Results and Discussions Fig. 2 presents the theoretical results for three types of vortex light intensity distributions and average OAM values. It shows that as the area of SSOO increases, the dark hollow structure inside the intensity disappears, which means that in the propagation with a large-area SSOO, the TC of the input beam cannot be recognized according to the distribution of the intensity of the beams. Figs. 3 and 4 are the theoretical results for the Fourier intensity distributions and phase distributions, respectively. It can be seen that when the obstructed area of the SSOO is larger than one half, methods based on the Fourier intensity and the phase distributions lost their efficiency, while the average OAM always keep around the value of the input TC. It means that in the case of a large-area SSOO, the method based on the averaged OAM works well. Fig. 7 presents the experimental results of the Fourier-transformed intensity distribution and the average OAM of three types of off-axis vortex beams. The input TC equals 1. The comparison between Fig. 7 and Fig. 3 shows that the experimental results agree with theoretical predictions. Table 1 is the theoretical results of the average OAM of the UIS beam and the RIS beam under the obstruction of SSOO when different spiral offsets are considered. It shows the robustness of the average OAM method in perturbations of spiral offset. Table 2 is the results of the average OAM of two types of off-axis vortex beams under SSOO at a TC value of -1. It indicates that this method can recognize the magnitude as well as the sign of the TC. Table 3 is the results of the average OAM under SSOO at different off-axis perturbations. It shows that this method can give the right value of TC when the wandering perturbation is as high as 1.2w.

Topological charge (TC) is a key factor to characterize the orbital angular momentum (OAM) of vortex beams. Accurate determination of TC is an essential prerequisite for the applications of OAM beams in optical communication and sensing. In complex environments, wandering perturbation and opaque obstacles destroy the amplitude and phase of the optical field and put challenges to accurate measurements. In this paper, we propose a method to determine the TC of obstructed wandering vortex beams. According to the theoretical results of the propagation of two uniform off-axis multi-center vortex beams and a random off-axis wandering vortex beam through a sector-shaped opaque obstacle (SSOO), it is observed that the averaged OAM of wandering vortex beams can reveal the TC of the input field, and even wandering perturbation and large-angle SSOO are encountered. Experimental measurement of the averaged OAM is carried out based on a single cylindrical lens (CL), and the results agree with the theoretical predictions. Results show that the method proposed here works well when the methods of light intensity, Fourier transform, and phase distribution fail to determine the TC in extremely complex environment such as the angle of SSOO of 270° and large wandering perturbations.

In this work, three types of multi-center off-axis vortex beams, namely, the beams of uniform off-axis coherent superposition (UCS), uniform off-axis incoherent superposition (UIS), and random off-axis incoherent superposition (RIS), are studied both in theory and experiment. In the theory part, utilizing an equivalent matrix method (EMA), we obtain the propagation of the three types of vortex beams after the SSOO. Theoretical results of the distribution of the light intensity, Fourier intensity, and phase are presented. According to the ABCD propagation law of a tilted CL, the average OAM of the three types of beams is calculated for various input topological charges. In the experimental part, the intensity distribution is detected, and the average OAM is obtained using a single 45^o-tilted CL.

In this paper, the characteristics of the average OAM, optical intensity, Fourier intensity, and phase distribution of the wandering vortex beams obstructed by SSOO are investigated theoretically and experimentally. The results show that the detection method based on the average OAM can correctly reveals the absolute value and sign of the TC of the input beam when the wandering vortex beam is obstructed by the angle of SSOO of over 180°. Moreover, this method works robustly when off-axis perturbations and vortex center misalignment are encountered.