1. Introduction

Fiber lasers stimulate diverse industrial applications due to their prominent advantages, such as their high output intensity [1], high pulse repetition frequency [2], narrow pulse width [3] and narrow linewidth [4]. The investigation of the generation [5,6] and dynamics [7,8] of mode-locked soliton pulses in fiber lasers is of great significance and has attracted many researchers’ attention. Since artificial intelligence (AI) technology has been widely introduced into optics, the subject termed “intelligent photonics” has emerged [7–11]. AI methods, including evolutionary algorithms (EAs), genetic algorithms (GAs), neural networks (NNs), deep learning (DL) and human-like algorithms (HLAs), have been extensively employed for automatically triggering mode-locked pulse generation. In general, AI-induced mode locking is achieved with assistance from a single information domain, such as a second harmonic generation (SHG) trace [12], radio frequency (RF) [13] or spectrum [14]. However, these approaches suffer from slow optimization speeds and limited parameter control. On the other hand, manipulating the characteristics of mode-locked pulses based on AI technology has been another popular topic in the field of fiber optics. AI technology has been used to generate various mode-locked patterns, such as dissipative solitons [15] and breathing solitons [16], and manipulate pulse parameters based on the HLA-based Rosenbrock search method as human-like behavior [17]. Additionally, AI technology can dynamically control the dynamics of mode-locked pulses by optimal strategies [18,19]. However, the aforementioned manipulations were fulfilled by utilizing a single formation domain, such as spectrum information or temporal signals.

Although AI technology has made great achievements in triggering mode locking and manipulating mode-locked pulse parameters, the repeatability of the reproduction of unique pulse states is not available for the abovementioned studies. Considering that one frequency component of an arbitrary pulse wave function is denoted as $\omega _{_{1}}$, the computation of $\omega _{_{1}}$ can be expressed as:

Hence, we propose a novel spectrotemporal domain-informed deep learning algorithm for mode-locked pulse reproduction. This algorithm combines pulse information in both the spectral and temporal domains, providing a more comprehensive criterion for accurate reproduction. We first trained a fully connected neural network (FCNN) to describe the mode-locked fiber laser by using the parameters of a self-developed automated polarization controller (PC) as inputs and the predicted spectrotemporal data as outputs. After training the FCNN, we could further optimize the polarization settings within the cavity by employing the backpropagation (BP) and gradient descent (GD) algorithms. Then, the loss function, consisting of both the autocorrelation trace and spectrum as the temporal and spectral information domains, respectively, was designed for optimizing the reproduction process. Furthermore, we introduced a novel single-layer perceptron model to retrieve the initial phase distribution of the reproduced pulses. Thus, the uniqueness of the reproduced results can be demonstrated by the retrieval of the phase information. The reproduction matching similarity defined by the spectrotemporal MSE was $3.99\times {10}^{-5}$. Our work can stimulate the potential application of intelligent photonics in the ultrafast laser industry.

2. Principle of reproduction

The reproduction workflow can be summarized as follows: Firstly, a comprehensive data set including spectra and autocorrelation data for every PC setting is collected. Subsequently, an FCNN is iteratively trained with the collected data set until the loss error arrives at a predefined threshold. Then, a nonlinear mapping between the PC parameters and experimental outputs is established through the well-trained FCNN. Finally, the FCNN has an ability of manipulating the PC to experimentally reproduce the targeted pulse state by utilizing the trained mapping.

2.1 Training for the laser model

An FCNN was constructed to simulate an experimental fiber laser system to establish the mathematical relationship between the PC parameters and soliton states. As depicted in Fig. 1(b), the FCNN employs 2 parameters, i.e., the rotation angle $\theta$ and the squeeze level $\phi$ of the self-developed automated PC, in the input layer, represented as $\left (\theta,\phi \right )^\top$, where $\top$ indicates the transpose of the vector. The squeeze level is defined by the pressure on the optical fiber in the PC and measured by a stress sensor. The FCNN incorporates 3 hidden layers in an inverted pyramid shape, with the hyperbolic tangent function $\tanh \left (x\right )$ as the nonlinear activation function for better performance. All hidden layers together are essentially a function $\mathbf {F}$. The output layer is designed for predicting the spectrum data $\widehat {A}$ and the autocorrelation trace data $\widehat {I}$. Therefore, the mathematical relationship between inputs and outputs can be represented as follows:

 

Fig. 1. The reproduction process of spectrotemporal domain-informed deep learning. (a) Digital control system. (b) Well-trained FCNN. (c) Matching the optimized process between the experimental results and the targeted soliton state. (d) Experimental setup.

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The FCNN is trained with experimental data. The well-trained FCNN depicted in Fig. 1(b) can provide a well-fitted mathematical relationship between the fiber laser output and the PC settings, e.g., autocorrelation trace and the corresponding PC parameters.

2.2 Training for mode-locked pulse reproduction

After laser model training, the subsequent step is target reproduction training. The desired PC parameters are searched by GD. The well-trained FCNN is considered a special activation function. It should be noted that the manipulation of PC parameters for reproducing targeted pulses cannot be achieved without the mapping of PC parameters and laser output provided by FCNN, BP and GD. The input layer is then transformed into the layer to be optimized. As shown in Fig. 1(b), the reproduction procedure starts with the random initialization of the PC parameters $\left (\widehat {\theta },\widehat {\phi }\right )^\top$. Then, by inputting the initial PC parameters into the well-trained FCNN, the FCNN will generate the predicted output of both spectral and temporal information. On the other hand, as shown in Fig. 1(a), the PC parameters will also be transmitted to a programmable logic controller (PLC) to control the servo-based automated PC1 shown in Fig. 1(d) for manipulating the experimental laser system. To reduce the random fluctuations induced by the environment during the processes of data collection and model training, as illustrated in Fig. 1(c), the spectrotemporal outputs are also measured and involved in the model. Therefore, a new loss function is constructed by combining the FCNN prediction data, real-time experimental data and target mode-locked pulse data $\left (I,A\right )^\top$ in the spectrotemporal domains. As a result, a spectrotemporal domain-informed deep learning model is constructed to manipulate the reproduction of targeted solitons.

The optimization process is based on the GD algorithm, which can be mathematically represented as follows:

The optimizer depicted in Fig. 1 is developed by incorporating the aforementioned processes and algorithms. Through deductive reasoning and mathematical verification, we have established the mathematical operation of our proposed DL model. Thus, we confidently employ our model to reproduce designated mode-locked pulses.

2.3 Criteria for soliton reproduction

Generally, loss functions in deep learning, such as the mean absolute error (MAE), MSE and Huber loss, are used to evaluate the reproduction performance. Due to the lack of physical background, however, these loss functions cannot capture the inherent features presented in optical spectra and autocorrelation traces. Given the inherent distinctions between spectra and autocorrelation traces, it becomes imperative to employ distinct loss functions to optimize the network’s performance in terms of convergence speed, prediction accuracy and generalization capacity. Therefore, we propose constructing loss functions with physical properties to evaluate the matching quality of mode-locked pulse reproduction.

For the spectral criterion, we propose a novel algorithm, termed “optical spectrum matching absolute deviation variance (OSM-ADV)”. The OSM-ADV loss function, denoted by $S^{2}_{_{\mathrm {OSM-ADV}}}$, is specifically designed to capture key features of optical spectra, such as the positions and spike amplitudes of Kelly sidebands. Mathematically, $S^{2}_{_{\mathrm {OSM-ADV}}}$ is calculated as follows:

For the temporal criterion, we utilize the modified Pearson’s correlation coefficient, denoted as modified Pearson’s $r$, as the loss function for temporal pulse reproduction. Mathematically, Pearson’s $r$ is defined as:

In conclusion, the OSM-ADV and the modified Pearson’s ${\tilde {r}}_{_{\hat {I},I}}$ loss functions provide a more comprehensive and physically meaningful evaluation of the similarity between the predicted results and the targeted information. Then, the whole spectrotemporal loss function is defined as:

By incorporating these loss functions into the target reproduction training, our proposed model not only enables the evaluation of specific data differences but also captures a broader range of similarities in the overall trends for achieving more accurate and reliable pulse reproduction. Simultaneously, our spectrotemporal loss functions can be universally applied to all types of mode-locked lasers. These loss functions can effectively evaluate the reproduction performance of various output states.

3. Experimental reproduction of the targeted soliton

The experimental setup, depicted in Fig. 1(d), features a ring cavity mode locked by the nonlinear polarization rotation (NPR) technique. The laser is composed of a 4-meter segment of commercial erbium-doped fiber (EDF, YOFC EDF1022) pumped by a 980-nm laser, incorporating a 200-meter segment of conventional single-mode fiber (SMF) to accumulate nonlinear phases of the pulses for facilitating NPR mode-locking. The NPR mechanism is realized by strategically placing a pair of PC3 and PC4 and a single polarization-dependent isolator (PD-ISO) within the ring cavity, serving as both a mode locker and a filter. Additionally, a Lyot filter, consisting of a 0.25-meter segment of polarization-maintaining fiber (PMF) and two polarization controllers (PC2 and automated PC1), is integrated with the NPR-based filtering effects through a shared PD-ISO. The characteristics of mode-locked pulses can be manipulated by adjusting polarization settings induced by PCs in our NPR-based fiber laser. We aim to fully automatically activate mode-locking and search the target in fiber lasers by adjusting PCs assisted by our FCNN model. On the other hand, the composite filtering composed of NPR filtering and Lyot filtering can help generate diverse pulse patterns. Therefore, we practice reproducing various output states with our FCNN model by using composite filtering, which will validate the model’s ability to distinguish between different pulse states. The total length of the cavity is approximately 207.3 meters. The output of the laser system is obtained from the 10% port of a Y-type coupler. By configuring specific polarization settings and reaching a threshold pump power of 53 mW, we could achieve a self-started mode-locked soliton with inherent Kelly sidebands, which served as the target pulse to be faithfully reproduced.

We employed an OSA (Yokogawa AQ6375B) and an autocorrelator (Femtochrome FR-103XL) with an oscilloscope (Teledyne LeCroy WaveMaster 813Zi-B) for monitoring spectrotemporal signal information. The resolutions of the OSA and the autocorrelation were kept at the highest precision of 0.01 nm and 5 fs during the measurement processes, respectively. The oscilloscope as a screen was used to display the autocorrelation trace of the signal. Real-time data collection was facilitated using a MacBook laptop equipped with an Apple Silicon M1 CPU and 16 GByte of memory. The experimentally acquired data were utilized to construct a comprehensive spectrotemporal database as the training data set for our FCNN. The laptop then exports the predictions of our deep learning model to the PLC shown in Fig. 1(a), enabling the adjustment of the automated PC1 shown in Fig. 1(d) to ensure that the experimental spectrotemporal output aligns with the designated target for the precise reproduction of the mode-locked soliton state through the aforementioned automatic control algorithm.

We commenced the training for the laser model by collecting spectra and autocorrelation traces within the PC1 adjustment ranges of $0^\circ \le \theta \le {150}^\circ$ and $-5^\circ \le \phi \le 5^\circ$, respectively, with a step size of 1$^{\circ }$ for each degree of freedom. The acquired data were divided into a training data set (80%) used for training the FCNN and a test data set (20%) employed to evaluate the training performance. To address the issue of overfitting in deep learning, we employed dropout regularization during the training process. By randomly deactivating the neurons in the FCNN with a typical probability $p$ value of 0.5 [21], we aimed to prevent the model from relying too heavily on specific features or memorizing the training data [22]. This technique helps promote better generalization and reduces the risk of overfitting, enhancing the model’s ability to perform well on unseen data. The training of the FCNN was terminated when the training error reached a value below ${10}^{-7}$. The FCNN achieved a prediction accuracy of 99.0237% for the training data set and 95.3450% for the testing data set, indicating a well-trained FCNN.

Figures 2(a) and (d) illustrate the evolution of the 49 optimal cases in the process of searching and reproducing the targeted mode-locked pulses. Figure 2(a) depicts the evolution of the spectra during the search and reproduction processes, exhibiting a clear trend from longer wavelengths to shorter wavelengths due to the target spectrum being biased at the short wavelength band centered at 1542 nm. Figure 2(d) depicts the corresponding evolution of the autocorrelation traces, demonstrating the progression of the pulse peak intensity from weak to strong and the pulse width from broad to narrow. The provided target for pulse reproduction is represented by the red boxes or curves in Figs. 2(a), (b), (c) and (d), while the deep blue boxes or curves depict the tracked and captured experimental results through our proposed algorithm. These two algorithms employed in our system have a unique advantage in accurately evaluating spectrotemporal domain information, which are much different from the reported algorithms in the field of AI. Therefore, we additionally calculated the MSE loss to facilitate intuitive comparison during the optimization process, as shown in Fig. 2(e). The minimum MSE is $3.99 \times 10^{-5}$.

 

Fig. 2. The reproduction of a conventional soliton. (a) Spectral matching process. (b) Spectra. (c) Autocorrelation traces. (d) Temporal matching process. (e) Error curve.

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Figures 2(b) and (c) present the spectra and autocorrelation traces, respectively, corresponding to PC1 state index numbers of 0, 20, 40 and 49. When the index number is 0, the spectrum already exhibits Kelly sidebands, and the autocorrelation trace indicates the generation of a soliton. Since the central wavelength and intensity of the autocorrelation trace differ from the target, our reproduction system does not consider it as the final result. When the index number is 20, the central wavelength of the experimentally obtained soliton has been manipulated to a spectral region close to the target. Because there are differences in the spectral profile, spectrum width, pulse width and pulse intensity, it is still not identified as the target. At index number 40, the obtained pulse exhibits a high similarity with the target in terms of the spectral trend, central wavelength and pulse width. However, its MSE loss, as shown in Fig. 2(e), is only $7.31\times {10}^{-5}$, resulting in the tracked result not being considered the best result. This indicates that our proposed OSM-ADV algorithm can capture subtle differences in the spectra, while Pearson’s correlation coefficient, as the pulse similarity analysis algorithm, can capture differences in pulse intensity. Finally, the reproduction process halts at index number 49, corresponding to the 237th epoch. The subsequent 500 epochs can be regarded as the optimal reproduction results with a high degree of similarity in the whole spectral and temporal pulse parameters. Specifically, the spectrotemporal MSE loss is $3.99\times {10}^{-5}$ for the best reproduction.

To verify the validity of our model, we further reproduced another pulse pattern, dual-wavelength solitons. As shown in Fig. 3(a), The autocorrelation trace of dual-wavelength solitons has a small pedestal similar to that of noise-like pulses. However, due to the constraint of the spectral information, our model did not misjudge noise-like signals as reproducible objects. Then, the autocorrelation trace and spectrum of reproduced dual-wavelength solitons in orange is shown in Fig. 3, which have a much high similarity with the target in blue. The reproduction loss defined by MSE is $8.23\times 10^{-5}$. Thus, two different reproduction cases in Figs. 2 and 3 show that the validity and accuracy of our model.

 

Fig. 3. The reproduction of dual-wavelength solitons. (a) Autocorrelation trace. (b) Optical spectrum.

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4. Discussion

To verify the physical completeness of our reproduction approach, we will demonstrate how to achieve the reproduction of authentic mode-locked pulses. In accordance with the Fourier theorem, an arbitrary mode-locked pulse can be defined in a complex exponential summation form. As shown in the stereogram in Fig. 4(b), the best matched output mode-locked pulse, denoted as $\boldsymbol{\varPsi}(t)$ (depicted as the deep blue time-intensity graph in Fig. 4(b)), is actually composed of a linear superposition of monochromatic signals with different frequencies (represented by a series of sinusoidal curves in Fig. 4(b) ranging from purple to red in spectral order). Consequently, $\boldsymbol{\varPsi}(t)$ can be expressed as:

 

Fig. 4. The structure of PPA. (a) The physics-informed phase retrieval process. (b) The data-informed pulse output state measurement process.

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As depicted in Fig. 4(b), Eq. (13) characterizes the temporal amplitude $\boldsymbol{\varPsi}(t)$ of the mode-locked optical pulse field with $N$ distinct frequency components, in which each component is defined by individual amplitudes $A_{_{k}}$, frequencies $\nu _{_{k}}$ and initial phase $\varphi _{_{k}}$. The distribution of $A_{_{k}}^2$ at $\nu _{_{k}}$ measured by the OSA represents the spectral information of the pulse. The initial phase $\varphi _{_{k}}$ of each frequency component corresponds to the horizontal offset of that component’s signal, ranging from 0 to $2\pi$. For a single monochromatic sinusoid signal, $\varphi _{_{k}}$ determines the positions of its peaks and valleys. When the phases cyclically vary as the frequency increases or decreases, the superposition of these frequency components can produce high-intensity peaks at specific positions, thereby achieving mode-locked pulses.

As illustrated in Fig. 4(b), we performed an autocorrelation measurement to obtain the temporal data of $\boldsymbol{\varPsi}(t)$. The autocorrelation trace represents the self-convolution of $\boldsymbol{\varPsi}(t)$ and is obtained using an autocorrelator. An oscilloscope was also involved to measure the autocorrelation trace $I(t)$, which can be represented as:

As shown in Fig. 4(a), we constructed the PPA model by using 6666 sampling points of spectral data. The model’s weights were set as the frequency $\omega _{_{k}}=2\pi \nu _{_{k}}$, and the corresponding scaling factor was set as the amplitude $A_{_{k}}$. The biases were optimized by setting them as the parameter column $\varphi _{_{k}}$. However, the mathematical expression for the pulse phase cannot be directly deduced in terms of the spectra and autocorrelation traces. We can numerically calculate the phase distribution by an iterative phase-retrieval algorithm. The phase-retrieval process was experimentally carried out as follows: the phase-retrieval algorithm was to first introduce an initial phase for constructing an optical pulse field based on the experimentally achieved spectral data, and then to compare with the error of the similarity between the theoretically calculated autocorrelation trace and the experimental result. When the error arrived at a given target, the iterative calculation process would be terminated. The perceptron output, $\boldsymbol{\varPsi}(t)$, reconstructs the mode-locked pulse signal, which can be described by Eq. (13). To measure the discrepancy between the predicted and actual autocorrelation traces, we constructed a loss function based on the temporal loss proposed in Eq. (11), which is the modified Pearson’s ${\tilde {r}}_{_{\hat {I},I}}$. We used the complex exponential $\exp \left (ix\right )$ as the nonlinear activation function and employed BP and GD to optimize the parameter column $\varphi _{_{k}}$ according to the loss function. The optimization process enabled our proposed PPA model to achieve phase retrieval. And we also incorporated a dropout regularization into the PPA framework. Specifically, we randomly deactivated 20% neurons of PPA every iteration and recover their avidities subsequently. This innovative approach serves to mitigate the impact of high-amplitude components on the final results, thereby guaranteeing equal involvement of all spectral components. After optimization, we obtained the retrieved phase distribution schematically represented in Fig. 4(b).

By feeding with the experimental autocorrelation trace data and optical spectrum data of the best matched result, PPA is capable of retrieving the corresponding initial phase distribution of the measured mode-locked soliton. It is crucial to clarify that while frequency-resolved optical gating (FROG) is primarily employed to retrieve the temporal phase distribution, PPA is specifically designed for the retrieval of the initial spectral phase distribution.

According to Fig. 5(b), PPA directly introduces the best reproduced spectral data as the initial invariant parameter of the perceptron. On the other hand, as shown in Fig. 5(a), the corresponding autocorrelation trace data are introduced into the perceptron as fitting objects. Therefore, the orange, green and blue curves in Fig. 5(i) show the autocorrelation signals reconstructed from the same optical spectrum data (Fig. 5(b)) after phase distribution retrieval (Figs. 5(c), (d) and (e)) using PPA. Simultaneously, Figs. 5(c), (d) and (e) show periodic features in the initial phase distributions, which indicate the generation of mode-locked pulses. The initial phases determine the positions of spectral peaks and valleys within the mode-locked pulses. Figures 5(f), (g) and (h) show zoom-in images of Figs. 5(c), (d) and (e) for the initial phase periods of 0.298 nm, 0.833 nm and 1.000 nm, respectively. In other words, Figs. 5(c), (d) and (e) accordingly contain 84, 30, and 25 times of initial spectral phase periods. Figure 5(i) shows three different PPA-retrieved autocorrelation traces, which have evident differences in phase periodicity and spectral phase profile due to different training iterations. The red, orange, green and blue dashed lines in Fig. 5(i) represent the symmetry axes of the experimental autocorrelation trace, best PPA-reconstructed autocorrelation signal, and other PPA-reconstructed autocorrelation cases of 1 and 2, respectively. Notably, the best reconstructed autocorrelation signal is consistent with the targeted soliton in terms of the envelope profile and symmetry axis, thereby achieving reconstruction.

 

Fig. 5. Results of the PPA. (a) The best experimental reproduced autocorrelation trace. (b) The best experimental reproduced optical spectrum. (c) The best retrieved phase distribution. (d) and (e) Retrieved phase distributions with the same spectral information. (f), (g), and (h) Details of the retrieved initial phase distribution. (i) Comparison between the reconstructed autocorrelation signals and the best reproduced autocorrelation trace.

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It is worth noting that PPA operates without specific constraints. The fidelity of the phase retrieval will be enhanced for larger sampling points from the experimental instrument, higher data smoothness and reduced data noise. In essence, PPA is not limited to any particular pulse type and can effectively perform phase retrieval as long as autocorrelation data and spectral information are accessible. However, our model is only valid for the reproduction of a single polarized pulse state currently. According to a pulsation operation containing a series of pulses with different polarization states, our model is not accordingly trained for effectively reproducing mode-locked pulsation operation.

5. Conclusion

We have successfully reproduced targeted mode-locked solitons by spectrotemporal domain-informed deep learning. A robust reproduction approach is proposed, i.e., combining the OSM-ADV algorithm in the spectral domain and the modified Pearson’s ${\tilde {r}}_{_{\hat {I},I}}$ in the temporal domain, for evaluating the similarity of the spectra and the autocorrelation trace, respectively. The experimental results demonstrate perfect reproduction, with a total MSE of $3.99\times {10}^{-5}$. Furthermore, our designed PPA model enables the precise retrieval of the phase distribution of the measured pulses by using experimental optical spectra and autocorrelator traces, which validates the physical completeness of our proposed deep learning reproduction algorithm. Our research contributes to a deep understanding of the dynamic properties of mode-locked pulses, enabling advancements in various fields, including ultrafast laser technology, optical communication and nonlinear optics.