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Abstract
In this work, we utilize three parallel optical reservoir computers to model three optical dynamic systems, respectively. Here, the three laser-elements in the response laser array with both delay-time feedback and optical injection are utilized as nonlinear nodes to realize three optical chaotic reservoir computers (RCs). The nonlinear dynamics of three laser-elements in the driving laser array are predictively learned by these three parallel RCs. We show that these three parallel reservoir computers can reproduce the nonlinear dynamics of the three laser-elements in the driving laser array with self-feedback. Very small training errors for their predictions can be realized by the optimization of two key parameters such as the delay-time and the interval of the virtual nodes. Moreover, these three parallel RCs to be trained will well synchronize with three chaotic laser-elements in the driving laser array, respectively, even when there are some parameter mismatches between the response laser array and the driving laser array. Our findings show that optical reservoir computing approach possibly provide a successful path for the realization of the high-quality chaotic synchronization between the driving laser and the response laser when their rate-equations imperfectly match each other.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The 1-dimensional (1-D) lateral laser array (1D-LLA) with nearest-neighbor interactions proposed by Marom et al. has attracted significant research as ideal integrated light source. In contrast to uncoupled laser array, the 1D-LLA is advantageous by a simple structure and thus easy fabrication on a single chip, which can be accurately modelled by a set of ordinary differential equations (usually called coupled laser model). A 1D-LLA exhibits rich dynamics including stable continuous wave (CW) operation, oscillatory states, and chaos, as well as collective dynamical behaviors [1–4]. The light produced by the1D-LLA can be of narrow beam, stable and characterized by significant modulation bandwidth enhancement. With delay-time feedback, a 1D-LLA exhibit rich chaos dynamical behaviors in its emitted light. Due to these advantages, the 1D-LLA has many potential applications, such as multi-channel secure communication, complex neural networks, multi-target chaotic lidar synchronous ranging. For these applications, synchronization of chaotic dynamics is a necessary and challenging task that has attracted intensive research over the recent decades [5–8]. The works reported in literatures range from complete synchronization, generalized synchronization, isochronal synchronization to leader/laggard synchronization. However, all existing techniques for optical chaos synchronization rely on the assumptions that the drive chaotic laser system are identical to the response system, and the mathematic models for the drive and response lasers are known in advance [9–11]. These assumptions do not hold in practice, where only one or some observation signals provided by some known equations are usually available [12]. Besides, implementation of high-quality optical chaos synchronization is a difficult task in practice due to the inevitable imperfect match between the driving and response lasers.
Recently, significant progress has been made in model-free prediction of chaotic systems from data using a reservoir computing (RC) approach [13–16]. As an effective way of implementing neural network computing in hardware, the RC system based on a nonlinear node and a delay feedback loop (named as delay-based RC) was first proposed by Appeltant, et al.[17]. Many hardware implementations for delay-based RC have been experimentally and theoretically demonstrated, such as electronic system [17,18], opto-electronic system [19–22], all-optical system [23–25], and laser dynamical system [26–29]. Of special interest is that the delay RC based on nonlinear semiconductor lasers are promising for high-speed computing due to high relaxation oscillation frequency and the ability of transforming low dimensional data into high dimensional state space. A few more related studies [23–41] demonstrated that this technique has the advantages of fast-speed, high efficiency, parallel computing. This technique can be applied to many problems, such as time series prediction [30–33,41], optical packet header recognition [34], speech recognition [35], and nonlinear channel equalization [31,36,37]. Moreover, existing work reported in [13–16] has shown that this approach can predict chaotic dynamical behaviors, extract Lyapunov exponents from real data, capture the evolutionary rule of large spatiotemporally chaos, and even achieve chaos synchronization between the trained reservoir computer and the driving chaotic system. In 2017, Anonik et al. demonstrated that the experimental reservoir can be trained to yield similar dynamics to the original system (similar spectrum, Lyapunov exponents, etc.) [42]. These works demonstrated that a trained delay-based reservoir computer can emulate a priori chaotic dynamical systems.
In order for 1D-LLA with chaotic dynamics to be applied in such applications as multi-channel secure communication, complex neural networks, multi-target chaotic lidar synchronous ranging, it is necessary to overcome the limitations of optical chaos synchronization theory as described above [43–45]. The delay-based RC technology in training optical chaos synchronization provides a possible solution for this problem. However, there is still not sufficient work on the quality and accuracy of the trained optical chaos synchronization using delay-based RC, as well as its potential limitations. To address to this problem, we propose the three parallel optical chaotic RCs based on three laterally coupled semiconductor lasers with double delay-time feedbacks and optical injection, which are modeled with the coupled wave theory developed by our previous work [8]. Compared with reservoir computers in three uncoupled semiconductor lasers, these three RCs in three laterally coupled semiconductor lasers are advantageous by a simple structure and thus easy fabrication on a single chip. Moreover, they can deal with higher-rate predictive data since they are characterized by significant modulation bandwidth enhancement. In this paper, we show how three trained optical chaotic RCs can reproduce the nonlinear dynamics of three elements of the original laser array with self-feedback, respectively. We explore the influences of the delay-time and the interval of the virtual nodes for the training errors. We further demonstrate that three trained optical chaotic RCs can well synchronize with three chaotic driving laser-elements of the laser array with self-feedback, respectively. At last, we estimate the predictive performances of chaotic synchronization in different parameter spaces.
2. Chaotic synchronization learning scheme and theoretical model
Figure 1 depicts a schematic diagram of three parallel optical chaotic reservoir computers to train real-time chaotic synchronizations. Here, TLC-SL represents three laterally coupled semiconductor lasers, referred to a three-element laser array. The TLC-SL$_1$ and TLC-SL$_2$ are considered as the driving and response laser arrays, respectively. They both have three identical laser waveguides (LWGs), A, B and C with an edge-to-edge separation of 2d. Each LWG is of width 2a. The LWG A locates between the LWG B and the LWG C. For the convenience of discussions, the LWGs in TLC-SL$_1$ are called as the driving lasers A$_1$, B$_1$ and C$_1$, and those in TLC-SL$_2$ are named as the response lasers A$_2$, B$_2$ and C$_2$ in turn. Three laser-elements in the TLC-SL$_1$ with self-feedback are the original dynamical system to be learned. With double delay-time feedback and optical injection, three laser-elements in the TLC-SL$_2$ are utilized as nonlinear nodes to realize three parallel RCs. The six neutral density filters (NDFs) are used to control light strength. The variable attenuators (VAs) are used to control the feedback strengths. The optical isolators (ISs) are applied to avoid light feedback. Three fiber beam splitters (FBS) are used to separate the injection light into two identical components: FBS$_1$ separates the injection light from the laser B$_1$ into input layer 1 and the photodetector 1 (PD$_1$); FBS$_2$ divides the injection light from the laser A$_1$ into input layer 2 and the PD$_2$; FBS$_3$ splits the injection light from the laser C$_1$ into input layer 3 and the PD$_3$.
Fig. 1. Schematic diagram of three parallel optical chaotic reservoir computers to train real-time chaotic synchronizations based on the three-element laser array with double delay-time feedback and optical injection (see texts for the detailed description).
The system presented by Fig. 1 is composed of three input-layers, three parallel reservoirs and three output-layers. In the input layers, the chaotic waveforms emitted from lasers A$_1$, B$_1$ and C$_1$ in the TLC-SL$_1$, as three prediction targets, are injected into respectively the three input-layers. These injected chaos signals are firstly converted into the current signals by the PDs (the subscripts of 7-9), then sampled as input data by the discrete modules (DMs) (with subscripts of 1-3), which are denoted as u$_A$(n), u$_B$(n) and u$_C$(n) respectively. Here, n is the discrete time index. These sampled data holding for a period of T are firstly amplified by electric amplifiers (EAs), then multiplied by the mask signal (named as Mask) with the periodicity T. Here the Mask is a chaotic signal generated by two mutually-coupled SLs, which is presented in [46]. After being scaled by a scaling factor $\gamma$ using the scaling operation circuit (SC), these three masked input signals are denoted as S$_A$(t), S$_B$(t) and S$_C$(t), respectively. They are utilized to modulate the phase of the optical field of the output from the driving semiconductor laser (D-SL) by means of three phase modulators (PM$_1$, PM$_2$ and PM$_3$), where the D-SL is a distributed feedback semiconductor laser used to convert the masked input signals into an optical injection signal.
In the reservoir layers, the output chaotic dynamics of the response lasers A$_2$, B$_2$ and C$_2$ in the TLC-SL$_2$ with double delay-time feedback and optical injection are utilized as nonlinear nodes to realize three parallel reservoirs, which are denoted as R$_A$, R$_B$ and R$_C$, respectively. In output layers, the outputs of the response lasers A$_2$, B$_2$ and C$_2$ are extracted at an interval $\theta$ within the feedback delay time ($\tau _c$). The number N of the virtual nodes along any of the delay lines (DL$_1$, DL$_2$ and DL$_3$) satisfies N=T/$\theta$. Here, we consider the case of desynchronization scheme, i.e. $\tau _c$=T+$\theta$, due to the desynchronization of the reservoir loop with respect to the input and output layers of the reservoir computer allows for the coupling of each internal variable to one of its neighbors, as seen in [23,47]. The states of N virtual nodes along the DL$_1$, DL$_2$ and DL$_3$ are respectively weighted and linearly summed up, ensuring that reservoirs R$_A$, R$_B$ and R$_C$ are trained to have similar dynamics to lasers A$_1$, B$_1$ and C$_1$, respectively. Here, for the optimization of the weights, the linear least-squares method is used to minimize the mean-square error between the target function and the corresponding reservoir output [26,47]. In such a system, the trained reservoirs R$_A$, R$_B$ and R$_C$ can be synchronized with the chaotic driving lasers A$_1$, B$_1$ and C$_1$, respectively. These chaotic synchronizations can be observed by the oscilloscopes (OSs) with the numbers of 1-3.
The nonlinear dynamics of the three laser-elements in the TCL-SL$_1$ with self-feedback can be described by the coupled mode theory developed by our previous work [8] as follows.
In the reservoir layers, the dynamics of the three laser-elements in the TCL-SL$_2$ with both delay-time feedback and optical injection can be modeled as
Moreover, $\tau$ is the self-feedback delay; k$_{f1}$, k$_{f2}$ and k$_{f3}$ are the self-feedback strengths of lasers A$_1$, B$_1$ and C$_1$ in turn; k$_{r1}$, k$_{r2}$ and k$_{r3}$ are the feedback strengths for the delay lines DL$_1$, DL$_2$ and DL$_3$, respectively; k$_{inj}$ is injection strength. $\tau _n^{(1)}$ and $\tau _n^{(2)}$ are the carrier lifetime; $\tau _p^{(1)}$ and $\tau _p^{(2)}$ are the photon lifetime; $\alpha _H^{(1)}$ and $\alpha _H^{(2)}$ are the line-width enhancement factor that accounts for the phase-amplitude coupling in the electric field; $\omega _1$ and $\omega _2$ are the free-running angular frequencies of the total electric field of the TCL-SL$_1$ and TCL-SL$_2$, respectively; $\Omega _A$, $\Omega _B$ and $\Omega _C$ are respectively the waveguide frequencies of LWGs A, B and C in the TCL-SL$_1$ or the TCL-SL$_2$. The term $\Delta v$ is the frequency detuning between D-SL and TCL-SL$_2$. Q$^{(1)}$ and Q$^{(2)}$ are the normalized pumping rate, which can be expressed in terms of the ratio of pumping rate to its threshold value, P/P$_{th}$, as
where C$_Q$=a$_{dif\negthickspace f}$N$_0$/g$_{th}$, with a$_{dif\negthickspace f}$ as the differential gain, N$_0$ as the carrier density under transparency, and g$_{th}$ as the unit length gain under threshold. The injected slowly varying complex electric field E$_j$(t) can be written as [33] where E$_d$ is the amplitude of CW output from the D-SL; c is the speed of light in vacuum; n$_0$ is the refractive index of the laser waveguides A, B and C in the TCL-SL$_2$.In Eqs. (8)–(14), the masked input data S$_A$(t), S$_B$(t) and S$_C$(t) can be written as
where Mask express the masked signal, which is chaos signal and presented in [46]. u$_A$(n), u$_B$(n) and u$_C$(n) are the sampled and held input signals, which are from the outputs of the lasers A$_1$, B$_1$ and C$_1$, respectively; $\gamma$ is the scaling factor.3. Results and discussions
The key material parameter values for the TCL-SL$_1$ and TCL-SL$_2$ are identical and given as follows [48]: g$_{th}$=87.7cm$^{-1}$; W$_r$=1.26, W$_i$=0; C$_\eta$=83.6ns$^{-1}$, C$_\theta$=0, n$_0$=3.4. The half-width (a) of each LWG in the TCL-SL$_1$ or TCL-SL$_2$ is taken as 4$\mu$m. The other parameter values for calculation are given as follows [33,48]: a$_{dif\negthickspace f}$= 1$\times$10$^{-15}$ cm$^2$; N$_0$= 1$\times$10$^{18}$ cm$^{-3}$; k$_{inj}$=12.43ns$^{-1}$; $\Delta v$=-4GHz and E$_d$=2.5612$\times$10$^{10}$V/m; $\tau _n^{(1)}$, $\tau _n^{(2)}$= 1.0 ns; $\tau _p^{(1)}$, $\tau _p^{(2)}$= 1.53 ps; $\alpha _H^{(1)}$=3, $\alpha _H^{(2)}$=2; k$_{f1}$, k$_{f2}$, k$_{f3}$=3ns$^{-1}$; k$_{r1}$, k$_{r2}$, k$_{r3}$=7ns$^{-1}$; d$^{(1)}$/a=1.2, d$^{(2)}$/a=3.5; P$^{(1)}$/P$_{th}$=1.2, P$^{(2)}$/P$_{th}$=1.4. The waveguide frequencies $\Omega _A$, $\Omega _B$ and $\Omega _C$ are all equal to 1.4499$\times$10$^{15}$ rad/s, which central wavelengths are all 1310 nm.
Using the above-mentioned parameters, we calculate numerically Eqs. (1)–(14) using the fourth-order Runge-Kutta method with a step of 1 ps. For the numerically solving Eqs. (1)–(7) for the TCL-SL$_1$, 5000 samples of input data (u$_A$(n), u$_B$(n) and u$_C$(n)) are recorded under the sampling interval of 10ps. After discarding the first 1000 samples (to eliminate transient states), we use the 3000 points for training the three reservoirs (R$_A$, R$_B$, R$_C$), and take their remaining 1000 points to test these reservoirs. Moreover, three mask signals are all chaotic signals generated by two mutually-coupled SLs, as presented in [46]. The intervals of these mask signals $\theta$ are all set as 20ps. The amplitudes of the mask signals are adjusted, making their standard deviations to be 1 and mean values 0. The period T of the input data is set as 8ns, and hence the data processing speed is 125 Mb/s. The number of virtual nodes N is set as 400, where N=T/$\theta$. The delay time $\tau _c$=T+$\theta$. The scaling factor $\gamma$ is set as 1. For the prediction tasks of the nonlinear dynamics of the lasers A$_1$, B$_1$ and C$_1$, the training error, i.e., the jth normalized mean-square error (NMSE$_j$) between the jth input data (u$_j$(n)) and the jth reservoir (R$_j$) output y$_j$(n) , is calculated to measure the performance of the R$_j$, which is defined as
where n is the discrete time index of the input data; L is the total number of data in the testing data set; the term var represents the variance. The NMSE with subscripts of j indicates how far the time series y$_j$(n) generated by the jth reservoir (R$_j$) deviates from the jth target time series u$_j$(n). NMSE$_j$ =0 means that y$_j$(n) is perfectly matched with u$_j$(n). When NMSE$_j$ =1 indicates that they do not match at all. Moreover, when NMSE$_j$ is less than 0.1, the trained reservoirs (R$_A$, R$_B$ and R$_C$) can infer the output chaotic trajectories from the driving lasers A$_1$, B$_1$ and C$_1$, respectively. In other word, the jth input chaotic time series u$_j$(n) from the laser-j element in the TCL-SL$_1$ wishes to synchronize with the trained predicted values y$_j$(n) from the R$_j$ output, which can be characterized by using the correlation coefficient as follows:3.1 Prediction performances for the nonlinear dynamics
In the parameter space of d$^{(1)}$/a and $\alpha _H^{(1)}$, we first calculate the dynamical evolutions of the lasers A$_1$, B$_1$ and C$_1$, and those of the trained reservoirs (R$_A$, R$_B$ and R$_C$), as displayed in Fig. 4. Here, $\tau$=4ns; $\tau _c$=8.02ns; T=8ns; $\theta$=20ps; $\alpha _H^{(2)}$=2; k$_{f1}$, k$_{f2}$, k$_{f3}$=3ns$^{-1}$; k$_{r1}$, k$_{r2}$, k$_{r3}$=7ns$^{-1}$; d$^{(2)}$/a=3.5; P$^{(1)}$/P$_{th}$=1.2, P$^{(2)}$/P$_{th}$=1.4. As seen from Fig. 2, the dynamical evolutions from the driving laser A$_1$, B$_1$ and C$_1$ are identical to those of the trained reservoirs R$_A$, R$_B$ and R$_C$ outputs,respectively. For example, if $\alpha _H^{(1)}$>1.5, u$_A$(n), u$_B$(n) and u$_C$(n) that respectively come from the outputs of the lasers A$_1$, B$_1$ and C$_1$ are chaotic state. The output states of the trained reservoirs R$_A$, R$_B$ and R$_C$, i.e., y$_A$(n), y$_B$(n) and y$_C$(n), are also chaotic. When $\alpha _H^{(1)}$ is between 1 and 1.5, and d$^{(1)}$/a ranges from 1 to 3.5, the laser A$_1$ output experiences the following dynamical evolution: from chaos (CO), single period (P$_1$), double period (P$_2$), quasi period (QP), CO, continuous wave (CW), CO, CW, CO to CW (see Fig. 2(a$_1$)). The output state of the trained reservoir R$_A$ has the same evolution with the output of the laser A$_1$ (see Fig. 2(a$_2$)). The output states of the lasers B$_1$ and C$_1$ both go through the dynamical changes from CO, QP, P$_1$, P$_2$, CO to CW (see Figs. 2(b$_1$) and 2(b$_2$)), which are identical to the output states of the trained reservoirs R$_B$ and R$_C$, respectively (see Figs. 2(c$_1$) and 2(c$_2$)). To further observe the similarity of the output of the driving laser-element to the corresponding trained reservoir output, Fig. 3 displays the samples of the chaotic time series data (u$_A$(n), u$_B$(n) and u$_C$(n)), and the output values of the reservoirs (R$_A$, R$_B$ and R$_C$) when $\alpha _H^{(1)}$=3, d$^{(1)}$/a=1.2, and the other parameters are presented in Fig. 2. It is found from Fig. 3(a) that the target value u$_A$(n) from the chaotic output of the driving laser A$_1$ is highly similar to the output value y$_A$(n) the trained reservoir R$_A$. The NMSE between them is 0.058 when $\tau _c$=8.02ns and $\theta$=20ps (see Fig. 5(a) and Fig. 6(a)). As seen from Fig. 3(b), the target value u$_B$(n) from the driving laser B$_1$ output is almost identical to the output value y$_B$(n) the trained reservoir R$_B$, and their NMSE is 0.057 at $\tau _c$=8.02ns and $\theta$=20ps (see Fig. 5(b) and Fig. 6(b)). One sees from Fig. 3(c) that the target value u$_C$(n) rom the driving laser C$_1$ is almost consistent with the output value y$_C$(n) the trained reservoir R$_C$, and their NMSE is 0.057 while $\tau _c$=8.02ns and $\theta$=20ps (see Fig. 5(c) and Fig. 6(c)). Figure 4 presents the chaotic attractors of three laser-elements in the TCL-SL$_1$ and those of three trained reservoirs, where $\alpha _H^{(1)}$=3; $\alpha _H^{(2)}$=2; d$^{(1)}$/a=1.2; d$^{(2)}$/a=3.5; $\tau$=4ns; $\tau _c$=8.02ns; T=8ns; $\theta$=20ps; k$_{f1}$, k$_{f2}$, k$_{f3}$=3ns$^{-1}$; k$_{r1}$, k$_{r2}$, k$_{r3}$=7ns$^{-1}$; P$^{(1)}$/P$_{th}$=1.2, P$^{(2)}$/P$_{th}$=1.4. As shown in Figs. 4(a$_1$) and 4(a$_2$), the chaotic attractors of the driving laser A$_1$ have almost the same type of the chaotic attractors of the trained reservoir R$_A$. Compared with Figs. 4(b$_1$) and 4(b$_2$), we found that the chaotic attractors of the driving laser B$_1$ are highly consistent with those of the trained reservoir R$_B$. As displayed in Figs. 4(c$_1$) and 4(c$_2$), the chaotic attractors of the laser C$_1$ are highly similar to those of the trained reservoir R$_C$. In addition, as observed from Figs. 2(b$_2$)–2(c$_2$) and 4(b$_2$)–4(c$_2$), the dynamical trajectories and chaotic attractors of the reservoir R$_B$ are similar with those of the reservoir R$_C$. The reason is that the dynamical trajectories and chaotic attractors of these training reservoirs depend on those of the driving lasers (laser A$_1$, B$_1$ and C$_1$) to be predicted. Under the structural symmetry between the driving lasers B$_1$ and C$_1$ their dynamical trajectories and chaotic attractors to be predicted are identical (see Figs. 2(b$_1$)–2(c$_1$), and 4(b$_1$)–4(c$_1$)).
Fig. 2. (a$_1$) Map of the dynamics from the laser A$_1$ output in the parameter space of d$^{(1)}$/a and $\alpha _H^{(1)}$; (a$_2$) Map of the dynamics of the trained reservoir R$_A$ output in the parameter space of d$^{(1)}$/a and $\alpha _H^{(1)}$; (b$_1$) Map of the dynamics from the laser B$_1$ output in the parameter space of d$^{(1)}$/a and $\alpha _H^{(1)}$; (b$_2$) Map of the dynamics of the trained reservoir R$_B$ output in the parameter space of d$^{(1)}$/a and $\alpha _H^{(1)}$; (c$_1$) Map of the dynamics from the laser C$_1$ output in the parameter space of d$^{(1)}$/a and $\alpha _H^{(1)}$; (c$_2$) Map of the dynamics of the trained R$_C$ output in the parameter space of d$^{(1)}$/a and $\alpha _H^{(1)}$. Here, CO: Chaotic state; QP: Quasi-periodic oscillation; P$_2$: Period-two oscillation; P$_1$: Period-one oscillation; CW: Stable operation. Here, $\tau$=4ns; $\tau _c$=8.02ns; T=8ns; $\theta$=20ps; $\alpha _H^{(2)}$=2; k$_{f1}$, k$_{f2}$, k$_{f3}$=3ns$^{-1}$; k$_{r1}$, k$_{r2}$, k$_{r3}$=7ns$^{-1}$; d$^{(2)}$/a=3.5; P$^{(1)}$/P$_{th}$=1.2, P$^{(2)}$/P$_{th}$=1.4.
Fig. 3. (a) A sample of the chaotic time series data u$_A$(n) from the driving laser A$_1$ output (the blue solid line ) and the trained reservoir R$_A$ output (the red dashed line); (b) A sample of the target chaotic time series data u$_B$(n) from the driving laser B$_1$ output (the red dashed line) and the trained reservoir R$_B$ output (the blue solid line); (c) A sample of the target chaotic time series data u$_C$(n) from the driving laser C$_1$ output (the blue solid line) and the trained reservoir and the trained reservoir R$_C$ output (the red dashed line). Here, $\alpha _H^{(1)}$=3, d$^{(1)}$/a=1.2, and the other parameter values are the same as those in Fig. 2.
Fig. 4. (a$_1$) and (a$_2$) Chaotic attractors as a function of (M$_A^{(1)}$, u$_A$) from the driving laser A$_1$ and those as a function of (M$_A^{(2)}$, y$_A$) from the trained reservoir R$_A$, respectively; (b$_1$) and (b$_2$) Chaotic attractors as a function of (M$_B^{(1)}$, u$_B$) from the driving laser B$_1$ and those as a function of (M$_B^{(2)}$, y$_B$) from the trained reservoir R$_B$, respectively; (c$_1$) and (c$_2$) Chaotic attractors as a function of (M$_C^{(1)}$, u$_C$) from the driving laser C$_1$ and those as a function of (M$_C^{(2)}$, y$_C$) from the trained R$_C$, respectively. Here, $\alpha _H^{(1)}$=3, d$^{(1)}$/a=1.2, and the other parameter values are the same as those in Fig. 2.
Fig. 5. (a) The NMSE$_A$ as a function of the delay-time $\tau _c$ for the prediction of the reservoir R$_A$ to the chaotic time series from the laser A$_1$ output; (b) the NMSE$_B$ as a function of $\tau _c$ for the prediction of the reservoir R$_B$ to the chaotic time series from the laser B$_1$ output; (c) the NMSE$_C$ as a function of $\tau _c$ for the prediction of the reservoir R$_C$ to the chaotic time series from the laser C$_1$ output. Here, the parameters except for $\tau _c$ are the same as those in Fig. 2.
Fig. 6. (a) The NMSE$_A$ as a function of the virtual node interval $\theta$ for the prediction of the reservoir R$_A$ to the chaotic time series from the laser A$_1$ output; (b) NMSE$_B$ as a function of $\theta$ for the prediction of the reservoir R$_B$ to the chaotic time series from the laser B$_1$ output; (c) NMSE$_C$ as a function of $\theta$ for the prediction of the reservoir R$_C$ to the chaotic time series from the laser C$_1$ output. Here, the parameters except for $\theta$ are the same as those in Fig. 2.
3.2 Training errors for the nonlinear dynamics
To further explore the predictive performances of the trained reservoirs R$_A$, R$_B$ and R$_C$ to the dynamics of the driving lasers A$_1$, B$_1$ and C$_1$, respectively, Fig. 5 displays three training errors (NMSE$_A$, NMSE$_B$ and NMSE$_C$) as a function of the delay-time $\tau _c$ under $\theta$=20ps. It can be clearly seen from Fig. 5 that in the region of $\tau _c$ between 0.1ns and 9ns, these three training errors are less than 0.065, but they show a rise in oscillation with the increase of $\tau _c$. The reason that a longer delay-time $\tau _c$ results in an oscillation rising training error may be explained as follows. In this work, when $\theta$ is fixed at 20ps, T=$\tau _c$-$\theta$ and N=$\tau _c$/$\theta$ −1, a larger N is accompanied by a larger $\tau _c$, indicating that a higher dimension state space, which can lead to the case that the predictions of the trained reservoirs to the dynamics of the driving laser-elements become unsteady and more difficult. Therefore, this case results in a larger NMSE. Moreover, for the predictions of the chaotic time series of three driving laser-elements, Fig. 6 shows their training errors as a function of the virtual node interval $\theta$. From this figure, one sees that with T fixed at 8ns, the training errors appear cliff-like decline when $\theta$ increases from 4ps to 15ps, then gradually stabilize to 0.048 with the increase of $\theta$ from 15ps to 200ps. The reason is explained as follows. A very small $\theta$ leads to the reduction of the trained reservoir responses, which indicates larger training errors. If $\theta$ further increases from 15 ps to 200 ps, the responses of the trained reservoirs are further enhanced, which induces less training errors. In addition, although the dimension of state space is further lowered with the increase of $\theta$, there are still enough virtual nodes for training reservoirs, which makes these training errors to be small and changed between 0.048 and 0.058.
3.3 Predictive learning of three-channel chaotic synchronization
By the observations of Figs. 2–6, we find that the trained reservoirs can accurately infer the dynamical trajectories of the driving laser-elements by the optimization of some key parameters such as $\tau _c$, $\theta$, k$_{f1}$, k$_{f2}$, k$_{f3}$,k$_{r1}$, k$_{r2}$ and k$_{r3}$. In the other words, three trained reservoir computers based on the driving lasers A$_2$, B$_2$ and C$_2$ can reproduce the nonlinear dynamics of the driving lasers A$_1$, B$_1$ and C$_1$, respectively, and their training errors are very small if $\theta$ is more than 15ps and $\tau _c$ is less than 10ns. These findings indicate that the three trained reservoirs can well synchronize with three chaotic driving laser-elements, respectively, i.e., u$_A$(n)/y$_A$(n)$\approx$1, u$_B$(n)/y$_B$(n)$\approx$1 and u$_C$(n)/y$_C$(n)$\approx$1. To realize their chaos synchronizations, in the output layer, the jth time-dependent output y$_A$(n)from the jth reservoir R$_j$ is taken to be a linear function of the reservoir sate and input data such that
where the term W$_{j,i}$ is the ith-element of the output weight matrix W$_j$; the term b$_{out}$ is a constant and equal to 1; the variable $Y_{j,i}^{(2)}(n)$ represents the ith internal states of the jth reservoir R$_j$. Previous studies have shown that the output weight matrix W$_j$ can be analytically given by [21,22] where the subscript j means the jth reservoir R$_j$; the superscript T represents transpose; $\Pi$ is an identity matrix; $\lambda$ is the ridge regression parameter for avoiding overfitting, which is set as 10$^{-6}$. X$_j$ is the matrix whose kth column is [b$_{out}$;u$_j$(k);$Y_{j,i}^{(2)}(k)$] and Z$_j$ is the matrix whose kth column is [u$_j$(k+1)].To further look into the quality of their synchronizations, based on Eqs. (23)–(25), we presents the evolutions of the correlation coefficients ($\rho _A$, $\rho _B$ and $\rho _C$) in parameter spaces of d$^{(1)}$/a and $\alpha _H^{(1)}$, d$^{(1)}$/a and P$^{(1)}$/P$_{th}$, d$^{(1)}$/a and $\gamma$, k$_{f1}$ and k$_{r1}$, k$_{f2}$ and k$_{r2}$, k$_{f3}$ and k$_{r3}$, which are displayed Fig. 7. As seen from Figs. 7(a$_1$), 7(b$_1$) and 7(c$_1$), where $\alpha _H^{(1)}$ is between 1 and 4, $\rho _A$ is not less than 0.975, $\rho _B$ and $\rho _C$ are greater than or equal to 0.975. One sees from Figs. 7(a$_2$), 7(b$_2$) and 7(c$_2$), $\rho _A$, $\rho _B$ and $\rho _C$ are all more than 0.97 when P$^{(1)}$/P$_{th}$ ranges from 1.1 to 4. As shown in Figs. 7(a$_3$), 7(b$_3$) and 7(c$_3$), these correlation coefficients are all more than 0.98 when $\gamma$ is between 0.1 and 2. As further observed from Figs. 7(a$_4$)–7(c$_4$), they are all more than 0.99 when the feedback strengths(k$_{r1}$, k$_{r2}$ and $k_{r3}$) and the self-feedback strengths(k$_{f1}$, k$_{f2}$ and $k_{f3}$) are all between 1ns$^{-1}$ and 10ns$^{-1}$.
Fig. 7. Maps of the evolutions of the correlation coefficients ($\rho _A$, $\rho _B$ and $\rho _C$) in different parameter spaces. Here, $\tau$=4ns; $\alpha _H^{(2)}$=2; d$^{(2)}$/a=3.5; P$^{(2)}$/P$_{th}$=1.4; $\tau _c$=8.02ns and T=8ns; $\theta$=20ps; (a$_1$)-(c$_1$): the parameter space of $\alpha _H^{(1)}$ and d$^{(1)}$/a where $\gamma$= 1, k$_{f1}$, k$_{f2}$, k$_{f3}$=3ns$^{-1}$; k$_{r1}$, k$_{r2}$, k$_{r3}$=7ns$^{-1}$ and P$^{(1)}$/P$_{th}$=1.2; (a$_2$)-(c$_2$): the parameter space of P$^{(1)}$/P$_{th}$ and d$^{(1)}$/a where $\alpha _H^{(1)}$=3, k$_{f1}$, k$_{f2}$, k$_{f3}$=3ns$^{-1}$; k$_{r1}$, k$_{r2}$, k$_{r3}$=7ns$^{-1}$ and $\gamma$ = 1; (a$_3$)-(c$_3$): that of $\gamma$ and d$^{(1)}$/a where $\alpha _H^{(1)}$ =3, k$_{f1}$, k$_{f2}$, k$_{f3}$=3ns$^{-1}$; k$_{r1}$, k$_{r2}$, k$_{r3}$=7ns$^{-1}$ and P$^{(1)}$/P$_{th}$ =1.2; (a$_4$): that of k$_{f1}$ and k$_{r1}$ where $\alpha _H^{(1)}$ =3, $\gamma$ = 1, P$^{(1)}$/P$_{th}$=1.2, d$^{(1)}$/a=1.2, k$_{r1}$, k$_{r2}$= k$_{r3}$ and k$_{f1}$, k$_{f2}$=k$_{f3}$; (b$_4$): that of k$_{f2}$ and k$_{r2}$ where $\alpha _H^{(1)}$ =3, $\gamma$ = 1, P$^{(1)}$/P$_{th}$=1.2, d$^{(1)}$/a=1.2, k$_{r1}$, k$_{r2}$= k$_{r3}$ and k$_{f1}$, k$_{f2}$=k$_{f3}$; (c$_4$): that of k$_{f3}$ and k$_{r3}$ where $\alpha _H^{(1)}$ =3, $\gamma$ = 1, P$^{(1)}$/P$_{th}$=1.2, d$^{(1)}$/a=1.2, k$_{r1}$, k$_{r2}$= k$_{r3}$ and k$_{f1}$, k$_{f2}$=k$_{f3}$.
These results show that even when there are mismatches between some key parameters of the driving laser-element and the response laser-element, high-quality chaotic synchronization between one driving laser and its corresponding trained RC can still be achieved. The reason is that under very small training errors, these three well trained RCs can accurately infer the chaotic trajectories of three driving laser-elements, respectively.
4. Conclusions
We proposed to use a machine-learning technique by means of three optical chaotic reservoir computers for the modeling of three optical nonlinear dynamic systems, respectively. The three optical chaotic reservoir computers are based on the trained three-element laser array subject to double delay-time feedback and optical injection simultaneously, and the three optical nonlinear dynamic systems to be modeled are driving three-element laser array with self-feedback. It is found that the three trained reservoir computers can accurately predict the dynamical trajectories of three driving laser-elements, respectively, where very small training errors for their nonlinear dynamic predictions are obtained by the optimization of the delay-time and the interval of the virtual nodes. Moreover, these three trained reservoir computers can well synchronize with the corresponding driving chaotic laser-elements, respectively.
In general, the previously reported works on optical reservoir computers are usually applied in the benchmark tasks, such as Santa-Fe time series prediction, channel equalization, speech recognition, and so on. Compared with these works, the advantages of our presented reservoir computers are as follows: accurate prediction for all dynamical behaviors (including chaos, period-one, quasi-period, period-two, stable operation, etc) of nonlinear lasers; predictive learning of multi-channel high-quality chaotic synchronization under larger parameters mismatch. Moreover, our findings show that these chaotic synchronizations are very robust even despite existence of some key parameter mismatches between the response three-element laser array and the driving three-element laser array. Our work shows that the optical reservoir computing approach is an effective technology to achieve high-quality chaotic synchronization between the driving laser and the response laser when their rate-equations imperfectly match each other. Our results provide potential applications in real multichannel secure communications, real complex neural network, and real synchronized lidar ranging for multi-target, where the high-quality synchronization is required under the imperfect matching for many parameters.
Funding
National Natural Science Foundation of China (62075168); GuangDong Basic and Applied Basic Research Foundation (2020A1515010088); Major Project of Basic Research and Applied Research for Natural Science in GuangDong Province (2017KZDXM086); Special project in key fields of Guangdong Provincial Department of Education of China (2019KZDZX1025, 2020ZDZX3052).
Disclosures
The authors declare no conflicts of interest.
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