Optical orbital angular momentum (OAM) is an important fundamental property of the light field. Beams carrying orbital angular momentum are known as vortex beams, which have a wide range of potential applications in various fields such as optical micromanipulation, super-resolution imaging, and optical communications. Fractional vortex beams have novel properties different from those of intrinsic vortex beams, which are expected to further expand the application range of vortex beams and realize some unique functions. Mode sorters that can effectively categorize different vortex modes are the core of vortex beam applications. However, since fractional vortex modes are not eigenmodes, existing vortex mode classification methods are not applicable to classify fractional vortex modes. This shortcoming limits the application of fractional vortex modes in many scenarios. Here, we propose a new method to realize the effective sorting of fractional vortex modes by using the coordinate transformation method.

Figure 1(a) Schematic of the coordinate transformation；(b) Theoretical prediction of the coordinate transformation

In this study, the coordinate transformation method is firstly utilized to realize the transformation from fractional vortex mode to integer vortex mode. The schematic and theoretical prediction diagram of the coordinate transformation method is shown in Fig. 1. The coordinate transformation method is derived from the generalized Snell's law, which can realize the one-to-one mapping of the incident light field between two planes according to a specific rule. The transformation rule used in this study is the spiral rule, which is characterized by the existence of a multiplicative relationship between the azimuthal coordinates of the coordinate points before and after the mapping, thus realizing the interconversion of vortex modes. Due to the existence of discontinuities in the phase distribution of the fractional vortex modes, it is necessary to introduce additional corrective phases to obtain the desired integer vortex modes. The effective classification of the fractional vortex modes after the coordinate transformation is realized by means of inverse topological charge matching.

Figure 2 Numerical simulation

Figure 2 illustrates the numerical simulation results of this scheme. The first two columns show the intensity and phase distribution of the converted beam. The phase distribution of the outgoing beam results in agreement with the target integer vortex beam. The last five columns are the results of inverse topological load matching. Under the condition of topological charge matching, the outgoing light field is a Gaussian spot. Under the condition of mismatching, the outgoing light field is a halo significantly larger than the spot. Such results indicate that the topological charges of the outgoing beam are as expected, confirming the feasibility of the present scheme.

Further, we performed classification experiments on two different sets of fractional vortex beams. The experimental results and the corresponding mode purities are shown in Fig. 3. As expected, the outgoing light field exhibits a helical vortex distribution in each set of experiments. The results of the topological charge matching are also in good agreement with the numerical simulations. It can be clearly seen that the light field distribution exhibits a Gaussian spot pattern only when the topological charges are matched. Under the condition of mismatching, the light field distribution is a halo slightly larger than the Gaussian spot. The agreement between the experimental and simulation results confirms the effectiveness of the present scheme. In addition, we analyzed the purity of the classified patterns. The purities of the experimentally obtained modes are all greater than 86%, which verifies that the present scheme can achieve effective classification of fractional vortex modes with low crosstalk.

Figure 3 (a)-(b) Experimental results；(c) Mode purity analysis

This work is published in “Zhengyang Mao, Haigang Liu, and Xianfeng Chen, Effective sorting of fractional optical vortex modes, Advanced Photonics Nexus, 3(6), 066001 (2024)”.