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 [1119].

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 [1119]. 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.

 

Fig. 1. Schematic diagram of reservoir computing based on an optical feedback semiconductor laser under electrical information injection. SL: semiconductor laser; OC: optical circulator; VA: variable attenuator.

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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 jm(t), and then, jm(t) and the bias current jb are combined to pump the SL.

The output yout of the RC system can be obtained by calculating a linear combination of virtual node states, i.e., yout= WoutX, where Wout is the readout weight and can be obtained by training. The purpose of training is to minimize the normalized mean square error (NMSE) between yout and the desired values yd, and the NMSE can be expressed as [25]:

$$NMSE = \frac{1}{L}\sum\limits_{n = 1}^L {{{({y_{out}}(n) - {y_d}(n))}^2}} /var({y_d}),$$
where 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 Wout can be obtained via the regression algorithm: Wout= (XTX)-1XTyd. However, in practical applications, overfitting often occurs, where XTX approaches singular matrix and the amplitude of Wout 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, Wout= (XTX + CI)-1XTyd, where CR+ 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]:

$$\frac{{dE(t)}}{{dt}} = \frac{1}{2}(1 + i\alpha )\left[ {\frac{{{G_N}({N(t )- {N_0}} )}}{{1 + \varepsilon {{|{E(t)} |}^2}}} - \frac{1}{{{\tau_p}}}} \right]E(t)\; + \;kE({t - \tau } ){e^{ - i{\omega _0}\tau }} + F(t)\;,$$
$$\frac{{dN(t)}}{{dt}} = {j_b}(1 + \gamma {j_m}(t)){J_{th}} - \frac{{N(t)}}{{{\tau _s}}} - \frac{{{G_N}({N(t )- {N_0}} )}}{{1 + \varepsilon {{|{E(t)} |}^2}}}{|{E(t)} |^2},$$
where E(t) is the slowly varying complex electric field and N(t) is the carrier density. GN is the gain coefficient, N0 is the carrier density at transparency, τp is the photon lifetime, τs is the carrier lifetime, 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 ω0 is the angular frequency of the free running laser. F(t) (= (2βN)1/2ξ(t)) is the Langevin noise source, modeling the spontaneous emission noise, where we set β = 5×10−5; ξ(t) is Gaussian white noise. Jth is the injection current at the threshold, jb is the normalized bias current, jm(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]: GN = 8.40×10−13 m3s-1, N0 = 1.4 × 1024 m-3, τp = 1.927 ×10−12 s, τs = 2.04×10−9 s, α = 3.0, Jth = 1.037 × 1033 m-3s-1, ω0 =1.226 × 1015 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 = H0 T0, where H0< 1 and T0 is the characteristic time scale of the nonlinear node [6]. For the SL-based RC system, T0 refers to the characteristic time scale of the laser’s relaxation oscillation Tro. In [10], H0 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 θ of 0.2 ns. As is well known, the relaxation oscillation frequency fro increases with the increasing SL current, and therefore, Tro (=1/ fro) decreases with the increasing SL current. The dependence of fro of an SL on the bias current can be described by [28]:

$${f_{ro}} = \sqrt {{G_N}{J_{th}}({j_b} - 1)} /(2\pi ).$$
Under GN and Jth given above, the variations of fro and Tro with increasing SL bias current are shown in Fig. 2. When jb increases from 1.1 to 3.5, fro increases from 1.45 GHz to 7.2 GHz, and Tro correspondingly decreases from 0.69 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.

 

Fig. 2. Relaxation oscillation frequency fro and relaxation oscillation characteristic time Tro of the free-running SL as a function of the normalized bias current jb.

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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 [2933]. 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 jb = 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 kBP. It is well known that the value of kBP depends on not only the SL bias current jb but also the feedback delay time τ. In our system, τ = 50H0Tro, where Tro can be read out from Fig. 2 for a given jb. Under this circumstance, for a given jb and H0, the corresponding kBP can be determined. Figure 3(b) shows the variation of kBP with the increase in jb under different H0 values (H0 = 0.1, 0.2, 0.3, 0.4, and 0.5). From this diagram, one can see that the value of kBP increases with jb and that kBP decreases with increasing H0 for a fixed jb. As a result, to avoid the system operating in an oscillating state without input, we set k to no more than 0.9 kBP in the following discussion.

 

Fig. 3. (a) Bifurcation diagram as a function of feedback strength k for jb = 2.0 and τ = 1.5 ns; (b) Feedback strength at the bifurcate point (kBP) as a function of the normalized bias current jb under θ = H0 Tro, where H0 takes 0.1, 0.2, 0.3, 0.4 and 0.5.

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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 Wout, 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 jb = 1.5, 2.5, and 3.0, are taken as examples. The dependence of the NMSEs on k under θ = H0 Tro (H0 =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 k, the prediction error NMSE first decreases, reaches a minimum value, and then gradually increases. For jb = 1.5 (as shown in Fig. 4(a)), the NMSEs are always higher than 0.01 for 0.1kBPk ≤0.9kBP when H0 = 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 θ. For too large θ (H0 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 H0 is set at 0.2, 0.3, and 0.4, the NMSEs are relatively low. Generally, a relatively large 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 H0 = 0.2 in the following discussion. Under this case, for jb = 1.5 (as shown in Fig. 4(a)), good performances can be obtained within 0.2kBPk ≤0.8kBP, and the minimal value emerges at k = 0.7kBP. For jb = 2.5 (as shown in Fig. 4(b)), a good performance can be achieved within 0.25kBPk ≤0.65kBP, and the minimal value is located at 0.4kBP. For jb = 3.0 (as shown in Fig. 4(c)), good performances can be realized within 0.2kBPk ≤0.65kBP, and the minimal value is located at 0.45 kBP. Taking into account the above three cases under different jb, we set the optimized value of k to 0.5kBP.

 

Fig. 4. NMSEs as a function of feedback strength k, where (a) jb = 1.5, (b) jb = 2.5, and (c) jb = 3.0.

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The above results demonstrate that good performances can be achieved for this RC system under the three cases of jb by adopting θ = 0.2Tro and k = 0.5kBP. Next, we analyze the case in which jb 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 Tro are different for different jb values (as shown in Fig. 2); therefore, the feedback time should be varied with jb, as shown in Fig. 5(b). Similarly, the values of k ( = 0.5kBP) should also be varied with jb since kBP is dependent on jb (as shown in Fig. 3(b)). For comparison, the NMSEs obtained under 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 ≤ jb ≤ 3.5, i.e., good prediction performances can be achieved. Moreover, as shown in Fig. 5(b), the delay time τ decreases with the increasing jb; therefore, the information processing rate can be increased from 0.15GSa/s to 0.73GSa/s for jb values from 1.1 to 3.5.

 

Fig. 5. (a) NMSEs as a function of jb, where the blue line is for the case of θ = 0.2Tro and k = 0.5kBP and the red line is for the case of θ = 0.2 ns and k = 0.15ns-1; (b) Delay time (blue line) and information processing rate (red line) under θ = 0.2Tro as a function of jb.

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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:

$$\begin{aligned}q(n) &= 0.08d(n + 2) - 0.12d(n + 1) + d(n) + 0.18d(n - 1)\\ & \quad - 0.1d(n - 2) + 0.091d(n - 3) - 0.05d(n - 4)\\ & \quad + 0.04d(n - 5) + 0.03d(n - 6) + 0.01d(n - 7). \end{aligned}$$
Then, q(n) passes through a nonlinear channel with Gaussian white noise, and the final signal u(n) received by the receiver is given by:
$$u(n) = q(n) + 0.036{q^2}(n) - 0.011{q^3}(n) + {\xi _e}(n),$$
where ξe(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 Wout, and another 50000 points are used to test the performance of the RC system.

To test this task, the value of θ is still set as 0.2Tro. Under this condition, the feedback strength has little influence on system performance for 0.1kBP ≤ k ≤ 0.9kBP, which is shown in Fig. 6(a). Moreover, under three different values of jb, the SERs obtained under SNR = 24 dB fluctuate around 3×10−3. Figure 6(b) shows the SERs obtained under continually changing jb within [1.1, 3.5], where θ = 0.2Tro, k = 0.5kBP, and SNR = 24 dB. One can observe that the SERs are smaller than 4×10−3 in the range of 1.1 ≤ jb ≤ 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 jb in Fig. 6(c). Obviously, as the SNR increases, the SERs decrease. Moreover, under a relatively large SNR, a larger jb is helpful for obtaining a smaller SER. Under SNR = 32 dB, the SERs obtained are 7.4×10−5 ± 2.0×10−5, 5.2×10−5 ± 3.0×10−5 and 4.0×10−5 ± 2.0×10−5 for jb =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 θ = 0.2Tro and k = 0.5kBP.

 

Fig. 6. SERs of nonlinear channel equalization task with θ = 0.2Tro. (a) SERs as a function of k under SNR = 24 dB; (b) SERs as a function of jb under k = 0.5kBP and SNR = 24 dB; (c) SERs as a function of SNR under k = 0.5kBP.

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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, fro at different jb can be measured, and then the interval θ and corresoponding delay time under a given jb can be obtained, which means the optimalized length of feedback loop can be determined. Furthermore, kBP can be experimentally measured, and then the appropriate feedback strength can be determined. As a result, this proposed RC system has experimental feasibility.

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 ≤ jb ≤ 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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28. A. Uchida, Optical Communication with Chaotic Lasers, Applications of Nonlinear Dynamics and Synchronization (Wiley-VCH, 2012).

29. H. F. Liu and W. F. Ngai, “Nonlinear dynamics of a directly modulated 1.55 μm InGaAsP distributed feedback semiconductor laser,” IEEE J. Quantum Electron. 29(6), 1668–1675 (1993). [CrossRef]  

30. J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, “Modulation bandwidth, noise, and stability of a semiconductor laser subject to strong injection locking,” IEEE Photonics Technol. Lett. 9(10), 1325–1327 (1997). [CrossRef]  

31. G. Q. Xia, S. C. Chan, and J. M. Liu, “Multistability in a semiconductor laser with optoelectronic feedback,” Opt. Express 15(2), 572–576 (2007). [CrossRef]  

32. F. Y. Lin and J. M. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum Electron. 39(4), 562–568 (2003). [CrossRef]  

33. N. Q. Li, L. Zunino, A. Locquet, B. Kim, D. Choi, W. Pan, and D. S. Citrin, “Multiscale ordinal symbolic analysis of the Lang-Kobayashi model for external-cavity semiconductor lasers: a test of theory,” IEEE J. Quantum Electron. 51(8), 1–6 (2015). [CrossRef]  

34. A. S. Weigend and N. A. Gershenfeld, “Time series prediction: forecasting the future and understanding the past,” http://www-psych.stanford.edu/∼andreas/Time-Series/SantaFe.html (1993).

35. H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304(5667), 78–80 (2004). [CrossRef]  

References

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  1. G. Van der Sande, D. Brunner, and M. C. Soriano, “Advances in photonic reservoir computing,” Nanophotonics 6(3), 561–576 (2017).
    [Crossref]
  2. H. Jaeger, “The ‘echo state’ approach to analysing and training recurrent neural networks,” Technical Report GMD Report 148, German National Research Center for Information Technology (2001).
  3. W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable States: A new framework for neural computation based on perturbations,” Neural Comput. 14(11), 2531–2560 (2002).
    [Crossref]
  4. D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks 20(3), 391–403 (2007).
    [Crossref]
  5. D. Brunner, B. Penkovsky, B. A. Marquez, M. Jacquot, I. Fischer, and L. Larger, “Tutorial: Photonic neural networks in delay systems,” J. Appl. Phys. 124(15), 152004 (2018).
    [Crossref]
  6. L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2(1), 468 (2011).
    [Crossref]
  7. Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2(1), 287 (2012).
    [Crossref]
  8. L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20(3), 3241–3249 (2012).
    [Crossref]
  9. F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and S. Massar, “All-optical reservoir computing,” Opt. Express 20(20), 22783–22795 (2012).
    [Crossref]
  10. D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4(1), 1364 (2013).
    [Crossref]
  11. K. Hicke, M. A. Escalona-Morán, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1501610 (2013).
    [Crossref]
  12. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast photonic information processing using semiconductor lasers with delayed optical feedback: Role of phase dynamics,” Opt. Express 22(7), 8672–8686 (2014).
    [Crossref]
  13. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Simultaneous computation of two independent tasks using reservoir computing based on a single photonic nonlinear node with optical feedback,” IEEE Trans. Neural Netw. Learn. Syst. 26(12), 3301–3307 (2015).
    [Crossref]
  14. Y. S. Hou, G. Q. Xia, W. Y. Yang, D. Wang, E. Jayaprasath, Z. F. Jiang, C. X. Hu, and Z. M. Wu, “Prediction performance of reservoir computing system based on a semiconductor laser subject to double optical feedback and optical injection,” Opt. Express 26(8), 10211–10219 (2018).
    [Crossref]
  15. J. Nakayama, K. Kanno, and A. Uchida, “Laser dynamical reservoir computing with consistency: an approach of a chaos mask signal,” Opt. Express 24(8), 8679–8692 (2016).
    [Crossref]
  16. Y. S. Hou, G. Q. Xia, E. Jayaprasath, D. Z. Yue, W. Y. Yang, and Z. M. Wu, “Prediction and classification performance of reservoir computing system using mutually delay-coupled semiconductor lasers,” Opt. Commun. 433, 215–220 (2019).
    [Crossref]
  17. J. Bueno, D. Brunner, M. C. Soriano, and I. Fischer, “Conditions for reservoir computing performance using semiconductor lasers with delayed optical feedback,” Opt. Express 25(3), 2401–2412 (2017).
    [Crossref]
  18. Y. Kuriki, J. Nakayama, K. Takano, and A. Uchida, “Impact of input mask signals on delay-based photonic reservoir computing with semiconductor lasers,” Opt. Express 26(5), 5777–5788 (2018).
    [Crossref]
  19. J. Vatin, D. Rontani, and M. Sciamanna, “Enhanced performance of a reservoir computer using polarization dynamics in VCSELs,” Opt. Lett. 43(18), 4497–4500 (2018).
    [Crossref]
  20. Q. Vinckier, F. Duport, A. Smerieri, K. Vandoorne, P. Bienstman, M. Haelterman, and S. Massar, “High performance photonic reservoir computer based on a coherently driven passive cavity,” Optica 2(5), 438–446 (2015).
    [Crossref]
  21. L. Appeltant, G. Van der Sande, J. Danckaert, and I. Fischer, “Constructing optimized binary masks for reservoir computing with delay systems,” Sci. Rep. 4(1), 3629 (2015).
    [Crossref]
  22. F. Duport, A. Smerieri, A. Akrout, M. Haelterman, and S. Massar, “Fully analogue photonic reservoir computer,” Sci. Rep. 6(1), 22381 (2016).
    [Crossref]
  23. M. C. Soriano, S. Ortín, D. Brunner, L. Larger, C. R. Mirasso, I. Fischer, and L. Pesquera, “Optoelectronic reservoir computing: tackling noise-induced performance degradation,” Opt. Express 21(1), 12–20 (2013).
    [Crossref]
  24. M. Tezuka, K. Kanno, and M. Bunsen, “Reservoir computing with a slowly modulated mask signal for preprocessing using a mutually coupled optoelectronic system,” Jpn. J. Appl. Phys. 55(8S3), 08RE06 (2016).
    [Crossref]
  25. H. Zhang, X. Feng, B. X. Li, Y. Wang, K. Y. Cui, F. Liu, W. B. Dou, and Y. D. Huang, “Integrated photonic reservoir computing based on hierarchical time-multiplexing structure,” Opt. Express 22(25), 31356–31370 (2014).
    [Crossref]
  26. R. S. Zimmermann and U. Parlitz, “Observing spatio-temporal dynamics of excitable media using reservoir computing,” Chaos 28(4), 043118 (2018).
    [Crossref]
  27. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
    [Crossref]
  28. A. Uchida, Optical Communication with Chaotic Lasers, Applications of Nonlinear Dynamics and Synchronization (Wiley-VCH, 2012).
  29. H. F. Liu and W. F. Ngai, “Nonlinear dynamics of a directly modulated 1.55 μm InGaAsP distributed feedback semiconductor laser,” IEEE J. Quantum Electron. 29(6), 1668–1675 (1993).
    [Crossref]
  30. J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, “Modulation bandwidth, noise, and stability of a semiconductor laser subject to strong injection locking,” IEEE Photonics Technol. Lett. 9(10), 1325–1327 (1997).
    [Crossref]
  31. G. Q. Xia, S. C. Chan, and J. M. Liu, “Multistability in a semiconductor laser with optoelectronic feedback,” Opt. Express 15(2), 572–576 (2007).
    [Crossref]
  32. F. Y. Lin and J. M. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum Electron. 39(4), 562–568 (2003).
    [Crossref]
  33. N. Q. Li, L. Zunino, A. Locquet, B. Kim, D. Choi, W. Pan, and D. S. Citrin, “Multiscale ordinal symbolic analysis of the Lang-Kobayashi model for external-cavity semiconductor lasers: a test of theory,” IEEE J. Quantum Electron. 51(8), 1–6 (2015).
    [Crossref]
  34. A. S. Weigend and N. A. Gershenfeld, “Time series prediction: forecasting the future and understanding the past,” http://www-psych.stanford.edu/∼andreas/Time-Series/SantaFe.html (1993).
  35. H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304(5667), 78–80 (2004).
    [Crossref]

2019 (1)

Y. S. Hou, G. Q. Xia, E. Jayaprasath, D. Z. Yue, W. Y. Yang, and Z. M. Wu, “Prediction and classification performance of reservoir computing system using mutually delay-coupled semiconductor lasers,” Opt. Commun. 433, 215–220 (2019).
[Crossref]

2018 (5)

2017 (2)

2016 (3)

F. Duport, A. Smerieri, A. Akrout, M. Haelterman, and S. Massar, “Fully analogue photonic reservoir computer,” Sci. Rep. 6(1), 22381 (2016).
[Crossref]

M. Tezuka, K. Kanno, and M. Bunsen, “Reservoir computing with a slowly modulated mask signal for preprocessing using a mutually coupled optoelectronic system,” Jpn. J. Appl. Phys. 55(8S3), 08RE06 (2016).
[Crossref]

J. Nakayama, K. Kanno, and A. Uchida, “Laser dynamical reservoir computing with consistency: an approach of a chaos mask signal,” Opt. Express 24(8), 8679–8692 (2016).
[Crossref]

2015 (4)

Q. Vinckier, F. Duport, A. Smerieri, K. Vandoorne, P. Bienstman, M. Haelterman, and S. Massar, “High performance photonic reservoir computer based on a coherently driven passive cavity,” Optica 2(5), 438–446 (2015).
[Crossref]

L. Appeltant, G. Van der Sande, J. Danckaert, and I. Fischer, “Constructing optimized binary masks for reservoir computing with delay systems,” Sci. Rep. 4(1), 3629 (2015).
[Crossref]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Simultaneous computation of two independent tasks using reservoir computing based on a single photonic nonlinear node with optical feedback,” IEEE Trans. Neural Netw. Learn. Syst. 26(12), 3301–3307 (2015).
[Crossref]

N. Q. Li, L. Zunino, A. Locquet, B. Kim, D. Choi, W. Pan, and D. S. Citrin, “Multiscale ordinal symbolic analysis of the Lang-Kobayashi model for external-cavity semiconductor lasers: a test of theory,” IEEE J. Quantum Electron. 51(8), 1–6 (2015).
[Crossref]

2014 (2)

2013 (3)

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4(1), 1364 (2013).
[Crossref]

K. Hicke, M. A. Escalona-Morán, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1501610 (2013).
[Crossref]

M. C. Soriano, S. Ortín, D. Brunner, L. Larger, C. R. Mirasso, I. Fischer, and L. Pesquera, “Optoelectronic reservoir computing: tackling noise-induced performance degradation,” Opt. Express 21(1), 12–20 (2013).
[Crossref]

2012 (3)

2011 (1)

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2(1), 468 (2011).
[Crossref]

2007 (2)

D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks 20(3), 391–403 (2007).
[Crossref]

G. Q. Xia, S. C. Chan, and J. M. Liu, “Multistability in a semiconductor laser with optoelectronic feedback,” Opt. Express 15(2), 572–576 (2007).
[Crossref]

2004 (1)

H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304(5667), 78–80 (2004).
[Crossref]

2003 (1)

F. Y. Lin and J. M. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum Electron. 39(4), 562–568 (2003).
[Crossref]

2002 (1)

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable States: A new framework for neural computation based on perturbations,” Neural Comput. 14(11), 2531–2560 (2002).
[Crossref]

1997 (1)

J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, “Modulation bandwidth, noise, and stability of a semiconductor laser subject to strong injection locking,” IEEE Photonics Technol. Lett. 9(10), 1325–1327 (1997).
[Crossref]

1993 (1)

H. F. Liu and W. F. Ngai, “Nonlinear dynamics of a directly modulated 1.55 μm InGaAsP distributed feedback semiconductor laser,” IEEE J. Quantum Electron. 29(6), 1668–1675 (1993).
[Crossref]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Akrout, A.

F. Duport, A. Smerieri, A. Akrout, M. Haelterman, and S. Massar, “Fully analogue photonic reservoir computer,” Sci. Rep. 6(1), 22381 (2016).
[Crossref]

Appeltant, L.

L. Appeltant, G. Van der Sande, J. Danckaert, and I. Fischer, “Constructing optimized binary masks for reservoir computing with delay systems,” Sci. Rep. 4(1), 3629 (2015).
[Crossref]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20(3), 3241–3249 (2012).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2(1), 468 (2011).
[Crossref]

Bienstman, P.

Brunner, D.

D. Brunner, B. Penkovsky, B. A. Marquez, M. Jacquot, I. Fischer, and L. Larger, “Tutorial: Photonic neural networks in delay systems,” J. Appl. Phys. 124(15), 152004 (2018).
[Crossref]

G. Van der Sande, D. Brunner, and M. C. Soriano, “Advances in photonic reservoir computing,” Nanophotonics 6(3), 561–576 (2017).
[Crossref]

J. Bueno, D. Brunner, M. C. Soriano, and I. Fischer, “Conditions for reservoir computing performance using semiconductor lasers with delayed optical feedback,” Opt. Express 25(3), 2401–2412 (2017).
[Crossref]

M. C. Soriano, S. Ortín, D. Brunner, L. Larger, C. R. Mirasso, I. Fischer, and L. Pesquera, “Optoelectronic reservoir computing: tackling noise-induced performance degradation,” Opt. Express 21(1), 12–20 (2013).
[Crossref]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4(1), 1364 (2013).
[Crossref]

K. Hicke, M. A. Escalona-Morán, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1501610 (2013).
[Crossref]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20(3), 3241–3249 (2012).
[Crossref]

Bueno, J.

Bunsen, M.

M. Tezuka, K. Kanno, and M. Bunsen, “Reservoir computing with a slowly modulated mask signal for preprocessing using a mutually coupled optoelectronic system,” Jpn. J. Appl. Phys. 55(8S3), 08RE06 (2016).
[Crossref]

Chan, S. C.

Chen, H. F.

J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, “Modulation bandwidth, noise, and stability of a semiconductor laser subject to strong injection locking,” IEEE Photonics Technol. Lett. 9(10), 1325–1327 (1997).
[Crossref]

Choi, D.

N. Q. Li, L. Zunino, A. Locquet, B. Kim, D. Choi, W. Pan, and D. S. Citrin, “Multiscale ordinal symbolic analysis of the Lang-Kobayashi model for external-cavity semiconductor lasers: a test of theory,” IEEE J. Quantum Electron. 51(8), 1–6 (2015).
[Crossref]

Citrin, D. S.

N. Q. Li, L. Zunino, A. Locquet, B. Kim, D. Choi, W. Pan, and D. S. Citrin, “Multiscale ordinal symbolic analysis of the Lang-Kobayashi model for external-cavity semiconductor lasers: a test of theory,” IEEE J. Quantum Electron. 51(8), 1–6 (2015).
[Crossref]

Cui, K. Y.

D’Haene, M.

D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks 20(3), 391–403 (2007).
[Crossref]

Dambre, J.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2(1), 287 (2012).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2(1), 468 (2011).
[Crossref]

Danckaert, J.

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Simultaneous computation of two independent tasks using reservoir computing based on a single photonic nonlinear node with optical feedback,” IEEE Trans. Neural Netw. Learn. Syst. 26(12), 3301–3307 (2015).
[Crossref]

L. Appeltant, G. Van der Sande, J. Danckaert, and I. Fischer, “Constructing optimized binary masks for reservoir computing with delay systems,” Sci. Rep. 4(1), 3629 (2015).
[Crossref]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast photonic information processing using semiconductor lasers with delayed optical feedback: Role of phase dynamics,” Opt. Express 22(7), 8672–8686 (2014).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2(1), 468 (2011).
[Crossref]

Dou, W. B.

Duport, F.

F. Duport, A. Smerieri, A. Akrout, M. Haelterman, and S. Massar, “Fully analogue photonic reservoir computer,” Sci. Rep. 6(1), 22381 (2016).
[Crossref]

Q. Vinckier, F. Duport, A. Smerieri, K. Vandoorne, P. Bienstman, M. Haelterman, and S. Massar, “High performance photonic reservoir computer based on a coherently driven passive cavity,” Optica 2(5), 438–446 (2015).
[Crossref]

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2(1), 287 (2012).
[Crossref]

F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and S. Massar, “All-optical reservoir computing,” Opt. Express 20(20), 22783–22795 (2012).
[Crossref]

Escalona-Morán, M. A.

K. Hicke, M. A. Escalona-Morán, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1501610 (2013).
[Crossref]

Feng, X.

Fischer, I.

D. Brunner, B. Penkovsky, B. A. Marquez, M. Jacquot, I. Fischer, and L. Larger, “Tutorial: Photonic neural networks in delay systems,” J. Appl. Phys. 124(15), 152004 (2018).
[Crossref]

J. Bueno, D. Brunner, M. C. Soriano, and I. Fischer, “Conditions for reservoir computing performance using semiconductor lasers with delayed optical feedback,” Opt. Express 25(3), 2401–2412 (2017).
[Crossref]

L. Appeltant, G. Van der Sande, J. Danckaert, and I. Fischer, “Constructing optimized binary masks for reservoir computing with delay systems,” Sci. Rep. 4(1), 3629 (2015).
[Crossref]

M. C. Soriano, S. Ortín, D. Brunner, L. Larger, C. R. Mirasso, I. Fischer, and L. Pesquera, “Optoelectronic reservoir computing: tackling noise-induced performance degradation,” Opt. Express 21(1), 12–20 (2013).
[Crossref]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4(1), 1364 (2013).
[Crossref]

K. Hicke, M. A. Escalona-Morán, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1501610 (2013).
[Crossref]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20(3), 3241–3249 (2012).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2(1), 468 (2011).
[Crossref]

Gershenfeld, N. A.

A. S. Weigend and N. A. Gershenfeld, “Time series prediction: forecasting the future and understanding the past,” http://www-psych.stanford.edu/∼andreas/Time-Series/SantaFe.html (1993).

Gutierrez, J. M.

Haas, H.

H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304(5667), 78–80 (2004).
[Crossref]

Haelterman, M.

F. Duport, A. Smerieri, A. Akrout, M. Haelterman, and S. Massar, “Fully analogue photonic reservoir computer,” Sci. Rep. 6(1), 22381 (2016).
[Crossref]

Q. Vinckier, F. Duport, A. Smerieri, K. Vandoorne, P. Bienstman, M. Haelterman, and S. Massar, “High performance photonic reservoir computer based on a coherently driven passive cavity,” Optica 2(5), 438–446 (2015).
[Crossref]

F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and S. Massar, “All-optical reservoir computing,” Opt. Express 20(20), 22783–22795 (2012).
[Crossref]

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2(1), 287 (2012).
[Crossref]

Hicke, K.

K. Hicke, M. A. Escalona-Morán, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1501610 (2013).
[Crossref]

Hou, Y. S.

Y. S. Hou, G. Q. Xia, E. Jayaprasath, D. Z. Yue, W. Y. Yang, and Z. M. Wu, “Prediction and classification performance of reservoir computing system using mutually delay-coupled semiconductor lasers,” Opt. Commun. 433, 215–220 (2019).
[Crossref]

Y. S. Hou, G. Q. Xia, W. Y. Yang, D. Wang, E. Jayaprasath, Z. F. Jiang, C. X. Hu, and Z. M. Wu, “Prediction performance of reservoir computing system based on a semiconductor laser subject to double optical feedback and optical injection,” Opt. Express 26(8), 10211–10219 (2018).
[Crossref]

Hu, C. X.

Huang, Y. D.

Jacquot, M.

D. Brunner, B. Penkovsky, B. A. Marquez, M. Jacquot, I. Fischer, and L. Larger, “Tutorial: Photonic neural networks in delay systems,” J. Appl. Phys. 124(15), 152004 (2018).
[Crossref]

Jaeger, H.

H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304(5667), 78–80 (2004).
[Crossref]

H. Jaeger, “The ‘echo state’ approach to analysing and training recurrent neural networks,” Technical Report GMD Report 148, German National Research Center for Information Technology (2001).

Jayaprasath, E.

Y. S. Hou, G. Q. Xia, E. Jayaprasath, D. Z. Yue, W. Y. Yang, and Z. M. Wu, “Prediction and classification performance of reservoir computing system using mutually delay-coupled semiconductor lasers,” Opt. Commun. 433, 215–220 (2019).
[Crossref]

Y. S. Hou, G. Q. Xia, W. Y. Yang, D. Wang, E. Jayaprasath, Z. F. Jiang, C. X. Hu, and Z. M. Wu, “Prediction performance of reservoir computing system based on a semiconductor laser subject to double optical feedback and optical injection,” Opt. Express 26(8), 10211–10219 (2018).
[Crossref]

Jiang, Z. F.

Kanno, K.

J. Nakayama, K. Kanno, and A. Uchida, “Laser dynamical reservoir computing with consistency: an approach of a chaos mask signal,” Opt. Express 24(8), 8679–8692 (2016).
[Crossref]

M. Tezuka, K. Kanno, and M. Bunsen, “Reservoir computing with a slowly modulated mask signal for preprocessing using a mutually coupled optoelectronic system,” Jpn. J. Appl. Phys. 55(8S3), 08RE06 (2016).
[Crossref]

Kim, B.

N. Q. Li, L. Zunino, A. Locquet, B. Kim, D. Choi, W. Pan, and D. S. Citrin, “Multiscale ordinal symbolic analysis of the Lang-Kobayashi model for external-cavity semiconductor lasers: a test of theory,” IEEE J. Quantum Electron. 51(8), 1–6 (2015).
[Crossref]

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Kuriki, Y.

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Larger, L.

Li, B. X.

Li, N. Q.

N. Q. Li, L. Zunino, A. Locquet, B. Kim, D. Choi, W. Pan, and D. S. Citrin, “Multiscale ordinal symbolic analysis of the Lang-Kobayashi model for external-cavity semiconductor lasers: a test of theory,” IEEE J. Quantum Electron. 51(8), 1–6 (2015).
[Crossref]

Lin, F. Y.

F. Y. Lin and J. M. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum Electron. 39(4), 562–568 (2003).
[Crossref]

Liu, F.

Liu, H. F.

H. F. Liu and W. F. Ngai, “Nonlinear dynamics of a directly modulated 1.55 μm InGaAsP distributed feedback semiconductor laser,” IEEE J. Quantum Electron. 29(6), 1668–1675 (1993).
[Crossref]

Liu, J. M.

G. Q. Xia, S. C. Chan, and J. M. Liu, “Multistability in a semiconductor laser with optoelectronic feedback,” Opt. Express 15(2), 572–576 (2007).
[Crossref]

F. Y. Lin and J. M. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum Electron. 39(4), 562–568 (2003).
[Crossref]

J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, “Modulation bandwidth, noise, and stability of a semiconductor laser subject to strong injection locking,” IEEE Photonics Technol. Lett. 9(10), 1325–1327 (1997).
[Crossref]

Locquet, A.

N. Q. Li, L. Zunino, A. Locquet, B. Kim, D. Choi, W. Pan, and D. S. Citrin, “Multiscale ordinal symbolic analysis of the Lang-Kobayashi model for external-cavity semiconductor lasers: a test of theory,” IEEE J. Quantum Electron. 51(8), 1–6 (2015).
[Crossref]

Maass, W.

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable States: A new framework for neural computation based on perturbations,” Neural Comput. 14(11), 2531–2560 (2002).
[Crossref]

Markram, H.

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable States: A new framework for neural computation based on perturbations,” Neural Comput. 14(11), 2531–2560 (2002).
[Crossref]

Marquez, B. A.

D. Brunner, B. Penkovsky, B. A. Marquez, M. Jacquot, I. Fischer, and L. Larger, “Tutorial: Photonic neural networks in delay systems,” J. Appl. Phys. 124(15), 152004 (2018).
[Crossref]

Massar, S.

F. Duport, A. Smerieri, A. Akrout, M. Haelterman, and S. Massar, “Fully analogue photonic reservoir computer,” Sci. Rep. 6(1), 22381 (2016).
[Crossref]

Q. Vinckier, F. Duport, A. Smerieri, K. Vandoorne, P. Bienstman, M. Haelterman, and S. Massar, “High performance photonic reservoir computer based on a coherently driven passive cavity,” Optica 2(5), 438–446 (2015).
[Crossref]

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2(1), 287 (2012).
[Crossref]

F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and S. Massar, “All-optical reservoir computing,” Opt. Express 20(20), 22783–22795 (2012).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2(1), 468 (2011).
[Crossref]

Meng, X. J.

J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, “Modulation bandwidth, noise, and stability of a semiconductor laser subject to strong injection locking,” IEEE Photonics Technol. Lett. 9(10), 1325–1327 (1997).
[Crossref]

Mirasso, C. R.

M. C. Soriano, S. Ortín, D. Brunner, L. Larger, C. R. Mirasso, I. Fischer, and L. Pesquera, “Optoelectronic reservoir computing: tackling noise-induced performance degradation,” Opt. Express 21(1), 12–20 (2013).
[Crossref]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4(1), 1364 (2013).
[Crossref]

K. Hicke, M. A. Escalona-Morán, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1501610 (2013).
[Crossref]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20(3), 3241–3249 (2012).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2(1), 468 (2011).
[Crossref]

Nakayama, J.

Natschläger, T.

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable States: A new framework for neural computation based on perturbations,” Neural Comput. 14(11), 2531–2560 (2002).
[Crossref]

Ngai, W. F.

H. F. Liu and W. F. Ngai, “Nonlinear dynamics of a directly modulated 1.55 μm InGaAsP distributed feedback semiconductor laser,” IEEE J. Quantum Electron. 29(6), 1668–1675 (1993).
[Crossref]

Nguimdo, R. M.

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Simultaneous computation of two independent tasks using reservoir computing based on a single photonic nonlinear node with optical feedback,” IEEE Trans. Neural Netw. Learn. Syst. 26(12), 3301–3307 (2015).
[Crossref]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast photonic information processing using semiconductor lasers with delayed optical feedback: Role of phase dynamics,” Opt. Express 22(7), 8672–8686 (2014).
[Crossref]

Ortín, S.

Pan, W.

N. Q. Li, L. Zunino, A. Locquet, B. Kim, D. Choi, W. Pan, and D. S. Citrin, “Multiscale ordinal symbolic analysis of the Lang-Kobayashi model for external-cavity semiconductor lasers: a test of theory,” IEEE J. Quantum Electron. 51(8), 1–6 (2015).
[Crossref]

Paquot, Y.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2(1), 287 (2012).
[Crossref]

Parlitz, U.

R. S. Zimmermann and U. Parlitz, “Observing spatio-temporal dynamics of excitable media using reservoir computing,” Chaos 28(4), 043118 (2018).
[Crossref]

Penkovsky, B.

D. Brunner, B. Penkovsky, B. A. Marquez, M. Jacquot, I. Fischer, and L. Larger, “Tutorial: Photonic neural networks in delay systems,” J. Appl. Phys. 124(15), 152004 (2018).
[Crossref]

Pesquera, L.

Rontani, D.

Schneider, B.

Schrauwen, B.

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2(1), 287 (2012).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2(1), 468 (2011).
[Crossref]

D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks 20(3), 391–403 (2007).
[Crossref]

Sciamanna, M.

Simpson, T. B.

J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, “Modulation bandwidth, noise, and stability of a semiconductor laser subject to strong injection locking,” IEEE Photonics Technol. Lett. 9(10), 1325–1327 (1997).
[Crossref]

Smerieri, A.

F. Duport, A. Smerieri, A. Akrout, M. Haelterman, and S. Massar, “Fully analogue photonic reservoir computer,” Sci. Rep. 6(1), 22381 (2016).
[Crossref]

Q. Vinckier, F. Duport, A. Smerieri, K. Vandoorne, P. Bienstman, M. Haelterman, and S. Massar, “High performance photonic reservoir computer based on a coherently driven passive cavity,” Optica 2(5), 438–446 (2015).
[Crossref]

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2(1), 287 (2012).
[Crossref]

F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and S. Massar, “All-optical reservoir computing,” Opt. Express 20(20), 22783–22795 (2012).
[Crossref]

Soriano, M. C.

G. Van der Sande, D. Brunner, and M. C. Soriano, “Advances in photonic reservoir computing,” Nanophotonics 6(3), 561–576 (2017).
[Crossref]

J. Bueno, D. Brunner, M. C. Soriano, and I. Fischer, “Conditions for reservoir computing performance using semiconductor lasers with delayed optical feedback,” Opt. Express 25(3), 2401–2412 (2017).
[Crossref]

M. C. Soriano, S. Ortín, D. Brunner, L. Larger, C. R. Mirasso, I. Fischer, and L. Pesquera, “Optoelectronic reservoir computing: tackling noise-induced performance degradation,” Opt. Express 21(1), 12–20 (2013).
[Crossref]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4(1), 1364 (2013).
[Crossref]

K. Hicke, M. A. Escalona-Morán, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1501610 (2013).
[Crossref]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20(3), 3241–3249 (2012).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2(1), 468 (2011).
[Crossref]

Stroobandt, D.

D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks 20(3), 391–403 (2007).
[Crossref]

Takano, K.

Tezuka, M.

M. Tezuka, K. Kanno, and M. Bunsen, “Reservoir computing with a slowly modulated mask signal for preprocessing using a mutually coupled optoelectronic system,” Jpn. J. Appl. Phys. 55(8S3), 08RE06 (2016).
[Crossref]

Uchida, A.

Van der Sande, G.

G. Van der Sande, D. Brunner, and M. C. Soriano, “Advances in photonic reservoir computing,” Nanophotonics 6(3), 561–576 (2017).
[Crossref]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Simultaneous computation of two independent tasks using reservoir computing based on a single photonic nonlinear node with optical feedback,” IEEE Trans. Neural Netw. Learn. Syst. 26(12), 3301–3307 (2015).
[Crossref]

L. Appeltant, G. Van der Sande, J. Danckaert, and I. Fischer, “Constructing optimized binary masks for reservoir computing with delay systems,” Sci. Rep. 4(1), 3629 (2015).
[Crossref]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast photonic information processing using semiconductor lasers with delayed optical feedback: Role of phase dynamics,” Opt. Express 22(7), 8672–8686 (2014).
[Crossref]

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2(1), 468 (2011).
[Crossref]

Vandoorne, K.

Vatin, J.

Verschaffelt, G.

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Simultaneous computation of two independent tasks using reservoir computing based on a single photonic nonlinear node with optical feedback,” IEEE Trans. Neural Netw. Learn. Syst. 26(12), 3301–3307 (2015).
[Crossref]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast photonic information processing using semiconductor lasers with delayed optical feedback: Role of phase dynamics,” Opt. Express 22(7), 8672–8686 (2014).
[Crossref]

Verstraeten, D.

D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks 20(3), 391–403 (2007).
[Crossref]

Vinckier, Q.

Wang, D.

Wang, Y.

Weigend, A. S.

A. S. Weigend and N. A. Gershenfeld, “Time series prediction: forecasting the future and understanding the past,” http://www-psych.stanford.edu/∼andreas/Time-Series/SantaFe.html (1993).

Wu, Z. M.

Y. S. Hou, G. Q. Xia, E. Jayaprasath, D. Z. Yue, W. Y. Yang, and Z. M. Wu, “Prediction and classification performance of reservoir computing system using mutually delay-coupled semiconductor lasers,” Opt. Commun. 433, 215–220 (2019).
[Crossref]

Y. S. Hou, G. Q. Xia, W. Y. Yang, D. Wang, E. Jayaprasath, Z. F. Jiang, C. X. Hu, and Z. M. Wu, “Prediction performance of reservoir computing system based on a semiconductor laser subject to double optical feedback and optical injection,” Opt. Express 26(8), 10211–10219 (2018).
[Crossref]

Xia, G. Q.

Yang, W. Y.

Y. S. Hou, G. Q. Xia, E. Jayaprasath, D. Z. Yue, W. Y. Yang, and Z. M. Wu, “Prediction and classification performance of reservoir computing system using mutually delay-coupled semiconductor lasers,” Opt. Commun. 433, 215–220 (2019).
[Crossref]

Y. S. Hou, G. Q. Xia, W. Y. Yang, D. Wang, E. Jayaprasath, Z. F. Jiang, C. X. Hu, and Z. M. Wu, “Prediction performance of reservoir computing system based on a semiconductor laser subject to double optical feedback and optical injection,” Opt. Express 26(8), 10211–10219 (2018).
[Crossref]

Yue, D. Z.

Y. S. Hou, G. Q. Xia, E. Jayaprasath, D. Z. Yue, W. Y. Yang, and Z. M. Wu, “Prediction and classification performance of reservoir computing system using mutually delay-coupled semiconductor lasers,” Opt. Commun. 433, 215–220 (2019).
[Crossref]

Zhang, H.

Zimmermann, R. S.

R. S. Zimmermann and U. Parlitz, “Observing spatio-temporal dynamics of excitable media using reservoir computing,” Chaos 28(4), 043118 (2018).
[Crossref]

Zunino, L.

N. Q. Li, L. Zunino, A. Locquet, B. Kim, D. Choi, W. Pan, and D. S. Citrin, “Multiscale ordinal symbolic analysis of the Lang-Kobayashi model for external-cavity semiconductor lasers: a test of theory,” IEEE J. Quantum Electron. 51(8), 1–6 (2015).
[Crossref]

Chaos (1)

R. S. Zimmermann and U. Parlitz, “Observing spatio-temporal dynamics of excitable media using reservoir computing,” Chaos 28(4), 043118 (2018).
[Crossref]

IEEE J. Quantum Electron. (4)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

H. F. Liu and W. F. Ngai, “Nonlinear dynamics of a directly modulated 1.55 μm InGaAsP distributed feedback semiconductor laser,” IEEE J. Quantum Electron. 29(6), 1668–1675 (1993).
[Crossref]

F. Y. Lin and J. M. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum Electron. 39(4), 562–568 (2003).
[Crossref]

N. Q. Li, L. Zunino, A. Locquet, B. Kim, D. Choi, W. Pan, and D. S. Citrin, “Multiscale ordinal symbolic analysis of the Lang-Kobayashi model for external-cavity semiconductor lasers: a test of theory,” IEEE J. Quantum Electron. 51(8), 1–6 (2015).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

K. Hicke, M. A. Escalona-Morán, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1501610 (2013).
[Crossref]

IEEE Photonics Technol. Lett. (1)

J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, “Modulation bandwidth, noise, and stability of a semiconductor laser subject to strong injection locking,” IEEE Photonics Technol. Lett. 9(10), 1325–1327 (1997).
[Crossref]

IEEE Trans. Neural Netw. Learn. Syst. (1)

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Simultaneous computation of two independent tasks using reservoir computing based on a single photonic nonlinear node with optical feedback,” IEEE Trans. Neural Netw. Learn. Syst. 26(12), 3301–3307 (2015).
[Crossref]

J. Appl. Phys. (1)

D. Brunner, B. Penkovsky, B. A. Marquez, M. Jacquot, I. Fischer, and L. Larger, “Tutorial: Photonic neural networks in delay systems,” J. Appl. Phys. 124(15), 152004 (2018).
[Crossref]

Jpn. J. Appl. Phys. (1)

M. Tezuka, K. Kanno, and M. Bunsen, “Reservoir computing with a slowly modulated mask signal for preprocessing using a mutually coupled optoelectronic system,” Jpn. J. Appl. Phys. 55(8S3), 08RE06 (2016).
[Crossref]

Nanophotonics (1)

G. Van der Sande, D. Brunner, and M. C. Soriano, “Advances in photonic reservoir computing,” Nanophotonics 6(3), 561–576 (2017).
[Crossref]

Nat. Commun. (2)

L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, “Information processing using a single dynamical node as complex system,” Nat. Commun. 2(1), 468 (2011).
[Crossref]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4(1), 1364 (2013).
[Crossref]

Neural Comput. (1)

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable States: A new framework for neural computation based on perturbations,” Neural Comput. 14(11), 2531–2560 (2002).
[Crossref]

Neural Networks (1)

D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Networks 20(3), 391–403 (2007).
[Crossref]

Opt. Commun. (1)

Y. S. Hou, G. Q. Xia, E. Jayaprasath, D. Z. Yue, W. Y. Yang, and Z. M. Wu, “Prediction and classification performance of reservoir computing system using mutually delay-coupled semiconductor lasers,” Opt. Commun. 433, 215–220 (2019).
[Crossref]

Opt. Express (10)

J. Bueno, D. Brunner, M. C. Soriano, and I. Fischer, “Conditions for reservoir computing performance using semiconductor lasers with delayed optical feedback,” Opt. Express 25(3), 2401–2412 (2017).
[Crossref]

Y. Kuriki, J. Nakayama, K. Takano, and A. Uchida, “Impact of input mask signals on delay-based photonic reservoir computing with semiconductor lasers,” Opt. Express 26(5), 5777–5788 (2018).
[Crossref]

Y. S. Hou, G. Q. Xia, W. Y. Yang, D. Wang, E. Jayaprasath, Z. F. Jiang, C. X. Hu, and Z. M. Wu, “Prediction performance of reservoir computing system based on a semiconductor laser subject to double optical feedback and optical injection,” Opt. Express 26(8), 10211–10219 (2018).
[Crossref]

J. Nakayama, K. Kanno, and A. Uchida, “Laser dynamical reservoir computing with consistency: an approach of a chaos mask signal,” Opt. Express 24(8), 8679–8692 (2016).
[Crossref]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Fast photonic information processing using semiconductor lasers with delayed optical feedback: Role of phase dynamics,” Opt. Express 22(7), 8672–8686 (2014).
[Crossref]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20(3), 3241–3249 (2012).
[Crossref]

F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and S. Massar, “All-optical reservoir computing,” Opt. Express 20(20), 22783–22795 (2012).
[Crossref]

M. C. Soriano, S. Ortín, D. Brunner, L. Larger, C. R. Mirasso, I. Fischer, and L. Pesquera, “Optoelectronic reservoir computing: tackling noise-induced performance degradation,” Opt. Express 21(1), 12–20 (2013).
[Crossref]

H. Zhang, X. Feng, B. X. Li, Y. Wang, K. Y. Cui, F. Liu, W. B. Dou, and Y. D. Huang, “Integrated photonic reservoir computing based on hierarchical time-multiplexing structure,” Opt. Express 22(25), 31356–31370 (2014).
[Crossref]

G. Q. Xia, S. C. Chan, and J. M. Liu, “Multistability in a semiconductor laser with optoelectronic feedback,” Opt. Express 15(2), 572–576 (2007).
[Crossref]

Opt. Lett. (1)

Optica (1)

Sci. Rep. (3)

L. Appeltant, G. Van der Sande, J. Danckaert, and I. Fischer, “Constructing optimized binary masks for reservoir computing with delay systems,” Sci. Rep. 4(1), 3629 (2015).
[Crossref]

F. Duport, A. Smerieri, A. Akrout, M. Haelterman, and S. Massar, “Fully analogue photonic reservoir computer,” Sci. Rep. 6(1), 22381 (2016).
[Crossref]

Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar, “Optoelectronic reservoir computing,” Sci. Rep. 2(1), 287 (2012).
[Crossref]

Science (1)

H. Jaeger and H. Haas, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science 304(5667), 78–80 (2004).
[Crossref]

Other (3)

A. S. Weigend and N. A. Gershenfeld, “Time series prediction: forecasting the future and understanding the past,” http://www-psych.stanford.edu/∼andreas/Time-Series/SantaFe.html (1993).

A. Uchida, Optical Communication with Chaotic Lasers, Applications of Nonlinear Dynamics and Synchronization (Wiley-VCH, 2012).

H. Jaeger, “The ‘echo state’ approach to analysing and training recurrent neural networks,” Technical Report GMD Report 148, German National Research Center for Information Technology (2001).

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Figures (6)

Fig. 1.
Fig. 1. Schematic diagram of reservoir computing based on an optical feedback semiconductor laser under electrical information injection. SL: semiconductor laser; OC: optical circulator; VA: variable attenuator.
Fig. 2.
Fig. 2. Relaxation oscillation frequency fro and relaxation oscillation characteristic time Tro of the free-running SL as a function of the normalized bias current jb.
Fig. 3.
Fig. 3. (a) Bifurcation diagram as a function of feedback strength k for jb = 2.0 and τ = 1.5 ns; (b) Feedback strength at the bifurcate point (kBP) as a function of the normalized bias current jb under θ = H0 Tro, where H0 takes 0.1, 0.2, 0.3, 0.4 and 0.5.
Fig. 4.
Fig. 4. NMSEs as a function of feedback strength k, where (a) jb = 1.5, (b) jb = 2.5, and (c) jb = 3.0.
Fig. 5.
Fig. 5. (a) NMSEs as a function of jb, where the blue line is for the case of θ = 0.2Tro and k = 0.5kBP and the red line is for the case of θ = 0.2 ns and k = 0.15ns-1; (b) Delay time (blue line) and information processing rate (red line) under θ = 0.2Tro as a function of jb.
Fig. 6.
Fig. 6. SERs of nonlinear channel equalization task with θ = 0.2Tro. (a) SERs as a function of k under SNR = 24 dB; (b) SERs as a function of jb under k = 0.5kBP and SNR = 24 dB; (c) SERs as a function of SNR under k = 0.5kBP.

Equations (6)

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N M S E = 1 L n = 1 L ( y o u t ( n ) y d ( n ) ) 2 / v a r ( y d ) ,
d E ( t ) d t = 1 2 ( 1 + i α ) [ G N ( N ( t ) N 0 ) 1 + ε | E ( t ) | 2 1 τ p ] E ( t ) + k E ( t τ ) e i ω 0 τ + F ( t ) ,
d N ( t ) d t = j b ( 1 + γ j m ( t ) ) J t h N ( t ) τ s G N ( N ( t ) N 0 ) 1 + ε | E ( t ) | 2 | E ( t ) | 2 ,
f r o = G N J t h ( j b 1 ) / ( 2 π ) .
q ( n ) = 0.08 d ( n + 2 ) 0.12 d ( n + 1 ) + d ( n ) + 0.18 d ( n 1 ) 0.1 d ( n 2 ) + 0.091 d ( n 3 ) 0.05 d ( n 4 ) + 0.04 d ( n 5 ) + 0.03 d ( n 6 ) + 0.01 d ( n 7 ) .
u ( n ) = q ( n ) + 0.036 q 2 ( n ) 0.011 q 3 ( n ) + ξ e ( n ) ,

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