Objective Vortex beams carrying orbital angular momentum (OAM) mode have important application prospects in improving communication capacity, multiple-channel information modulation with the same frequency and time. As the vortex beam propagates through complex medium environments, the beam field amplitude and phase may be disturbed, causing channel crosstalk and high bit error rate. Several studies proved that the OAM spectrum disturbance caused by a complex medium environment is one of the key problems that need to be solved in the OAM multiplexing technology. The distribution of the vortex beam’s OAM spectrum can easily be obtained through the numerical method. However, the accurate inversion of the OAM spectrum distribution according to the measured field in the actual measurement system is still worth investigating and discussing. Existing methods for classifying OAM mode mainly focus on the main mode. This study proposes a method for identifying the OAM spectrum distribution of the measured image of the disturbed field using the ResNeXt network. Converting the problem of OAM spectrum distribution to the probability problem discriminates test samples belonging to training samples of each type in convolutional neural network(CNN), and the percentage of the main module and side lobes of the OAM modal can be obtained.

Methods By optimizing the ResNeXt network, the fitting function of the OAM spectrum distribution of the vortex beam’s transmission field calculated from the spectrum analysis theory of vortex beams is set as the self-defined loss function of the network. The total loss function of the network L_total consists of three parts. L₁ is the cross-entropy function, which calculates the main model classification loss of the test sample. L₂+L₃ are the loss functions for predicting the OAM spectrum distribution, and the custom loss function is included in L₂. L₃ is the cross-entropy function, the same as L₁.The training samples are the light intensity pattern of a Bessel vortex beam with l=4--12,which does not propagate through the glass medium without aberration. The test samples are the light density diagrams of a Bessel vortex beam with l=4, propagating through an isotropic glass plate with a thickness of 4 mm under different incident angles. The optimized ResNeXt network must first test the category of test samples, called the main mode discrimination. Then, it sets the network study according to the preset fitting function of the OAM spectrum distribution (the self-defined loss function) and obtains the probability that test samples belong to training samples in each category, achieving the purpose of reconstructing the OAM spectrum distribution of the beam’s transmission field.

Results and Discussions The discriminating results of the OAM spectrum of the Bessel vortex beam’s transmission field are compared with the numerical results based on the ResNeXt network by using a fixed recognition process (Table 1), and light density diagrams of the eighth-order Bessel vortex beam propagate through a glass medium under different incident angles as test samples (Fig.3). Additionally, the relative errors are calculated (Fig.4). Since the diffusion of the OAM spectrum distribution increases with an increase in the incident angle, the main mode loses its dominance when the incident angle increases to a certain degree. The proportion of side lobes exceeds the main mode, and the spectrum distribution’s regularity is weakened. The ResNeXt network’s discriminating error also has an increasing tendency with an increase in the incident angle. However, the maximum relative error does not exceed 22.86% compared with numerical results, and the absolute error of side lobes does not exceed 22.23% (Table 4). There are three main reasons. First, since the Bessel beam has the characteristics of multi-topological charges, its OAM spectrum distribution extends infinitely to both sides with the position of the main mode in the center. However, the training set types are limited and can only be discussed in the main modes. Second, since the regularity of the OAM spectrum distribution of the Bessel vortex beam’s transmission field decreases with an increase in the incident angle, the dispersion becomes more obvious, decreasing the recognition accuracy. Finally, since the OAM spectrum recognition problem is different from the conventional classification problem, the CNN feature of test samples is relatively related to the main mode and weakly related to the side lobes. Although the custom loss function contributes to reducing the error, it cannot determine the recognition result of the network. Thus, the side lobe recognition error is greater than the main mode.

Conclusions Based on the ResNeXt network and by introducing self-defined loss function, this study discriminates the OAM spectrum distribution of the Bessel vortex beam’s transmission field, which propagates through the glass medium with an incident angle ranging from 0°~45°, the comparison is made, and the error of the numerical result is analyzed. The results show that the relative error of the main mode increases along with the increase in the wave beam incident angle. However, it does not exceed 22.86%, and the discriminating error of side lobes is bigger than that of the main mode. The proposed method is not limited to the category discrimination of the main mode, but discriminates the percentage of the main mode in the OAM spectrum distribution in detail. This study provides a new idea for the OAM mode discrimination; however, how to improve the discriminating accuracy of side lobes will be the next challenge.