## 2 Theoretical Framework and Network
### Mode Classifier  
In response to the problems of large sample size, low detection accuracy, and weak model generalization ability in OAM modal detection datasets, this paper designs a Transfer OAM modal classifier. Figure 2 shows the scale variation of distorted beam images in the network. The network consists of sixteen layers, as shown in Table 2. The network structure is divided into two main modules: feature extraction module and OAM modal detection module. The feature extraction module is used to learn multi-level feature representations and extract complexity and abstract features of input data. The module consists of seven similar convolution blocks, represented as Conv1 to Conv7, each consisting of a convolutional layer (Conv), a maximum pooling layer (MP), an activation function, or a combination of convolutional and activation functions. Due to its faster convergence speed in solving gradient vanishing problems compared to traditional nonlinear functions such as Sigmoid and Tanh, Rectified Linear Unit (ReLU) is chosen as the nonlinear activation function to improve detection performance. The OAM modal detection module maps the high-dimensional features learned by the feature extraction module to a low-dimensional space and outputs the predicted distribution of category labels. This module consists of three similar classification blocks, represented as FC1 to FC3, each consisting of a fully connected layer (FC), a dropout layer (Dropout), an activation function, or a combination of a fully connected layer and an activation function. This module introduces a dropout layer that randomly removes some neurons with a probability of 0.5 during forward propagation to prevent the network from overly relying on local features, thereby forcing the network to learn multiple different sub-networks and improving its generalization ability. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Specification** \\ \hline \(\lambda\) & 632.8 nm \\ \(w_{0}\) & 0.05 m \\ \(N\) & 512 \\ \(\Delta z\) & 200 m \\ \(l_{0}\) & 0.005 m \\ \(L_{0}\) & 3 m \\ \(C_{n}^{2}\) & \(C_{n}^{2}=1\times 10^{-16}\,\text{m}^{-2/3}\sim C_{n}^{2}=1\times 10^{-14}\, \text{m}^{-2/3}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Simulation Parameters. Figure 2: Schematic diagram of Transfer OAM classifier. where * represents multiplication sign, and * 2 represents repeating the module twice. Our distortion intensity image detection method is summarized in Figure 3. Firstly, the distorted LG beam intensity images of different modalities are labeled and used as network inputs; Secondly, in order to prevent overfitting, data augmentation methods such as ToTensor, Normalize, Resize_224, RandomHorizontalFlip, and RandomVerticalFlip are used to perform data augmentation operations on the training and testing sets. These operations help the model adapt to the distribution of the dataset, increase the diversity of the training data, and better adapt to vortex light mode detection tasks in more turbulent environments; Subsequently, due to the fact that datasets with stronger turbulence strengths typically contain more complex and diverse features, AlexNet models trained on 15,000 large-scale datasets under strong turbulence strengths were selected as pre-training models, and their convolutional layer weights and parameters were loaded into Transfer OAM as the initialization weight parameters of the model. It is worth noting that the pre-trained dataset of 15,000 images does not overlap with the target dataset. By continuously iterating and updating the network, the features learned from the source domain are gradually transferred and adapted to the target domain, thereby preserving the common features in the pre-trained model and improving the training speed and generalization ability of the model.