## Abstract

We numerically investigate reservoir computing based on the consistency of a semiconductor laser subjected to optical feedback and injection. We introduce a chaos mask signal as an input temporal mask for reservoir computing and perform a time-series prediction task. We compare the errors of the task obtained from the chaos mask signal with those obtained from other digital and analog masks. The performance of the prediction task can be improved by using the chaos mask signal due to complex dynamical response.

© 2016 Optical Society of America

## 1. Introduction

Artificial neural networks of information processing systems that mimic the structure of the brain have been studied for a long time. The relationship between the input and output is expressed by a nonlinear mapping in artificial neural networks, and allows a variety of information processing through training. Recurrent neural networks are very effective to process empirical data, since they are systems with feedback and some memory capacity. Recently, neuron computers composed of optical elements have attracted much attention [1]. The optical neuron computing has a potential to achieve high-speed information processing and lower power consumption. Recurrent neural networks require huge amounts of calculation and complex algorithms, because the connections in the network have to be trained [2].

In the early 2000s, “reservoir computing” (RC) has been proposed as a new approach of information processing system [3,4]. The concept of RC is based on a mapping of an input signal into a high dimensional space in order to facilitate classification and time-series prediction. In RC, the connections of each node of the network are kept fixed, and the network is referred to as “reservoir”. RC is composed of three parts: the input layer, the reservoir and the output layer. The connection between the input layer and the reservoir is fixed randomly, as well as the reservoir internal connection, and only the training of the output weights is required [3,4]. Compared to conventional recurrent neural networks, this approach has the advantage that the learning algorithm is simple and that small calculation power is required. Reservoir computing has been demonstrated in several biological approaches [5] and optical nonlinear dynamical systems [6–8].

In 2011, delay-based RC using a single nonlinear system has been proposed [9]. This approach has realized easy hardware implementation of a virtual network based on a nonlinear dynamical system with time-delayed feedback. The outputs within the time-delayed feedback loop are equidistantly sampled and considered as the internal states of the virtual network, which results in a simple implementation of a complex network. Several implementations have been reported in delay-based RC, such as a Mackey-Glass electronic circuit [9], optoelectronic systems based on the Ikeda model [10–12], all-optical systems [13–15], laser dynamical systems [16], and Boolean logic elements [17]. Numerical simulations of all-optical RC have also been reported [18–21]. In particular, semiconductor lasers with time-delayed feedback are very promising for high-speed implementation of RC and high dimensional transformation of the input signal [16,18–21]. A small feedback time and a small node interval can be achieved in the semiconductor laser systems in order to increase the processing speed.

One of the important properties of RC is consistency [22], where the same response output can be observed by using a repeated drive signal. Consistency can be achieved in a semiconductor laser with optical feedback and injection, and has been observed numerically [23] and experimentally [24]. The property of consistency is determined by the susceptibility of the laser output to the input drive signal, and is dependent on the nature of the input signals. In RC, a temporal mask is applied to each input data in order to introduce complex transient response under consistency conditions, and the masked input signal is sent to the reservoir. In most cases, a binary random signal is used as the input temporal mask, consisting of a piecewise constant function with a randomly-modulated binary sequence. Studies on the design of the input mask signal have been reported in RC, such as a six-level digital mask [25] and a binary mask with optimized combination [26] in order to reduce the influence of noise. Recently, a sinusoidal analog mask signal has been implemented to RC [27]. It is expected to improve the performance of RC by using a more complex mask signal, such as a chaos mask, since the laser dynamics induced by the chaos mask signal could be more complex than those induced by the binary mask signal. It has been shown that the input weights determined by a chaotic time series could outperform random weights in an echo state network [28].

In this study, we show the effectiveness of a chaos mask signal used for RC based on the consistency of a semiconductor laser subjected to optical feedback and injection. We compare the errors of a time-series prediction task obtained from the chaos mask signal with those obtained from other digital and analog mask signals. We show that the performance of the prediction task can be improved by using the chaos mask signal due to complex dynamical response.

## 2. Scheme of reservoir computing

#### 2.1 Delay-based reservoir computing

Traditional RC requires a network of many nonlinear nodes in order to achieve variable node states in the reservoir. On the other hand, delay-based RC consists of a virtual network, using a single dynamical device with time-delayed feedback [9].

Figure 1 shows our scheme of RC. The RC is composed of three parts: the input layer, the reservoir and the output layer. An input data is discretized and denoted as *u*(*n*), where *n* is a discrete time index, as a pre-processing step in the input layer. *u*(*n*) is hold for a time *T*, and a temporal mask is applied for each duration of *T*. The value of the mask is set to vary at each interval *θ*, corresponding to the virtual-node interval in the reservoir. For example, the mask consists of a piecewise constant function with a randomly-modulated binary sequence {-1, 1} with equal probabilities [9,10,13]. The value of *θ* is set smaller than the transient response of the nonlinear system (i.e., the relaxation oscillation frequency of the laser), so that the system can generate a complex behavior. The feedback delay time *τ* in the reservoir is set nearly equal to the input holding time *T*, which is determined by the product of *N* nodes and the node interval *θ* (i.e., *T = Nθ*). We use de-synchronization scheme of the input holding time *T* and the feedback delay time *τ* (i.e., *τ* = *T* + *θ*) [11,13,14]. The input sampling time *T* is set to 40.0 ns in our numerical simulations. The reservoir is composed of 400 virtual nodes (*N* = 400) with node interval of *θ =* 0.1 ns.

In the reservoir, virtual nodes are determined from the transient response of the laser system (see Sec. 2.2 in detail). The reservoir is composed of the virtual nodes *x _{i}*(

*n*), (

*i*= 1,2,…,

*N*) for the

*n*-th input data, and individual virtual nodes indicate different values to achieve high dimensional space mapping. For post-processing in the output layer, the output

*y*(

*n*) for the

*n*-th input data is calculated as a linear combination of virtual nodes

*x*(

_{i}*n*) with the output weights

*W*for every temporal-mask periodicity

_{i}*T*as follows.

*W*are optimized by minimizing the mean-square error between the target function $\overline{y}(n)$ and the RC output

_{i}*y*(

*n*) as follows.

*W*with the training data.

_{i}#### 2.2 All optical RC using semiconductor laser

We investigate RC based on the consistency of a semiconductor laser subjected to optical delayed-feedback and injection, as shown in Fig. 1. Our scheme of RC uses two semiconductor lasers, referred to as a drive laser and a response laser. The dynamics of the response laser is used as the reservoir. The virtual nodes are defined as the outputs of the response laser at the each interval *θ* within the feedback delay time *τ*. The drive laser is used to achieve consistency of the response laser, as well as to convert the input signal into an optical injection signal. The dynamics of the response laser is calculated by using the Lang-Kobayashi equations as follows [29,30].

*E*and

_{d}*E*are the electric-field amplitudes of the drive and response lasers, and

_{r}*N*is the carrier density of the response laser.

_{r}*α*is the linewidth enhancement factor,

*G*is the gain coefficient,

_{N}*N*

_{0}is the carrier density at transparency, $\epsilon $ is the saturation coefficient,

*τ*

_{p}_{,}

*are the photon and carrier lifetimes,*

_{s}*κ*is the feedback strength of the response laser,

*κ*is the injection strength from the drive to response laser.

_{inj}*ω*

_{d}_{,}

*are the optical angular frequency of the drive and response lasers.*

_{r}*Δω*is the angular frequency detuning (

*Δω*= 2

*πΔf =*2

*π*(

*f*)).

_{d}- f_{r}*J*is the injection current of the response laser.

_{r}*j*is the injection current normalized by the lasing threshold

_{r}*J*.

_{th}*τ*is the feedback delay time of the response laser. These parameter values are summarized in Table 1.

A white Gaussian noise *ξ*(*t*) is added to the electric-field amplitude to include spontaneous emission. The signal-to-noise ratio of the response laser output is set to 20 dB in our numerical simulations. The relaxation oscillation frequencies of the response laser without and with optical injection are 4.2 and 5.7 GHz, respectively. The parameters of *κ* and *Δω* are selected so that injection locking can be achieved between the drive and response lasers [31].

The RC time scale is determined by the inverse of the relaxation oscillation frequency (~0.18 ns) of the response laser under optical injection. The transient response is crucial for the RC [12], and we selected the node interval *θ* = 0.1 ns to maintain the transient response between the node dynamics.

In the input layer, the masked input signal *M*(*t*) is generated by multiplying the input data *u*(*n*), the mask signal *mask*(*t*) with periodicity *T*, and the scaling factor *γ*, as follows.

*M*(

*t*). After the phase modulation, the electric field of the drive laser is described by the following equation.Where

*E*is the electric field of the drive laser,

_{d}*I*is the light intensity of the steady state of the drive laser [30]. The phase-modulated drive signal is injected into the response laser to achieve consistency, as well as to modulate the response laser output for RC.

_{d}## 3. Performance of Santa Fe time-series prediction task

To evaluate the performance of our RC scheme, we use the Santa Fe time-series prediction task [32]. The aim of this task is to perform single-point-prediction of chaotic data. This chaotic data is generated from a far-infrared laser. We use 3000 steps for training and 1000 steps for testing. The amplitude of the Santa Fe time-series is normalized so that the input signal *u*(*n*) of the Santa Fe time-series can be ranged from 0 to 1.

#### 3.1 RC with chaos and binary mask signals

In this section, we use a digital binary mask signal and an analog chaos mask signal to compare the performances of RC. The binary mask signal is composed of a binary sequence {-1, 1} which varies randomly at each interval *θ* [9,10,13]. The standard deviation of the binary mask signal is set to 1. The chaos mask signal is generated from another semiconductor laser with optical feedback [29,30]. The amplitude of the chaos mask signal *mask*(*t*) is rescaled so that the standard deviation of the chaos mask signal is set to 1, and the mean value is set to 0. The scaling factor is set to *γ* = 1.0 for both of the masks. In this case, the maximum and minimum values of the chaos mask signal become 3.59 and −2.40, respectively.

Figures 2(a) and 2(b) show the temporal waveforms of the masked input signals *M*(*t*) and the output of the response laser for the binary and chaos mask signals, respectively. In the case of the binary mask signal [Fig. 2(a)], some nodes in the response signal show similar values when the mask value is constant. On the other hand, the response laser generates complex response in the case of the chaos mask signal in Fig. 2(b). In this case, a variety of node states can be obtained in the complex and fast dynamical response signal. Figure 2(c) shows the frequency spectra of the chaos mask signal and the response laser output without the input signal under optical injection. The peak frequency of the chaos mask signal is 4.76 GHz, which is close to the relaxation oscillation frequency of the response laser under optical injection (5.70 GHz). The RF spectrum of the chaos mask signal is selected to be similar to that of the response laser output under optical injection, so that complex dynamics can be induced in the response laser cavity.

We investigate the performance of the time-series prediction task by using these mask signals. Figure 3 shows the results for the binary and chaos mask signals. In both cases, the prediction results are similar to the original signals. However, smaller errors are observed in the case of the chaos mask signal in Fig. 3(b).

The performance of the time-series prediction task is quantitatively evaluated by using the normalized mean-square error (NMSE) as follows [10,19].

*n*is the index of the input data and

*L*is the total number of the data.

*y*is the RC output that is compared to the original value

*ȳ*as a target.

*var*represents the variance. In the case of the binary mask, the minimum NMSE is 0.064 in Fig. 3(a), which is comparable to other delay-based RC systems in the literature [18–21]. In the case of the chaos mask signal, the minimum NMSE is 0.008 in Fig. 3(b). Better performance of the time-series prediction task can be achieved by using the chaos mask signal.

To interpret the mechanism of the improvement of RC performance using the chaos mask signal, we investigate the histograms of the node states from the response laser outputs in the cases of the binary and chaos mask signals, as shown in Fig. 4. For the binary mask in Fig. 4(a), the histogram has a peak shape and the node states are concentrated in a narrow region. The value of the node states at the peak of the histogram corresponds to the steady state of the response laser, which indicates that many node states do not show transient dynamics. For the chaos mask in Fig. 4(b), however, a broader distribution is obtained. The variety of the node states for the chaos mask signal is larger than that for the binary mask signal. We speculate that this variability of the nodes states results in higher dimensional mapping of the input signal and higher performance of RC using the chaos mask signal.

#### 3.2 RC with digital mask signals

In this section, we use several digital mask signals for RC and compare their prediction errors with the case of the chaos mask signal. The values of the digital mask signal are changed randomly with the constant interval *θ*, similar to the binary mask signal [9,25,26]. In addition to the binary mask, we use a six-level mask signal { ± 1.0, ± 0.6, ± 0.3} and a random-level mask signal {-1 ~1}. Figures 5(a) and 5(b) show the temporal waveforms of the six-level and random-level mask signals, and their corresponding outputs of the response laser. The values of *γ* are set to 1.3 and 1.5 for the six- and random-level mask signals, respectively. Complex temporal waveforms of the response laser output are obtained in both cases of Figs. 5(a) and 5(b).

We investigate the performance of the time-series prediction task when the standard deviation of the amplitude of the mask signal is changed by varying the scaling factor *γ*, as shown in Fig. 6. The minimum error can be obtained when the standard deviation is close to 1 for all the mask signals. We speculate that the existence of the minimum error is due to the phase modulation of the drive signal, whose modulation amplitude is bounded by 2π. We found that the NMSEs for the binary mask signal is larger than those for the other digital mask signals. The NMSEs for the six-level mask signal are comparable to those for the random mask signal. The NMSEs for the chaos mask signal are smaller than for all the digital mask signals, and the minimum NMSE is 0.008. The prediction error can therefore be improved by using the chaos mask signal.

We select the best scaling factor *γ* to perform the minimum NMSE in Fig. 6 for each mask signal, and change the value of one of the laser parameters. Figure 7 shows the performance of the time-series prediction task when the feedback strength *κ* of the response laser is varied for the digital and chaos mask signals. Consistency of the response laser is achieved in the region of 0 ≤ *κ* ≤ 19 ns^{−1} under injection locking. The consistency region is estimated from the cross-correlation value of temporal waveforms between two response lasers with the same optical injection, under the condition of different initial values and different noise signals (The definition of the consistency region will be explained in Fig. 8). For larger *κ*, consistency is not achieved since the optical frequencies are mismatched between the two lasers. The NMSEs for all the mask signals are decreased as the feedback strength is increased within the consistency region. Smaller NMSEs are obtained at the vicinity of the consistency region (*κ* ~16 ns^{−1}.), which is close to the neutral stability of the laser dynamical system (also known as the edge of chaos [33]). Compared with the NMSEs between the digital and chaos mask signals, smaller NMSEs are obtained for the chaos mask signal.

Figure 8(a) shows the cross-correlation values of the temporal waveforms between the two response lasers with the same optical injection under the condition of different initial conditions and different noise signals when the feedback strength is changed. Figure 8(b) show the conditional Lyapunov exponents of the response laser under optical injection. The consistency region is defined as negative conditional Lyapunov exponents and corresponds to 0 ≤ *κ* ≤ 19 ns^{−1}. In this region, the cross correlation value exceeds ~0.9. Compared Fig. 8 with Fig. 7, the smallest error of the prediction task is obtained at *κ* = 16 ns^{−1} when the conditional Lyapunov exponent is λ_{c} = −0.002 ns^{−1}, which is a negative value and close to zero. This result indicates that the performance of RC is enhanced near the neutral stability condition (negative and almost zero conditional Lyaupnov exponents) in the consistency region.

We speculate that the chaos mask signal is more effective to induce complex behaviors in the response laser output than the digital mask signals, since it is generated from a laser system similar to the response laser (i.e., a semiconductor laser subjected to optical feedback with a similar relaxation oscillation frequency). The analog property of the chaos mask signal may also be useful to generate smooth and complex temporal waveforms of the response laser output, compared with the artificial digital mask signals consisting of square waveforms. The use of low-pass-filtered digital mask signals may be effective for better performance of RC and would be an interesting topic in the future work.

The amplitude and the frequency of the mask signal need to be optimized for the comparison between the binary and chaos mask signals. We adjusted the amplitude of the mask signal with the standard deviation, as seen in Fig. 6. For the frequency optimization, we changed the mask interval *θ _{m}* for the binary mask signal to reduce the prediction error. We obtained the minimum prediction error of 0.037 for

*θ*= 0.035 ns. This value is still larger than the prediction error of 0.008 for the chaos mask signal. This result indicates that the chaos mask signal is effective to improve the performance of RC.

_{m}To show the validity of our work, we tested the nonlinear channel equalization task [11,13,14,20] with our scheme as classification rather than prediction. We found that the chaos mask signal is more effective than the binary mask signal even for the nonlinear channel equalization task.

#### 3.3 RC with analog mask signals

To investigate the effectiveness of irregular and complex analog mask signals, we introduce noise mask signals for the comparison of the chaos mask signal. We investigate the effect of the frequency bandwidth of the noise mask signal required for good performance of RC by using both white-Gaussian noise and colored-noise signals. The cut-off frequency of the colored noise mask signal is set near the relaxation oscillation frequency of the response lasers under optical injection (5.70 GHz) to enhance the resonance between the mask signal and the laser dynamical response.

The white Gaussian noise is calculated from the Box-Muller transform. The colored noise is calculated from the Ornstein-Uhlenbeck process using the white Gaussian noise and a low-pass filter in numerical simulations [34]. The amplitudes of these noise mask signals are rescaled so that the standard deviation of the mask signal is set to 1, and the mean value is set to 0. The scaling factor is set to *γ* = 1.0 for both of the masks. In this case, the maximum and minimum values of the white-Gaussian noise mask signal become 3.99 and −3.95, respectively. The maximum and minimum values of the colored-noise mask signal become 3.32 and −3.07 with the cut-off frequency of 6.0 GHz, respectively.

Figure 9 shows the temporal waveforms of the white Gaussian noise mask signal, the colored-noise mask signal, and the corresponding response outputs. The amplitude of the response output is very small in the case of the white Gaussian noise mask in Fig. 9(a). The dynamical response of the laser output cannot follow too fast oscillations of the input signal with the white Gaussian noise mask. On the contrary, a large and complex response laser output is generated from the input signal for the colored-noise mask signal in Fig. 9(b). The cut-off frequency of the colored-noise mask signal is set to 6.0 GHz in this case, which is close to the relaxation oscillation frequency of the response laser.

Figure 10 shows the performance of the time-series prediction task when the standard deviation of the amplitude of the mask signal is changed by varying the scaling factor *γ*. Smaller NMSEs can be obtained where the standard deviation is close to 1 for all the cases, as seen in Fig. 6. The NMSEs for the white Gaussian noise mask are larger than those for the other analog mask signals. The NMSEs for the colored noise mask are comparable to those for the chaos mask signal.

Figure 11 shows the performance of the time-series prediction task when the feedback strength *κ* of the response laser is varied for the analog mask signals. The consistency region is the same as seen in Fig. 7 (0 ≤ *κ* ≤ 19 ns^{−1}), where small NMSEs are obtained. The minimum error is obtained at the vicinity of the consistency region, as in the case of the digital mask signals in Fig. 7. The NMSEs for the white Gaussian noise mask are larger than those for the colored-noise and chaos mask signals. However, the NMSEs for the colored-noise mask signal are very similar to those for the chaos mask signal in Fig. 11. This result indicates that the colored-noise mask signal is as effective as the chaos mask signal to induce complex dynamical response for RC. Complex analog mask signals would be suitable for input temporal masks of RC.

To clarify the conditions of the analog mask signals for better performance of RC, we change the characteristic frequency of the mask signals and investigate the performance of the time-series prediction task. We change the cut-off frequency of the colored-noise mask signal. We also change the peak frequency of the chaos mask signal by varying the injection current of the semiconductor laser used for the mask generation. The peak frequency of the chaos mask signal can be tuned from 0 to 5 GHz in the optical-feedback method. We also use a scheme for bandwidth enhancement of chaos in two unidirectionally-coupled semiconductor lasers [35] to generate chaos with peak frequencies over 5 GHz. The peak frequency of bandwidth-enhanced chaos can be tuned from 5 to 35 GHz by varying the detuning of the optical frequencies between the two coupled semiconductor lasers.

Figure 12(a) shows the NMSEs of the time-series prediction task when the peak frequency of the chaos mask signal is changed. The minimum NMSE is obtained when its value is about 6 GHz, which is close to the relaxation oscillation frequency of the response laser under optical injection (5.70 GHz). Figure 12(b) shows the NMSE for the colored-noise mask signal when the cut-off frequency is changed. The minimum NMSE is obtained for the cut-off frequency of 10 GHz. It is worth noting that the performance of RC can be improved by using the chaos (or colored-noise) mask signal with the peak (cut-off) frequency near the relaxation oscillation frequency of the response laser.

We speculate that the characteristics of the frequency dependence of the mask signal are related to the property of consistency in laser systems. It has been reported that consistency can be achieved very effectively in laser systems by using the chaos drive signal generated from the same laser system, or a colored-noise drive signal with the cut-off frequency near the relaxation oscillation frequency of the response laser [22]. In other words, consistency can be obtained with the smallest amplitude of the input signal when the characteristic frequency of the irregular input signal matches that of the response laser due to resonance effect. A better condition for consistency is thus related to a better performance of RC. However, the best performance of RC does not always correspond to the best condition of consistency, since the reservoir also needs to have separation and approximation properties. For example, the best performance of RC is obtained near the condition of the neutral stability of the response laser in the consistency region, as shown in Figs. 7 and 8. The stability analysis based on the conditional Lyapunov exponents and the dimensionality analysis with Lyapunov spectrum would be our future work for deeper understanding of the performance improvement of RC.

## 4. Conclusions

We investigated RC based on the consistency of a semiconductor laser subjected to optical delayed-feedback and injection in numerical simulations. We evaluated the performance of RC by using a time-series prediction task. We found that the prediction is successful in the region of consistency of the response laser. We also investigated performance improvement with an approach using an analog chaos mask signal, and compared the errors of the prediction task for the chaos mask signal with those for other digital and analog mask signals. We found that the performance can be improved by using the chaos or colored-noise mask signal when the characteristic frequency of the mask is close to the relaxation oscillation frequency of the response laser under optical injection.

## Acknowledgments

We thank Ingo Fischer and Laurent Larger for their fruitful comments. We acknowledge support from Grants-in-Aid for Scientific Research from Japan Society for the Promotion of Science, and Management Expenses Grants from the Ministry of Education, Culture, Sports, Science and Technology in Japan.

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