IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 12, DECEMBER 2015

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Simultaneous Computation of Two Independent Tasks Using Reservoir Computing Based on a Single Photonic Nonlinear Node With Optical Feedback Romain Modeste Nguimdo, Guy Verschaffelt, Jan Danckaert, and Guy Van der Sande

Abstract— In this brief, we numerically demonstrate a photonic delay-based reservoir computing system, which processes, in parallel, two independent computational tasks even when the two tasks have unrelated input streams. Our approach is based on a single-longitudinal mode semiconductor ring laser (SRL) with optical feedback. The SRL emits in two directional optical modes. Each directional mode processes one individual task to mitigate possible crosstalk. We illustrate the feasibility of our scheme by analyzing the performance on two benchmark tasks: 1) chaotic time series prediction and 2) nonlinear channel equalization. We identify some feedback configurations for which the results for simultaneous prediction/classification indicate a good performance, but with slight degradation (as compared with the performance obtained for single task processing) due to nonlinear and linear interactions between the two directional modes of the laser. In these configurations, the system performs well on both tasks for a broad range of the parameters.

Index Terms— Delay-based reservoir computing (RC), parallel computation, semiconductor ring lasers (SRLs). I. I NTRODUCTION Reservoir computing (RC) is a training concept for recurrent neural networks inspired by the way that the brain processes information [1], [2]. It can potentially perform computationally hard tasks, such as pattern recognition, chaotic time series prediction, and classification [1]–[4]. In the RC concept, a reservoir consisting of an ensemble of randomly connected neurons (referred to, by some authors, as nodes) is excited by an input. The responses of all the neurons to the input stimulus are weighted and combined through a linear readout function at the output layer, which is explicitly separated from the rest of the system. The values of the weights are optimized through a linear training procedure. For a reservoir to be useful for computational tasks, the number of neurons should be sufficiently large such that the associated state space has a sufficiently high dimensionality. With a reservoir of 102 –103 neurons, state-of-the-art performance in recurrent neural networks has been achieved [1]–[4]. Recently, Appeltant et al. [5] demonstrated that a single nonlinear node (i.e., one neuron) with delay achieves a comparable performance as very large recurrent neural networks. This simplification of the RC architecture was a conceptual breakthrough, which provided a technological pathway toward experiments at high speeds [6]–[11]. These implementations of RC systems focused on one input data stream per RC system. However, our brain has a great ability to Manuscript received June 5, 2014; revised February 11, 2015; accepted February 13, 2015. Date of publication March 3, 2015; date of current version November 16, 2015. This work was supported in part by the Research Foundation Flanders, in part by the Research Council through the Vrije Universiteit Brussel (VUB), Brussels, Belgium, in part by the Hercules Fundation, and in part by the Interuniversity Attraction Poles Program through the Belgian Science Policy Office under Grant IAP P7-35. The authors are with the Applied Physics Research Group, Vrije Universiteit Brussel, Brussels 1050, Belgium (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2015.2404346

process simultaneously several independent tasks, including those that require unrelated input streams. It is obvious that parallel computation requiring different input streams is highly desired and even necessary in some applications. In communication networks, for example, a single RC system could be used to equalize several bitstreams simultaneously. Of course, several independent tasks can be simultaneously processed using separated similar RC systems. This will be costly, energy inefficient, and will probably lead to complex experimental control and finally bulky devices. To mimick the parallelism of the brain and also to satisfy the aforementioned technological needs, there has recently been an attempt for parallel computation: a single input data signal is injected into the reservoir and different tasks are calculated in parallel on the same reservoir responses [11], [12]. These computations are done, however, starting from the same input stream. In [11], for example, spoken digit data are injected into a single reservoir and two parallel calculations are done on the transient responses for both digit recognition and speaker recognition. In [12], different Boolean logical operations (e.g., OR, XOR, AND, and NAND) are performed in parallel on a single signal fed into the reservoir. While these results are very successful, the simultaneous computation of independent machine learning tasks with unrelated input streams using the same reservoir remains very challenging. It is to be expected that all the input streams will be mixed in the reservoir causing crosstalk. The reservoir’s nodes will, therefore, respond more to the global stimulus rather than to the specific stimuli, and successful computation of each individual task in parallel will be compromised. To the best of our knowledge, simultaneous computation of several independent tasks requiring different types of input streams has never been investigated in RC. Fortunately, the parallelism offered by photonics may allow to process simultaneously different data signals using a single nonlinear node. Specific tasks could be processed, for example, at different wavelengths of a multilongitudinal mode laser or/and in different directional modes of a laser provided that the effect of the coupling between the modes is negligible. This coupling is not caused by the parallelism of the photonic scheme, but it is rather due to device imperfections and material effects. Semiconductor ring lasers (SRLs) or microring lasers are such a class of lasers, which can support both multilongitudinal modes [13], [14] and two directional modes (i.e., in the clockwise (CW) and in the counterclockwise (CCW) directions [15], [16]). They are scalable, can be easily implemented on chip, and do not require distributed feedback or distributed Bragg reflector mirrors, and multiple output ports (for coupling out the light) can easily be implemented. In addition, RC schemes with very short delay times are possible when using semiconductor lasers [17]. Therefore, SRLs are also promising candidates for on-chip reservoir computers as both an SRL and a short delay can be easily integrated on the same chip. In this contribution, we numerically show that, even when considering linear and nonlinear couplings between modes, SRLs with delayed optical feedback are excellent prototypes of

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Fig. 1. SRL with double (a) self-feedback and (b) cross-feedback. (c) Scheme for demonstrating parallel information processing using an SRL with double self-feedback. CW: clockwise mode. CCW: counterclockwise mode. MZM: Mach–Zehnder modulator. S1 (t) and S2 (t) are the input data streams of the tasks to be processed in CW and CCW modes, respectively. We use a single random mask generator (i.e., identical masks) for the two tasks. Red: optical path. Blue: electric path.

reservoir computers, capable of solving simultaneously several tasks with unrelated input streams. To demonstrate this, we use numerical simulations of a single-longitudinal mode SRL and use each directional mode to process individual data signals independently. We investigate different feedback configurations and we illustrate the performance of our system using a Santa-Fe chaotic time series prediction and a nonlinear channel equalization task. This brief is structured as follows. Section II deals with the theoretical model of SRLs with optical feedback and injection. Section III briefly describes the tasks used to benchmark our system. In Section IV, we characterize the reservoir state in the absence of the input data. In Section V, we address the system performance for a single task processed either in the CW mode or in the CCW mode. In Section VI, we discuss in detail the results for simultaneous processing of two different machine learning tasks and compare the results with those shown in Section V. Finally, the conclusion is drawn in Section VII. II. M ODEL OF SRLs W ITH O PTICAL D ELAYED F EEDBACK FOR RC Under appropriate conditions, an SRL simultaneously emits light in two directional modes with the same wavelength. The two modes share the same cavity and, therefore, can interact linearly and nonlinearly due to reflections and saturation effects. The ability of these lasers to emit simultaneously in two directional modes offers the possibility to implement several feedback configurations, such as optical self- or cross-feedback configurations. In the selffeedback configuration [Fig. 1(a)], a part of the output signal of one directional mode is injected back in the same directional mode, while for the cross-feedback configuration [Fig. 1(b)], a part of the output signal of one directional mode is injected in the opposite directional mode. When only one directional mode is subjected to feedback, the system is referred to as a single feedback configuration while it is called a double feedback configuration when the two modes are subjected to feedback. Here, we consider

a single-longitudinal mode SRL with double optical self- or cross-feedback. Fig. 1 shows a schematic representation of an SRL with double self-feedback and double cross-feedback. Each configuration in Fig. 1 is a double feedback configuration as the two directional modes are simultaneously subjected to feedback. Fig. 1(c) shows one potential SRL-based RC scheme, including the injection and the readout layer. The operating principle is as follows: we continuously feed the input (original) data S1,2 (t) into the nonlinear node (i.e., the SRL) and the readout consists in recording simultaneously the output power in each directional mode of the SRL. Here, S1 (t) is injected in CW and the training/postprocessing is done based on the recorded output signal |E cw |2 while S2 (t) is injected in CCW and the training/postprocessing is done based on the recorded output signal |E ccw |2 . In each case, the recorded power is subsequently sampled to construct an N × M matrix allowing for the calculation of the weight to be assigned to each readout connection. Here, N and M are, respectively, the number of the nodes constructed by sampling the optical delay line (these nodes are usually referred to, by the researchers working in the delay-based RC systems, as virtual nodes) and the number of input samples used for training. In semiconductor lasers, we have shown that the computation relies on the nonlinear response of the optical field to a fast-injected optical signal. This nonlinearity is expected to involve only the mode that is injected (be it defined by injection direction or injection wavelength). Nevertheless, the modes in a semiconductor laser can couple via other processes, such as backscattering, gain saturation, and gain competition, which are not related to this computational nonlinearity. It is, therefore, the goal, here, to investigate whether such effects allow simultaneous computation of tasks. For SRLs without feedback, a theoretical model describing the dynamics was proposed in [15]. This model considers the mean-field slowly varying complex amplitudes of the electric field associated with the two counter-propagating modes E cw and E ccw , and the carrier number N . Later on, it was extended for SRLs subject to an optical feedback [18]–[21]. Here, we further extend the model to

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include optical injection in each directional mode so that the reservoir of our RC consists of the SRL, the feedback path, and the bias injection (injection without input data) in each direction. Considering the effect of spontaneous emission noise, the model of concern is given by [18]–[21] E˙ cw = κ(1 + iα)[Gcw N − 1]E cw − (kd + ikc )E ccw  + ηcw Fcw (t) + Dcw ξcw (t) + k1 E1 (t) E˙ ccw = κ(1 + iα)[Gccw N − 1]E ccw − (kd + ikc )E cw  + ηccw Fccw (t) + Dccw ξccw (t) + k2 E2 (t) N˙ = γ [μ − N (1 + Gcw |E cw |2 + Gccw |E ccw |2 )]

(1) (2) (3)

where the parameters are the linewidth enhancement factor α, renormalized bias current μ, field decay rate κ, carrier inversion decay rate γ, feedback strengths ηcw and ηccw , and the backscattering coefficients kd + ikc , where kc and kd are the conservative and the dissipative couplings, respectively. The differential gain functions are given by Gcw = 1 − s|E cw |2 − c|E ccw |2 and Gccw = 1 − s|E ccw |2 − c|E cw |2 , where s and c account for the phenomenological self- and cross-saturations, respectively. Fcw (t) and Fccw (t) are the feedback terms which can be explicitly defined depending on the feedback configuration. For the cross-feedback configuration, Fcw (t) = E ccw (t − Tccw )e−iω0 Tccw and Fccw (t) = E cw (t − Tcw )e−iω0 Tcw , where ω0 is the solitary laser frequency, Tcw and Tccw are delay times, and ω0 Tcw and ω0 Tccw are the constant feedback phases. For the self-feedback configuration, Fcw (t) = E cw (t − Tcw )e−iω0 Tcw and Fccw (t) = E ccw (t − Tccw )e−iω0 Tccw . The fourth terms at the righthand sides of (1) and (2) represent the effect of spontaneous emission noise coupled to the CW/CCW modes. It can be explicitly written as Dcw,ccw = Dm (N + G 0 N0 /κ), where Dm is the noise strength, G 0 is the gain parameter, and N0 is the transparency carrier density. ξi (t) (i = cw, ccw) are two independent complex Gaussian white noises with zero mean and correlation ξi (t)ξ ∗j (t  ) = δi j (t − t  ). The last terms in (1) and (2) are the injected fields containing the data of the different tasks to be processed in the two modes (e.g., CW for task 1 and CCW for task 2), k1,2 being the injection strengths. Typical benchmark tasks to test the RC performance include signal classification, isolated spoken digit recognition, nonlinear channel equalization, and Santa-Fe prediction [6], [7], [9]. The data can be injected electrically into the reservoir by modulating directly the bias pump current of the laser [10], [11]. It can also be injected optically by coupling an optical signal containing the original data (to be processed) to the CW or the CCW mode [10], [11], [17]. Here, we only consider optical injection as a means to inject the data in the reservoir. In order to ensure variability of the input data over different virtual nodes in the delay line, the input data are first convoluted with a mask generated randomly before being sent to the reservoir. The mask is generated such that it is periodic over one loop delay time and constant over a time corresponding to the timeinterval separating two consecutive virtual nodes on the delay line, i.e., [5]. Often masks with two to six predefined discrete levels are considered [5], [22]. Here, we use a mask with four discrete values (−1, −0.25, 0.25, 1) generated randomly with equal probability. Note that the choice of generating the mask randomly is motivated by the random connectivity in reservoirs in classical neural networks. Nonetheless, other ways to construct masks yielding similar results have been suggested in [23]. For practical reasons, we assume that data are fed into the reservoir via a Mach–Zehnder modulator (MZM) as in [10], [11], and [17]. The data convoluted with the mask are used in this case to modulate a continuous-wave input power of an MZM by modulating the phase of one of the MZM arms. The MZM output is coupled to the directional

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mode of the SRL in which the task has to be processed. Here, task 1 is injected and processed in the CW mode while a similar processing is achieved in the CCW mode for task 2. It is worth noting that the preprocessing and the postprocessing of the signals in the two modes are independent. The masks, N and can, therefore, differ from one mode to another if desired. In this brief, we will particularly consider identical masks, as our goal is to use an as-small-as possible number of components. Considering the convoluted data S1,2 (t), the field at the MZM output E1,2 (t) can be written as |E0 | (4) {1 + ei[S1,2 (t )+ 0 ] }ei ω1,2 t 2 where ω1,2 is the detuning between E cw,ccw and E1,2 , |E0 | is the amplitude of the injection, and 0 is the normalized bias voltage of MZM. E1,2 (t) =

III. B ENCHMARK TASKS We use nonlinear channel equalization [2], [7], [24] and Santa-Fe time series prediction [5], [6], [17] as benchmark tasks in this brief to demonstrate the parallel computational abilities of SRLs. Three different cases are considered: 1) a simultaneous prediction of two different Santa-Fe time series; 2) a simultaneous computation of two independent nonlinear channel equalization tasks; and 3) a simultaneous computation of two different tasks with different input signals, e.g., a Santa-Fe time series prediction and a nonlinear channel equalization. A. Santa-Fe Data Prediction The Santa-Fe data are intensity time series experimentally recorded from a far-infrared laser operating in a chaotic state [25]. Our Santa-Fe data set contains 10 000 points and we use the first 4000 points. Of these 4000 points, the first 75% is used for training while the remaining 25% is used for testing. In Section VI-A, the last 4000 points are also used for the simultaneous prediction in the CCW mode (task 2). This means that the 2000 intermediate points in the data set are left out to ensure that the two tasks are different. The goal for the Santa-Fe task is to predict the next sample in the chaotic time trace before it has been injected into the reservoir computer (one-step ahead prediction). The performance of this task is typically evaluated based on the normalized mean square error (NMSE) defined as NMSE(y, ytarget ) =

||y(n) − ytarget (n)||2  ||ytarget (n) − ytarget (n)||2 

(5)

where y is the predicted value, whereas ytarget is the expected value, n is a discrete time index, and ·  and ·  stand for the norm and the average, respectively. Note that NMSE = 0 means perfect prediction while NMSE = 1 indicates no prediction at all. In the literature, this task has been widely used to analyze the performance of different RC systems. Here, we will consider that the system performs well when NMSE ≤ 0.106. This value is the lowest NMSE experimentally obtained for single processing using a similar photonic system [10], [11]. Better results have been obtained in other systems (e.g., in digital RC implementations), but we want to compare our results with those obtained in similar delay-based RC systems that also contain a similar amount of optical noise. B. Nonlinear Channel Equalization The equalization of a wireless communication channel is a way for one party to communicate a symbol to another party. To do that, the symbol is first converted into a piecewise envelope signal d(n) and modulated on a high-frequency carrier signal before being transmitted. It is subsequently demodulated by the recipient into

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an analog signal S(n), which is a corrupted version of d(n) [26]. The main sources of corruption are noise, channel effects (dispersion, distortion, and so on), and interaction with other transmitted signals. The equalization of a wireless communication channel has been used as a benchmark task for RC in [2], [7], and [24]. Mathematically, this task is formulated as follows. The original signal is a random sequence d(n) with values from {−3, −1, 1, 3}. To account for intersymbol interference during the transmission, the model considers interaction between 10 consecutive symbols described by a linear equation with memory length of 10 as follows: Q(n) = 0.08d(n + 2) − 0.12d(n + 1) + d(n) + 0.18d(n − 1) − 0.1d(n − 2) + 0.091d(n − 3) − 0.05d(n − 4) (6) + 0.04d(n − 5) + 0.03d(n − 6) + 0.01d(n − 7). The second-order and the third-order nonlinear distortions, and additive Gaussian white noise, which may result from the channel, are accounted for by applying a nonlinear transformation on Q(n) to obtain the final input signal to the reservoir S(n) S(n) = Q(n) + 0.036Q(n)2 − 0.011Q(n)3 + ξe (n)

Fig. 2. Bifurcation diagram as a function of the pump current. It is obtained by extracting the local extrema from the recorded power |E cw |2 . The same bifurcation diagram (not shown) is also obtained from |E ccw |2 as the same parameters are considered in both modes. Tcw = Tccw = 4 ns and ηcw = ηccw = 10 ns−1 . (a) Self-feedback configuration. (b) Cross-feedback configuration.

(7)

where ξe (t) is an independent Gaussian white noise with zero mean and correlation ξe (t)ξe∗ (t  ) = De δ(t − t  ). De is typically chosen such that the signal-to-noise ratio (SNR) yields values between 12 and 32 dB. In a typical channel equalization task, the goal is to reconstruct d(n) when S(n) is presented as input data [7]. Similarly to [2] and [24], we will compute d(n − 2) starting from S(n). This implies a delay in the equalizer response of four time steps as the signal S(n) depends on d-values up to time n + 2. We consider a sequence of 3000 values belonging to {−3, −1, 1, 3} for training and 1000 others for testing. We use as a performance metric for this task the symbol error rate (SER), which is the fraction of d(n − 2) values that is misclassified Number of misclassified values . (8) SER = Total number of tested values Here, the perfect classification is obtained for SER = 0. For delaybased RC systems, this task has been used to analyze the performance of the system in [7] and [8]. A SER value of ≈1.3 × 10−4 is obtained in [7], which is the state-of-the-art in delay-based RC systems for this task. We will, therefore, compare our results with that in [7]. IV. Q UALIFYING THE DYNAMICAL S TATE OF THE R ESERVOIR IN THE A BSENCE OF THE I NPUT DATA We consider typical parameters derived from previous experiments using SRLs [15], [18], [19], [21]: 1) α = 3.5; 2) s = 0.005; 3) c = 0.01; 4) κ = 100 ns−1 (corresponding to the photon lifetime of 10 ps); 5) γ = 0.2 ns−1 ; 6) kd = 0.033 ns−1 ; and = 0.44 ns−1 . The noise parameters are [27]: 7) kc 1) Dm = 5 × 10−6 ns−1 ; 2) G 0 = 10−12 m3 s−1 ; and 3) N0 = 1.4 × 1024 m−3 . We also consider k1 = k2 = 9 ns−1 , ω0 Tcw = ω0 Tccw = π/2, N = 200 nodes, = 20 ps, |E0 | = 2, ω1 = 0 GHz, ω2 = −10 GHz, and 0 = 0. With = 20 ps, the overall delay loop is Tcw = Tccw = N = 4 ns. This short delay loop leads to a processing speed of 0.25 GSample/s. For all the results, we integrate (1)–(3) using the second-order Runge–Kutta method for stochastic differential equations, with the integration step of 2 ps. In order to consider the transient time, all the results obtained after integrating over a time ≤100 ns are left out. For RC to work well, the reservoir needs to be in a proper dynamical regime. The steady state has been identified in many delay systems to be suitable for RC [5]–[7], [9], [11]. Fig. 2 shows the bifurcation diagram showing the state of the reservoir for S1 (t) = S2 (t) = 0 [absence of input data but still with bias injection k1,2 |E0 |(1 + ei 0 )ei ω1,2 t /2]

Fig. 3. Performance of SRL-based RC on a single task as a function of the pump current. The task is (a) Santa-Fe time series prediction and (b) nonlinear channel equalization. The task is processed in the CW mode meaning that S2 (t) = 0. Each experiment to obtain the value of NMSE and SER is evaluated 10 times with different randomly generated masks and the displayed results are the mean values over these 10 runs. Vertical bars: standard deviation around the mean values.

when the pump current is gradually increased. Despite the moderate feedback strengths considered in Fig. 2, the rest point remains stable in a broad range of the pump current in both configurations because of the optical injection. More precisely, the output of the system is a stable steady state up to μ ≈ 1.55 and μ ≈ 1.47 for the self-feedback and the cross-feedback, respectively. Beyond this bifurcation point, the output of the system oscillates periodically leading to two branches in the bifurcation diagram. The upper branch corresponds to the maximum value in the oscillations while the lower branch indicates the minimum value. V. R ESULTS FOR S INGLE TASK C OMPUTATION For comparison purposes, we show in this section the results for processing a single task, meaning that S1 (t) = 0 and S2 (t) = 0. Although the injection in only one directional mode of the SRL is sufficient for a single task, we consider a reservoir in which we inject a dc bias signal in each mode of the SRL (k1,2 = 0) while an input signal is only added to one of the modes. In this way, we can use the same reservoir throughout this brief. The input data are rescaled such that −π ≤ S1 (t) ≤ π and S2 (t) = 0 for the Santa-Fe time series task while −π/2 ≤ S1 (t) ≤ π/2 and S2 (t) = 0 for the nonlinear channel equalization task. These boundary values were chosen because they yield the best performance. We will also consider throughout this brief De = 0.2 (leading to SNR = 31 dB) nonlinear channel equalization task. Fig. 3(a) shows the prediction error as a function of the pump current μ for self- and cross-feedback configurations. The values shown in the figure are the mean values over 10 runs with different masks in each run. The vertical bars indicate the standard deviation around the mean value for the 10 runs. The results point out that

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Fig. 4. NMSE for (•) task 1 and () task 2 obtained with SRL with (a) double self-feedback and (b) double cross-feedback. Each experiments to obtain the value of NMSE is evaluated 10 times with different randomly generated masks and the shown results are the mean values over these 10 runs. In each run, identical random mask is used for the two tasks.

Fig. 5. SER for (•) task 1 and () task 2 obtained with SRL with double (a) self-feedback and (b) cross-feedback. The values of SER are the mean values over 10 runs implying different random masks, but identical for the two tasks.

there is a broad range of the pump current for which our system can successfully predict the next step in the Santa-Fe time series with an NMSE ≤ 0.106. The smallest prediction error of ≈0.02 ± 0.006 is obtained around μ ≈ 1.3 for the self-feedback configuration while for the cross-feedback configuration, NMSE ≈ 0.022 ± 0.008 is obtained around μ ≈ 1.5. Note that these results are obtained while we use a realistic amount of spontaneous emission noise in the reservoir (Dm = 5 × 10−6 ns−1 ). The optimum performance of the system for the Santa-Fe time series prediction [Fig. 3(b)] is similar to that numerically obtained in other RC schemes based on semiconductor lasers with optical feedback [10], [17]. While in these schemes the prediction error drastically increases above the bifurcation point, it can be noticed here that the error increases very slowly after the bifurcation point, i.e., at μ ≈ 1.55 and μ ≈ 1.47 for the self- and cross-feedback configuration, respectively. As a consequence, an acceptable prediction error is still obtained above, but near the bifurcation point. It is worth noting that despite the fast processing speed, a realistic amount of noise in the readout layer (i.e., noise generated by photodetectors) will not significantly degrade the system performance [17]. In Fig. 3(b), we show the performance of our RC scheme for the nonlinear channel equalization task for different pump currents. The smallest SER for the nonlinear channel equalization task is 4 ×10−4 and 7×10−4 for self- and cross-feedback configurations, respectively, despite the realistic values for the spontaneous emission noise (Dm = 5 × 10−6 ns−1 ) and for the noise added to the channel equalization task (De = 0.2 leading to SNR = 31 dB). These error rates are slightly larger than that previously reported for a delay-based RC scheme for this task [7]. In the self-feedback configuration, however, it can be seen that SER ≈ 4×10−4 is obtained for relatively broad values of the pump current μ, namely, 1  μ  1.4. This range of parameter values also belongs to that yielding good performance for the Santa-Fe time series prediction, i.e., μ  1.6.

Santa-Fe time series for different values of the pump current considering an SRL with double self-feedback and cross-feedback. The same mask is considered for both tasks meaning that a single mask generator is used. For the self-feedback configuration, it turns out that there is still a broad range of pump currents for which the mean value of NMSE over 10 runs stays below or is equal to 0.106 for both the tasks. The minimum NMSE is 0.031 ± 0.01 for task 1 and 0.040 ± 0.015 for task 2 [Fig 4(a)]. This performance, while being good, is slightly degraded as compared with the minimum NMSE obtained for single prediction [Fig. 4(a)]. As shown in Fig. 4(b), the results for simultaneous one-step prediction in the two chaotic time series are degraded more strongly and the standard deviations around the mean values are also larger for the cross-feedback configuration. More precisely, the smallest prediction errors for simultaneous prediction are 0.086 ± 0.032 and 0.058 ± 0.02 for task 1 and task 2, respectively. This is about four times larger as compared with the minimum prediction error found when only one task is processed considering the same configuration [Fig. 3(a)]. The large deviations around the mean values already illustrate the difficulty for this configuration to perform well when computing two tasks in parallel. The worse performance obtained in this configuration is because the cross-feedback configuration introduces additional coupling between the two counter-propagating modes. The amount of information transferred from the CW mode to the CCW mode and vice versa is, therefore, larger as compared with the self-feedback configuration.

VI. R ESULTS FOR S IMULTANEOUS C OMPUTATION OF T WO D IFFERENT TASKS A. Simultaneous Prediction of Two Different Santa-Fe Time Series We aim at investigating, here, whether a single nonlinear delay-based RC can be used for simultaneous computation of several independent tasks, which all require a nonlinearity and a memory. For this purpose, we consider in our Santa-Fe data set the first 4000 points for task 1 to be processed in the CW mode of the SRL and the last 4000 points for task 2 to be processed simultaneously in the CCW mode of the same SRL. The input data are rescaled so that −π ≤ S1,2 (t) ≤ π. Fig. 4 shows the performance of the system for a simultaneous prediction of the future sample in each

B. Simultaneous Processing of Two Independent Nonlinear Channel Equalization Tasks Here, the two independent tasks to be processed in the CW and the CCW modes of the SRL are nonlinear channel equalizations. The original signals d1 (n) processed in the CW mode and d2 (n) processed in the CCW mode are independently constructed as explained in Section III-B. The corresponding input signals S1 (n) and S2 (n) are rescaled so that −π/2 ≤ S1,2 (t) ≤ π/2. The SNR is set to 31 dB. Fig. 5 shows the performance of our system for simultaneous computation of two independent nonlinear channel equalization tasks as a function of the pump current μ considering the same reservoir parameters as in Section VI-A. Again the plotted value, for each μ, is the average over 10 SER-values obtained from 10 different random masks. In each case, tasks 1 and 2 are convoluted with the same random mask. Compared with the results in Fig. 3(b), our results shown in Fig. 5(a) evidence a certain degradation of the performance both for tasks 1 and 2. In particular, it is found that the minimum classification error of ≈1.4 × 10−3 for task 1 and ≈1.5×10−3 for task 2 is obtained around μ ≈ 1.2 for the

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degradation of the performance when two tasks are simultaneously processed using the cross-feedback configuration. D. Effect of the Mode Coupling

Fig. 6. NMSE (•) and SER () representing, respectively, the error on tasks 1 and 2 obtained with SRL with double (a) and (b) self-feedback and (c) and (d) cross-feedback configurations.

self-feedback configuration. At this point, we would like to point out that, considering different random masks for tasks 1 and 2, we obtained a minimum classification error of ≈10−3 for both tasks. For the cross-feedback configuration shown in Fig. 5(b), it is also clear that the simultaneous computation of two different nonlinear channel equalization tasks shows a worse performance as compared with the results for a single task [Fig. 3(b)] and self-feedback configuration [Figs. 3(b) and 5(a)]. The best classification error stays around 10−2 , obtained for pump currents μ  1.4. This shows that the information transferred from one directional mode to the counter-propagating mode (and which is seen as additional noise to the spontaneous emission noise) is degrading the system performance. C. Simultaneous Prediction of Santa-Fe Time Series and Classification of Nonlinear Channel Equalization Here, we show the results for simultaneous computation of two different tasks requiring different types of input data signals, i.e., a Santa-Fe time series prediction processed in the CW mode and a nonlinear channel equalization task processed in the CCW mode. We set −π ≤ S1 (t) ≤ π and −π/2 ≤ S2 (t) ≤ π/2. The results are summarized in Fig. 6. Compared with Figs. 4(a) and 5(a), it can be seen for the self-feedback configuration that the prediction performance of the system is similar to that found for the single prediction task [Fig. 6(a)] while the classification errors are similar to that found for two simultaneous classification tasks [Fig. 6(b)]. More precisely, the lowest errors for prediction and classification are 0.022 and 2.3 × 10−3 , respectively, obtained around μ ≈ 1.1. This means that the prediction performance of the system is improved at the cost of the classification performance for two tasks with structurally different input streams. For the cross-feedback configuration, Fig. 6(c) and (d) shows that the performance of the system both for the prediction and for the classification task is very similar to that found when two tasks with the same structural input streams are processed [Figs. 4(b) and 5(b)]. By the way of illustration, the minimum error rates of 0.06 and 4.26 × 10−3 are obtained for the prediction and classification, respectively, around μ ≈ 1.1. Compared with the minimum error rates obtained for single prediction or classification task, these values indicate a strong degradation for the prediction task as well as for the classification task processed simultaneously. This further confirms the

The unwanted coupling due to the saturation effects and linear coupling due to backscattering can be made smaller and are device dependent, e.g., the saturation effects depend on the quantum well structure and can, for example, be lowered using longitudinal modes with large wavelength difference. The linear coupling is due to fabrication imperfections and can be improved by optimizing the laser fabrication. In order to demonstrate that the degradation for two simultaneous tasks results from the coupling effects only (and not from the RC nonlinearity), we have performed simulations considering the optimal pump current μ = 1.2 when the coupling parameter values are 10 times smaller (i.e., s = 0.0005, c = 0.001, kd = 0.0033 ns−1 , kc = 0.044 ns−1 ). For two simultaneous Santa-Fe time series predictions, we obtain an NMSE of 0.025 and 0.03 for tasks 1 and 2, respectively. For two simultaneous nonlinear channel equalization tasks, we obtain a minimum SER of 6 × 10−4 for both tasks processed in parallel. For simultaneous prediction of a Santa-Fe time series and a nonlinear channel equalization, we obtain an NMSE ≈ 0.02 for task 1 and an SER ≈ 9 × 10−4 for task 2. In all cases, it can be noticed that the system performance is improved and approaches that obtained for the single task processing. This shows that the performance increases when we lower the mode coupling effects and we expect the same performance as single task processing if the coupling terms are set to zero. Remark that decreasing the coupling parameters s, c, kd , and kc does not decrease the gain competition through the shared carrier reservoir. Thus, as we obtain similar results as for the single task processing, this implies that the gain competition does not lead to a significant degradation of the RC performance. We also note that longitudinal modes of lasers are typically coupled through nonlinear effects only while directional modes are coupled through both nonlinear and linear effects. Moreover, the longitudinal modes have slightly different wavelengths whereas the directional modes have the same wavelength, which typically leads to a stronger nonlinear interaction between the directional modes than between the longitudinal modes. The coupling can, therefore, be expected to be stronger between directional modes than between longitudinal modes. VII. C ONCLUSION We have numerically investigated the capabilities of SRLs subject to optical feedback and optical injection to compute simultaneously two tasks with different input data signals. In particular, we have considered two different optical feedback configurations: 1) self-feedback and 2) cross-feedback configurations. Using Santa-Fe time series prediction and nonlinear channel equalization as benchmark tests, we have shown that two tasks can be simultaneously processed using the CW and the CCW modes of the SRLs, however, with some degradations. For self-feedback configuration, we found that the system performance, while being good, is slightly degraded for two simultaneous Santa-Fe time series prediction while the degradation is stronger for two simultaneous nonlinear channel equalization tasks. In this configuration, we found for simultaneous computation of two structurally different tasks (e.g., simultaneous computation of Santa-Fe time series prediction and nonlinear channel equalization tasks) that the system performance remains unaltered for one task while the second task is degraded. We have shown that the degradation of the system performance is essential due to the unwanted coupling effects between the directional modes that can be reduced in the device design and fabrication. Therefore, we conclude that a single photonic

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RC system using SRL as the nonlinear node can process two tasks in parallel provided that the coupling effects are negligible. For the cross-feedback configuration, the system performance is significantly degraded both for prediction and classification. This indicates that the extra coupling between the two modes introduced by the cross-feedback is significantly degrading the system. On the broad perspective, we expect delay-based RC schemes using multilongitudinal mode lasers to lead to better performance since the longitudinal modes with well-separated wavelengths will only experience limited nonlinear couplings. It should be noted that the maximum number of tasks, which can be solved simultaneously is equal to the number of modes that the photonic hardware supports. As single-longitudinal mode SRLs can be used to process a maximum of two tasks, multilongitudinal mode lasers will be investigated in the future for the simultaneous processing of a higher number of tasks. R EFERENCES [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., vol. 14, no. 11, pp. 2531–2560, 2002. [2] H. Jaeger and H. Haas, “Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication,” Science, vol. 304, no. 5667, pp. 78–80, 2004. [3] D. Verstraeten, B. Schrauwen, M. D’Haene, and D. Stroobandt, “An experimental unification of reservoir computing methods,” Neural Netw., vol. 20, no. 3, pp. 391–403, 2007. [4] J. J. Steil, “Backpropagation-decorrelation: Online recurrent learning with O(N) complexity,” in Proc. IEEE IJCNN, Jul. 2004, pp. 843–848. [5] L. Appeltant et al., “Information processing using a single dynamical node as complex system,” Nature Commun., vol. 2, pp. 468–472, Sep. 2011. [6] L. Larger et al., “Photonic information processing beyond Turing: An optoelectronic implementation of reservoir computing,” Opt. Exp., vol. 20, no. 3, pp. 3241–3249, 2012. [7] Y. Paquot et al., “Optoelectronic reservoir computing,” Sci. Rep., vol. 2, p. 287, Feb. 2012. [8] F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and S. Massar, “All-optical reservoir computing,” Opt. Exp., vol. 20, no. 20, pp. 22783–22795, 2012. [9] R. Martinenghi, S. Rybalko, M. Jacquot, Y. K. Chembo, and L. Larger, “Photonic nonlinear transient computing with multiple-delay wavelength dynamics,” Phys. Rev. Lett., vol. 108, no. 24, pp. 244101-1–244101-4, 2012. [10] K. Hicke, M. A. Escalona-Moran, 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. Topics Quantum Electron., vol. 19, no. 4, Jul./Aug. 2013, Art. ID 1501610. [11] D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nature Commun., vol. 4, p. 1364, Jan. 2013.

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[12] K. Vandoorne et al., “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nature Commun., vol. 5, p. 3541, Mar. 2014. [13] M. Khoder, G. Verschaffelt, R. M. Nguimdo, J. Bolk, X. Leijtens, and J. Danckaert, “Controlled multiwavelength emission using semiconductor ring lasers with on-chip filtered optical feedback,” Opt. Lett., vol. 38, no. 14, pp. 2608–2610, 2013. [14] A. Peréz-Serrano, J. Javaloyes, and S. Balle, “Directional reversals and multimode dynamics in semiconductor ring lasers,” Phys. Rev. A, vol. 89, no. 2, pp. 023818-1–023818-14, 2014. [15] M. Sorel, G. Giuliani, A. Sciré, R. Miglierina, P. J. R. Laybourn, and S. Donati, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: Experiment and model,” IEEE J. Quantum Electron., vol. 39, no. 10, pp. 1187–1195, Oct. 2003. [16] J. Javaloyes and S. Balle, “Emission directionality of semiconductor ring lasers: A traveling-wave description,” IEEE J. Quantum Electron., vol. 45, no. 5, pp. 431–438, May 2009. [17] 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. Exp., vol. 22, no. 7, pp. 8672–8686, 2014. [18] L. Gelens et al., “Exploring multistability in semiconductor ring lasers: Theory and experiment,” Phys. Rev. Lett., vol. 102, no. 19, pp. 193904-1–193904-4, 2009. [19] I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: Bifurcations and stability,” Commun. Nonlinear Sci. Numer. Simul., vol. 17, no. 12, pp. 4767–4779, Dec. 2012. [20] R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Loss of time-delay signature in chaotic semiconductor ring lasers,” Opt. Lett., vol. 37, no. 13, pp. 2541–2543, 2012. [21] R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, and G. Van der Sande, “Fast random bits generation based on a single chaotic semiconductor ring laser,” Opt. Exp., vol. 20, no. 27, pp. 28603–28613, 2012. [22] M. C. Soriano et al., “Optoelectronic reservoir computing: Tackling noise-induced performance degradation,” Opt. Exp., vol. 21, no. 1, pp. 12–20, 2013. [23] L. Appeltant, G. Van der Sande, J. Danckaert, and I. Fischer, “Constructing optimized binary masks for reservoir computing with delay systems,” Sci. Rep., vol. 4, p. 3629, Jan. 2014. [24] A. Rodan and P. Ti˘no, “Minimum complexity echo state network,” IEEE Trans. Neural Netw., vol. 22, no. 1, pp. 131–144, Jan. 2011. [25] A. S. Weigend and N. A. Gershenfeld. (1993). Time Series Prediction: Forecasting the Future and Understanding the Past. [Online]. Available FTP: ftp://ftp.santafe.edu/pub/Time-Series/Competition [26] V. J. Mathews and J. Lee, “Adaptive algorithms for bilinear filtering,” Proc. SPIE, vol. 2296, pp. 317–327, Jul. 1994. [27] S. Sunada et al., “Random optical pulse generation with bistable semiconductor ring lasers,” Opt. Exp., vol. 19, no. 8, pp. 7439–7450, 2011. [28] M. C. Soriano et al., “Delay-based reservoir computing: Noise effects in a combined analog and digital implementation,” IEEE Trans. Neural Netw. Learn. Syst., vol. 26, no. 2, pp. 388–393, Feb. 2015.

Simultaneous Computation of Two Independent Tasks Using Reservoir Computing Based on a Single Photonic Nonlinear Node With Optical Feedback.

In this brief, we numerically demonstrate a photonic delay-based reservoir computing system, which processes, in parallel, two independent computation...
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