## 3 Results and Discussion
### Analysis of Multi-Mode OAM Mode Recognition Based on AP-CNN  
In order to compare the performance of the AP-CNN with that of ResNet18, the OAM recognition accuracy, confusion matrix, and BER of both models are numerically analyzed in this section. Variations in OAM mode recognition accuracy are shown in Figures 7 and 8 under different turbulence intensities for transmission distances 2000 m and 3000 m, respectively. As shown, the detection accuracy increases gradually with an increase in the training epoch. Additionally, the comparisons show that the detection accuracy increases significantly more slowly in strong turbulence than in weak ones. When the turbulence is strong, the training process becomes more problematic because the image suffers more serious distortion. Taking a transmission distance of 2000 m as an example, as shown in Figure 7, after 100-times training in medium turbulence (\(C_{n}^{2}=1.0\times 10^{-14}\)m\({}^{-2/3}\), \(C_{n}^{2}=3.0\times 10^{-14}\)m\({}^{-2/3}\), \(C_{n}^{2}=5.0\times 10^{-14}\)m\({}^{-2/3}\)),  
Figure 8: Accuracy of OAM mode recognition based on AP-CNN under different turbulence intensities. (**a**) transmission distance: 2000 m (**b**) transmission distance: 3000 m.  
Figure 6: Plane wave interferogram of single-mode vortex beam with large topological charge. Figure 7: Accuracy of OAM mode recognition based on ResNet18 at different turbulence intensities (**a**) transmission distance: 2000 m (**b**) transmission distance: 3000 m.  
ResNet18 can still achieve an OAM mode detection accuracy of more than 96.9%, whereas under strong turbulence \(C_{n}^{2}=3.0\times 10^{-13}\)m\({}^{-2/3}\) and \(C_{n}^{2}=5.0\times 10^{-13}\)m\({}^{-2/3}\)), the detection accuracy decrease to 91.7% and 87.3%, respectively. The detection accuracy is further reduced from 87.3% to 83.2% when the transmission distance is increased to 3000 m under strong turbulence (\(C_{n}^{2}=5.0\times 10^{-13}\)m\({}^{-2/3}\)). In other words, the greater the turbulence intensity and the farther the transmission distance, the lower the detection accuracy. A comparison of Figures 7 and 8 reveals that the accuracy of OAM mode recognition based on AP-CNN is superior to that of ResNet18. When the turbulence is weak, the optimization effect of the AP-CNN is minimal, while when the turbulence is strong, it is more obvious. For example, when the transmission distance is 2000 m for \(C_{n}^{2}=3.0\times 10^{-14}\)m\({}^{-2/3}\), as shown in Figure 8, the accuracy of OAM mode recognition based on the AP-CNN can reach up to 98.1% after 100 training epochs, which is only a 0.6% improvement compared with ResNet18. Additionally, under these circumstances, the optimization effect is not obvious. Meanwhile, when \(C_{n}^{2}=3.0\times 10^{-13}\)m\({}^{-2/3}\), the recognition accuracy based on ResNet18 is only 92.1%. The accuracy of OAM mode recognition based on AP-CNN, on the other hand, is significantly improved, and the recognition accuracy of the best model reaches 94.4%, showing an improvement of about 2.3%. When the transmission distance is large, the improvement of the recognition accuracy based on AP-CNN is limited in a highly turbulent environment. For example, when \(C_{n}^{2}=5.0\times 10^{-13}\)m\({}^{-2/3}\), the accuracy of OAM mode recognition based on the AP-CNN can reach up to 90.5% after 100 training epochs when the transmission distance is 2000 m, which is about a 3.1% improvement compared to ResNet18, as shown in Figure 8. However, when the transmission distance extends to 3000 m, the recognition accuracy of the best model can only reach 84.2% after AP-CNN, i.e., 1.2% higher than ResNet18. The reason for this is that in strong turbulence, when the distance is too great, the transmission of the OAM beam is greatly affected by the turbulent disturbance distortion. The light intensity distribution captured by the CCD camera at the receiving end is seriously distorted, and the image features used for recognition are compromised.