## Abstract

Semiconductor lasers subject to delayed optical feedback have recently shown great potential in solving computationally hard tasks. By optically implementing a neuro-inspired computational scheme, called reservoir computing, based on the transient response to optical data injection, high processing speeds have been demonstrated. While previous efforts have focused on signal bandwidths limited by the semiconductor laser’s relaxation oscillation frequency, we demonstrate numerically that the much faster phase response makes significantly higher processing speeds attainable. Moreover, this also leads to shorter external cavity lengths facilitating future on-chip implementations. We numerically benchmark our system on a chaotic time-series prediction task considering two different feedback configurations. The results show that a prediction error below 4% can be obtained when the data is processed at 0.25 GSamples/s. In addition, our insight into the phase dynamics of optical injection in a semiconductor laser also provides a clear understanding of the system performance at different pump current levels, even below solitary laser threshold. Considering spontaneous emission noise and noise in the readout layer, we obtain good prediction performance at fast processing speeds for realistic values of the noise strength.

© 2014 Optical Society of America

## 1. Introduction

Traditional computers can be inefficient when trying to solve highly complex computational tasks such as speech recognition or facial recognition. Novel computational techniques are therefore highly desired [1, 2]. New brain-inspired information processing methods based on artificial neural networks have shown great potential in solving computationally hard tasks such as pattern recognition, time series prediction and classification at which the brain typically excels [3–6]. In such neural networks, a given task can be performed by first appropriately adjusting the strengths of the network connections, which is done through learning by example or in a training procedure. Training such a recurrent network is a highly nonlinear problem and requires a large amount of computational power. This problem is avoided in reservoir computing (RC) in which an artificial neural network is split into three separate layers: the input layer, the reservoir layer and the output layer, with the reservoir layer typically being a large recurrent network. The output layer is explicitly separated from the rest of the network and only the connections from the reservoir to the output layer are trained. As a result, a linear training algorithm can suffice. The computational power of the RC concept lies in the complex nonlinear transient response to an input signal of the very-high dimensional nonlinear system, that is the reservoir. Photonics, besides its application for super-computation [7], has been identified as a highly suitable technology for enabling the implementations of networks suited for RC [8]. The training of the system is performed in the output layer alone. Therefore this training does not alter the dynamical behavior of the reservoir itself. This means that the exact implementation of the reservoir is not constrained to a network of a large number (10^{2} − 10^{3}) of nodes and any high-dimensional nonlinear system could be a suitable candidate for RC. Recently, it has been demonstrated that the RC architecture can be drastically simplified by relying on a single dynamical nonlinear node subject to delayed-feedback [9]. Delay systems are very attractive from an implementation point of view as only very few components are required to build them. This breakthrough has therefore paved the way for photonic implementations based on electro-optics [10–14]. Also, and this is the focus of this paper, an all-optical RC scheme based on a semiconductor laser (SL) with delayed optical feedback and using optical data injection has been shown to achieve state-of-the-art computational performances while operating at high bit rates [15, 16].

To ensure that the optical input signal can effectively access the full dimensionality of the delay-based reservoirs and that the system remains in a transient regime, a pre-processing procedure relying on the use of a temporal mask is required [9, 11]. This masking procedure is illustrated in Fig. 1. The input data is always discrete with a sampling time matching the delay time *T _{D}* [see Fig. 1(a)]. A temporal mask

*M*(

*t*) is defined in Fig. 1(b), which is a piecewise constant function that is only non-zero over a temporal interval of the same length as the delay time

*T*(i.e.

_{D}*t*∈ [0,

*T*[). This interval is divided into

_{D}*N*sub-intervals of length Θ, referred to as the node separation during which the mask is kept constant. The constant value of the temporal mask within one node separation is randomly drawn from a pre-defined set of the suitable values. The full input signal is then constructed by convoluting the discrete time series of the input data with this newly defined temporal mask [see Fig. 1(c)]. As a consequence, the node separation Θ defines the positions of so-called virtual nodes along the delay line. When one input sample (of length

*T*) has been completely injected into the system, the delay line is tapped at the virtual nodes and the local intensities recorded. A linear combination of these virtual nodes will constitute the output of the system. The goal is that this output matches the desired target response of the RC. This can be achieved by performing a training procedure on the linear weights. It is clear that

_{D}*N*and Θ which define the delay length

*T*=

_{D}*N*Θ, and the mask properties limit the processing speed.

*N*= 50 – 400 is typically sufficient and defined by the task at hand. The node separation Θ should be chosen not too large ensuring that the system is permanently maintained in a transient regime and not too small as the system would filter out the input data. The importance of the ratio between Θ and the characteristic timescale of the nonlinear node was pointed out before [9]. Most often, the optimized value of Θ is somewhat smaller than the intrinsic time scale of the nonlinear node. Given the relatively slow time scales of optoelectronic RC systems discussed in the literature, delay times of about 20

*μ*s (i.e Θ ∼ 10 ns) have been used [10–14]. Shorter delay times of ≈ 80 ns have been also used in all-optical RC systems based on SLs [15, 16]. In all these schemes, the delay time is implemented experimentally using an optical fiber [10, 11] or with electronic delay lines based on a first-in first-out memory [9, 12].

In photonics, long delay lengths will limit the processing speed. In addition, they are not suitable for potential on-chip implementations of RC schemes as on-chip waveguide lengths are limited by absorption and chip real estate. Concerning semiconductor lasers with delayed optical feedback, previous works [15, 16] have targeted a node separation Θ ≈ 200 ps. This separation was determined from the relaxation oscillation (RO) period. In this contribution, we numerically demonstrate that in exactly the same system it is possible to achieve 10 times faster processing speeds with a 10 times shorter overall delay length as compared to [15, 16]. In particular we show that, thanks to the combined effect of optical feedback and injection, the phase dynamics which is much faster than the RO dynamics is also suitable for processing. We also demonstrate that one is free to define the node separation Θ in a broad band ranging from the fastest time scale of the system (i.e photon lifetime) to the intensity relaxation time without degrading computing power. Furthermore, we show that semiconductor lasers are suited for RC in a wide range of operation points. Considering spontaneous emission noise and noise in the readout layer, we obtain good prediction performance at fast processing speeds for realistic values of the noise strength.

## 2. Model

We consider a quantum well SL operating in a single-longitudinal mode with delayed optical feedback. We extend this setup to include a Mach-Zehnder modulator (MZM) seeded by a contineous-wave laser. The MZM will be used to inject the data optically. This ensemble constitutes the reservoir for our RC scheme. It is modeled by the so-called Lang-Kobayashi equations [17,18,27] extended to include optical injection. We describe the reservoir’s dynamical behavior in terms of the mean-field slowly varying complex electric field amplitudes of both the parallel (*E*_{1} = |*E*_{1}|*e*^{iφ1}) and the perpendicular (*E*_{2} = |*E*_{2}|*e*^{iφ2}) polarization direction, and the carrier number *N*:

*𝒢*

_{1,2}=

*g*

_{m}_{1,2}(

*N*−

*N*

_{0})/(1 +

*ε*|

*E*

_{1,2}|

^{2}) stands for the optical gain,

*ε*being the saturation factor. The parameters are the linewidth enhancement factor

*α*, the pump current

*I*

_{0}, photon decay rates

*γ*

_{1,2}, electron decay rate

*γ*, detuning ΔΩ between

_{e}*E*

_{1}and

*E*

_{2}, carrier number at transparency

*N*

_{0}, differential gains

*g*

_{m}_{1,2}, loop delay time

*T*=

_{D}*N*Θ, feedback strengths

*η*

_{1,2}and injection strength

*k*. Ω

_{inj}_{0}is the solitary laser angular frequency.

*ξ*

_{1,2}are complex Gaussian white noise terms with zero mean and $\u3008{\xi}_{i}(t){\xi}_{i}^{*}({t}^{\prime})\u3009=2{\beta}_{i}{\gamma}_{e}N\delta (t-{t}^{\prime})$ where

*i*= 1, 2. The last term

*k*(

_{inj}ℰ_{inj}*t*) in Eq. (1) represents the optical injection from the MZM. The optical gains of the two modes are usually not equal. We assume that

*g*

_{1}>

*g*

_{2}such that the signal in

*E*

_{1}is dominant in the solitary laser. Note that each polarization mode can be subjected to a feedback either from the node itself [polarization maintained optical feedback (PMOF)] or from the other mode [polarization rotated optical feedback (PROF)]. In this contribution, we neglect the feedback from

*E*

_{2}as in ref. [16]. The feedback in only one mode can be experimentally implemented using linear polarizers and Faraday rotators for PMOF and PROF configurations, respectively. These two configurations are implemented in the model through the parameters

*η*

_{1}and

*η*

_{2}:

*η*

_{2}= 0 for PMOF configuration and

*η*

_{1}= 0 for PROF configuration.

In practice, the data can be added optically to the nonlinear node via a MZM [13, 15, 16]. In this case, the input data convoluted with the mask is used to modulate the optical signal through the rf electrode of the MZM. The output of the MZM, i.e *ℰ _{inj}*(

*t*) is subsequently fed into the dominant polarization direction

*E*

_{1}. Then

*ℰ*(

_{inj}*t*) can be written as

*ω*is the detuning between

*E*

_{1}and

*ℰ*, |

_{inj}*ℰ*

_{0}| is the field amplitude of the injection.

*S*(

*t*) represent the normalized input data fed in the rf electrode, while Φ

_{0}is the bias voltage of the MZM. Typically,

*S*(

*t*) results from the input signals after the pre-processing procedures by convoluting the input data with the mask

*M*(

*t*) as mentioned in the introduction. Thus the temporal structure of

*S*(

*t*) depends on Θ (see Fig. 1). In order to identify suitably Θ, it is necessary to first identify the time scales which influence the transient dynamics of the system.

## 3. Characterization of the dynamical behavior of SL with delayed feedback

This section aims at providing some features regarding the dynamics of the SL with delayed feedback in the absence of any injection, i.e *k _{inj}* = 0 ns

^{−1}. Other parameters considered are as stated in Table 1 and in the figure captions [16]. With these parameters, the threshold of the pump current is

*I*= 9 mA. The numerical results are obtained by integrating the rate equations using the 2

_{th}*-order Runge-Kutta method for stochastic equations with an integration step of 2 ps. We perform a pre-integration over a period of 0.5*

^{nd}*μ*s (which is much longer than the longest time scale of the model) to account for transients. For

*k*= 0 ns

_{inj}^{−1}and

*η*

_{1}= 20 ns

^{−1}, the PMOF is chaotic in the whole range of pump currents we explored (i.e

*I*

_{0}≤ 1.5

*I*) while the PROF configuration remains stable in the same range of the pump current.

_{th}Several time scales such as the relaxation oscillation period *τ _{RO}* and the delay time

*T*can play a role in the dynamical behavior of a laser with feedback. If a time scale influences the intrinsic system dynamics, its signature can manifest itself in the intensity or in the phase dynamics. The relevant time scales can be typically revealed through the computation of statistical quantifiers for time scale identification such as the delayed mutual information, autocorrelation function and spectrum [20–24]. In order to identify the time scales present in the system, we consider a chaotic regime because in this regime, different time scales will be present in a single time trace.

_{D}Figure 2(a) displays the results for the spectra as computed from intensity time series |*E*_{1}(*t*)|^{2}(black) and from the phase *φ*_{1}(*t*) recovered within the interval [−*π*, *π*] (gray, blue) for *I*_{0} = 1.3*I _{th}* and

*T*= 4 ns considering the PMOF configuration. The power spectrum (black) reveals the relaxation period

_{D}*τ*

_{R}_{0}of the free-running SL through a clear peak located at the inverse of

*τ*

_{R}_{0}≈ 0.48 ns as expected (see refs. [22, 24]). However, the phase spectrum (gray, blue) shows that the phase relaxes at a different time scale which is much faster than

*τ*

_{R}_{0}attached to the intensity. The phase spectrum is considerably broader with a peak at ≈ 18 GHz. The intensity relaxation period and the phase response time, i.e,

*τ*

_{R}_{0}and

*T*, respectively are further quantified in Fig. 2(b) through the computation of the autocorrelation from the intensity and the phase dynamics, respectively. Damped oscillations are seen in both cases, however, with a much shorter period for the phase. A damped oscillation period

_{phase}*T*

_{phase}*approx*0.055 ns corresponding to the phase response frequency of 18 GHz can be identified in Fig. 2(b).

In Fig. 3, we decrease *I*_{0} to just under the threshold and compute again the intensity and the phase spectra [Fig. 3(a)], as well as the autocorrelation from the phase dynamics [Fig. 3(b)]. *τ _{RO}* vanishes as expected while the amplitude of the intensity signal becomes very small. This small amplitude existing below, but near the threshold current is caused by the feedback. It is therefore expected to increase when increasing the feedback. While

*τ*ceases to exist below the threshold, the phase still relaxes almost at the same frequency as that observed for

_{RO}*I*

_{0}>

*I*, i.e ≈ 18 GHz [Fig 3(a)]. This relaxation is also confirmed from the autocorrelation which shows damped oscillations with a period ≈ 0.055 ns [Fig. 3(b)]. These results show that the phase response seems to be independent of the pump current and thus of

_{th}*τ*. However, it is related to the frequency of the laser’s external cavity modes which strongly depend on the feedback strength

_{RO}*η*

_{1}. The nonlinear feedback terms sin [

*φ*

_{1}(

*t*−

*T*) −

_{D}*φ*

_{1}(

*t*)] in the phase equation can indeed induce a change in frequency and create oscillations related to the external cavity modes, which are feedback strength (but not pump current) dependent. For strong feedback, the phase relaxes at the frequency which is the closest to the feedback strength, i.e ${T}_{\mathit{phase}}^{-1}\approx \eta $. By way of illustration, Fig. 4 shows the phase response dependence on the feedback strength for

*η*

_{1}= 20 ns

^{−1}(gray, blue) and

*η*

_{1}= 30 ns

^{−1}(black). As can be seen, the peak in the phase spectrum is situated at ≈ 20 Ghz for

*η*

_{1}= 20 ns

^{−1}and at ≈ 30 Ghz for

*η*

_{1}= 30 ns

^{−1}. These results are further confirmed in the autocorrelation shown in Fig. 4(b) which, in each case, shows damped oscillations at a period corresponding to the inverse of the feedback strength. In the next section, we investigate how optical injection influences the dynamics of the reservoir.

## 4. Characterization of the dynamical behavior of the reservoir without the input data

As pointed out in ref. [25], the external injection into a SL with delayed optical feedback can influence its dynamical regime. In the RC scheme presented here, a bias injection |*ℰ*_{0}| (1 + *e*^{iΦ0}) *e ^{i}*

^{Δ}

*/2 remains even in the absence of the input data [*

^{ω}^{t}*S*(

*t*) = 0] when

*k*≠ 0 and therefore this bias injection needs to be considered as a part of the reservoir. For

_{inj}*k*= 50 ns

_{inj}^{−1}(as will be considered later), the PROF configuration which was stable without the injection becomes unstable above

*I*

_{0}≈ 1.4

*I*for

_{th}*S*(

*t*) = 0. On the other hand, the PMOF which was unstable becomes stable up to

*I*

_{0}≈ 1.15

*I*due to the injection for

_{th}*S*(

*t*) = 0. This means that, in some instances, the bias injection stabilizes the SL’s output [25] while in other instances, it destabilizes the output. We focus here only on the parameter region for which the output is stable because it will be the most suitable regime for reservoir computing. To investigate the time scales in this regime, we compute again the autocorrelation function from the intensity time series considering

*k*= 50 ns

_{inj}^{−1}. Note that the intrinsic noise, present in Eqs. (1)–(3) induces small excursions around the stable output in the time trace.

In Fig. 5, we show for the PMOF configuration the results of the autocorrelation computed from the intensity time series for a value just under threshold (dashed) and for a value just above threshold (solid line). Below the threshold, the intensity relaxation consists of oscillations with period ≈ 0.075 ns and which are completely damped after a lag time < 0.2 ns [Fig. 5(a), dashed line]. Above the threshold, the oscillation period in the autocorrelation is ≈ 0.15 ns. It is clear that this value is still far away *τ _{RO}* ≈ 0.83 ns. In addition, the damped oscillations span over a time > 0.6 ns [Fig. 5(a), solid line]. In all cases studied, we found that the intensity and the phase (not shown here) relax at the same frequency in the presence of the bias injection.

The stable output regime corresponds to a locking of the SL with delayed optical feedback to the injected signal. The locking relaxation frequency, which will define the damped oscillations observed in Fig. 5(a), is related to the detuning between the frequency of the injected signal and the frequency of the system without injection. Without injection (see Section 3) the laser will try to operate in a frequency range defined by the external cavity modes. Therefore, the locking oscillation frequency and consequently the oscillatory response of intensity and phase, can be fast and will be defined by the phase response time observed in Section 3.

To gain further insight, we show in Fig. 5(b) the autocorrelation computed from the carrier number considering *I*_{0} = 0.9*I _{th}* (dashed line) and

*I*

_{0}= 1.1

*I*(solid line). It turns out in both cases that the correlation decays to zero after $\approx 2{\gamma}_{e}^{-1}$ for

_{th}*I*

_{0}= 0.9

*I*and ≈ 2

_{th}*τ*for

_{RO}*I*

_{0}= 1.1

*I*. This proves that, above threshold, the normal relaxation oscillations of the solitary laser, have been replaced by these locking oscillations, but that the damping of these locking oscillations is still driven by the slower carrier dynamics. In the next sections, we investigate the suitability of these time scales for RC.

_{th}## 5. Performance of delay-based RC using SLs

The above results have illustrated two main features of the reservoir dynamics which may be useful for RC: the phase and the intensity relax at a fast time scale in the presence of the bias injection and the transient time spans over several oscillation periods when *I*_{0} > *I _{th}*. This suggests that the system can successfully perform RC tasks at different values of Θ, which we will analyze now. Typical benchmark tasks to test the RC performance are Signal Classification, Nonlinear Channel Equalization, Isolated Spoken Digit Recognition and Santa-Fe Time Series Prediction [8,10–12]. The latter is particularly challenging because it requires both nonlinearity and memory suggesting that feedback plays a role. Therefore, we use this task throughout the rest of the paper in order to quantify the RC performance.

The Santa-Fe data used are intensity time series recorded from a real far-infrared laser operating in a chaotic state [26]. Our Santa-Fe data set contains 10000 points and we use the first 4000 points (first 3000 points for training and the next 1000 for testing). The target 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 on this task is typically evaluated based on the normalized mean square error (NMSE) defined as

*y*is the predicted value while

*y*is the expected value,

_{target}*n*is a discrete time index, ‖.‖ and 〈.〉 stand for the norm and the average, respectively. Typically, the system is considered to be performing well when the NMSE stays below 10%.

We consider a reservoir with *N* = *T _{D}*/Θ = 200 virtual nodes and a random mask with four discrete levels (0, 0.25, 0.75, 1) [27]. Procedures to construct masks for optimal performance are not considered here, because they are limited to two-valued masks [28]. The number of nodes

*N*is kept constant throughout the rest of this paper, meaning that any change in Θ will be accompanied by a change in the delay length

*T*and in the temporal structure of

_{D}*S*(

*t*). The mask is used to preprocess the Santa-Fe data and the resulting signal is rescaled so that 0 ≤

*S*(

*t*) ≤

*π*/2. The data is injected in the rf input of the MZM after the reservoir has reached its steady state, i.e after the transient time.

Figure 6 displays the temporal profiles of the reservoir response both in the intensity (red) and the phase (blue), as well as the data input at the rf electrode of the MZM (black) for both Θ = 200 ps (a) and Θ = 20 ps (b). We can now relate the results from Fig. 5 to the transient response of the reservoir to the masked input. For longer node distances [as in Fig. 6(a)], each time the input signal (black) jumps to a new level, the laser responds initially fast both in intensity and phase. This oscillatory response is then slowly damped as predicted by Fig. 5. As explained in Ref. [9], to achieve a good computational performance, it is essential that the transient response to an input level is not completely damped before the system is subjected to the next new input level. As a result, Θ = 200 ps, can be considered as an adequate choice for the node distance. Again according to Ref. [9], the transient response also needs time to develop to measurable levels. In other words, the node distance should not be too small. As seen in Fig. 6(a), the node distance can still be reduced due to the very fast transient response. In Fig. 6(b), for Θ = 20 ps, the data is injected at a speed ${T}_{D}^{-1}=0.25$ GSamples/s. Despite the high speed at which the data is injected into the reservoir, it can be seen that both the intensity and the phase respond well to the external stimulus.

For very small node distances Θ, an appreciable transient response can be observed and this is mainly due to the very fast (phase-driven) locking relaxation, while for longer node distances Θ the transient response is still measurable thanks to the much slower carrier relaxation. Therefore, we can expect good computational performance in a broad range of Θ spanning at least one order of magnitude.

In order to identify the most suitable parameter regimes for which our system can successfully predict the chaotic input signal one-time step ahead in the future, we show in Fig. 7 the performance expressed by the NMSE as a function of the pump current *I*_{0} for two different values of Θ: one is close to 2*T _{phase}*/3 and another is close to 2

*τ*/3 (i.e Θ = 20 ps and Θ = 200 ps) both considering the PMOF configuration for

_{RO}*η*

_{1}= 10 ns

^{−1}[Fig. 7(a)] and PROF configuration for

*η*

_{2}= 10 ns

^{−1}[Fig. 7(b)]. For Θ = 200 ps (▪) which was experimentally used in refs [15,16] for the same system as studied here, we find a good agreement with experimental results. In particular, as experimentally shown in Fig. 3(a) in ref. [15] and Fig. 11 in ref. [16], a good performance (characterized by a small NMSE) is obtained close to the threshold current for the PMOF configuration (▪). This performance rapidly degrades for pump currents well above the threshold.

Remarkably, when decreasing Θ to a small value of Θ = 20 ps, similar results are obtained(•) although this value is far below the RO period and has therefore not been considered in previous studies on the topic. It can even be seen that these results are consistently better than or equal to those obtained for Θ = 200 ps. For the PROF configuration, it can be seen in Fig. 7(b) that the system has a good performance over a broader range of pump current values as compared to the PMOF configuration. The good performance in a large range of the pump current is because, in the absence of the input data, the PROF configuration is stable for these pump currents. It is worth noting that in the absence of input (*S*(*t*) = 0), the external bias injection, i.e |*ℰ*_{0}| (1 + *e*^{iΦ0}) *e ^{i}*

^{Δ}

*/2 favors the emergence of steady state emission for parameter values for which acceptable NMSE values have been found for PMOF configuration.*

^{ω}^{t}Considering Θ = 20 ps we further explore, in both configurations, the optimized parameters by simulating the NMSE as a function of *I*_{0} for different values of the feedback strengths (Fig. 8). In particular for *η*_{1} = 1 ns^{−1}, the PMOF configuration becomes stable for a broad range of pump currents leading thus to acceptable NMSE in this range. However, the range of *I*_{0} for which small NMSE values are obtained below the threshold is reduced due to the fact that the amplitude completely vanishes for a noise-free system (or falls into the background noise when noise is taken into account). As the feedback strength is gradually increased, the region of small NMSE values is increasingly confined to small values of the pump current. For the PROF configuration, it is seen that the increase of the feedback strength tends to improve the system performance. The PROF configuration is, in fact, very stable so that it may be difficult for the system to discriminate between similar inputs belonging to different classes (separability property of RC). Such a stable state becomes easily perturbed as the feedback is gradually increased and therefore the system can discriminate between such inputs. As a consequence, the NMSE is improved. By way of illustration, the NMSE values for *η*_{2} = 30 ns^{−1} are slightly better than those obtained for *η*_{2} = 20 ns^{−1}. In particular, the NMSE is smaller than 0.03 for 1.2*I _{th}* ≤

*I*

_{0}≤ 1.4

*I*, even with the realistic level of noise that is taken into account. We note that we found a similar error considering a noise-free reservoir. For a comprehensive study of noise effects in delay-based reservoir computers, we refer the reader to section 6 and to Ref. [29]. Interestingly, NMSE≈ 0.03 obtained for

_{th}*I*

_{0}≈ 1.35

*I*and

_{th}*η*

_{1}= 1 ns

^{−1}(i.e PMOF configuration) is of great practical importance for on-chip implementations: the above threshold pump current values lead to higher output powers which are easy to detect; PMOF does not need any polarization rotation; the low feedback levels can be obtained without a strong amplification even when there is a lot of absorption. We would also like to mention that similar results as those shown in Fig. 8 have been obtained for other values of Δ

*ω*and ΔΩ while the other parameters are kept unchanged.

For further insight, we next consider the parameter sets for which the best performance is obtained in Fig. 8 for each configuration and we investigate the dependence of the reservoir performance on Θ. Figure 9 shows the NMSE values for *η*_{1} = 10 ns^{−1} and *I*_{0} = 1.1*I _{th}* for the PMOF configuration (a), and

*η*

_{2}= 30 ns

^{−1}and

*I*

_{0}= 1.4

*I*for the PROF configuration (b). In both configurations it can be seen that, similar as for the phase dynamics of Figs. 2 and 3, the NMSE does not significantly change with Θ when the system operates above the threshold current provided that the fixed point is stable for

_{th}*S*(

*t*) = 0 (no input data). This good performance can also be explained by the fact that the reservoir is in the transient state for all the values of Θ we explored as shown in Fig. 5(a). The node separation Θ can be therefore freely chosen between the fastest intrinsic time scale of the model (i.e ${\gamma}_{1,2}^{-1}$) and the RO period

*τ*without significantly degrading the performance when the system operates above the threshold. However for

_{RO}*I*

_{0}<

*I*, the RC performance does depend on the value of Θ. In particular for the PMOF configuration, NMSE< 0.1 is obtained only for Θ near ≈ 2

_{th}*T*/3 and ≈ 2

_{phase}*τ*/3. This further evidences that all the coexisting intrinsic time scales play a role to maintain the system in a transient state useful for RC. From this point of view, the coexisting of several intrinsic time scales allows for a broad range of Θ values to be suitable for RC. For the PROF configuration, good performance is restricted around Θ ≈ 2

_{RO}*T*/3 when

_{phase}*I*

_{0}<

*I*.

_{th}## 6. Effects of noise

Reservoir computers based on physical systems are typically subjected to noise. The main noise contributions in SL-based RC schemes are the intrinsic noise from the spontaneous emission and the noise in the readout layer of the reservoir generated by the photodetectors or/and analog-to digital converters. This section further addresses the influence of these noise sources on the RC performance of the system under study.

#### 6.1. Intrinsic noise to the laser

The level of the intrinsic noise in the reservoir as modeled in Eqs. (1) and (2) through the spontaneous emission factors *β*_{1,2} can typically differ from one system to another. Figure 10 illustrates the change in the system performance for different values of the SL spontaneous emission noise from the very low levels to the relatively high levels. More precisely, we show the influence of the reservoir’s intrinsic noise when the system operates just above threshold [Fig. 10(a)] and just under threshold [Fig. 10(b)].

For *I*_{0} = 1.1*I _{th}*, it can be seen that the NMSE gradually increases for both configurations as the noise strength increases. It is also clear that noise degrades the performance of the RC. But for realistic levels of the noise (i.e 10

^{−7}≤

*β*

_{1,2}≤ 10

^{−5}), the increase of NMSE is not significant for the PMOF configuration (NMSE ≲ 5%). For the PROF configuration, however, we notice a profound increase in the NMSE from 5% ≲ to ≲ 20% when the noise strength

*β*

_{1,2}is scanned from 10

^{−7}to 10

^{−5}. Note that the PROF configuration can be further improved by using other values for the pump current and the feedback strength.

For *I*_{0} = 0.9*I _{th}*, the role of the intrinsic noise is very different [Fig. 10(b)]. When noise is too low (

*β*

_{1,2}≲ 10

^{−7}), the reservoir completely fails to predict the next sample in the chaotic time trace. When 10

^{−7}≲

*β*

_{1,2}≲ 10

^{−5}, the intrinsic noise is rather beneficial as acceptable NMSE values are obtained. The noise in this range can be therefore viewed as equivalent to ridge regularization often used in numerical simulations to increase the robustness of the reservoir [15, 16]. Above a certain noise strength, the effect of noise becomes detrimental as found for

*I*

_{0}>

*I*.

_{th}#### 6.2. Noise in the readout layer

In the previous sections, the simulations were done neglecting the noise in the readout layer in order to focus purely on the effects of the major design parameters of the reservoir. At high speeds, however, noise in the readout layer could become important as high speed photodetectors and/or analog-to-digital converters are usually more noisy. We model this noise by adding an extra noise term to the reservoir output signal such that it is read as |*E*_{1}(*t*)|^{2} + *D _{out}ξ_{out}* (

*t*) instead of |

*E*

_{1}(

*t*)|

^{2}, where

*D*is the noise amplitude while

_{out}*ξ*(

_{out}*t*) is a Gaussian white noise with zero mean and correlation $\u3008{\xi}_{\mathit{out}}(t){\xi}_{\mathit{out}}^{*}({t}^{\prime})\u3009=\delta (t-{t}^{\prime})$. Here

*D*= 0 refers to noiseless photodetectors and/or analog-to-digital converters.

_{out}To illustrate the detrimental effects of the noise in the readout layer, we evaluate again the system performance under the conditions of Fig. 7 and considering the value for *D _{out}* such that the signal-to-noise ratio (SNR) after the detection yields to ≈ 20 dB for

*I*

_{0}= 1.1

*I*in the PMOF configuration. Here we use the variance of the output signal and that of the noise to determine the SNR. This value is the smallest pump current at which we find very good performance (NMSE ≲ 4%). We keep the strength

_{th}*D*unchanged as the noise in the readout layer does not depend on the configuration nor on the pump current. In Figure 11, we show the NMSE for different values of the pump current taking into account noise in the readout layer. We take

_{out}*η*

_{1}= 10 ns

^{−1}for PMOF configuration and

*η*

_{2}= 30 ns

^{−1}for PROF configuration which are the feedback strengths for which the smallest NMSE is obtained for the PMOF and the PROF configurations, respectively (see Fig. 8). For comparison, we repeat the simulations for these feedback strengths and for

*D*= 0 and show the results in the same plot. In both configurations, it is obvious that noise in the readout layer is very detrimental when the reservoir operates below threshold. This is because the amplitude of the signal is small below threshold and therefore the SNR becomes too small (SNR≲ 2 dB). The effect of such noise is mitigated above the threshold because the SNR gradually increases as the pump current increases. As a result, the NMSE is very similar to that obtained for

_{out}*D*= 0.

_{out}## 7. Conclusions

We have studied the properties of a delay-based RC system with SLs. For the first time, we have investigated the parameter region where the node separation is much shorter than the relaxation oscillation period. Such short node separations are particularly interesting as they allow to increase the data processing speed. The results indicate that, thanks to the phase response which relaxes faster than the relaxation oscillations, good RC performance can be obtained even when the system is evaluated considering read out signals in the intensity |*E*_{1}(*t*)|^{2}. We also found that due to optical injection, which is an integral part of the setup, the intensity reacts at speeds related to the phase dynamics. As phase dynamics exists even below the threshold, good performance can be obtained below the threshold current provided that Θ is suitably chosen. More precisely, the system can still respond to an external stimulus below the threshold through the phase when the intensity does not completely vanish. Due to different time scales which coexist in SLs, we found that any value of the node separation between the fastest time scale of the model and the RO period can be used above the threshold. This leads to an overall delay length between 1 ns and 420 ns for *N* = 200 virtual nodes. It is worth noting that small Θ values and, thus small delay lengths, are useful for compact on-chip implementations. Furthermore, a short delay also allows to increase the processing speed as the data is fed at the delay period. In addition, considering spontaneous emission noise and noise in the readout layer (i.e noise in detectors and analog-to-digital converters), we have found that good RC performance can be obtained at fast processing speeds for realistic values of the noise strength. As a final remark, we also note that our RC scheme is robust to both intrinsic noise and noise in the readout layer. These results are promising for implementing compact on-chip delay-based RC schemes with fast processing speeds.

## Acknowledgments

The authors thank Dr. M. C. Soriano, Profs. S. Massar and I. Fischer for helpful discussions. The authors acknowledge the Research Foundation Flanders (FWO) for project support, the Research Council of the VUB and the Interuniversity Attraction Poles program of the Belgian Science Policy Office, under grant IAP P7-35 photonics@be.

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