Ultrafast lasers have fundamentally transformed the field of photonics, enabling diverse applications in material processing, biomedicine, and scientific research^[1–3]. Among various ultrafast lasers, ultrafast Raman fiber lasers (URFLs) demonstrate superior performance due to their unique capabilities. URFLs provide high pulse energy and short pulse duration, with their most distinctive advantage being wavelength agility that far exceeds the limited spectral range of conventional rare-Earth-doped fiber lasers^[4]. This broad spectral flexibility enables precise wavelength selection crucial for various advanced applications, including two-photon microscopy, optical frequency comb generation, and specialized material processing^[5–7].

URFLs can be generated via three primary approaches: mode locking, synchronous pumping, and nonlinear optical gain modulation (NOGM). Mode-locked and synchronous pumped URFLs are both based on oscillators to generate ultrafast Raman pulses. To achieve stable Raman solitons, gain, loss, dispersion, and nonlinearity inside the oscillator need to meet a delicate balance^[8]. Therefore, the output pulse energy is often restricted to tens of nanojoules^[9–11]. Besides, other disadvantages like the requirement for long Raman gain fiber or a complex synchronization unit greatly limit the practical application of URFLs.

In contrast, NOGM presents a simpler and more promising approach to generate high-energy Raman pulses by replacing the oscillator with an amplifier structure. This technique employs ultrafast pulse pumping to modulate the fiber amplifier’s gain, enabling direct conversion of continuous-wave (CW) single-frequency (SF) seed light into ultrashort pulses^[12,13]. The absence of a traditional resonant cavity not only simplifies the laser structure but also facilitates the generation of higher-energy, shorter-duration pulses by minimizing the deleterious effects of dispersion and nonlinearity typically associated with a long Raman fiber cavity^[4].

Despite the significant advantages of NOGM, the numerical simulation of such systems presents challenges. The underlying physics involves multiple nonlinear effects in optical fibers, including stimulated Raman scattering (SRS), self-phase modulation (SPM), cross-phase modulation (XPM), and self-steepening, necessitating sophisticated numerical models to accurately describe the pulse envelope evolution along the propagation path. These physical processes can be comprehensively described by the nonlinear Schrödinger equation (NLSE), which incorporates various effects such as gain, loss, dispersion, and multiple nonlinear terms. Traditional numerical methods for solving the NLSE, such as the split-step Fourier method and fourth-order Runge–Kutta method, suffer from substantial computational overhead. Although advanced numerical algorithms, such as those based on the differential quadrature method (DQM), have been developed for solving NLSE, such as implementations using improved cubic B-Spline functions as test functions^[14], these primarily enhance computational accuracy rather than efficiency. The computational constraint poses a significant challenge: the prolonged calculation times preclude their integration into systems that rely on real-time computation for dynamic control and operation.

Recently, deep learning techniques have emerged as powerful tools for analyzing and optimizing complex physical systems. In terms of ultrafast fiber lasers, pioneering works have demonstrated the capability of neural networks to predict system behaviors and learn from various data with remarkable accuracy and speed. Boscolo et al. employed fully connected neural networks (FCNNs) to predict self-similar parabolic pulse propagation in optical fibers^[15]. Their model considered gain, loss, group velocity dispersion (GVD), and SPM, successfully predicting output pulse intensities and spectra at various propagation distances. Jiang et al. utilized physics-informed neural networks (PINNs) to solve the NLSE, predicting ultrashort pulse propagation under the combined effects of GVD and SPM^[16]. Salmela et al. demonstrated FCNN-based prediction of supercontinuum generation^[17]. By offering superior computational speed and the capability to learn from experimental data, deep learning approaches provide potential solutions to the aforementioned challenge. Yet, numerical simulations of NOGM Raman fiber amplifiers based on neural networks have not been reported so far.

In this work, we introduce a deep learning approach to learn and predict the first-order Stokes light generation in NOGM systems. Our study targets a NOGM Raman fiber amplifier employing a 1121 nm SF-CW seed source and a 1064 nm pulsed pump source, where the first-order Stokes pulses at 1121 nm are generated through SRS gain modulation. We generated training datasets based on the NLSE by constructing pump pulse parameter sets with experimental relevance, incorporating parameters from different types of Raman gain fibers. An FCNN architecture was developed and trained on these datasets, achieving accurate and rapid predictions of the first-order Stokes light evolution. This approach demonstrates significant computational acceleration compared to conventional methods. Furthermore, the model exhibits effective generalization capabilities, accurately predicting outputs for untrained parameter combinations even when trained on imperfect, noise-affected data without complete knowledge of all influencing factors.

Numerical simulations are performed to study the pulse evolution in a fiber-based NOGM system, the schematic of which is shown in Fig. 1. Gaussian-shaped pump pulses at 1064 nm are launched into a piece of Raman gain fiber, seeded by a 1121 nm SF-CW laser^[14]. The NOGM process is accomplished in the Raman fiber, which eventually transforms the CW seed into ultrafast Raman pulses. A numerical model based on the generalized nonlinear Schrödinger equation (GNLSE) is used to simulate the pump and Raman pulse evolutions along the optical fiber through the NOGM process: $$\frac{\partial A}{\partial z}+\frac{i{\beta }_2}{2}\frac{{\partial }^{2}A}{\partial T^2}=i\gamma A(z,T){\int }_{-\infty }^{\infty }R(T^{\prime }){|A(z,T-T^{\prime })|}^2\mathrm{d}{T}^{\prime }.$$(1)

The left side of Eq. (1) builds a model of linear propagation effects, where $A$ is the slowly varying pulse envelope, ${\beta }_2$ accounts for the GVD, $z$ denotes the propagation distance, and $T$ is the pulse local time. The right side of Eq. (1) describes nonlinear optical effects including SPM, XPM, and SRS. Here, $\gamma$ is the nonlinear coefficient, and $R(T)$ represents the nonlinear response function, which is modeled as $$R(t)=(1-f_R){\delta }_t+f_R{h}_R(t),$$(2)where $f_R=0.18$ is the fractional contribution of delayed Raman response and $\delta (t)$ is the Dirac delta function. The Hollenbeck vibrational model is used to describe the Raman response function $h_R$. The spectral window considered in the model extended from 1100 to 1140 nm with the central wavelength at 1120 nm, which is the wavelength of the first-order Raman light in the case of a 1064 nm pump in silica fiber. The temporal window was equal to 450 ps. Two types of commercial single-mode fibers were used in the simulation to serve as the Raman gain fiber, which is PM 980 fiber by Coherent Corp. and PM Raman fiber by OFS LLC. For PM 980 fiber, ${\beta }_2$ and $\gamma$ are set as $0.024{\mathrm{ps}}^2/\mathrm{m}$ and 0.0053 m/W, respectively. For PM Raman fiber, ${\beta }_2$ and $\gamma$ are set as $0.018{\mathrm{ps}}^2/\mathrm{m}$ and 0.0278 m/W, respectively.

To accurately simulate the pulse evolution along the fiber, the pump pulses are coupled with the seed laser through a WDM at the fiber input ($z=0$). The fiber of length $L$ is discretized into multiple segments, with pulse characteristics sampled at each segment to track the spectral evolution of both pump and Raman pulses. At the fiber output ($z=L$), the Raman pulses are obtained along with residual pump pulses and higher-order Raman components.

Based on previous studies^[17], the FCNN was selected as our neural architecture for predicting Raman pulse evolution in the NOGM system, owing to its simple structure and excellent convergence properties compared with other network structures like convolutional neural networks (CNNs)^[18] or recurrent neural networks (RNNs)^[19]. Through extensive testing of architectures with varying widths and depths, we observed that, while increasing the network’s width and depth could improve overall accuracy, the enhancement in spectral prediction was not obvious. Therefore, as shown in Fig. 2, a relatively concise four-layer FCNN structure is selected and designed with hidden layer widths of 256, 256, 512, and 512, respectively. This architecture maintains prediction accuracy while reducing network complexity. The rectified linear unit (ReLU) activation function was implemented for all layers.

The FCNN was trained using raw data obtained from numerical solutions of the GNLSE described in Sec. 2. The training dataset was constructed with pulse widths uniformly distributed between 10 and 30 ps. For PM 980 fiber, initial pump pulse energies ranged from 20 to 500 nJ, while for PM Raman fiber, they varied from 20 to 300 nJ. The initial pump pulse width was varied in steps of 2 ps, and the initial pump pulse energy was incremented in steps of 20 nJ. The power of the seed light is 0.01 W. According to the ratio of their SRS gain coefficient, the total propagation distances were set to 100 and 20 cm for PM 980 and PM Raman fibers, respectively, with training data sampled at 1 cm intervals along the propagation length. Through iterative generation, we obtained 27,775 training samples for PM 980 fiber and 3465 samples for PM Raman fiber, which were subsequently organized into input–output pairs for network training.

The hidden layers were initialized using the Glorot method, and training was conducted with a batch size of 64. We employed mean square error (MSE) as the loss function, incorporating L2 regularization with a weight of 0.001 to ensure more uniform weight matrices. The learning rate was initially set to 0.001 and subsequently decreased following the cosine annealing schedule. Training was performed using the Adam optimizer over thousands of epochs. To prevent overfitting, 10% of the training data was randomly reserved for validation, with validation checks performed every 50 epochs. An early stopping strategy was implemented with a patience of 100 epochs, using validation loss as the optimization criterion. Prior to network input, the output data underwent logarithmic preprocessing to emphasize high-energy spectral features.

As shown in Fig. 3, both training and validation losses decrease steadily during the training process, eventually converging to stable values, indicating successful network optimization without overfitting.

Figures 4 and 5 illustrate the propagation evolution of a Gaussian pulse in PM 980 fiber (initial pump pulse energy $E=400\mathrm{nJ}$, initial pump pulse width 10 ps) and PM Raman fiber (initial pump pulse energy $E=300\mathrm{nJ}$, initial pump pulse width 10 ps), respectively. According to results shown in Figs. 4(a)–4(c) and 5(a)–5(c), the FCNN predicted results are very consistent with the GNSLE simulation results, except for some regions with strong spectral modulation induced by SPM. Increasing the network size, training sample number, and training time could address this issue. Yet, to avoid excessive increase in computational overhead, we chose to focus on achieving good overall prediction accuracy for Stokes pulses’ spectral evolution while accepting some averaging effects in cases with strong SPM.

Analysis of the error distribution reveals two primary sources of discrepancy. First, the red colored areas in Figs. 4(c) and 5(c) indicate errors in low-energy regions (particularly intensity below $-30\mathrm{dB}$), which stem from the training strategy that deliberately assigned lower priority to accuracy in these regions. Second, in regions dominated by strong SPM-induced spectral modulation, the predicted spectra tend to average the spectral oscillation, which are shown in Figs. 4(d)–4(i) and 5(d)–5(i).

To quantitatively evaluate the discrepancy between neural network predictions and GNLSE numerical simulations, we employed the normalized root mean square error (NRMSE) as our metric, $$\mathrm{NRMSE}=\sqrt{\frac{1}{n}\sum _{i=1}^n\frac{{|A_i-{\widehat{A}}_i|}^2}{{|A_i|}^2}},$$(3)where $A_i$ and ${\widehat{A}}_i$ represent the sample points of spectral intensities obtained from GNLSE simulation and neural network prediction, respectively. Considering practical applications where high-energy spectral regions are of primary interest, we calculated NRMSE specifically for regions above $-10\mathrm{dB}$ relative to the maximum intensity of the generated spectrum since this $-10\mathrm{dB}$ threshold region represents the core energy-containing portion of the spectrum. Hereafter, this specific NRMSE calculation for regions above $-10\mathrm{dB}$ is denoted as NRMSE ($-10\mathrm{dB}$).

Figure 6 presents the NRMSE ($-10\mathrm{dB}$) values under various conditions of the initial pump pulse width, initial pump pulse energy, and propagation distance for both PM 980 and PM Raman fibers. The observed NRMSE ($-10\mathrm{dB}$) trends with respect to these three parameters validate our previous analysis from Figs. 4 and 5. Specifically, the neural network achieves optimal prediction accuracy when Stokes generation becomes the dominant effect. However, in the initial propagation stage where noise and SPM-induced spectral modulation dominate, the network exhibits relatively large prediction errors. Furthermore, after a sufficient propagation distance, the prediction accuracy decreases as more complex nonlinear effects emerge across broader spectral ranges, as shown in the case of PM 980 fiber with high peak power and long interaction length.

Figure 7 compares the prediction NRMSE of three neural networks with different sizes. Net 2, with its architecture illustrated in Fig. 2, serves as our primary network structure. Net 1, which contains fewer parameters than Net 2, exhibits degraded prediction performance. Although Net 3, with more parameters than Net 2, achieves slightly higher accuracy, it shows no substantial advantages over Net 2 in predicting spectral evolution characteristics for most parameter combinations. This comparison demonstrates that our selected network architecture (Net 2) achieves an optimal balance between prediction performance and computational efficiency.

To investigate the generalization ability of the neural network model, we evaluated its prediction performance across various input parameters (pulse width and pulse energy). For each parameter combination, we calculated the NRMSE ($-10\mathrm{dB}$) values at each propagation step (0.01 m intervals from 0 to 0.20 m). The mean value of these NRMSEs ($-10\mathrm{dB}$) across all propagation distances was used as a comprehensive metric, reflecting the model’s overall prediction performance for each parameter combination.

Figure 8 presents a heat map of the mean NRMSE ($-10\mathrm{dB}$) metric across different combinations of input pulse parameters in PM Raman fiber. The regime within the red box contains the 3465 pieces of original training data, while the generalization performance of 56,928 pieces of other data for both inside and outside this regime is verified. Through analyzing the full-distance spectral evolution process for various parameter combinations, we observed distinct patterns in the model’s generalization capability. From the perspective of the initial pump pulse width, the model shows poor generalization for pulse widths below 10 ps, where the range of reliable energy predictions rapidly narrows with the decreasing pulse width. For pulse widths between 10 and 30 ps, the model maintains excellent prediction capability across various initial pump pulse energy levels, performing comparably to the training set results. In the 30–40 ps range, although prediction errors increase, the model still correctly captures the physical characteristics of the spectra, demonstrating moderate generalization ability. From the perspective of initial pump pulse energy, the model’s reliable prediction capability extends to 500 nJ for initial pump pulse widths above 13 ps. As shown in Fig. 9, the neural network successfully predicts the spectral evolution for parameter combinations outside the training range, such as at 460 nJ pulse energy and 36 ps pulse width.

To evaluate the computational efficiency of our neural network model, we compared the computation time between the deep learning approach and conventional GNLSE numerical solving under various conditions. All calculations were performed on a computer equipped with a 12th Generation Intel Core i7-12650 H processor (2.30 GHz) and an NVIDIA GeForce RTX 4060 Laptop GPU. The computation time was measured by averaging 100 consecutive calculations for each condition. For instance, when simulating a 20 cm propagation with an initial pump pulse energy of 280 nJ and an initial pump pulse width of 12 ps, 100 GNLSE numerical calculations took approximately 258 s (2.58 s per calculation) for PM 980 fiber and 1295 s (12.9 s per calculation) for PM Raman fiber, while 100 neural network predictions were completed in about 15 s (0.15 s per prediction) for both fiber types. Notably, the computation time of the GNLSE increases significantly with propagation distance, whereas the neural network’s computation time, primarily determined by the network parameters, remains largely independent of the propagation distance. This computational advantage makes the neural network model particularly suitable for practical applications requiring rapid predictions.

In this work, we have successfully developed and demonstrated a deep learning approach for predicting pulse evolution in an NOGM Raman fiber amplifier. Our four-layer FCNN architecture effectively captures the complex nonlinear dynamics involved in the NOGM process, achieving accurate predictions for both PM 980 and PM Raman fibers. Within the primary operational parameters (pulse widths at 10–30 ps, energies up to 500 nJ for PM 980 fiber and 300 nJ for PM Raman fiber), the neural network model demonstrates excellent prediction accuracy with the NRMSE ($-10\mathrm{dB}$) typically below 0.1 in the regions of interest. The model also exhibits robust generalization capabilities, maintaining reliable predictions even for parameter combinations outside the training dataset, as demonstrated in PM Raman fiber where accurate predictions were achieved for pulse widths up to 40 ps and energies up to 500 nJ. Notably, computational efficiency is significantly improved, with the neural network achieving predictions approximately 17 times faster than conventional GNLSE calculations for PM 980 fiber and 86 times faster for PM Raman fiber, while maintaining prediction accuracy. The model shows strength in predicting Stokes generation, which is crucial for practical NOGM applications, although some limitations exist in regions dominated by SPM or higher-order nonlinear effects. This work represents a significant step toward real-time prediction and control of NOGM systems, with future work potentially focusing on extending the model to incorporate additional physical effects and broader parameter ranges, enabling more comprehensive optimization of NOGM-based ultrafast fiber laser systems. Our next goal is to realize fast prediction of the spectrum evolution in an NOGM system using experimental training data. Precise control of multiple experimental parameters and reducing the impact of noise could be two primary challenges for achieving good prediction accuracy.