## Abstract

Via Santa Fe time series prediction and nonlinear channel equalization tasks, the performances of a reservoir computing (RC) system based on an optical feedback semiconductor laser (SL) under electrical information injection are numerically investigated. The simulated results show that the feedback delay time and strength seriously affect the performances of this RC system. By adopting a current-related optimized feedback delay time and strength, the RC can achieve a good performance for an SL biased within a wide region of 1.1–3.5 times its threshold. The prediction errors are smaller than 0.01 when implementing the Santa Fe tests, and the symbol error rates (SERs) are very low on the order of 10^{−5} for accomplishing nonlinear channel equalization tests under a signal-to-noise ratio (SNR) of 32 dB. Moreover, under a given RC performance level, the information processing rate of the RC can be improved by increasing the SL current.

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

## 1. Introduction

Reservoir computing (RC), derived from the recurrent neural network (RNN) [1], is a new machine learning method. Because it employs randomly generated and fixed connection weights at the input layer and the reservoir layer, the training process in RC is much easier than that in the RNN. This simplified training method was first used in the echo state network [2] and the liquid state machine [3] and then was collectively referred to as RC (currently also known as conventional RC) by Verstraeten *et al*. in 2007 [4]. Since RC possesses some unique benefits, such as a simple algorithm, high computational efficiency and easy implementation [5], it has attracted widespread attention in recent years. In 2011, Appeltant *et al*. proposed a novel scheme for constructing RC based on a nonlinear time-delay system [6]. In this scheme, only one Mackey Glass-type nonlinear node is utilized to replace a large number of nonlinear nodes in conventional RC, and the nonlinear transient responses of the system are sampled with an equal time interval from the feedback loop and are taken as the virtual node states, which play a similar role to that of the nonlinear node states used in conventional RC. Since this type of RC is based on a nonlinear time-delay system, it is also called a delay-based RC [1]. Due to the reduction in requirements in hardware implementation, the delay-based RC has been a research hotspot in the RC field. In 2012, a delay-based RC was implemented in an optoelectronic system based on the nonlinearity of the Mach-Zehnder intensity modulator (MZM) [7,8]. In the same year, Duport *et al*. realized an all optical delay-based RC via a semiconductor optical amplifier [9]. In 2013, a semiconductor laser (SL) was first utilized as the nonlinear node in the experimental implementation of RC by Brunner *et al*. [10]. Benefiting from the fast response of the SL, the data processing rate was on the order of gigabytes per second. Subsequently, some gratifying progress has been made in terms of increasing the data processing rate and improving the performance in SL-based delay RC systems [11–19].

In SL-based delay RC systems, information can be loaded either optically by modulating the intensity (or phase) of the injected light or electrically by modulating the SL current. In the past few years, SL-based delay RC systems under optical information injection have been well studied, and some significant progress has been made [11–19]. Comparatively, the SL-based delay RC system under electrical information injection has received little attention, although such a system may possess a relatively simple structure due to the fewer components used in the electrical injection system. In [10], Brunner *et al*. first presented and implemented an SL-based delay RC experiment under electrical information injection, where a good performance was achieved for an SL biased near its threshold but a degraded performance was achieved for an SL biased higher than its threshold. Following this original exploration in [10], we believe that this scheme is promising and that the performance of this system can be optimized. As a result, we undergo this research. For an SL-based delay RC system, when the SL is biased near its threshold, its output intensity is relatively low, and then, a weak external perturbation may result in degradation of the RC performance. Moreover, adopting an SL operating near its threshold as the nonlinear node is not conducive to achieving high-speed RC due to its relatively slow response. Based on these considerations, we consider how to enable an SL-based delay RC system under electrical injection to realize good performance when the SL is biased at a high current.

In this work, based on the rate equation model for an SL subject to optical feedback and current modulation, we numerically analyze the performance of an RC system based on an optical feedback SL under electrical information injection via a Santa Fe time series prediction task and a nonlinear channel equalization task. Different from the fixed feedback delay time in [10], the feedback delay time and strength are optimized according to the SL current level. The simulated results show that after adopting the current-related feedback delay time and strength, the RC system can present a good performance even for an SL biased within the wide range of 1.1 to 3.5 times its threshold, and the data processing rate can be accelerated by increasing the SL current.

## 2. Reservoir computing system

Figure 1 shows a schematic diagram of the RC system based on an SL subject to optical feedback, in which information injection is implemented by directly modulating the SL current. In this RC system, the SL is a nonlinear element taken as the reservoir, and its transient response to optical feedback and current modulation maps the input information into a high-dimensional state space, which is one of the necessary requirements for RC [1]. An optical circulator (OC) and a fiber constitute a feedback loop to provide a certain degree of fading memory for RC [20]. A variable attenuator (VA) is utilized to control the feedback strength, and the feedback delay time can be adjusted by varying the fiber length.

In the delayed RC system, the feedback time *τ* is divided into *n* equal parts, and then the time interval *θ* is *θ* = *τ/n*. The output of the SL in an interval of *θ* is interpreted as a state of a virtual network node, and the SL’s responses, with consecutive intervals of *θ*, accordingly correspond to the states of consecutive virtual nodes. The state matrix X constituted by these virtual node states is utilized to train and test the RC. To enrich the states of virtual nodes, an input mask *M*(*t*) is introduced. Here, *M*(*t*) is a piecewise constant function with a period of *τ* and remains constant over the interval *θ*, which means that each virtual node is assigned a mask code. There are several ways to design the mask, such as a binary mask [21], sinusoidal mask [22], six-level mask [23], chaotic mask [15,18], random mask [7], and slowly modulated mask [24]. The mask used in this work is a binary mask, where two mask values are randomly extracted from {0.1, 1}. The input data are sampled with a period of *T* resulting in a discrete sequence *u*(*k*); here, we set *T* = *τ*. The discrete sequence *u*(*k*) multiplied by the mask *M*(*t*) forms the modulation current *j _{m}*(

*t*), and then,

*j*(

_{m}*t*) and the bias current

*j*are combined to pump the SL.

_{b}The output *y _{out}* of the RC system can be obtained by calculating a linear combination of virtual node states,

*i.e*.,

*y*= W

_{out}*X, where W*

_{out}*is the readout weight and can be obtained by training. The purpose of training is to minimize the normalized mean square error (NMSE) between*

_{out}*y*and the desired values

_{out}*y*, and the NMSE can be expressed as [25]:

_{d}*L*is the length of the input data and

*var*denotes the variance. Since the training involves only the readout layer, it can be reduced to a linear regression problem, and W

*can be obtained via the regression algorithm: W*

_{out}

_{out}_{ }= (X

^{T}X)

^{-1}X

^{T}

*y*However, in practical applications, overfitting often occurs, where X

_{d}.^{T}X approaches singular matrix and the amplitude of W

*is very large. By adding a small amount of noise to the states of the reservoir at the readout layer [26], we can effectively avoid overfitting. Under this case, W*

_{out}

_{out}_{ }= (X

^{T}X

*+ C*I)

^{-1}X

^{T}

*y*, where

_{d}*C*∈

*R*

^{+}and I is the identity matrix. To stabilize the training procedure in our simulation, we set

*C*= 1×10

^{−8}.

## 3. System model description

The nonlinear dynamic behaviors of an SL subject to optical feedback and current modulation can be described by the following equations [27]:

*E*(

*t*) is the slowly varying complex electric field and

*N*(

*t*) is the carrier density.

*G*is the gain coefficient,

_{N}*N*is the carrier density at transparency,

_{0}*τ*is the photon lifetime,

_{p}*τ*is the carrier lifetime,

_{s}*α*is the line width enhancement factor, and

*ε*is the gain saturation coefficient. The second term of Eq. (2) refers to the feedback term, where

*k*is the feedback strength,

*τ*is the delay time of the feedback loop, and

*ω*is the angular frequency of the free running laser.

_{0}*F*(

*t*) (= (

*2βN*)

^{1/2}

*ξ*(

*t*)) is the Langevin noise source, modeling the spontaneous emission noise, where we set

*β*= 5×10

^{−5};

*ξ*(

*t*) is Gaussian white noise.

*J*is the injection current at the threshold,

_{th}*j*is the normalized bias current,

_{b}*j*(

_{m}*t*) =

*u*(

*k*)

*M*(

*t*) is the masked data, and

*γ*is the input scaling factor. The intensity of the SL can be expressed as

*I*(

*t*) = |

*E*(

*t*)|

^{2}.

We use the fourth-order Runge-Kutta method to solve the rate equations. Generally, during calculations, the smaller the selected integration step is, the higher the numerical simulation accuracy is but the longer the computing time is. In this work, after taking into account computational accuracy and time comprehensively, we set the integration step to 2 *ps*. During the simulation, the values of the parameters used are as follows [15]: *G _{N}* = 8.40×10

^{−13}

*m*

^{3}

*s*

^{-1},

*N*= 1.4 × 10

_{0}^{24}

*m*

^{-3}, τ

_{p}= 1.927 ×10

^{−12}

*s*, τ

_{s}= 2.04×10

^{−9}

*s*,

*α*= 3.0,

*J*= 1.037 × 10

_{th}^{33}

*m*

^{-3}

*s*

^{-1},

*ω*=1.226 × 10

_{0}^{15}

*s*

^{-1}, and

*γ*= 0.4. It should be noted that in this work, the number

*n*of virtual nodes is fixed at 50.

## 4. Results and discussion

#### 4.1 Dynamical characterization

Generally, for successful information processing, a delay-based RC system should be kept sufficiently far from the steady state during its dynamical response. This requirement can be fulfilled by setting *θ *= *H _{0} T_{0}*, where

*H*

_{0}_{ }< 1 and

*T*is the characteristic time scale of the nonlinear node [6]. For the SL-based RC system,

_{0}*T*refers to the characteristic time scale of the laser’s relaxation oscillation

_{0}*T*. In [10],

_{ro}*H*is set to 0.2, and the obtained results show that good performance can be achieved when the SL current is biased near the threshold with a constant

_{0}*θ*of 0.2

*ns*. As is well known, the relaxation oscillation frequency

*f*increases with the increasing SL current, and therefore,

_{ro}*T*(=1/

_{ro}*f*) decreases with the increasing SL current. The dependence of

_{ro}*f*of an SL on the bias current can be described by [28]:

_{ro}*G*and

_{N}*J*given above, the variations of

_{th}*f*and

_{ro}*T*with increasing SL bias current are shown in Fig. 2. When

_{ro}*j*increases from 1.1 to 3.5,

_{b}*f*increases from 1.45 GHz to 7.2 GHz, and

_{ro}*T*correspondingly decreases from 0.69

_{ro}*ns*to 0.14

*ns*. By further incorporating the suggestion in [10], we estimate that adopting a current-related

*θ*may be helpful in achieving a good performance of the RC system over a wide range of SL bias currents. As a result, for an SL biased at a higher current, the value of

*θ*should be smaller to achieve good RC performance. Correspondingly, the feedback delay time should be shortened in the case of a fixed number (

*n*= 50) of virtual nodes.

Previous studies have demonstrated that SLs can exhibit rich nonlinear behaviors such as single-periodic, double-periodic, quasi-periodic, and chaotic states under one or more external perturbations [29–33]. However, to obtain a good computational performance, the RC system should operate in an appropriate dynamic range to fulfill the properties of separation and approximation [11]. This dynamic range can be characterized as a stable state but not too far from the bifurcation point (BP) in the absence of input data [6]. In our system, the nonlinear state can be varied by adjusting the feedback strength *k*. Figure 3(a) shows the bifurcation diagram as a function of feedback strength *k* for *j _{b}* = 2.0 and

*τ*= 1.5

*ns*. As shown in this diagram, the system operates at a stable state while

*k*is smaller than 3.9

*ns*

^{-1}. Once

*k*exceeds 3.9

*ns*

^{-1}, a single-periodic oscillation appears, and then, the oscillation becomes more complicated as

*k*further increases. The value of

*k*at the bifurcation point is denoted by

*k*. It is well known that the value of

_{BP}*k*depends on not only the SL bias current

_{BP}*j*but also the feedback delay time

_{b}*τ*. In our system,

*τ*= 50

*H*, where

_{0}T_{ro}*T*can be read out from Fig. 2 for a given

_{ro}*j*. Under this circumstance, for a given

_{b}*j*and

_{b}*H*, the corresponding

_{0}*k*can be determined. Figure 3(b) shows the variation of

_{BP}*k*with the increase in

_{BP}*j*under different

_{b}*H*values (

_{0}*H*= 0.1, 0.2, 0.3, 0.4, and 0.5). From this diagram, one can see that the value of

_{0}*k*increases with

_{BP}*j*and that

_{b}*k*decreases with increasing

_{BP}*H*for a fixed

_{0}*j*. As a result, to avoid the system operating in an oscillating state without input, we set

_{b}*k*to no more than 0.9

*k*in the following discussion.

_{BP}#### 4.2 Santa Fe chaotic time series prediction task

In this section, we evaluate the performance of this reservoir via the Santa Fe chaotic time series prediction task, which is a benchmark task in the machine learning domain. Santa Fe time series is experimentally obtained from a far-infrared laser operating in a chaotic state [34]. The target of this task is to predict the trajectory of chaotic time series one step ahead of the input data, and the prediction error is evaluated by using the NMSE. In this test, the input data are normalized, and the first 1000 points are used to warm up the system. The following 3000 points are used to train W* _{out}*, and another 1000 points are used to test the performance. In this work, the performance is considered good when NMSE ≤ 0.01. The results shown in the following figures are the mean values over five tests.

We first analyze how to optimize the values of *k* and *θ* to achieve good prediction performance in the RC system. Here, three difference cases, in which the SL is biased at *j _{b}* = 1.5, 2.5, and 3.0, are taken as examples. The dependence of the NMSEs on

*k*under

*θ*=

*H*(

_{0}T_{ro}*H*=0.1, 0.2, 0.3, 0.4 and 0.5) is presented in Fig. 4. From this diagram, it can be seen that with increasing

_{0}*k*, the prediction error NMSE first decreases, reaches a minimum value, and then gradually increases. For

*j*= 1.5 (as shown in Fig. 4(a)), the NMSEs are always higher than 0.01 for 0.1

_{b}*k*≤

_{BP}*k*≤0.9

*k*when

_{BP}*H*= 0.1, which indicates that the prediction performance is not good due to the insufficient response of the SL to the input data for too small

_{0}*θ*. For too large

*θ*(

*H*takes a value of 0.5), the NMSEs are also relatively large since the coupling between virtual nodes may be insufficient. This phenomenon is more obvious for higher currents (as shown in Fig. 4(b) and 4(c)). However, when

_{0}*H*is set at 0.2, 0.3, and 0.4, the NMSEs are relatively low. Generally, a relatively large

_{0}*θ*results in a long feedback delay time, which reduces the data processing rate. Therefore, considering the trade-off between the prediction performance and the data processing rate, we set

*H*= 0.2 in the following discussion. Under this case, for

_{0}*j*= 1.5 (as shown in Fig. 4(a)), good performances can be obtained within 0.2

_{b}*k*≤

_{BP}*k*≤0.8

*k*, and the minimal value emerges at

_{BP}*k*= 0.7

*k*. For

_{BP}*j*= 2.5 (as shown in Fig. 4(b)), a good performance can be achieved within 0.25

_{b}*k*≤

_{BP}*k*≤0.65

*k*, and the minimal value is located at 0.4

_{BP}*k*. For

_{BP}*j*= 3.0 (as shown in Fig. 4(c)), good performances can be realized within 0.2

_{b}*k*≤

_{BP}*k*≤0.65

*k*, and the minimal value is located at 0.45

_{BP}*k*. Taking into account the above three cases under different

_{BP}*j*, we set the optimized value of

_{b}*k*to 0.5

*k*.

_{BP}The above results demonstrate that good performances can be achieved for this RC system under the three cases of *j _{b}* by adopting

*θ =*0.2

*T*and

_{ro}*k*= 0.5

*k*. Next, we analyze the case in which

_{BP}*j*is continuously varied within [1.1, 3.5]; the corresponding results are displayed in Fig. 5(a). It should be noted that the values of

_{b}*T*are different for different

_{ro}*j*values (as shown in Fig. 2); therefore, the feedback time should be varied with

_{b}*j*, as shown in Fig. 5(b). Similarly, the values of

_{b}*k*( = 0.5

*k*) should also be varied with

_{BP}*j*since

_{b}*k*is dependent on

_{BP}*j*(as shown in Fig. 3(b)). For comparison, the NMSEs obtained under

_{b}*θ*= 0.2

*ns*and

*k*= 0.15

*ns*

^{-1}are also presented in Fig. 5(a). As shown in this diagram, for

*θ*= 0.2

*ns*and

*k*= 0.15

*ns*

^{-1}, a good performance can be obtained only under the case where the SL is biased near its threshold. After adopting current-related

*θ*and

*k*, NMSEs are always smaller than 0.01 for 1.1 ≤

*j*≤ 3.5,

_{b}*i.e*., good prediction performances can be achieved. Moreover, as shown in Fig. 5(b), the delay time

*τ*decreases with the increasing

*j*; therefore, the information processing rate can be increased from 0.15GSa/

_{b}*s*to 0.73GSa/

*s*for

*j*values from 1.1 to 3.5.

_{b}#### 4.3 Nonlinear channel equalization task

In this section, we further evaluate the performance of this reservoir via a nonlinear channel equalization task. This task focuses on the property of the reservoir in classifying input data, which originates from a practical application of wireless communication [35]. During the propagation of a wireless signal, the message is disturbed by various noises and the echo of the signal itself. Therefore, the signal needs to be restored at the receiving end. The target of this task is to reconstruct the original message from the disturbed signal. The original message *d*(*n*) is randomly selected from the set {−3, −1, 1, 3}. The message first passes through a linear channel with memory and is transferred into *q*(*n*), which can be expressed as:

*q*(

*n*) passes through a nonlinear channel with Gaussian white noise, and the final signal

*u*(

*n*) received by the receiver is given by: where

*ξ*(

_{e}*n*) denotes white noise, the amplitude of which can be adjusted according to the SNR. Note that the input of the RC system in this task is a distorted signal

*u*(

*n*), and the desired output should be the original message

*d*(

*n*). The performance is evaluated on the basis of the symbol error rate (SER), which is the ratio of misclassified points to the total tested points. In this test, the input data are normalized, and the first 1000 points are used in the system warm-up. The subsequent 3000 points are used to train W

*, and another 50000 points are used to test the performance of the RC system.*

_{out}To test this task, the value of *θ* is still set as 0.2*T _{ro}*. Under this condition, the feedback strength has little influence on system performance for 0.1

*k*≤

_{BP}*k*≤ 0.9

*k*, which is shown in Fig. 6(a). Moreover, under three different values of

_{BP}*j*, the SERs obtained under SNR = 24 dB fluctuate around 3×10

_{b}^{−3}. Figure 6(b) shows the SERs obtained under continually changing

*j*within [1.1, 3.5], where

_{b}*θ*= 0.2

*T*,

_{ro}*k*= 0.5

*k*, and SNR = 24 dB. One can observe that the SERs are smaller than 4×10

_{BP}^{−3}in the range of 1.1 ≤

*j*≤ 3.5. To investigate the performance of this reservoir under different SNRs, we calculate the SERs as a function of the SNR for three different values of

_{b}*j*in Fig. 6(c). Obviously, as the SNR increases, the SERs decrease. Moreover, under a relatively large SNR, a larger

_{b}*j*is helpful for obtaining a smaller SER. Under SNR = 32 dB, the SERs obtained are 7.4×10

_{b}^{−5}± 2.0×10

^{−5}, 5.2×10

^{−5}± 3.0×10

^{−5}and 4.0×10

^{−5}± 2.0×10

^{−5}for

*j*=1.5, 2.5 and 3.0, respectively. Therefore, for a nonlinear channel equalization task, the RC system can still yield a good performance under relatively high bias currents by setting

_{b}*θ*= 0.2

*T*and

_{ro}*k*= 0.5

*k*.

_{BP}By the way, it should be pointed out that though the above results are numerical simulation, this scheme proposed in this work can be experimentally implemented. For a given SL, *f _{ro}* at different

*j*can be measured, and then the interval

_{b}*θ*and corresoponding delay time under a given

*j*can be obtained, which means the optimalized length of feedback loop can be determined. Furthermore,

_{b}*k*can be experimentally measured, and then the appropriate feedback strength can be determined. As a result, this proposed RC system has experimental feasibility.

_{BP}## 5. Conclusion

In summary, after taking into account the potential advantage and current research insufficiency of RC based on an optical feedback semiconductor laser (SL) under electrical information injection, we have numerically investigated the performances of this RC system via a Santa Fe time series prediction task and a nonlinear channel equalization task with the aim of optimizing the performance of the system. By adopting current-related *θ* and *k*, the RC system can maintain a good performance when addressing high-speed data tasks even if the SL is biased far above its threshold. The simulated results show that the prediction errors are smaller than 0.01 for 1.1 ≤ *j _{b}* ≤ 3.5 for the Santa Fe time series prediction task, and the SER values achieved are on the order of 10

^{−5}under SNR = 32 dB for the nonlinear channel equalization task. Moreover, a higher information processing rate can be achieved by increasing the SL current. Compared with the previously reported experimental results [10] where a good performance can be realized near the SL threshold current with a data processing rate of about 13MSa/s, this proposed RC system has the potentials in extending the SL bias current range and enhancing the information processing rate. We believe that this work would be helpful for setting up a simple and high performance RC system.

## Funding

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

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