我们还发现该方法还可用于估计模式分量，光束强度的比率以及任意两个叠加的LG光束的相位关系。