1. Introduction

Reservoir computing (RC) as a brain-inspired computational paradigm originated from recurrent neural network has been investigated theoretically and experimentally [1–10]. RC has been successfully applied in solving computationally hard tasks, such as speech recognition systems [3–6], financial forecasting [7], handwritten digit recognition [8,9], and chaotic systems prediction [10]. In a RC system, sequential input is nonlinearly mapped to a high dimensional space. Connection weights between input and node in the reservoir are fixed. Then the input-driven dynamic responses of all nodes in the reservoir are weighted and combined by the linear readout function of the output layer. Hence, only output weights need to be trained using simple algorithm, such as linear regression [11]. Usually, in the experimental implementation of a RC, a large number of randomly connected physical nodes are used as a standard reservoir to provide a sufficiently high dimensionality [11–13].

Recently, a simplified reservoir structure consisting of a single nonlinear node subject to delayed feedback was proposed by Appltant et al. [5]. After that, the delay-based RC has been implemented in optoelectronic [14,15], all-optical [6,16], and laser dynamical systems [17–25]. Among them, the laser dynamical system is a promising candidate to realize delay-based RC with ultra-high information processing rate, because of its inherently fast operation, parallelism, high bandwidth and ultra-low consumption. For example, in 2013, Brunner et al. experimentally demonstrated a semiconductor laser (SL) with delayed feedback as a reservoir. The transient responses of SL were employed for realizing RC with low classification and prediction error rates, and the information processing rate was up to 1 Gbps [17]. In 2017, Romain et al. experimentally and numerically demonstrated a RC system based on an optically pumped laser with optical delayed feedback. In this system, the input signal was directly coupled to the optical delayed feedback beam without additional external injection [18]. In 2019, they further proposed a RC system based on a single quantum cascade laser (QCL). It was found that, even without external injection, the QCL subject to delayed optical feedback could extend stable parameter area contributing to good RC performances [24].

In these delay-based RC systems, it is easy to achieve a large number of virtual nodes by using time multiplexing [5]. The information processing rate in the delay-based RC associates with the number of virtual nodes. A small number of the virtual nodes is required for increasing the information processing rate when interval between the adjacent virtual nodes is fixed. Note that, to ensure good performance of the delay-based RC, more variability of virtual node state is desired for a RC system with a small number of the virtual nodes. However, previous investigations of the delay-based RC using laser dynamical systems have mainly concentrated on a single nonlinear node, the variability of virtual node state is limited [17–24]. Therefore, the information processing rate of the delay-based RC is also limited.

Lately, a new delay-based RC architecture consisting of several nonlinear nodes is proposed to solve problems more efficiently [25–27]. Interestingly, a single vertical cavity surface emitting laser (VCSEL) has been successfully employed to realize RC with enhanced computation performance [23]. It is well known that VCSEL is a popular candidate for various potential applications, such as optical computation, optical communication, and optical signal processing [28,29]. VCSEL exhibits many advantages, including high energy efficiency, low manufacturing cost, easy to integrate into 2D arrays [28,29]. In particular, two orthogonal polarization modes, x-polarization(XP) and y-polarization(YP) modes, may coexist in a VCSEL, and much richer polarization dynamics can be achieved in a mutually delay coupled (MDC) VCSELs system [30–35]. However, it is still open to explore the multi-channels RC based on MDC-VCSELs, particularly the effects of the polarization dynamics of MDC-VCSELs on the RC performance.

In this paper, a novel Four-channels RC system based on polarization dynamics in MDC-VCSELs is proposed and demonstrated numerically. The four channels are realized in two orthogonal polarization modes of two VCSELs for the first time. A Santa-Fe chaotic time series prediction task is utilized to evaluate our system. For the purpose of comparison, One-channel RC system based on MDC-VCSELs is also considered. The remainder of this paper is structured as follows. In Section 2, the system design and operation principle diagram of Four-channels RC system based on MDC-VCSELs, the rate equation model and the task are introduced in detail. In Section 3, by numerical bifurcation analysis, the polarization dynamical states and stable regions of MDC-VCSELs are identified. Then the prediction performance has been examined and compared for both the Four-channels RC and the One-channel RC systems. The influences of injection current, external injection strength, frequency detuning, coupling strength, as well as internal parameters on the prediction performance of Four-channels RC based on MDC-VCSELs are also taken into account. Moreover, the roles of sampled period of input signal, and the number of virtual nodes are also considered. Finally, the conclusions are given in Section 4.

2. System model

2.1 System design and operation principle diagram

Here, the system design and operation principle diagram of the proposed Four-channels RC system based on MDC-VCSELs is shown in Fig. 1. Two VCSELs with four channels are used as a reservoir. The four channels are realized in two orthogonal polarization modes of the two VCSELs. The number of virtual nodes of each channel is $Z,$ and the total number of virtual nodes is $M=4Z.$ The interval between the adjacent virtual nodes is $\theta .$ The sampled period of input signal is $T=Z\theta ,$ and the information processing rate is $R_4=1/T.$ It can be seen that, four inputs $S_{1x,1y,2x,2y}$ are coupled to the XP and YP modes of VCSEL1 and VCSEL2 through four Mach–Zehnder modulators (MZMs), respectively. The XP and YP modes of VCSEL1 (VCSEL2) are injected to the XP and YP modes of VCSEL2 (VCSEL1) in parallel. Finally, four transient responses$E_{1x,2x,1y,2y}$ from each of the four channels are extracted for post-processing, respectively. We only consider one input signal under four random binary masks (−1, 1), leading to four different transient responses. Thus, as shown in Fig. 2, in the post-processing phase, the virtual node states extracted from the four channels are put into four matrices V_12x, V_12y, V_21x and V_21y, respectively. Then the four matrices are merged into one matrix V for post-processing, which can lead to more abundant transient responses than that of a single matrix. For One-channel RC, we only set one masked input signal, and only one matrix V_12x is employed for post-processing.

 

Fig. 1 System design and operation principle diagram of the Four-channels RC based on MDC-VCSELs with three layers: input layer, reservoir and output layer. VCSEL: vertical cavity surface emitting laser, MZMs: Mach–Zehnder modulators, the pink lines indicate the electric path and the black lines represent the optical path.

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Fig. 2 The virtual node states matrix for each channel.

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2.2 Rate equation model

In this paper, the MDC-VCSELs system with parallel polarization injection is considered. The rate equations are derived based on the well-known spin-flip model [36,37]. The slowly varied complex field amplitudes of XP and YP modes are $E_{1x,2x}$ and $E_{1y,2y}.$ The difference between carrier inversions with opposite spins are $n_{1,2},$ and the total population inversions are$N_{1,2}.$ The modified rate equations incorporating the mutual coupling and external injections are described as follows [36,37]:

2.3 Santa-Fe chaotic time series prediction task

We employ the Santa-Fe chaotic time series prediction task as a benchmark test to evaluate the prediction performance of Four-channels RC system based on MDC-VCSELs. The goal for the task is to predict the next step of the chaotic time series gained from a far-infrared laser experimentally [39]. The first 3000 points in the Santa-Fe data set are used for training and the next 1000 are adopted for testing. The normalized mean square error ($NMSE$) is defined to quantitatively evaluate the prediction performance of this benchmark task [38]:

3. Numerical results

In this section, the prediction performance of Four-channels RC based on MDC-VCSELs is evaluated quantitatively in detail. For the purpose of comparison, we also consider the One-channel RC based on MDC-VCSELs. At first, the polarization dynamical states and stable regions of MDC-VCSELs are analyzed and identified. Then, we study the influences of injection current, external injection strength, frequency detuning, coupling strength, and internal parameters on the prediction performance of Four-channels RC based on MDC-VCSELs. Moreover, the roles of sampled period of input signal, and the number of virtual nodes are also considered. In our simulation, the following parameter values are adopted:$\alpha =3,$$\kappa =300{\text{ns}}^{\text{-1}},$${\gamma }_a=0.1{\text{ns}}^{-1},$${\gamma }_p=10{\text{ns}}^{\text{-1}},$${\gamma }_N=1{\text{ns}}^{\text{-1}},$${\gamma }_s=50{\text{ns}}^{\text{-1}},$${\mu }_{}\text{=}1.01,$${\omega }_1=2\pi c/\lambda$ and${\omega }_2={\omega }_1\text{-2}\pi \Delta f_{12}$with$\lambda =850\text{nm,}$$k_r{\text{=10ns}}^{-1},$$k_{inj}\text{=}30{\text{ns}}^{\text{-}1},$$\Delta f_{12\text{,21}}\text{=}0\text{GHz,}$$\tau \text{=}0.5\text{ns,}$${\beta }_{}\text{=}1\times {10}^{\text{-}6},$$\left|{\epsilon }_0\right|=1,$${\Phi }_0=0,$$\Delta f\text{=}0\text{GHz,}$$\theta \text{=}25\text{ps,}$$\text{Z=}20,$ $M\text{=}80$ [21,22,31]. Note that those parameters are fixed, unless otherwise noted.

To begin with, the polarization-resolved intensities as a function of the normalized injection current $\mu$ for free-running VCSEL is shown in Fig. 3. Here, the mean intensities are calculated as $〈I_{x,y}〉=〈{\left|E_{x,y}(t)\right|}^2〉$ for the XP and YP modes. As can be seen, when$1<\mu \le 1.55$, the XP mode is the dominant polarization mode while the YP mode is suppressed completely. When$\mu >1.55$, the YP mode starts to oscillate. As the injection current continues to increase, the polarization-resolved intensities of both modes increase. However, the XP mode is still the dominant polarization mode.

 

Fig. 3 The polarization-resolved intensities as a function of the normalized injection current $\mu$ for free-running VCSEL.

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The bifurcation diagrams as a function of the normalized injection current $\mu$ are also shown in Fig. 4. Here, we set $s_{1x}=s_{1y}=s_{2x}=s_{2y}=0$ and $k_{inj}\ne 0.$ $I_{x,y}={\left|E_{x,y}(t)\right|}^2$ are the intensities of XP and YP modes. It can be seen that the two VCSELs exhibit similar dynamical state and two orthogonal polarization modes coexist. More precisely, when ${\mu }_{}\le 1.47,$ XP and YP modes of both VCSELs remain stable due to injection locking effect. As $\mu$ gradually increases, for XP mode, it can be seen from Figs. 4(a1) and (a2), the outputs of the two VCSELs oscillate with multiple periods resulting in several branches in the bifurcation diagrams. For YP mode, as can be seen from Figs. 4(b1) and (b2), the outputs of the two VCSELs oscillate periodically resulting in two branches. Note that, these stable regions are associated with the good performance of a RC system [5].

 

Fig. 4 Numerical bifurcation diagrams as a function of ${\mu }_{}$ for (the first column) VCSEL1 and (the second column) VCSEL2, for (a1, a2) XP mode, for (b1, b2) YP mode, with $k_r{\text{=10ns}}^{-1}$and $k_{inj}\text{=}30{\text{ns}}^{\text{-}1}.$

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Next, the $NMSE$values for Four-channels RC and One-channel RC based on MDC-VCSELs as a function of normalized injection current $\mu$ are shown in Fig. 5. Here, virtual node interval is fixed at$\theta =25\text{ps}\text{.}$ For Four-channels RC based on MDC-VCSELs, the number of virtual nodes of each channel is $Z=20,$ and the total number of virtual nodes is $M=4Z=80.$ Thus, the information processing rate is $R_4=1/(Z\theta )=2\text{Gbps}\text{.}$ For comparison, for One-channel RC ($k_{inj21x}=k_{inj21y}=k_{inj1x}=k_{inj1x}\ne 0,$ $s_{21x}\ne 0$and $s_{21y}=s_{12x}=s_{12y}=0$), the number of virtual nodes of one channel is set as $Z=\text{8}0$ corresponding to $R_1=1/(Z\theta )=0.5\text{Gbps}\text{.}$ That is to say, the information processing rate for Four-channels RC based on MDC-VCSELs is four times higher than that for One-channel RC. As can be seen that, the $NMSE$ curves for Four-channels RC and One-channel RC are almost overlapped for ${\mu }_{}\le \mathrm{1.45.}$ Then the $NMSE$ values of Four-channels RC (One-channel RC) based on MDC-VCSELs increase significantly for ${\mu }_{}>1.45$ (${\mu }_{}>1.52$), because of the onset of instabilities, which is in agreement with the experimental result in [40]. Besides, a good prediction performance of Four-channels RC (One-channel RC) based on MDC-VCSELs can be obtained when ${\mu }_{}\le 1.9$(${\mu }_{}\le 1.55$). Hence, compared with the One-channel RC based on MDC-VCSELs, the Four-channels RC based on MDC-VCSELs can produce comparable prediction performance with much faster information processing rate.

 

Fig. 5 The $NMSE$ values of RC system based on MDC-VCSELs as a function of ${\mu }_{}$for Four-channels RC with $Z=20$ and $R_4=2\text{Gbps,}$ for One-channel RC with $Z=80$and$R_1=0.5\text{Gbps,}$ with $k_r{\text{=10ns}}^{-1}$and $k_{inj}\text{=}30{\text{ns}}^{\text{-}1}.$

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In the following, we only focus on Four-channels RC based on MDC-VCSELs. At first, we consider the transient responses of the XP and YP modes of Four-channels RC based on MDC-VCSELs. Here, we set $k_{inj}=30{\text{ns}}^{\text{-1}},$$k_d=10{\text{ns}}^{\text{-1}}$and $\mu =\mathrm{1.2.}$ When 1000 testing data are injected into the Four-channels RC system, the spatio-temporal plots for the four matrices V_12x, V_12y, V_21x and V_21y are shown in Fig. 6, respectively. The transient responses for input signals are represented by column-shaped patterns throughout the entire spatio-temporal plot. It is indicated that rich information has been preserved.

 

Fig. 6 The transient responses of the XP and YP modes of Four-channels RC based on MDC-VCSELs, with $k_{inj}=30{\text{ns}}^{\text{-1}},$$k_d=10{\text{ns}}^{\text{-1}}$and $\mu =\mathrm{1.2.}$

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To obtain the dependence of the prediction performance, the two-dimensional maps of $NMSE$ values in the parameter space of external injection strength $k_{inj}$ and frequency detuning $\Delta f_{}$ are presented for different $k_r$ and $\Delta f_{12}$ in Fig. 7. Here, $\Delta f_{12}$ is fixed at 0 for Figs. 7(a1)-(a3). A better prediction performance region with $NMSE\le 0.05,$ is denoted as region A by the contour lines. As shown in Fig. 7(a1), for $k_r=5{\text{ns}}^{\text{-1}},$ the region A mainly locates at larger $k_{inj}$ for all the considered $\Delta f,$ which is similar to the injection locking area in [41]. That is to say, the good performance of RC is closely related to the injection locking effect. For $k_r=10{\text{ns}}^{\text{-1}}$ as shown in Fig. 7(a2), regions A becomes narrow. For larger coupling strength $k_r=20{\text{ns}}^{\text{-1}}$as shown in Fig. 7(a3), regions A becomes even narrower. Besides, higher $k_r$ leads to the shift of the region A toward negative$\Delta f.$ It is indicated that a larger $k_{inj},$ smaller $k_r$ and negative$\Delta f_{}$are desired for better prediction performance. Without loss of generality, we also consider other cases of $\Delta f_{12},$ and obtain similar results. A representative case for$\Delta f_{12}\text{=5GHz}$ is shown Figs. 7(b1)-(b3). It can be seen that good prediction performance of Four-channels RC based on MDC-VCSELs can be also obtained.

 

Fig. 7 Two dimensional maps of the $NMSE$ values of Four-channels RC system based on MDC-VCSELs in the parameter space of $\Delta f_{}$and$k_{inj}$,with $\mu =\mathrm{1.01.}$ (a1, a2, a3) for $k_r=5{\text{ns}}^{\text{-1}},$ $k_r=10{\text{ns}}^{\text{-1}},$ $k_r=20{\text{ns}}^{\text{-1}}$with $\Delta f_{12}=0\text{GHz;}$ (b1, b2, b3) for $k_r=5{\text{ns}}^{\text{-1}},$ $k_r=10{\text{ns}}^{\text{-1}},$ $k_r=20{\text{ns}}^{\text{-1}}$with $\Delta f_{12}=5\text{GHz}\text{.}$

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Then two dimensional maps of $NMSE$ values in the parameter space of $k_r$ and $k_{inj}$ are further presented in Fig. 8 for different$\Delta f.$ It is found that, a larger $\Delta f_{}$leads to a narrower region A. Besides, for a relatively large $\Delta f,$ the region A mainly covers the range with $k_{inj}>k_r,$ which is also attributed to the injection locking effect.

 

Fig. 8 Two dimensional maps of the $NMSE$ values of Four-channels RC system based on MDC-VCSELs in the parameter space of $k_r$and$k_{inj},$with $\mu =1.01$and $\Delta f_{12}=0.$ (a) for $\Delta f=-15\text{GHz,}$ (b) for $\Delta f=0\text{GHz,}$ (c)$\Delta f_{}=15\text{GHz}\text{.}$

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Subsequently, the $NMSE$values as a function of coupling strength $k_r$ for different $k_{inj}$ and three cases of $\mu$ are shown in Fig. 9. Here, $\Delta f$and $\Delta f_{12}$ are fixed at 0GHz, and $k_r$ is varied from $0{\text{ns}}^{\text{-1}}$ to $30{\text{ns}}^{\text{-1}}.$ For$\mu =1.01,$ as can be observed from Fig. 9(a) and inset, for $k_{inj}=10{\text{ns}}^{\text{-1}},$ the $NMSE$ values decrease with $k_r$ firstly when $0{\text{ns}}^{\text{-1}}\le k_r\le 14{\text{ns}}^{\text{-1}},$ and then increase with the further increment of $k_r$. Similar results can be obtained for $k_{inj}=15{\text{ns}}^{\text{-1}}$ and $k_{inj}=20{\text{ns}}^{\text{-1}}.$ Obviously, a wider range of $k_r$ contributing to $NMSE\le 0.001$ can be obtained for a larger $k_{inj}.$ It is indicated that, a larger $k_{inj}$ contributes to better prediction performance of Four-channels RC based on MDC-VCSELs. For $\mu =1.2,$ similar trends are obtained in Fig. 9(b). But for a given $k_{inj},$ the maximum $k_r$ leading to $NMSE\le 0.001$ is smaller than that for the case of $\mu =\mathrm{1.01.}$ Besides, when in the range of $k_r$ contributing to $NMSE<0.01$ for the case of $\mu =1.2,$ the fluctuation of the error bars of the $NMSE$ becomes larger compared to the case of $\mu =1.01$. This may be because that an injection current that is close to the threshold current makes the RC system based on MDC-VCSELs more stable [40]. For a much larger$\mu ,$ such as $\mu =1.5,$ as can be seen from Fig. 9(c), for $k_{inj}=10{\text{ns}}^{\text{-1}},$ the values of $NMSE$ are basically greater than 0.1 for all the considered$k_r.$ However, for some larger$k_{inj},$ such as $k_{inj}=20{\text{ns}}^{\text{-1}},$$k_{inj}=25{\text{ns}}^{\text{-1}},$ and $k_{inj}=30{\text{ns}}^{\text{-1}},$ similar results to the trend for the cases of $\mu =1.01$ and $\mu =1.2$ can be obtained. Besides, with the growth of $k_{inj},$ the maximum $k_r$ leading to $NMSE\le 0.001$ increases. That is to say, increasing $k_{inj}$ can improves the performance of Four-channels RC system when using larger currents.

 

Fig. 9 The $NMSE$ values of Four-channels RC system based on MDC-VCSELs as a function of the $k_r$for different $k_{inj},$ with $\Delta f_{}=0$and$\Delta f_{12}=0,$ (a) for $\mu =1.01,$ (b) for $\mu =1.2,$ (c) for $\mu =\mathrm{1.5.}$

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The effects of internal parameters on the prediction performance of Four-channels RC based on MDC-VCSELs are also considered. Figure 10(a) shows the variation of $NMSE$ values with the coupling strength $k_r$ under some typical values of the linear dichroism ${\gamma }_a.$ It can be seen that, for all the considered ${\gamma }_a,$ the prediction performance of Four-channels RC based on MDC-VCSELs is good and is hardly affected by the ${\gamma }_a.$ Besides, the effect of linear birefringence ${\gamma }_p$ on the prediction performance is also examined. The $NMSE$ curves are further shown in Fig. 10(b). It can be observed that the $NMSE$ curves are almost overlapped for different ${\gamma }_p.$ That is to say, the prediction performance of Four-channels RC based on MDC-VCSELs is hardly affected by the internal parameters of VCSELs.

 

Fig. 10 The $NMSE$ values of Four-channels RC system based on MDC-VCSELs as a function of $k_r$for different internal parameters with $\mu =1.01,$ $k_{inj}\text{=}30{\text{ns}}^{\text{-}1},$ $\Delta f_{12}=0,$$\Delta f_{}=0,$, (a) for ${\gamma }_a=-0.1{\text{ns}}^{\text{-1}},0.1{\text{ns}}^{\text{-1}},1{\text{ns}}^{\text{-1}},$ (b) for ${\gamma }_p=3{\text{ns}}^{\text{-1}},6{\text{ns}}^{\text{-1}},10{\text{ns}}^{\text{-1}}.$

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At last, to discuss the information processing rate of the proposed RC system, we further consider the influences of sampled period $T$for different virtual node interval $\theta$ in Fig. 11. Here, the coupling delay $\tau$ is also set as $T.$ Note, as $T$is changed, the information processing rate $R_4=1/T$ will also be varied. It can be seen from Fig. 11(a), for all the considered $\theta ,$ $NMSE$curves generally present a decrease trend with the increment of $T.$ It is indicated that the prediction performance of Four-channels RC is improved with the growth of $T.$ In addition, a smaller $\theta$ leads to a lower $NMSE$ for a given $T.$ More precisely, for $\theta =10\text{ps,}$ it can be observed that $NMSE$ values are below 0.1 for $T\ge 0.05\text{ns}\text{.}$ That is to say, the fastest information processing rate $R_4$ in our proposed RC system can be up to 20Gbps, which is much faster than the information processing rate reported in delay-based RC systems based on single nonlinear node [17–25]. Besides, for another case when the coupling delay $\tau$ is fixed at 0.5ns, it can be seen from Fig. 11(b), similar trends can be achieved. Even for a much larger $\tau ,$ such as $\tau =10\text{ns,}$ as can be seen from Fig. 11(c), obviously, similar results also can be observed.

 

Fig. 11 The$NMSE$ values of Four-channels RC system based on MDC-VCSELs as a function of the $T$ for different $\theta$for (a) $\tau =T,$ (b) $\tau =0.5\text{ns,}$ (c) $\tau =10\text{ns,}$ with $\mu =1.01,$$k_{inj}\text{=}30{\text{ns}}^{\text{-}1},$ $k_r\text{=}10{\text{ns}}^{\text{-}1},$$\Delta f_{12}=0,$$\Delta f=0.$

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For the case of the mentioned highest information processing rate ($R_4=20\text{Gbps}$ obtained from Fig. 11(a)), the variation of $NMSE$ values with the number of virtual nodes of each channel $Z$ is further shown in Fig. 12. Here, the sampled period $T=Z\theta =0.05\text{ns,}$ and $\tau =T.$ It can be seen that, when $Z\le 8,$$NMSE$values firstly decrease drastically. Then with the further increment of $Z,$ $NMSE$values decrease slowly. Interestingly, the trend of $NMSE$curve for Four-channels RC based on MDC-VCSELs is similar to that for the single channel RC system based on the conventional SLs [22].

 

Fig. 12 The$NMSE$ values of Four-channels RC system based on MDC-VCSELs as a function of Z for $T=Z\theta \text{=}{\tau }_{}\text{=0}{\text{.05ns}}^{}$with $\mu =1.01,$$k_{inj}\text{=}30{\text{ns}}^{\text{-}1},$ $k_r\text{=}10{\text{ns}}^{\text{-}1},$$\Delta f_{12}=0,$$\Delta f=0.$

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4. Conclusion

In summary, a Four-channels RC system based on polarization dynamics in mutually delay coupled VCSELs is proposed. The four channels are realized in two orthogonal polarization modes of two VCSELs for the first time. It is shown that, Four-channels RC based on MDC-VCSELs can produce comparable prediction performance with One-channel RC, and four times information processing rate can be achieved for Four-channels RC. The parameter regions leading to better prediction performance of Four-channels RC based on MDC-VCSELs are greatly broadened for smaller coupling strength, larger external injection strength and negative frequency detuning. Besides, the prediction performance is hardly affected by the internal parameters of VCSEL. Moreover, the fastest information processing rate in our proposed RC system can be even up to 20Gbps, which is much faster than the information processing rate reported in delay-based RC systems based on single nonlinear node. This work may be valuable for the realization of higher rate and multichannel RC.