## Abstract

In this work, we propose a scheme of reservoir computing (RC) for processing a Santa-Fe time series prediction task and a signal classification task in parallel, and the performances of the RC have been numerically investigated. For this scheme, a vertical-cavity surface-emitting laser (VCSEL) simultaneously subject to double optical feedback and optical injection is utilized as a nonlinear node, and the parallel information processing of the RC system is implemented based on the dynamical responses of X polarization component (X-PC) and Y polarization component (Y-PC) in the VCSEL. Considering that two different feedback frames (polarization-preserved optical feedback (PP-OF) or polarization-rotated optical feedback (PR-OF)) may be adopted in two feedback loops, four feedback combination cases are numerically analyzed. The simulated results show that the parallel processing ability of the proposed RC system depends on the feedback frames adopted in two loops. After comprehensively evaluating the parallel processing performances of the two tasks under different feedback combinations, the best parallel processing performance can be achieved by adopting PP-OFs in both two feedback loops. Under some optimized operation parameters, this proposed RC system can realize the lowest prediction error of 0.0289 and the lowest signal classification error of 2.78 × 10^{−5}.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Nonlinear systems with one or more delay feedback have been attracting considerable attention due to their numerous applications in science and engineering [1–6]. The simplest frame of a delay system only contains a nonlinear device and a delay loop. Recently, such a simple frame provides an exciting prospect for implementing reservoir computing (RC), where RC is a newly-proposed and brain-inspired computational conception [7–9]. In 2011, a RC system based on a nonlinear node and a delay feedback loop (named as delay-based RC) was firstly electrically implemented to process time-dependent signals, and its performance can be comparable to those based on complex network structures [10]. Since then, some related hardware implementations have been reported, such as optoelectronic systems [11–14], all-optical systems [15–17], and laser dynamical systems [18–27]. Particularly, semiconductor lasers (SLs) with time-delay feedback are very promising for high-speed implementation of RC due to their high relaxation oscillation frequency [28]. Brunner *et al*. experimentally demonstrated a RC system based on an SL subject to both optical feedback and optical injection for performing a time series prediction task, and the lowest prediction error is about 10% for a data rate beyond 1 GByte/s [18]. Uchida *et al*. theoretically and experimentally investigated the influences of the mask signal format on the performances of a RC based on an SL subject to both optical feedback and optical injection, and the results showed that a better prediction performance can be obtained by adopting a chaotic signal as the mask signal [22,23]. Recently, our group proposed a RC system based on an SL under double optical feedback and optical injection, and the performance of a Santa-Fe time series prediction task can be further improved [26].

In the schemes mentioned above, the used lasers are almost edge-emitting semiconductor lasers (EESLs). Compared with EESLs, vertical-cavity surface-emitting lasers (VCSELs) possess some unique virtues such as small size, low threshold current, single-longitudinal-mode emission, and easiness to photonic integration [29–31]. In particular, VCSELs are capable of emitting two orthogonal polarization components (X-PC and Y-PC) under certain conditions [32]. In recent, based on a VCSEL under an optical feedback and an optical injection, Vatin *et al.* proposed a RC system, in which the data to be processed was injected into a VCSEL along the main lasing polarization axis. The simulated results demonstrated that, under the case of dual polarization modes co-existing in a VCSEL, the RC system can yield a better performance for processing a single task [29].

In principle, for a VCSEL-based RC system, if the VCSEL can simultaneously output X-PC and Y-PC, each PC can be used to process a task and then the RC system can realize the parallel processing of two independent tasks via the nonlinear dynamical responses of X-PC and Y-PC, which may possess great prospects in today’s big data age. Based on this consideration, in this paper, we propose a RC system based on a VCSEL simultaneously subject to double optical feedback and optical injection to deal with a Santa-Fe time series prediction task and a signal classification task in parallel by using two PCs in the VCSEL. Considering that two different feedback frames (polarization-preserved optical feedback (PP-OF) or polarization-rotated optical feedback (PR-OF)) can be adopted in two feedback loops, there exist four feedback combination cases. For the RC system under four different feedback combinations, the parallel processing ability is examined numerically and compared with each other, and the influences of some typical operation parameters on the RC performance have also been investigated.

## 2. System model

Figure 1 depicts a schematic diagram of our proposed RC system for processing two independent tasks in parallel, which is based on a VCSEL subject to double optical feedback and optical injection. This system is composed of three main parts: an input layer, a reservoir and an output layer. In the reservoir, a VCSEL subject to double optical feedback supplied by two feedback loops is utilized as a nonlinear node. For convenience, the feedback loop with a shorter length is named as feedback loop 1, and the other loop is named as feedback loop 2. Correspondingly, the delay feedback times for loop 1 and loop 2 are denoted as *τ*_{1} and *τ*_{2}, respectively. Two variable optical attenuators (VOA1 and VOA2) and two polarization controllers (PC1 and PC2) are used to control the feedback strengths and the polarization directions of the feedback beams in loop 1 and loop 2, respectively. Here, we define a combination of a polarization-preserved optical feedback (PP-OF) adopted in loop 1 and a polarization-rotated optical feedback (PR-OF) adopted in loop 2 as the case of PP-PR-OF. Accordingly, the other three combination cases of PP-PP-OF, PR-PP-OF and PR-PR-OF correspond to adopting PP-OFs in both two loops, PR-OF in loop 1 and PP-OF in loop 2, PR-OFs in both two loops, respectively. In the input layer, input data 1 is firstly sampled and denoted as *U _{1}*(

*n*), where

*n*is a discrete time index and then

*U*(

_{1}*n*) is held within a period of

*T*. Next, a random sequence with a period of

*T*is applied as a mask signal and multiplied by

*U*(

_{1}*n*)

_{.}After scaled by a scaling factor

*γ*, the masked input signal is denoted as

*S*

_{1}(

*t*). Input data 2 experiences a similar process and is transferred into

*S*

_{2}(

*t*).

*S*

_{1}(

*t*) and

*S*

_{2}(

*t*) are utilized to respectively modulate the phase of the optical field output from SL1 and SL2 via two phase modulators (PM1 and PM2). The polarization directions of the two modulated optical fields are adjusted to match with X-polarization component (X-PC) and Y-polarization component (Y-PC) of the VCSEL, respectively, and then injected into the VCSEL. In the output layer, via a polarization beam splitter (PBS), X-PC and Y-PC are split. The output of X-PC (|

*E*|

_{x}^{2}) and Y-PC (|

*E*|

_{y}^{2}) is respectively extracted at an interval of

*θ*and interpreted as a state of a virtual node, thus the number

*N*of virtual nodes along loop 1 satisfies

*N*=

*τ*

_{1}/

*θ*. Here, we employ a synchronization scheme,

*i. e*.

*T*=

*τ*

_{1}[20,21]. The states of

*N*virtual nodes are finally weighted and linearly summed up to obtain the task output. Through minimizing the mean-square error between the target function and the RC output via a linear least-square method, the weights can be optimized [12,22].

Based on the spin-flip model (SFM), nonlinear dynamics of a VCSEL subject to double optical feedback and optical injection can be described by [29,33]:

*x*and

*y*represent X-PC and Y-PC of the VCSEL, respectively.

*E*is the slowly varied complex amplitude of the electric field,

*N*accounts for the total carrier inversion between conduction and valence bands,

*n*is the difference between carrier inversions for the spin-up and spin-down radiation channels,

*κ*accounts for the decay rate of field,

*α*describes the linewidth enhancement factor,

*γ*stands for the decay rate of total carrier population,

_{e}*γ*expresses the spin-flip rate,

_{s}*γ*and

_{a}*γ*denote the linear anisotropies representing dichroism and birefringence, respectively.

_{p}*μ*is the normalized injection current (

*μ*takes the value 1 at threshold).

The third terms on the right of Eqs. (1) and (2) describe the injection fields containing two independent input signals required to be processed via the dynamical responses of X-PC and Y-PC, respectively. *k _{inj}* is the injection strength, and the injected complex electric fields

*ε*

_{1}and

*ε*

_{2}are represented as [22]:

*P*is the power of CW output of SL. ∆

_{inj}*f*

_{1}(∆

*f*

_{2}) is the frequency detuning between the lasing frequency of SL1 (SL2) and the central frequency of the VCSEL.

*S*

_{1,2}stands for masked input data of two tasks and is written as:where

*γ*is the input scaling factor,

*Mask*

_{1,2}(

*t*) and

*U*

_{1,2}(

*n*) express the mask signals and the sampled and held input signals of two independent tasks, respectively.

The fourth and fifth terms on the right of Eqs. (1) and (2) describe the effects of optical feedback, where *η*_{1} (*η*_{2}) is the feedback strength for the loop 1 (loop 2). For four different feedback combination cases, the feedback electric fields are described as follows:

*ω*

_{0}= (

*ω*+

_{x}*ω*)/2 is defined as the central frequency of the VCSEL, and

_{y}*ω*(

_{x}*ω*) is the angular frequency of X-PC (Y-PC). In this work, an 850 nm VCSEL is considered, and then

_{y}*ω*

_{0}is about 2.22 × 10

^{15}Hz.

The last terms on the right of Eqs. (1) and (2) represent Langevin noise sources, which can be expressed as

*ξ*

_{1}and

*ξ*

_{2}are the complex Gaussian white noises with unit variance and zero mean, and

*β*is the spontaneous emission coefficient to be set at 10

_{sp}^{−6}ns

^{−1}after referring to [33].

## 3. Results and discussion

Equations (1)-(4) can be numerically solved by employing a fourth-order Runge-Kutta method with a step of 2 ps via MATLAB software. During the simulation, the following parameter values are used [33]: *μ* = 1.1, *α* = 3, *κ* = 300 ns^{−1}, *γ _{e}* = 1 ns

^{−1},

*γ*= 50 ns

_{s}^{−1},

*γ*= 0.1 ns

_{a}^{−1},

*γ*= 10 ns

_{p}^{−1},

*k*= 50 ns

_{inj}

^{-}^{1}. In this work, the mask signals are randomly chosen from a six-level sequence {-1, −0.6, −0.3, 0.3, 0.6, 1} as

*Mask*

_{1}for a Santa-Fe time series prediction task and

*Mask*

_{2}for a signal classification task since the mask with six possible values can offer enough variability of responses for computation [14].

*η*

_{1}and

*η*

_{2}are set 10 ns

^{−1}to guarantee that the system operates at a steady state in absence of input signals for the injected power

*P*varied within a discussed region of (0.2, 10) in this work. The delay time

_{inj}*τ*

_{1}of loop 1 is fixed at 8 ns, corresponding to a processing rate of 125 Mb/s. The time difference ∆

*τ*(

*= τ*

_{2}

*– τ*

_{1}) between two feedback loops is set as a half of the reciprocal of relaxation oscillation frequency (${f}_{ro}=\sqrt{2\kappa {\gamma}_{e}(\mu -1)}/2\pi $) of the VCSEL at free-running. Under the parameters given above, ∆

*τ =*0.406 ns [26,34].

Here, X-PC and Y-PC is utilized to execute a Santa-Fe time series prediction task (denoted as TASKx) and a signal classification task (denoted as TASKy), respectively. For TASKx, it needs to predict the next step of a Santa-Fe time series recorded experimentally from a far-infrared laser operating at a chaotic state [35]. A Santa-Fe time series data set contains 9000 points, where consecutive 4000 points (the first 75% is used for training and the remaining 25% is for testing) are selected in this simulation. For TASKy, input data is a random concatenation of sine and square waves, each of which is discretized into 12 points. 250 waves (3000 points) are selected for training and 83 waves (1000 points) for testing, with the target value being 0 for a sine wave and 1 for a square wave. For both tasks, the Normalized Mean Square Error (*NMSE*) between the target value *ӯ*(*n*) and the reservoir output *y*(*n*) is calculated to measure the performance of the reservoir, which is defined as [22]:

*n*is the discrete time index of input data and

*L*is the total number of data in the testing data set,

*var*(

*y*) represents the variance. Typically, a RC system can be considered to perform well when

*NMSE*is less than 0.1 for TASKx and 2 × 10

^{−4}for TASKy, respectively.

First, we investigate the influence of the number *N* of virtual nodes on the parallel processing ability of the VCSEL-based RC system. Figure 2 presents the variations of the mean *NMSE* of ten calculations for TASKx (Fig. 2(a)) and TASKy (Fig. 2(b)) with the number *N* of virtual nodes under four different feedback combination cases. It can be seen that, for four different cases, the *NMSE* values for processing two tasks firstly decrease rapidly and then maintain at a relatively stable level with the increase of *N*. The reason that a less virtual node results in a larger mean *NMSE* value may be explained as follows. In this work, *τ*_{1} = *Nθ* is fixed at 8 ns, a smaller *N* is accompanied by a larger node interval *θ*. On one hand, a smaller *N* means a lower dimension of the state space, which makes the training of the system be implemented more difficult and therefore results in a larger *NMSE*. On the other hand, a large *θ* may leads to an insufficient coupling between two adjacent nodes. As a result, the joint action of above two factors may cause a relatively bad performance. A relatively stable level of *NMSE* value maintained under a large *N* can be explained as follows. As demonstrated in [10], for a fixed *N*, with the decrease of *θ* the processing performance is degraded due to the increase of the correlation between two adjacent nodes. However, in this work, *Nθ* ( = *τ*_{1}) is fixed at 8 ns, *i.e*. a decrease of *θ* is accompanied by an increase of *N*. As a result, the degradation originating from the decreasing *θ* can be compensated through the increase of *N*, and then the performance of the RC system can maintain at a relatively stable level. Through comparing the four feedback cases, it can be found that the *NMSE* values for the RC under PP-PP-OF are always lower than those for the RC under the other three cases. The different processing ability of the RC system is owing to the complex coupling between X-PC and Y-PC under different feedback frames. For *N* ≥ 500, all the *NMSE* values of TASKx for the RC under the four feedback cases are lower than 0.1, while the *NMSE* values of TASKy for the RC are lower than 2 × 10^{−4} except the case of PP-PR-OF. Additionally, for comparison, the performance of the RC system under PP-PP-OF for processing a single task is also shown as dashed lines with square marks in this figure. Obviously, after adding another task, the system performance is degraded. However, the prediction performance of the parallel processing of the RC system is similar to that of a single prediction task of the system if *N* ≥ 500.

In order to intuitively observe the parallel processing ability of this proposed VCSEL-based RC system, the parallel processing results of TASKx and TASKy under PP-PP-OF with *N* = 500 are taken as an example and shown in Fig. 3. It can be seen that the output values of this proposed RC system are almost consistent with the target values for both tasks, and the *NMSE*s are 0.0364 for TASKx and 1.26 × 10^{−4} for TASKy, respectively.

Second, we investigate the *NMSE*s of two different tasks processed by this RC system when the frequency detuning ∆*f* is changed. For simplification, we set ∆*f*_{1} = ∆*f*_{2} = ∆*f*. Moreover, considering that the performance of the proposed RC system changes very slightly when *N* ϵ [500, 1000], in the following discussion we fix *N* = 500. Figure 4 displays the dependence of the *NMSE*s on ∆*f* for two tasks under different feedback cases. For TASKx as shown in Fig. 4(a), it can be seen that, within the frequency detuning region [-32 GHz, 16 GHz], almost all the *NMSE* values for the RC system under the three cases of PP-PP-OF, PP-PR-OF and PR-PP-OF are lower than 0.1, and the *NMSE* values for the RC under PP-PP-OF are almost the lowest. However, few *NMSE* values for the RC under PR-PR-OF are less than 0.1. As for TASKy (Fig. 4(b)), three feedback frames including PP-PP-OF, PR-PP-OF and PR-PR-OF possess similar *NMSE*s lower than 2 × 10^{−4} within −24 GHz ≤ ∆*f ≤* 20 GHz. Comparing Fig. 4(a) with Fig. 4(b), it can be seen that there exist some differences between TASKx and TASKy, which may originate from different properties of the two tasks and the complex coupling between X-PC and Y-PC. Generally speaking, the RC system adopting PP-PP-OF would be helpful for yielding a well parallel processing performance. In particular, for ∆*f* = −12 GHz, the values of *NMSE* for both two tasks arrive at their minima.

Next, the influence of the injection power *P _{inj}* on the parallel processing ability of the proposed RC system is examined. Figure 5 shows the

*NMSE*s for the RC under PP-PP-OF with ∆

*f*= −12 GHz and varying injection powers within (0.2, 10), where the error bars are the statistical results of 10 calculations. For processing TASKx (as shown in Fig. 5(a)), it can be obviously observed that, with the increase of

*P*, the

_{inj}*NMSE*values firstly decrease sharply, after passing through a minimum, and then increase slowly. As a result, there exists an optimal injected power, which is coincided with that reported in [29]. For processing TASKy (as shown in Fig. 5(b)), a similar variation trend with Fig. 5(a) can be observed. Therefore, a moderate injected power is more suitable for achieving a well RC performance.

Previous researches have demonstrated that the computational performance of a RC system is also dependent on the scaling factor *γ* [22,36]. Finally, we calculate the evolution of *NMSE*s for processing TASKx and TASKy in the parameter space of *P _{inj}* and

*γ*, as shown in Fig. 6. Here, we take the RC under PP-PP-OF as an example. In this diagram, different colors correspond to different

*NMSE*values, and the regions surrounded by white dashed lines satisfy

*NMSE*≤ 0.1 in Fig. 6(a) and

*NMSE*≤ 2 × 10

^{−4}in Fig. 6(b), in which the RC system presents a well processing ability. Obviously, the parameter region with

*NMSE*≤ 2 × 10

^{−4}for implementing TASKy is wider than that with

*NMSE*≤ 0.1 for implementing TASKx. In order to achieve a well parallel processing ability,

*P*and

_{inj}*γ*should be located at a overlapped region of the two regions surrounded by the white lines in Fig. 6(a) and 6(b). The minimal

*NMSE*value is 0.0289 for TASKx and 2.78 × 10

^{−5}for TASKy.

## 4. Conclusions

In summary, a RC system based on a vertical-cavity surface-emitting laser (VCSEL) subject to double optical feedback and optical injection is proposed and numerically investigated. For this system, the dynamical responses of two polarization components (named as X-PC and Y-PC) of a VCSEL are utilized to process a Santa-Fe time series prediction task and a signal classification task, respectively. Taking into account four different feedback combination frames adopted in two feedback loops including PP-PP-OF, PP-PR-OF, PR-PP-OF and PR-PR-OF, we simulate the dependence of the parallel processing ability of the RC on certain typical operation parameters such as the number *N* of virtual nodes, the frequency detuning ∆*f*, the injection power *P _{inj}*

_{,}and the input scaling factor

*γ*. The results indicate that, for these parameters varied within a relatively broad region, this VCSEL-based RC system can well process two independent tasks in parallel. Through comparing the processing performance of the RC under the four feedback combination cases, the best parallel processing ability can be achieved under a PP-PP-OF. As a result, for optimized parameters the RC system can process a Santa-Fe time series prediction task with a minimal error of 0.0289 and a signal classification task with a minimal error of 2.78 × 10

^{−5}, respectively.,

## Funding

National Natural Science Foundation of China (NSFC) (61575163, 61775184, 61875167).

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