## Abstract

A reservoir computing (RC) system based on a semiconductor laser (SL) with double optical feedback and optical injection is proposed, and the prediction performance of such a system is numerically investigated via Santa Fe Time-Series Prediction task. The simulation results indicate that the RC system can yield a good prediction performance. Through optimizing some relevant operating parameters, ultra-fast information processing rates up to Gb/s level can be realized for the prediction error is below 3%.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Rapidly increasing information to be processed, as well as higher demands on the processing techniques, has motivated the search for novel computational techniques [1–3]. Inspired from the way the brain processes information, artificial neural network (ANN) has been proposed as a new computational conception. Due to its strong nonlinear mapping ability, self-learning adaptability and parallel information processing ability, ANN has received considerable attention in the field of machine learning. As an effective network model of ANN, recurrent neural network (RNN) has been widely investigated. However, there still exists some drawbacks for practical applications of RNN such as hardly training connection weights, slowly converging, and gradually fading memory [4,5].

In the early 2000s, after modifying traditional training algorithm in RNN, Jaeger and Maass proposed Echo State Network (ESN) and Liquid State Machine (LSM) [5,6], respectively. Although the two methods are proposed from different perspectives, they are essentially the same and collectively named as reservoir computing (RC) [7]. Usually, RC is composed of three parts: input layer, reservoir and output layer. Different from conventional RNNs, the connection between the input layer and the reservoir together with the internal connection in the reservoir are fixed randomly, and only the output weights are required to train. As a result, this approach has such unique advantages that the learning algorithm is simple and the calculation power is low. RC has been demonstrated in several biological approaches [8] and optical nonlinear dynamical information processing systems [9].

In 2011, based on a single nonlinear component under delayed feedback, a novel RC system named as time delayed RC was proposed [10]. For time delayed RC, the nonlinear component under delayed feedback is used as the reservoir, and its outputs within the time delayed feedback loop are equidistantly sampled and taken as virtual nodes. Thus, such time delayed RC system possesses the superiority of easy hardware-implementation. Several time delayed RC systems have been reported such as a Mackey-Glass electronic circuit [10], optoelectronic systems based on the Ikeda model [11–13], all-optical systems [14–17], laser dynamical systems [18], and Boolean logic elements [19]. In particular, semiconductor lasers (SLs) with time delayed feedback, which have wide applications in chaotic secure communication and physical random number generation [20–22], are very promising for high-speed implementation of RC and high dimensional transformation of input signal [18, 23–26]. A small feedback time and a small node interval required for high processing speed, are easy to achieve in RC systems based on SLs with time delayed feedback.

Presently, reports on RC based on SLs with time delayed feedback almost focus on the case that only a single feedback loop is included in the system. However, related investigations in electronic circuit-based RC have demonstrated that introducing much more feedback loops is helpful to improve the prediction performance [27]. In this work, we propose a RC system in which a SL under double optical feedback is taken as the reservoir, and an analog chaos signal generated by mutually-coupled SLs is utilized to mask input signal. Via Santa Fe Time-Series Prediction task, the prediction performance of such RC system is investigated, and the influences of some typical parameters on the prediction performance have been analyzed.

## 2. System model

Figure 1 is the schematic diagram of our proposed all optical RC for predicating, which is different from the case of single optical feedback adopted in [28]. In this system, a response SL (R-SL) under double time-delayed optical feedback is taken as a reservoir, and a chaotic signal generated by two mutually-coupled SLs [29] is utilized as input temporal mask. After removing one of double optical feedback loops, this system can be transferred into that used in [28]. This system also consists of three parts: an input layer, a reservoir and an output layer. In input layer, through sampling input signal and holding an operation-time *T* for each sampling point, the input signal is transferred into *u*(*n*). Multiplied by a temporal mask signal *M*(*t*) and rescaled in a suitable range, *u*(*n*) is changed into masked input signal *S*(*t*), which is sent into the reservoir through loading into the injection light from drive SL (D-SL). The role of the mask is to ensure the variability of input signal over different virtual nodes where the information is read out [30]. In the reservoir, an output of R-SL in an interval of *θ* is interpreted as a state of a virtual network node, and therefore the R-SL’s responses with consecutive intervals *θ* will correspond to the states of consecutive virtual nodes. In particular, in this system, there are two delayed feedback lines. When *S*(*t*) transformed from the input digit *u*(*n*) is completely injected into the system, the shorter delay line only includes virtue nodes corresponding to currently completed input digit *u*(*n*) (green), while the longer delay line includes not only virtue nodes corresponding to currently completed input digit *u*(*n*) (green) but also some virtue nodes corresponding to previously completed input digit *u*(*n*-1) (orange). In the output layer, all responses of virtual nodes from the short delay line (green) are weighted and linearly summed up. In this work, the weights are optimized by minimizing the mean-square error between target function and RC output through using linear least-squares method [18,28]. There are three time scales in such system. One is the sampled time of input signal *T* (or the periodicity of mask signal), which determines the number of virtual nodes *N* ( = T/*θ*) for a given value of *θ*. The others are the delayed times for short feedback loop (*τ*_{1}) and long feedback loop (*τ*_{2}). Here, we consider the case of de-synchronization scheme, *i.e. τ*_{1} *= T* + *θ* [12,14,28].

For the system proposed in this work, the R-SL with double optical feedback and optical injection is used as the reservoir, and the outputs of R-SL at each interval *θ* within the feedback delay time *τ*_{1} are the states of virtual nodes. The optical injection is used to achieve the consistency of R-SL and load masked input signal. The nonlinear dynamics of R-SL under double optical feedback and optical injection can be modeled by [31]:

*E*is the slowly varying complex electric field, and

*N*is the average carrier density.

*N*0 is the carrier density at transparency.

*k*

_{1}and

*k*

_{2}denote the feedback strengths of external cavity 1 and cavity 2 of the R-SL, respectively.

*k*

_{inj}denotes the injection strength from the drive to R-SL.

*α*is the linewidth-enhancement factor,

*g*is differential gain coefficient, and

*ε*is the gain saturation coefficient.

*ν*is the frequency of the free-running R-SL, and Δ

*ν*is the frequency detuning between the D-SL and R-SL.

*τp*is the photon lifetime, and

*τs*is the carrier lifetime.

*J*is the injection current.

*ɵ*

_{1}(

*ɵ*

_{2}) is the phase difference between the feedback field from external cavity 1 (external cavity 2) and the intra-cavity field at the feedback injection point.

*F*(

*t*) is the white Gaussian noise. The signal-to-noise ratio of R-SL under free-running is set to 20 dB in our simulations.

Considering that the input signal multiplied with the mask is used to modulate the optical signal through phase modulation, the injected slowly varying complex electric field *E*_{inj} can be written as

*I*is the intensity of CW output from D-SL, and

_{d}*S*(

*t*) represents the masked input signal and is described as:where

*u*(

*n*) is obtained through sampling input signal and holding in an operation-time

*T*,

*M*(

*t*) is the mask signal with periodicity

*T*, and

*γ*is the scaling factor.

In this work, we will use Santa Fe Time-Series Prediction task to quantify RC performance. This task belongs to chaotic time series prediction, which is a particularly challenging task that aims to make a one-step time series prediction of chaotic data. The Santa Fe data are intensity time series generated from a far-infrared laser operating at a chaotic state [32,33]. Santa Fe data set contains 9000 points, in which we take the first 3000 points as a training set and the next 1000 points as a testing set. The performance of prediction is typically evaluated by calculating normalized mean square error (NMSE) between the target and the reservoir output:

where*y*is the predicted value, and

*ӯ*is the expected value.

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

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

*var*is the variance. Typically, the system is considered to be performed well if NMSE is below 10% [24].

## 3. Results and discussion

Equations (1) – (2) can be solved by using fourth-order Runge-Kutta method via Matlab software. For the convenience of comparison, the parameters are set as [28]: *α* = 3.0, *g* = 8.4 × 10^{−13} m^{3}s^{−1}, *N*_{0} = 1.4 × 10^{24} m^{−3}, *ɛ* = 2.0 × 10^{−23}, *τp* = 1.927 ps, *τs* = 2.04 ns, *k*_{inj} = 12.43 ns^{−1}, *ν =* 1.96 × 10^{14} Hz, Δ*ν* = – 4.0 GHz, *J* = 1.037 × 10^{33} m^{−3}s^{−1}, *I*_{d} *=* 6.56 × 10^{20}. *k*_{1} = 15.53 ns^{−1} and *k*_{2} = 0 for single optical feedback, and *k*_{1} = *k*_{2} = 7.765 ns^{−1} for double optical feedback. *γ* is taken as 1 without specific instructions.

Previous investigation has demonstrated that the prediction performance of time delayed optical RC can be improved by adopting chaos mask signal whose peak frequency nears to the relaxation oscillation frequency of R-SL [28]. Under above given parameters, the relaxation oscillation frequency of R-SL under double optical feedback and optical injection is calculated to be about 5.70 GHz, and then the peak frequency of used chaos mask signal generated by mutually-coupled SLs is set to be about 5.68 GHz. The amplitude of chaos mask signal is rescaled so that the standard deviation of chaos mask signal is equal to 1 meanwhile the mean value is 0.

Generally, the nonlinearity of R-SL considered as the reservoir directly affects the prediction performance of the RC system, and the prediction performance is relatively well when the R-SL operates at high nonlinearity state. Based on the results of [34], when a SL is subjected to double optical feedback whose delayed time difference is about a half of reciprocal of its free-running relaxation oscillation frequency, the laser may exhibit stronger nonlinearity than that obtained under single optical feedback. Considering the relaxation oscillation frequency for the free-running R-SL is about 1.5 GHz, we set τ_{2} – τ_{1} = 0.335 ns for calculating following Figs. 2 and 3.

Figure 2 shows NMSEs as a function of the virtual node interval (a) and the number of virtual nodes (b) under the sampling period T of input signal fixed at 40 ns, *i. e.* the data processing speed is 25 Mb/s. From this diagram, it can be clearly seen that the RC system with double optical feedback generally behaves better prediction performance than that with single optical feedback. As shown in Fig. 2(a), all NMSEs fluctuate around 0.01 under *θ* < 140 ps for the RC system with double optical feedback. Especially, under *θ* = 50 ps, NMSE is only about 0.0083. From Fig. 2(b), one can observe that for relatively small numbers of virtual nodes such as *N* = 50 and 80, NMSEs of the RC system with double optical feedback do not exceed 0.10, which is much smaller than those obtained for the RC system with single optical feedback. For larger values of *N*, NMSEs do not exceed 5%, which means that the RC system with double optical feedback possesses excellent prediction performance.

Usually, the RC system with double optical feedback may result in much more complex response of R-SL compared to a RC system with single optical feedback due to the introduction of another delayed feedback loop. Consequently, higher dimensional transformation of input signal and stronger independence between different virtual node states may be achieved for the proposed RC system. As a result, the RC system with double optical feedback possesses the potential for realizing faster information processing rate (reciprocal of the sample period *T* ( = *Nθ*)). Figure 3 displays the variation of NMSEs with *T* for several typical values of *N*. As shown in this diagram, for a given *N*, NMSE generally presents a decrease trend with the increase of *T*. In other words, the prediction performance is gradually improved with the decrease of data processing rate, which is easy to understand. As mentioned above, if a RC system behaves well prediction performance, NMSE should be smaller than 10%. Under this circumstance, there exists a minimum value for *T*, which corresponds to fastest data processing rate. Obviously, for *N* = 100, this minimum value of *T* is about 1 ns, and therefore the fastest data processing speed can be up to 1 Gb/s. However, for *N* = 200 and 400, the fastest data processing rate of the RC system is only 500 Mb/s for satisfying NMSE < 10%. Furthermore, from the viewpoint of application, for similar data processing rate, the smaller *N* is, the easier the experiment implementation will be. The reason is that, for a fixed *T* ( = *Nθ*), a smaller *N* means a larger *θ*, which will be helpful to reduce the demand for the instrument used to sample output signal. After taking into account these factors, in the following discussion, we set *N* = 100 and *θ* = 10 ps. Accordingly, *T* is 1 ns, which means the data stream is fed into the RC system with a rate of 1 Gb/s.

In above simulations, we refer to our previous work in [34] and set *τ*_{2} = *τ*_{1} + *τ*_{R}/2 (*τ*_{R} is the free-running relaxation oscillation periodicity of R-SL). However, different from [34], extra optical injection is further introduced since in this work the R-SL is taken as a reservoir, and then optimized feedback times may be different from that under only double optical feedback and without optical injection [34]. Figure 4(a) depicts NMSE as a function of *τ*_{2} for the RC with double feedback loops under a data processing rate of 1 Gb/s. Here, the step of *τ*_{2} is about 10 ps. From this diagram, one can see that NMSE fluctuates with the increase of *τ*_{2}, whose minimum (about 3.62%) appears at *τ*_{2} = *τ*_{2m} = 1.17 ns. Furthermore, the dependence of NMSE on the *τ*_{2} is also given in Fig. 4(b) in a much smaller scale, in which the phase offset to 2π*vτ*_{2m} induced by the variation of *τ*_{2} is from – 2π to 2π and the corresponding step of *τ*_{2} is about 0.16 fs. Such a small step may be achieved in practical implementation via high-precision (nm or below) controlled system based on piezoelectric ceramics. As shown in Fig. 4(b), a periodically varied trend of NMSE with *τ*_{2} can be observed.

Above results are obtained under the case of *γ* = 1. Previously related investigation has demonstrated that the scaling factor of mask signal is also an important parameter that affects the performance of RC system. The variation of NMSE with *γ* is given in Fig. 5, where the used parameter data are the same as those used in Fig. 4 except *τ*_{2} = 1.17 ns. With the increase of *γ*, NMSE generally shows a trend of firstly decreasing and then increasing accompanied by fluctuations. All the NMSEs are less than 10% when the scaling factor *γ* is within (0.4, 4). When *γ* = 1.12, minimal NMSE is obtained to be 3.41%. Similar variation trend has been reported in RC system with single optical feedback [28], where relevant physical mechanism has also been clarified in [28].

Finally, in order to systematically reveal the influences of *τ*_{2} and *γ* on the prediction performance of RC with double optical feedback, Fig. 6 displays a mapping of NMSE in the parameter space of *τ*_{2} and *γ* under *N* = 100 and *θ* = 10 ps, where different pseudo-color characterizes different value of NMSE. For blue and green regimes, RC system behaves well prediction performance since NMSEs < 10%. In particular, for *τ*_{2} and *γ* located at green regimes, the prediction performance of the RC is excellent since NMSEs ≤ 5%. Under *τ*_{2} = 1.52 ns and *γ* = 0.63, the best prediction performance can be realized to be NMSE = 2.93%.

Additionally, it should be pointed out that the above results are obtained under the case of *ɵ*_{1} = *ɵ*_{2} = 0. For *ɵ*_{1} and *ɵ*_{2} take different values, we recalculate NMSEs corresponding to Fig. 4(b), which is displayed in Fig. 7. From this diagram, it can be observed that NMSEs are sensitive to the values of *ɵ*_{1} and *ɵ*_{2}, which is due to strong dependence of the dynamics of this system on *ɵ*_{1} and *ɵ*_{2}.

## 4. Conclusions

In summary, a RC system based on a semiconductor laser (SL) subjected to double optical feedback and optical injection is proposed, and its prediction performance is numerically investigated. In this RC system, an analog chaos signal generated by mutually-coupled SLs is used to mask input signal, and Santa-Fe data set is adopted to evaluate the performance for one-step chaotic time series prediction. The simulation results show that, compared with the result obtained in similar time delayed RC system with single optical feedback, this proposed RC system possesses better prediction performance. Based on the mapping of NMSE in the parameter space of *τ*_{2} and *γ*, the optimized parameter regions for realizing well (or excellent) prediction performance can be specified, and the lowest NMSE is 2.93% under a information processing rate of 1 Gb/s. To the best of our knowledge, such an information processing rate is the fastest under similar prediction error level.

## Funding

National Natural Science Foundation of China (NSFC) (61475127, 61575163, 61775184).

## Acknowledgments

The authors would like express great appreciation to the reviewers for their constructive suggestions and Prof. Jia-Ming Liu (University of California, Los Angeles) for very useful discussion.

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