Results and Discussions Specifically, the prediction precision in the four cases is discussed. As training epochs increase, network weights are gradually optimized, and the prediction error of the deep convolutional neural network is gradually reduced. After training for 10000 rounds, the normalized errors on the testing sets in the four cases are all smaller than 1×10^-7 (Fig. 4 and Fig. 6). Even in the most complex case (different initial pulse power, width, and chirp), excellent visual agreement is achieved between the predicted pulse propagations and the real ones where all the temporal distributions include details. The prediction error is mainly concentrated in the propagation range after 350 m and is distributed in the range of the pulse peak, with a maximum value smaller than 10 mW (Fig. 7). In conclusion, the normalized root-mean-square errors of the 500 testing samples are smaller than 0.0584. The results show that the proposed network can predict the process of ultrashort CPA under complex initial pulse conditions with high precision. Furthermore, the computation efficiency of the proposed DL method is investigated and compared with that of the traditional split-step Fourier method. The computation time of the proposed DL method for 500 independent samples is less than 1/10 that of the traditional split-step Fourier method, demonstrating that the DL method has clear advantages over the conventional approach in computation efficiency.

Fiber optical parametric chirped pulse amplification (FOPCPA) is a widely studied ultrashort pulse amplification technique. The FOPCPA can provide excellent gain bandwidth and achieve ultrashort pulse amplification with a more compact and stable system design. The basic principle of the operation relies on a degenerate phase-matched four-wave mixing process involving one strong narrow-bandwidth pump wave, a weak stretched signal, and a generated idler wave. The FOPCPA process can be described by the nonlinear Schrodinger equation. However, the FOPCPA system is highly sensitive to the initial parameters and fiber parameters. Consequently, the traditional numerical methods (i.e., split-step Fourier method and finite-difference method) of analyzing the ultrashort CPA in an FOPCPA system require a huge amount of computation and become less efficient. Nowadays, deep learning (DL) methods have been developed to model and predict nonlinear pulse dynamics and thereby reap the benefits of purely data-driven methods without any underlying governing equations. This study focuses on modeling the ultrashort CPA in fiber by a DL method. The proposed method is expected to broaden the application of DL methods in the prediction of laser behavior and provide an alternative for studying the characteristics of ultrashort pulses in fiber.

A deep convolutional neural network is constructed in the present study. This network contains three parts: five convolutional blocks, a reshaping layer, and three fully connected layers (Fig. 3). Each convolutional block contains a one-dimensional (1d) convolutional layer, a batch normalization layer, a rectified linear unit activation function, and a 1d max pooling layer. The intensity distribution of the initial chirped pulse is used as the input of the neural network. After five convolutional blocks and three fully connected layers, the predicted ultrashort pulse propagation is obtained. For better feature extraction, the real and imaginary parts of the initial pulse are simultaneously used as the input of the deep convolutional neural network. The weights and biases of the proposed network are updated by the back-propagation of the root-mean-square error between the predicted pulse propagation intensity and the ground truth. In the training phase, this study uses the Adam optimizer and sets the learning rate of the network to 0.0001. The whole program is implemented in the Pytorch framework with a 2080Ti GPU. Four cases are considered to test the performance of the proposed network (Table 2). In all these cases, the training sets and testing sets are independent of each other, namely that no duplicate samples are used.

In this study, a DL method is employed to model ultrashort CPA in fiber. A deep convolutional neural network that consists of convolutional blocks and fully connected layers is designed to predict ultrashort pulse propagation under different initial parameters with high precision. Specifically, the paper analyzes the propagation characteristics of the chirped ultrashort pulse and the influence of initial chirp on pulse evolution. The prediction precision and computation efficiency of the proposed method are further studied under different initial pulse parameters. Without compromising generality, the study selects the case of different initial pulse power, width, and chirp to present the testing results. The results show that the neural network constructed performs well in both prediction precision and computation efficiency. On 500 independent testing samples, the proposed deep convolutional neural network achieves normalized root-mean-square errors smaller than 0.0584 and takes less than 1/10 the computation time of the traditional split-step Fourier method. The proposed method extends the application of DL methods in laser technologies and ultrafast optics and provides an alternative for modeling ultrashort pulse propagation in fiber.