Results and Discussions In the simulation, the first eight modes (LP₀₁, LP_11e, LP_11o, LP_21e, LP_21o, LP_31e, LP_31o, and LP₀₂) that support the selected RCF are used as examples with a wavelength of 1550 nm. Four mode-superposition cases (superposition of the first three, five, seven, and eight modes) are analyzed to test the performance of the PFCNN-MD. By comparing the PFCNN-MD with traditional VGG-MD schemes without pre-training, it can be found that even in the three-mode case, the loss function in the VGG-MD always oscillates at higher loss values and cannot converge, while that in the PFCNN-MD quickly converges to 0.007 (Fig. 3). Moreover, the designed PFCNN architecture can achieve convergence within about 55 epochs of training in the four mode superposition cases. The error results of ρ² and θ in the superposition case with three to eight modes are compared between the non-pre-trained PFCNN-MD and the VGG-MD pre-trained on the ImageNet dataset. It can be found that in the eight-mode case, compared with that by VGG-MD, the error of ρ² is reduced from 5.62% to 0.95%, and the accuracy is improved by more than five times. The error of θ drops from 11.86% to 1.92%, and the accuracy is improved by about eight times (Fig. 4). In the three-mode case, the average correlation coefficient between the reconstructed beam pattern and the original beam pattern reaches 99.99%, and one MD consumes 5 ms. When the mode number becomes eight, the average correlation remains at 99.60%, and the trained CNN completes one MD in only 9 ms (Fig. 5). In order to further characterize the performance of PFCNN-MD, the proposed method is compared with four CNN-MD methods (VGG, NFFL-CNN, DH-ResNet18, and DenseNet121) emerging in recent years on various indexes (Table 1). In the eight-mode case, even if the RCF has more complex modal coupling characteristics, the accuracy of the PFCNN is still higher than that of other CNN-MD schemes in the step-index FMF. The time consumed by PFCNN to perform one MD is on the order of ms. In addition, the PFCNN-MD doesn't require the pre-training process and only needs a near-field (NF) beam pattern for training and algorithm input. It has lower experimental operation complexity and equipment requirements. Much time for pre-training is also saved. Moreover, the PFCNN only uses a single GPU to achieve more accurate MD, saving many computer computing resources and reducing energy consumption. Although PFCNN needs to go through about 55 epochs of training to converge the network, it can complete in advance. In the experiment, an experimental setup based on an all-fiber device is used to collect real beam pattern to examine the practical performance of the PFCNN. The average correlation coefficient between the real and reconstructed patterns in the test dataset is 90.01% (Fig. 7).

The core of a ring-core few-mode fiber (RCF) is composed of a central refractive index (RI) depression region and an outer high RI ring. The RCF plays an important role in modal gain equalization, mode division multiplexing transmission, and vortex beam generation. In order to figure out the mode coupling and reveal the associated beam properties in few-mode fibers (FMFs), the mode decomposition (MD) techniques are required, which can obtain the modal weight (ρ²) and modal relative phase difference (θ) from modal superposition images. However, the RCF has complex modal overlap and mode coupling because the power of each mode is confined to the same high RI ring region. As a result, the mode coupling analysis of RCFs faces severe challenges. In this paper, we propose a pretraining-free CNN-MD algorithm (PFCNN-MD) based on a convolutional neural network (CNN) for high-accuracy characterization of complex couplings in RCFs, and the algorithm uses branch structures with different receptive fields to improve the learning ability of neural networks.

In the PFCNN-MD, the normalized field distribution of each supported mode is calculated based on the tested RCF's structural parameters first. After that, massive simulated grayscale beam patterns can be generated numerically with random modal coefficients as the dataset. The corresponding modal coefficient values are set as the label. The generated beam pattern dataset is divided into three parts: training set, validation set, and test set. The CNN is trained by using the training set, which helps the neural network learn the modal features from the beam patterns. During the training process, the training set is iteratively input into the network. The weight and bias parameters of the CNN are updated by minimizing the difference between the network output and the label until the CNN converges. The validation set is used to monitor network fitting and tune network hyperparameters. After the trained CNN has been examined for generalization based on the test set, MD can be implemented on the real beam pattern. The entire process can be completed with only one forward propagation calculation by the trained CNN. The designed PFCNN-MD architecture consists of eight blocks (Fig. 2). In blocks 3 to 7, the InceptionNet-type branch structures are set to increase the width and depth of the network. A variety of small-sized convolution kernels and pooling layers are combined on each branch to extract features in parallel. This not only enables the neural network to have a strong learning ability for features of different scales but also effectively avoids the defects of overfitting and inefficient use of computing resources. Therefore, the designed CNN structure does not require pre-training to enhance the network's ability. Instead, the extraction of complex modal features can be realized directly on the beam pattern training set.

In this paper, a high-precision PFCNN-MD algorithm is proposed to solve the problem of complex mode coupling characterizing in the RCF. The proposed algorithm can fast complete the training and obtain high-precision MD results without pre-training. The performance of PFCNN-MD is tested from both simulation and experiment. In the simulation, compared with that by the traditional CNN-MD, errors of ρ² and θ in the eight-mode case are lower than 0.95% and 1.92%, which are decreased by 80% and 87.5%, respectively. One MD consumes 9 ms. In the experiment, the correlation between the real and the reconstructed beam patterns is higher than 90%. The PFCNN-MD algorithm shows great potential in real-time MD and the characterization of the RCF's mode coupling properties.