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Abstract
Studying the chaotic dynamics of semiconductor lasers is of great importance for their applications in random bit generation and secure communication. While considerable effort has been expended towards investigating these chaotic behaviors through numerical simulations and experiments, the accurate prediction of chaotic dynamics from limited observational data remains a challenge. Recent advancements in machine learning, particularly in reservoir computing, have shown promise in capturing and predicting the complex dynamics of semiconductor lasers. However, existing works on laser chaos predictions often suffer from the need for manual parameter optimization. Moreover, the generalizability of the approach remains to be investigated, i.e., concerning the influences of practical laser inherent noise and measurement noise. To address these challenges, we employ an automated optimization approach, i.e., a genetic algorithm, to select optimal reservoir parameters. This allows efficient training of the reservoir network, enabling the prediction of continuous intensity time series and reconstruction of laser dynamics. Furthermore, the impact of inherent laser noise and measurement noise on the prediction of chaotic dynamics is systematically examined through numerical analysis. Simulation results demonstrate the effectiveness and generalizability of the proposed approach in achieving accurate predictions of chaotic dynamics in semiconductor lasers.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Chaotic semiconductor lasers have been studied for a variety of applications including random bit generation and secure communication [1–9]. The chaotic dynamics are usually generated by semiconductor lasers subject to various forms of perturbations such as optical injection, optical feedback, and mutual coupling [10–20]. To study the dynamical behaviors, chaotic waveforms have been investigated in both numerical simulations and experiments. For instance, the time-varying intra-cavity parameters can be obtained by considering the rate equation model in numerical simulations [21,22]. By contrast, in experiments the chaotic emission is usually converted into an electrical signal and then recorded as an intensity time series. It can be further processed for evaluating the autocorrelation function, power spectrum, and probability distribution of the chaotic emission [23–29]. Alternatively, by reconstructing the states from a chaotic time series in simulations and experiments, the diverging trajectories of neighboring states have been examined in the chaotic attractor, where the Lyapunov exponents have also been evaluated [30–33].
In recent years, significant advancements have been made in the field of computer science, particularly in the realm of machine learning. These advancements have sparked a growing interest in exploring the application of machine learning algorithms for predicting chaotic dynamics in semiconductor laser systems. One approach that has shown promise is reservoir computing [34–38], which leverages the power of neural networks with dynamic reservoirs. By using the knowledge of a single variable based on a simulated laser model, reservoir computing has been employed to reconstruct the dynamics of two out of the three variables that govern the laser’s dynamics [39]. Beyond the reconstruction of laser variables, various machine learning algorithms have been compared to predict the peak amplitudes of consecutive pulses in chaotic lasers [40]. Furthermore, a recent study on reservoir computing has demonstrated the capability to forecast the continuous intensity time series and the long-term dynamics of a chaotic optically injected semiconductor [41]. Additionally, the prediction has also been implemented using a single-node photonic reservoir computing approach for a laser with optical feedback [42]. These results have shown great potential of reservoir computing as a powerful tool in understanding and predicting the dynamical behavior of chaotic laser systems, even with limited observational data.
As a key limitation, to the best of our knowledge, the above-mentioned works for laser chaos prediction typically require manual optimization of reservoir parameters such as spectral radius and leakage rate. These manual tuning processes often require expert knowledge and trial-and-error iterations, hindering the efficiency of reservoir computing for laser chaos prediction. Exploring alternative approaches for selecting optimal reservoir parameters would alleviate these limitations and enhance the practicality of reservoir computing for laser chaos prediction.
Moreover, while the works described above primarily focus on predicting intensity time series in numerical simulations, two crucial factors were usually not considered, i.e., the inherent noise present in chaotic lasers and the measurement noise introduced during data acquisition. The effects of these noise sources on the accuracy and reliability of predictions remain to be investigated for general applications in real-world scenarios.
In this paper, we aim to address the aforementioned challenges in the prediction of chaotic dynamics in semiconductor lasers using reservoir computing. Specifically, our investigation focuses on overcoming the limitations associated with manual optimization of reservoir parameters. We introduce a genetic algorithm that leverages automated methods for parameter optimization, thereby enhancing the efficiency of reservoir computing for laser chaos prediction. Furthermore, we extend the existing prediction model to account for the inherent noise present in laser systems and the effects of measurement noise. By considering these noise sources, we aim to improve the reliability and generalizability of our proposed approach in accurately predicting chaotic dynamics in semiconductor laser systems.
Following this introduction, Section 2 presented the numerical model for optically injected semiconductor lasers, the concepts of reservoir computing, and the principles of genetic algorithm for selecting reservoir parameters. Section 3 investigates the details on the optimization of different parameters for efficient prediction of laser chaos. Section 4 investigates the prediction results by considering the laser spontaneous emission noise and measurement noise. These are followed by a conclusion in Section 5.
2. Simulation model
The reservoir computing and genetic algorithm used in this work are applicable to semiconductor lasers subject to different forms of external perturbations. Here we consider simulating an optically injected semiconductor laser in a conventional master-slave configuration. It should be noted that in practical experiment the chaotic emission is usually measured in the form of an intensity time series through optical-to-electrical conversion. Thus, in simulation we will focus on the generation of a chaotic intensity time series by a rate-equation model. The data is then fed into the reservoir network for training and prediction.
2.1 Optically injected semiconductor laser
Numerically, the slave laser is described by a complex intra-cavity field amplitude $a(t)$ and a charge carrier density $\tilde{n}(t)$. The injection is specified by the normalized injection strength ${\xi _\textrm{i}}$ and detuning frequency ${f_\textrm{i}}$ with respect to the free-running optical frequency of the slave laser. The rate equations for the slave laser are given by [26]:
2.2 Reservoir computing
Figure 1 shows the implementation of reservoir computing for laser chaos prediction [41]. In the training phase, a chaotic intensity time series is first recorded and converted into an input vector u(t). It is then fed into the reservoir network through an input weight matrix ${{\boldsymbol W}_{in\; }}$, in which the elements are randomly drawn within the range [-1, 1]. The reservoir network consists of N reservoir nodes properly connected through a weighted adjacency matrix A, where a spectral radius ρ is used to describe the largest magnitude of its eigenvalues.
Driven by the input vector u(t), the state r(t) of the reservoir system is updated as follows [34,41]:
To evaluate the accuracy of the prediction, the root-mean-squared (RMS) error is calculated by comparing the predicted and simulated intensity time series [34,41]. The RMS error is given by:
2.3 Genetic algorithm
Despite the effective training of reservoir network, a good performance of reservoir computing has been found dependent on the optimal choice of the reservoir parameters [44]. In particular, the careful selection of spectral radius $\rho$ and leakage rate $\delta$ has been found important for successful prediction of the laser chaotic dynamics [39,41]. However, reservoir parameter optimization using traditional manual methods often rely on heuristics or expert knowledge, which may lead to suboptimal results. The genetic algorithm, on the other hand, provides a robust global search capability [45–48]. It explores a diverse set of parameter combinations, enabling the discovery of optimal or near-optimal solutions that may not be readily apparent using manual approaches.
Genetic algorithm is a powerful optimization technique inspired by natural selection and evolution. It is well-suited for exploring a large parameter space for finding the optimal parameters of the reservoir network. To initialize the algorithm, an initial population of individuals (i.e., parameter sets ($\rho$, $\delta$)) is first created, representing potential solutions to the optimization problem. The genetic algorithm operates by iteratively applying genetic operations, including selection, crossover, and mutation, to evolve and produce new generations of individuals towards better solutions. To evaluate the fitness of each individual in the population, here the RMS error is calculated by comparing the predicted and simulated chaotic intensity time series according to Eq. (5).
In the selection step, individuals from the current population are carefully chosen to become parents for the next generation. The selection process is typically based on a tournament selection method. A number of k individuals are randomly collected and compared according to their fitness (e.g., RMS error), with the individual having the minimum RMS error being selected. The entire process is repeated until a complete population is formed. The selection process is then followed by a one-point crossover operation. Two individuals are randomly chosen from the current population, and the genetic information (i.e., $\rho$ and $\delta$) of them are swapped with each other. Note that crossover is invoked only if a randomly generated number in the range of 0 to 1 is smaller than a predetermined crossover rate pc. Otherwise, the genetic information of the individual remains unaltered. The crossover allows for the exploration of different combinations of genetic information (i.e., trying different combinations of $\rho$ and $\delta$) and introduces diversity into the population. After crossover, the mutation operation is applied to introduce random changes into the population, i.e., altering $\rho$ and $\delta$ values within a specified range. For each individual, either $\rho$ or $\delta$ is randomly chosen to mutate, while the probability of mutation is controlled by a mutation rate pm.
By repeating these steps over multiple generations, the genetic algorithm explores and searches the parameter space, gradually improving the quality and fitness of the population (i.e., reducing the RMS error). The overall goal is to iteratively evolve the population towards better solutions, following the principles of natural evolution. The process continues until it reaches a maximum number of iterations, where optimal reservoir parameters is usually obtained.
3. Results
To demonstrate the effectiveness of using genetic algorithm for reservoir computing, here we investigate the optimization of spectral radius $\rho$ and leakage rate $\delta$, which are two of the most important parameters for prediction of laser dynamics [39,41]. The genetic algorithm is first initialized by creating a number of 20 individuals. Each individual is a parameter combination of ($\rho$, $\delta$), in which the values are randomly chosen within $0 < \rho \le 1.5$ and $0 < \delta \le 1$. By setting the reservoir size to N = 5000, a reservoir network can be constructed for each individual ($\rho$, $\delta$). In the training phase, an intensity time series is first recorded with L = 30000 data points sampled every $\Delta t$=1.19ps from the simulation. It is fed into the reservoir network for data training and prediction. To evaluate the fitness of each individual, a RMS error is computed by comparing the predicted and simulated intensity time series. Genetic operations are then applied to generate a new set of individuals for the next generation, with a tournament size k = 3, a crossover rate pc = 0.7, and a mutation rate pm = 0.05. These steps are repeated over multiple iterations, while in each iteration a minimum RMS error is obtained for the 20 individuals.
Figure 2 shows the minimum RMS error versus the number of iterations. For an iteration number of zero, the genetic algorithm is initialized with randomly selected individual ($\rho$, $\delta$). The parameters are used in the reservoir network, while the resultant RMS error is found higher than 0.3. Therefore, careful optimization of the reservoir parameters is required for minimizing the RMS error. After one genetic iteration, the minimum RMS error is efficiently reduced to about 0.1. As the number of iteration further increases, the RMS error continues to decrease and it reaches a sufficiently low level of 0.03 for about 30 iterations. The optimal individual ($\rho$, $\delta$) is thus determined to be $\rho$=1.15 and $\delta$=0.35. Based on these reservoir parameters, Fig. 3 shows successful prediction of the continuous chaotic intensity time series for a time duration of longer than 0.6 ns, which is six times the reciprocal of the laser relaxation resonance frequency. The prediction time is also found comparable to that in a similar work, where careful tuning and optimization of the reservoir parameters were typically required [41].
Fig. 3. Chaotic emission intensity time series simulated by numerical modeling of an optically injected laser (solid black line) and predicted by reservoir computing (dashed red line).
The prediction of chaotic emission intensity time series adopts reservoir size N = 5000 and training data length L = 30000. These parameters are chosen according to previously reported work on predictions of laser chaos [41]. However, other choices of reservoir size and training data length are allowed as long as the RMS error remains at a sufficiently low level. Figure 4 plots the RMS error as a function of the reservoir size N and the training data length L. Each data point in Fig. 4 is the mean RMS error over 20 different random realizations of the reservoir network, and the error bars indicate the 95% confidence interval. In Fig. 4(a), the reservoir size N is varied from 2000 to 8000, while the training data length is fixed at L = 30000. As the reservoir size N increases, the resultant RMS error quickly decreases and it is lower than 0.05 for a reservoir size of larger than 4000. In Fig. 4(b), the length of the data points used in the training phase is varied for evaluating the RMS error. As L increases, the RMS error reduces monotonically and approaches a constant value of lower than 0.05 for L of larger than 20000. Adopting an excessively large L is found not effective for further reducing the RMS error. It should be noted that, the computation time is proportional to the reservoir size and training data length, thus the choice of N = 5000 and L = 30000 is acceptable under practical computational considerations.
To quantify the performance of the prediction, the RMS error is calculated by comparing the predicted and simulated intensity time series over l = 500 data points according to Eq. (5). The value of l is chosen such that reservoir parameters can be efficiently optimized by genetic algorithm. Figure 5 further investigates the choice of other l for evaluating the RMS error, where only one of the reservoir parameters ($\rho$, $\delta$) is varied in Fig. 5(a) and 5(b). The data are also averaged over 20 different random realizations of the reservoir network. In Fig. 5(a), the RMS error is plotted as a function of the spectral radius $\rho$, while the leakage rate is fixed to be $\delta$=0.35. For a small l = 300, the prediction is found successful for most choices of $\rho$, where the RMS error remains not much changed. By increasing l, the resultant RMS error is found more dependent on the value of $\rho$. In particular, for l = 500, the RMS error reaches a minimum value of less than 0.05 for $\rho$=1.15. If l further increases, e.g., for l = 700, the RMS errors are generally much higher and they are not much affected by varying $\rho$. This is because the prediction is no longer accurate with the increase of time. The prediction is usually failed for large l such that the resultant RMS error is very large and is not much changed for most choices of $\rho$. Figure 5(b) plots the RMS error versus the leakage rate $\delta$ for different choices of l, where the spectral radius is set to $\rho$=1.15. The results are quite similar to that in Fig. 5(a). In particular, for l = 500, the RMS error is found to be minimized at $\delta$=0.35. According to Fig. 5, other choices of l are allowed as long as the minimum RMS error can be efficiently identified for determining the optimal reservoir parameters $\rho$ and $\delta$. Nonetheless, the proper selection of l is found useful for efficient optimization of reservoir parameters.
Fig. 5. RMS error versus the (a) spectral radius $\rho$ and (b) leakage rate $\delta$. The RMS error is obtained by comparing predicted and simulated intensity time series over l = 300, 500, 700, and 900 data points.
Despite the successful optimization of reservoir parameters in Figs. 2–5, the generalizability of our proposed approach for laser chaos prediction remains to be investigated. In particular, it is of great interest whether these reservoir parameters determined by genetic algorithm optimization are suitable for other cases when a different intensity time series is fed into the reservoir network. Besides, the RMS error is also found dependent on the random initialization of the reservoir system according to above simulations. This is further investigated in Fig. 6 by evaluating the probability distributions of the calculated RMS errors for different cases. In Fig. 6(a), the probability distribution is obtained for 100 different random initializations of the reservoir systems. Despite the change of the random numbers for system initialization, about 90% of the resultant RMS errors are kept lower than 0.05. This shows that the prediction is not much affected by the random initializations of the reservoir systems, as long as the reservoir parameters are properly selected.
Fig. 6. Probability distributions of the calculated RMS error by considering (a) different random realizations of the reservoir system and (b) different emission intensity time series in training phase.
Figure 6(b) shows the probability distributions for the prediction of 100 different intensity time series, which are randomly collected from the simulated I(t) with injection parameters (${\xi _\textrm{i}}$, ${f_\textrm{i}}$) = (0.05, 8 GHz). The distribution of the RMS error is found similar to that in Fig. 6(a), though about 15% of the cases are found larger than 0.1. Thus, even though optimal reservoir parameters can be determined by training a given intensity time series, the prediction performance could be affected if the reservoir network is used for a different intensity time series. Indeed, the predictability of the chaotic intensity time series is related to the memory time, which is the time for the growth of entropy. Previous investigations demonstrated that the memory time is dependent on the initial state at t = 0, i.e., the position on the attractor [31]. For different intensity time series in our case, even though they are generated by using the same injection parameter, they actually correspond to different positions on the attractor. Thus, by considering different intensity time series the prediction time could be affected such that the RMS error in Fig. 6(b) cannot always maintain at a sufficiently low level. Nonetheless, the RMS error for more than 80% of the 100 cases is found lower than 0.1. The RMS error can be further reduced for each case by optimizing the reservoir parameters using the proposed genetic algorithm.
4. Effects of noise
Due to the effect of noise amplification by chaotic mixing, the prediction of chaotic dynamics can be affected by the inherent noise of the chaotic laser [30–33]. The Shannon entropy for a chaotic laser has been estimated by numerical perturbations using different noise series on an initial state. It was also shown that the growth of entropy (i.e., memory time) decreases logarithmically with the increase in noise strength [31]. Thus, the effects of laser inherent noise on the prediction performance is first investigated in detail in this session. The noise is in the form of spontaneous emission modeled using ${f_{\textrm{sp}}}$ in Eq. (1) at a strength specified by the free-running laser linewidth $\Delta \upsilon$.
Figure 7 shows the RMS error versus $\Delta \upsilon$ by averaging over 20 random initializations of the reservoir network. In Fig. 7(a), the RMS error is first calculated by setting l = 500 in Eq. (5). for $\Delta \upsilon = 0$, the prediction is performed using the reservoir parameters ($\rho$, $\delta$) = (1.15, 0.35), which are determined by using genetic algorithm. The resultant RMS error is lower than 0.05, which is consistent with the results in Section 3. The prediction is then performed on intensity time series generated by considering the laser inherent noise, while the reservoir parameters remain unchanged during the training phase. The red line in Fig. 7(a) shows the RMS error versus the laser linewidth $\Delta \upsilon$. As noise strengthens, the increasing $\Delta \upsilon$ always causes RMS error to increase, which is consistent with the reduction of memory time in previous works [31]. for $\Delta \upsilon > 1\textrm{kHz}$, the prediction is completely failed with RMS errors much higher than 0.5. The fast increment of the RMS error is in fact obtained as the laser states are continually mixed by the chaotic dynamics and constantly perturbed by noise. Therefore, if the reservoir parameters are used for a laser model where inherent noise is considered, the prediction accuracy could be significantly affected.
Fig. 7. RMS error versus the free-running laser linewidth $\Delta \upsilon$. The RMS error is calculated according to Eq. (5) by setting (a) l = 500 and (b) l = 200. Red line: the reservoir parameters are fixed to ($\rho$, $\delta$) = (1.15, 0.35) for each $\Delta \upsilon$. Black line: the reservoir parameters are optimized for each $\Delta \upsilon$.
In order to improve the prediction performance, the reservoir parameters are optimized for each $\Delta \upsilon$ by using genetic algorithm. The corresponding results are shown by the black line in Fig. 7(a). After optimization, the RMS error is found effectively suppressed for different $\Delta \upsilon$. In particular, the RMS error is reduced to merely 0.1 for $\Delta \upsilon = 1\textrm{kHz}$. Nonetheless, the RMS error is always increased due to noise enhancement. This shows the difficulty of predicting intensity time series for more than l = 500 data points, despite the optimization of reservoir parameters. This is further investigated by setting l = 200 in Fig. 7(b). Due to reduction of the data points for calculating RMS error in Eq. (5), the RMS error is found much lower than that in Fig. 7(a) for different $\Delta \upsilon$. By optimizing the reservoir parameters for each $\Delta \upsilon$, the RMS error is not much changed in Fig. 7(b). Moreover, the RMS errors for both cases are found lower than 0.1 for a laser linewidth of lower than 100 kHz. Therefore, by considering laser inherent noise, the optimization of reservoir parameters for each $\Delta \upsilon$ is helpful for reducing the RMS error.
In above figures, the reservoir parameters are determined based on an intensity time series which is ideally the squared magnitude of the complex laser field, i.e., $I(t) = {|{a(t)} |^2}$. However, it is of great interest whether the prediction will remain successful if such reservoir parameters are used for data contaminated by measurement noise. To evaluate the effects of measurement noise on predicting laser dynamics, in simulation, white Gaussian noise is added to the chaotic emission intensity I(t), where the strength of noise is characterized by the signal-to-noise ratio (SNR). Figure 8 shows the prediction of intensity time series by varying the SNR of the training data. The reservoir parameters are first determined to be ($\rho$, $\delta$) = (1.15, 0.35) by using an ideal emission intensity I(t) as the training data. This allows successful prediction of continuous intensity time series as shown in Fig. 8(a), which is consistent with the results in Fig. 3. In the following the training data is obtained by adding a measurement noise to I(t), while the reservoir parameters remain unchanged for each case. As noise strengthens in Figs. 8(b)–8(c), the SNR continues to decrease, while the prediction accuracy is not much affected. This is because for these cases the measurement noise is relatively weak, such that it has no obvious influences on the training data. When the SNR is reduced to 15 dB in Fig. 8(d), the intensity time series (solid black line) is dominated by strong noise fluctuations, leading to significant deterioration of prediction. Nonetheless, the prediction output waveform (dashed red line) remains irregular and smooth, where the first few laser pulses are found successfully reproduced. Therefore, the prediction of continuous intensity time series remains successful as long as the strength of measurement noise is maintained at a sufficiently low level.
Fig. 8. Predicted (dashed red line) and simulated (solid black line) intensity time series by varying the SNR.
The effect of measurement noise on attractor reconstruction is further investigated by varying the SNR in Fig. 9, where two-dimensional attractors are plotted with a delay time of $7\Delta t$[30]. If noise is absent, the chaotic attractor is accurately reconstructed in Fig. 9(a), as reported in a previous literature [41]. For relatively large SNR, attractor reconstruction remains successful according to Fig. 9(b)–9(c), which is also consistent with the results in Figs. 8(b)–8(c). For SNR = 15 dB, the attractor for chaotic emission is dominated by noise in Fig. 9(d-i), where no specific structures are observed. For comparison, in Fig. 9(d-ii), the attractor is obtained using the predicted intensity time series in Fig. 8(d). The envelope of the chaotic attractor is still reconstructed. Therefore, the prediction algorithm is efficient in learning the chaotic dynamics of semiconductor lasers despite that the emission intensity time series is contaminated by measurement noise.
Fig. 9. Reconstructed two-dimensional attractor based on the (a) simulated and (b) predicted intensity time series by varying the SNR.
5. Conclusion
In conclusion, we have investigated the prediction of chaotic dynamics in an optically injected semiconductor laser using reservoir computing. The reservoir parameters are efficiently selected through the application of an automated genetic algorithm optimization approach. The choice of other parameters for training the reservoir network is also investigated in detail. Simulation results demonstrate the flexibility of choosing appropriate operating parameters for prediction of continuous intensity time series. Moreover, by exploring the influence of practical laser inherent noise and measurement noise, the robustness and generalizability of the proposed approach is validated. As such, this research paves the way for understanding and predicting the chaotic dynamics of semiconductor lasers, which could be important for ensuring the security in communication and random bit generation.
Funding
National Natural Science Foundation of China (62204029); Natural Science Foundation of Liaoning Province (2022-MS-132).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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