Probing arbitrary Laguerre–Gaussian beams and pairs through a tilted biconvex lens

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 examining 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 components, the ratio of beam intensity and phase relationships of any two superposed LG beams.