## 2 Materials and Methods
### Recognizing OAM Modes Based on AP-CNN  
Figure 3 displays the light intensity distributions of two similar superposition mode vortex beams of OAM= {1, \(-\)2, 3, \(-\)5} and OAM= {\(-\)2, 3, \(-\)5}. The ROI, localized from low to high level, are slow shown. This approach can be used to identify the ROI located on different pyramid levels, and more detailed information can be captured at the low levels to distinguish different OAM modes. Compared with high-level image semantic information, after thinning image features, this low-level information is very helpful to improve the accuracy of OAM mode detection. The CCD camera at the receiving end captures the light intensity distribution of the vortex beam after atmospheric turbulence and inputs it into the AP-CNN to detect the OAM mode and retrieve the transmitted data. Here, we use the ResNet18 network as the backbone of the AP-CNN. The ResNet18 structure used in this paper differs slightly from the official ResNet18 [35] structure, as described below:  
1. The size of the input light intensity map is 128 \(\times\) 128 \(\times\) 3. To avoid the problems of the low resolution of the final feature map after multiple downsampling and the serious loss of semantic information, the maximum pooling layer of stage0 is removed. 2. To reduce the number of model parameters, the 7 \(\times\) 7 convolutional kernel of stage0 is replaced by a 3 \(\times\) 3 convolution kernel. The structure of the modified ResNet18 is shown in Table 1.  
The OAM mode recognition algorithm AP-CNN consists of two parts: the backbone network ResNet18 and the refinement network, as shown in Figure 4. The size of the simulated light intensity distribution map of the OAM beam is set to \(128\times 128\) and is input into the ResNet18 network for training. Firstly, it is input into stage0 for pre-processing, and only the features are extracted (64 convolutional kernels with a size of \(3\times 3\) and a step size of 2), and the 128 \(\times\) 128 feature map is output. Then, the feature map is fed into the next four layers of residual blocks, which reduces the size of the input feature map by half compared to the original size and doubles the number of channels. Next, the output feature maps of the third, fourth, and fifth layers of ResNet18 are denoted as \(B_{3}\), \(B_{4}\), \(B_{5}\), respectively, for subsequent building of the feature pyramid, as shown in Figure 4. Further refinement is carried out at the \(B_{3}\) level of the pyramid. We respectively assign anchors with single scales of 18, 36, and 72 and a 1:1 ratio for each pyramidal level and choose the top 5, 3, and 1 anchors with the highest activation values as potential refinement candidates. For the adaptive NMS, the cutoff threshold is set to 0.05, the merge threshold to 0.9, and the drop block probability to [30%, 30%, 0%]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Network Layer** & **Output Feature Map Size** & **ResNet18** \\ \hline conv1 & \(128\times 128\) & \(3\times 3,64\) \\ conv2\_x & \(64\times 64\) & \(\begin{bmatrix}3\times 3,64\\ 3\times 3,64\end{bmatrix}\times 2\) \\ conv3\_x & \(32\times 32\) & \(\begin{bmatrix}3\times 3,128\\ 3\times 3,128\end{bmatrix}\times 2\) \\ conv4\_x & \(16\times 16\) & \(\begin{bmatrix}3\times 3,256\\ 3\times 3,256\end{bmatrix}\times 2\) \\ conv5\_x & \(8\times 8\) & \(\begin{bmatrix}3\times 3,512\\ 3\times 5,512\end{bmatrix}\times 2\) \\  & \(1\times 1\) & Average pooling + full \\  & & connected + softmax \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Modified ResNet18 network structure. Figure 3: Regions of interest localized by AP-CNN at different levels for different OAM modes. During AP-CNN training, the initial learning rate is set to 0.01, decreasing by 10% for every 20 iterations; as such, a total of 100 epochs are trained. The random gradient descent algorithm with a momentum coefficient of 0.9 and a minipatch of 16 is used for parameter optimization, and the weight attenuation is set to \(5\times 10^{-4}\).