High-Speed Reservoir Computing Based on Circular-Side Hexagonal Resonator Microlaser with Optical Feedback

Abstract

In the current environment of the explosive growth in the amount of information, the demand for efficient information-processing methods has become increasingly urgent. We propose and numerically investigate a delay-based high-speed reservoir computing (RC) using a circular-side hexagonal resonator (CSHR) microlaser with optical feedback and injection. In this RC system, a smaller time interval can be obtained between virtual nodes, and a higher information processing rate (R_inf) can also be achieved, due to the ultra-short photon lifetime and wide bandwidth of the CSHR microlaser. The performance of the RC system was tested with three benchmark tasks (Santa-Fe chaotic time series prediction task, the 10th order Nonlinear Auto Regressive Moving Average task and Nonlinear channel equalization task). The results show that the system achieves high-accuracy prediction, even with a small number of virtual nodes (25), and is more feasible, with lower requirements for arbitrary waveform generators at the same rate. Significantly, at the high rate of 10 Gbps, low error predictions can be achieved over a large parameter space (e.g., frequency detuning in the interval 80 GHz, injected strength in the range of 0.9 variation and 2% range for feedback strength). Interestingly, it has the potential to achieve R_inf of 25 Gbps under technical advancements. Additionally, its shorter external cavity length and cubic micron scale size make it an excellent choice for large-scale photonic integration reservoir computing.

1. Introduction

Recurrent Neural Networks are one of today’s means of rapidly processing large amounts of information. To overcome their difficulty in weight training, reservoir computing (RC) is proposed, which randomly fixes the input weights and the internal weights of the reservoir, and only needs to train the output weights [1,2,3]. However, the nonlinearity of the reservoir in the traditional RC system is realized by the high number of nodes, which means that hardware implementation will be quite complicated. Therefore, Appeltant et al. used delay-based nonlinear nodes to implement RC systems with virtual nodes, which greatly simplified its structure [4].

At present, delay-based RCs are implemented in hardware based on electrical [5], optoelectronic [6,7,8] and photonic topologies [9,10,11,12,13,14,15,16]. Meanwhile, semiconductor lasers have become ideal candidates for nonlinear nodes, due to their advantages of high speed and low power consumption in optical information processing. High-speed information processing can maximize the use of information, which is convenient for daily life. Thus, the information processing rate (R_inf) of the RC system has always been widely concerned and researched. Brunner et al. realized the RC experimentally using semiconductor laser diodes as nonlinear nodes, reaching a 1 Gbps R_inf with 10% error in chaotic time series prediction tasks [12]. Guo et al. and Yu et al. used nanolaser and Fano lasers with an ultra-short photon lifetime to increase the R_inf to 10 Gbps, respectively [17,18]. In addition, Estébanez et al. proved the positive effect of bandwidth enhancement on R_inf in the RC system [19].

Therefore, the R_inf of a photonic RC system can be improved by the ultra-short photon lifetime and broad bandwidth of the nonlinear node. We found that the circular-side hexagonal resonator (CSHR) microlaser owns the features that satisfy these mentioned conditions: an ultra-short photon lifetime (picoseconds) and a high relaxation oscillation frequency and flat small signal modulation response under a high bias current [20]. Therefore, using a CSHR microlaser as a nonlinear node in the RC system has the potential to obtain a smaller time interval for virtual node θ and enrich the dynamics of the system.

In this study, we built a delay-based RC system using a CSHR microlaser and evaluated its performance using three benchmark tasks (Santa-Fe chaotic time series prediction task, 10th-order Nonlinear Auto-Regressive Moving Average (NARMA-10) task and nonlinear channel equalization (NCE) task). Furthermore, the impact of important parameters (bias current I, injection strength κ_inj, feedback strength κ_f, frequency detuning ∆ν, and the number of virtual nodes M) on the performance of the system was considered. The simulation results show that the system can still exhibit high-precision prediction and classification over a large parameter space at 10 Gbps. In addition, it is more feasible for the experimental system. In previous work on the RC system, the selected M is higher (usually 100) at R_inf of 10 Gbps, resulting in a smaller θ (1 ps), which requires a higher-speed arbitrary waveform generator to match it. However, in our system, the number of virtual nodes that can be achieved at a smaller value (25), and θ of 4 ps can be satisfied by the performance of existing arbitrary waveform generator products. However, the further development of technology makes it possible to reduce θ to achieve a higher R_inf.

The structure of this paper is as follows. Section 2 describes the theoretical model of the system and the characteristics of the nonlinear node. Section 3 includes a numeric analysis of the system’s influence on key parameters (κ_inj, κ_f, ∆ν, M) for the Santa-Fe chaotic time series prediction task, discusses the potential of processing speed improvements, and further tests the prediction performance of the system using the NARMA-10 task. In addition, the classification performance of the system is also evaluated by the NCE task. Section 4 concludes this paper.

2. System Model and Methods

2.1. Theory

A schematic of delay-based RC using a CSHR microlaser with optical feedback and injection is shown in Figure 1. The system includes three layers: input layer, reservoir layer and output layer. The input signal is multiplied by the mask signal (Mask), and then modulated by the Mach–Zehnder modulator (MZM) to the laser, which is generated from the distributed feedback (DFB) laser (driving laser). The laser passes through an optical isolator (OI), coupler and optical circulator (OC) into the CSHR microlaser (response laser). The delay fiber combines with the OC to form a feedback loop, which further forms a delay-based RC with the CSHR microlaser. Then, the transient response X_i(n) of the virtual node after the delay loop is linearly superimposed with the output layer weight W_i to obtain the output ŷ(n).

In the input layer, the input signal is sampled, and the holding time of each sampling point is T. The modulation signal is obtained by multiplying the input signal after sampling and holding with the mask signal, whose period is T and remains constant in the interval θ. In this paper, T was set to be consistent with τ. We used a chaos mask to improve the performance of the RC system. Previous research has shown that the use of a chaotic mask signal is prone to causing complex behavior in response to the laser output [21,22]. In the reservoir layer, a nonlinear conversion was performed on the input signal, and nonlinearity between the input and high-dimensional state space was realized. The DFB laser was used as the input light source, and the CSHR microlaser with a delay loop was used as the reservoir. τ = M × θ is the feedback delay time, where the number of virtual nodes is M, and the interval between two adjacent virtual nodes is θ. The R_inf is the reciprocal of τ [12].

2.2. Model

As can be seen from Figure 1, the dynamic characteristics of the response laser will be affected when the strength of the external optical injection is changed, thereby changing the performance of the RC system. Based on the Lang–Kobayashi equation, the equation of the CSHR microlaser subjected to optical feedback and injection is as follows [23,24]:

$$\begin{array}{l}\frac{dE\left(t\right)}{dt}=\frac{1+i\alpha }{2}\left[\mathsf{\Gamma }{v}_{\mathrm{g}}G\left(t\right)-{\alpha }_i{v}_{\mathrm{g}}-\frac{1}{{\tau }_{\mathrm{pc}}}\right]E\left(t\right)+\frac{{\kappa }_{\mathrm{f}}}{{\tau }_{\mathrm{L}}}E\left(t-\tau \right)\exp \left(-i{\omega }_0\tau \right)\\ +\frac{{\kappa }_{\mathrm{inj}}}{{\tau }_{\mathrm{L}}}{E}_{\mathrm{inj}}\left(t\right)\exp \left(i\mathsf{\Delta }\omega t\right)+\sqrt{D}\xi \left(t\right)\end{array}$$

(1)

$$\frac{dN\left(t\right)}{dt}=\frac{\eta I}{q{V}_{\mathrm{a}}}-AN\left(t\right)-B{N}^2\left(t\right)-C{N}^3\left(t\right)-v_{\mathrm{g}}G\left(t\right){\left|E\left(t\right)\right|}^2$$

(2)

where E and N are the slowly varying complex electric field and carrier density, respectively. ν_g = c/n_g is the light group speed, where c is the velocity of light in a vacuum and n_g is the mode group refractive index; the angular frequency ω₀ = 2πc/λ₀, where λ₀ is the wavelength; φ is the feedback phase, and it ranges from 0 to 2π. Δω = 2π∙Δν is the angular frequency detuning, where Δν is the frequency detuning between the driving laser and the response laser.

The gain coefficient G(t) and the average amplitude of the injected electrical field E_inj(t) are expressed as follows:

$$G\left(t\right)=g_0\cdot {\left[1+\epsilon {\left|E\left(t\right)\right|}^2\right]}^{-1}\cdot \ln \left(\frac{N\left(t\right)+N_{\mathrm{s}}}{N_{\mathrm{tr}}+N_{\mathrm{s}}}\right)$$

(3)

$$E_{\mathrm{inj}}\left(t\right)=E_{\mathrm{inj},0}\cdot \left[b_{\mathrm{bias}}+u(t)\cdot m\left(t\right)\right]$$

(4)

where u(t) represents the input signal and m(t) is the mask sequence.

It should be noted that the nonlinear gain coefficient, which is shown by Equation (3), makes the CSHR microlaser’s characteristics quite different from those of DFB lasers with a linear gain coefficient. In the simulation, we solved the above rate equation using the fourth order Runge–Kutta method with a step size of 0.5 ps. The definitions and values of the symbols used in the simulation are shown in

Table 1. Numerical Model Parameters.

Symbol	Description	Value
α	Linewidth enhancement factor	4
Γ	Confinement factor	0.25
αi	Internal loss factor	6 cm−1
c	Velocity of light in vacuum	3 × 108 m/s
ng	Mode group refractive index	3.5
τpc	Photon lifetime	8.9 × 10−12 s
λ0	Wavelength	1550 nm
τL	Internal cavity round-trip time	7.5 × 10−13 s
η	Current injection efficiency	0.8
Va	Volume of the active region	30 µm3
A	Defect recombination coefficient	1 × 108 s−1
B	Radiation recombination coefficient	1 × 10−10 cm3/s
C	Auger recombination coefficient	1 × 10−28 cm6/s
Ith	Threshold current	3.6 mA
g0	Material gain parameter	1500 cm−1
ε	Gain suppression factor	18/Ntr
Ntr	Transparency density	1.2 × 1018 cm−3
Ns	Logarithmic gain parameter	0.92 Ntr
Einj,0	Average amplitude of injected electrical field	4 × 1010
bbias	Bias term	0.5

[23].

2.3. Nonlinear Node Characteristics

In this paper, we refer to the parameters of the CSHR microlaser, which was designed and reported by Huang’s team. The CSHR microlaser is surrounded by SiNx and bisbenzocyclobutene (BCB) layers. The mode characteristics of the CSHR were simulated by the three-dimensional (3D) finite-difference time-domain (FDTD) method [25]. This is manufactured using planar semiconductor technology and exhibits the characteristics of low cost, small size (on the order of cubic microns), low power consumption, low threshold current I_th (about 2 mA), ultra-short photon lifetime (picosecond) and easy integration [25]. The above characteristics make it an excellent candidate for large-scale photonic integration. Therefore, we next analyze the dynamic characteristics of the laser and its effect as a nonlinear node in the RC.

2.3.1. Modulation Response

The Lang–Kobayashi equation is usually used to study the characteristics of DFB lasers. In addition to the aforementioned differences, we also use the modulation characteristic curve to further analyze the differences in the output characteristics of the two lasers by adding a sinusoidal current to the laser for modulation. The values of the DFB parameters are referred to in Reference [26]. As shown in Figure 2a, the small signal modulation response curves of the DFB laser and CSHR microlaser under different bias currents are studied, where the dashed line represents the case of the DFB laser, and the solid line represents the CSHR microlaser.

Although the DFB laser’s current in not at such a high level in practice, this simulation was still carried out to better compare the two lasers. In Figure 2a, when I is 3 I_th, 5 I_th and 10 I_th, respectively, obvious peaks appear at the relaxation oscillation frequency (6.3 GHz, 8.9 GHz, and 13.1 GHz) and the peak heights are 23.8 dB, 21.1 dB and 17.5 dB. It can be seen that the increase in I has a positive effect on the relaxation oscillation frequency of the DFB laser. Compared with the DFB laser, the small signal modulation response curve of the CSHR microlaser has less obvious peaks, and gradually flattens with the increase in the current. When I = 10 I_th, the flatness decreases to less than 3 dB and the relaxation oscillation significantly increases.

Correspondingly, we compared the curves under the same current (I = 30 mA) according to the actual situation. The small signal modulation response curves of the DFB laser (blue) and CSHR microlaser (red) are depicted in Figure 2b. The I at this time is 1.4-times and 12-times I_th of the two lasers, respectively. The results show that the small signal modulation response curve of the DFB laser has an obvious peak, while the curve of the CSHR microlaser is relatively flat, indicating that it has a wider bandwidth. This also means that the two have significantly different relaxation oscillation frequencies, which are 3.8 GHz (DFB laser) and 12 GHz (CSHR microlaser). Therefore, the RC based on the CSHR microlaser is beneficial to achieving a smaller θ. In addition, our research group Wang Yixuan et al. also obtained a high bandwidth with a flatness of less than 5 dB in a CSHR microlaser subjected to optical feedback under a high bias current [20].

2.3.2. Dynamics

To achieve a better performance in the RC system, the reservoir should operate in a stable regime, but not too far from the bifurcation point [4]. Figure 3 depicts the transition of different dynamic states of the CSHR microlaser with the continuous increase in κ_f. The blue and red points represent the maximum and minimum values in the time series waveform output by the CSHR microlaser under optical feedback, respectively. It can be seen from the figure that when κ_f is less than 1.2%, the blue and the red points basically coincide, which means the laser is working in a stable state at this time. With the increase in κ_f, period, period-doubling and a chaotic state appear, so κ_f = 1.2% is the Hopf bifurcation point of the laser dynamics.

3. Task Test Results and Discussion

In the research of on RC, a variety of recognized benchmark tasks were used to evaluate the information processing capability of the system. Different tasks have different requirements for the system. To evaluate the generality of the system in prediction tasks and classification tasks, we used three tasks: Santa-Fe chaotic time series prediction task, NARMA-10 task, and NCE task. The effect of the system performing these tasks is as follows:

3.1. Santa-Fe Chaotic Time Series Prediction Task

The data for Santa-Fe chaotic time series come from the records of far-infrared laser experiments running in chaos [27]. We selected 4000 points among them; the first 3000 points were used for training, and the rest were used for testing. The goal of this task was to predict the data before the next data were transmitted to the reservoir system, which is also called one-step prediction. We used a normalized mean square error (NMSE) as the standard to evaluate the RC’s prediction performance. The definition of NMSE is as follows [13]:

$$\mathrm{NMSE}=\frac{1}{L}\frac{{\displaystyle \sum _{i=1}^L{\left(\overline{y}\left(i\right)-y\left(i\right)\right)}^2}}{var\left(\overline{y}\right)}$$

(5)

where ӯ(i) is the target value, y(i) is the predicted value, and L represents the total number of test data. νar(ӯ) represents the variance in the target value. When NMSE ≤ 0.01, we believe that the prediction performance of the system is good [28]. A smaller NMSE value corresponds to a better prediction performance from the RC system.

In the RC system, the size of virtual node M plays an important role. When M is small, it is difficult to handle complex tasks due to the low dimensions of the RC. If τ is increased when θ is fixed, although M can be increased, R_inf will be decreased. In order to describe the influence of M on the RC system more intuitively and find a suitable value for M, we carried out the analysis shown in Figure 4. In the figure, Δν = −25 GHz, κ_inj = 0.4, κ_f = 1%, the dashed line represents the case of NMSE = 0.01, where τ is fixed at 0.1 ns, namely the R_inf of the system is 10 Gbps. As a result, the NMSE is above the dashed line when M is lower than 20. This is due to the insufficient coupling between virtual nodes caused by too large a θ and reduces system performance. With the increase in M, the prediction performance of the system gradually improves, but when M increases to 160, the NMSE does not decrease but shows an upward trend. The reason for this phenomenon is that under a smaller θ, there is too much averaging between nodes [4].

According to Equation (1), the E of CSHR microlaser is affected by κ_inj, κ_f and Δν, which will further influence the performance of the RC system based on a CSHR microlaser. In order to explore the internal connection, the next step is to find the optimal working area of the system by adjusting these key parameters.

Figure 5 describes the effects of κ_f and Δν on the predicted performance of the RC system at a moderate injection (κ_inj = 0.4) and strong injection (κ_inj = 0.8), respectively. The dashed line in the figure represents NMSE = 0.01, which divides the figure into two regions; the part greater than 0.01 corresponds to region A, and the rest belongs to region B. The case under the moderate injection is shown in Figure 5a; the red region (NMSE ≥ 1.8 × 10⁻²) is mainly distributed to the right of the Hopf bifurcation point of the CSHR microlaser (κ_f = 1.2%). The NMSE is almost always greater than 0.01 in the considered range of κ_f when Δν is less than −32 GHz, which means that the prediction error rate is higher at this time. When −30 GHz ≤ Δν ≤ −22 GHz, NMSEs belong to region B in the range of κ_f ≤ 1.4%. Moreover, the prediction performance gradually decreases with the increase in κ_f. The prediction performance of the system is basically independent of κ_f in the range of −12 GHz ≤ Δν ≤ −8 GHz and −2 GHz ≤ Δν ≤ 2 GHz, which means it has strong robustness. They are almost all located in region A, as Δν increases to 10 GHz and the performance of the system degrades significantly.

The area of good performance (NMSE ≤ 0.01) of the RC system with strong injection (κ_inj = 0.8) generally occupies Figure 5b. When κ_f exceeds the Hopf bifurcation point, the performance of the RC system slightly decreases. Meanwhile, region A only appears in part of the range of Δν from −40 GHz to 6 GHz if κ_f exceeds 2%. The area of area B in this figure is much larger than that in Figure 5a, and the values in area B are basically below 6.1 × 10⁻³, while in Figure 5a, almost all of them exceed 6.1 × 10⁻³. From the color bars, the system performs well in predicting the performance in the range of Δν from −40 GHz to −28 GHz and −4 GHz to 10 GHz.

One possible reason for this phenomenon, exhibited in Figure 5a,b, is that the response laser exhibits complete injection locking behavior in all considered Δν ranges when κ_f is small. As κ_f increases, three typical injection locking behaviors gradually appear: fully locked, partially locked and unlocked. This results in a change in the dynamics of the response laser, which further affects the performance of the RC system. In addition, a comparison of the two figures shows that the strong injection expands the range of Δν and κ_f to achieve a better prediction performance. Furthermore, it is evident from Figure 5a that this has asymmetry, which is caused by the usual symmetry of injection locking in semiconductor lasers with a larger linewidth enhancement factor.

According to the richness and effects in Figure 5, Δν was selected as −25 GHz and 25 GHz in the following analysis. Figure 6 analyzes the dependence of the predicted performance of the RC system on κ_inj and κ_f. The figure is divided into two regions by the dashed line, which indicates the case of NMSE = 0.01, where the area B corresponds to NMSE ≤ 0.01 and the remaining part is area A. In this analysis, the main parameters were set as I = 12 I_th, M = 25 and τ = 0.1 ns. When κ_inj is close to 0, the value of NMSE is around 1, which means that the RC system has little predictive effect. In Figure 6a, when κ_f ≤ 1.2%, the dashed line of NMSE = 0.01 is almost parallel to the vertical axis, and with the further increase in κ_f, the dashed line is obviously inclined. The left side of the figure belongs to region A, where κ_inj ≤ 0.32. The area of region B gradually increases in the considered range of κ_f when κ_inj varies from 0.32 to 0.51. NMSEs fully belong to region B in κ_inj between 0.51 and 1.2. In general, the area of region B is larger than that of region A, which means that a good prediction performance occupies a larger range. In Figure 6b, Δν is 25 GHz, which obtains a similar trend to that in Figure 6a but reduces the area where the system predicts a better performance. It can be seen from the figure that when κ_inj ≤ 0.4, the NMSE is basically less than 0.01. The prediction performance of the system improves with the increase in κ_inj. In terms of color, the regions with a better prediction performance are concentrated between a κ_f of 1% and 2%. Both figures show that with the increasing κ_f, the NMSE shows an upward trend. This means that a larger κ_inj and a smaller κ_f help to improve the prediction performance of the RC system.

The R_inf has always been a concern in RC systems, so we analyzed its impact on the RC system, as shown in Figure 7. The dashed line indicates the case of NMSE = 0.01. Considering the further development of technology, we can more easily obtain a smaller θ without being restricted by existing equipment, which will help to reach a much higher R_inf. In Figure 7, a different τ is obtained by changing θ under an M fixed at 25. The lowest NMSE = 3.4 × 10⁻³ is obtained when R_inf is 2 Gbps. Meanwhile, the value of NMSE fluctuates in the range of 3.4 × 10⁻³ to 1 × 10⁻² when R_inf is in the range below 25 Gbps. The reason for this phenomenon may be due to the coupling and averaging effects between nodes due to noise and θ. It is worth noting that the NMSE can still be lower than 0.01 when θ is 1.6 ps and R_inf reaches 25 Gbps. To more intuitively depict the prediction performance of the RC, we selected a higher R_inf of 25 Gbps and compared the predicted value with the target value, as shown in the inset. The NMSE is 6.1 × 10⁻³, the predicted value almost coincides with the target value, and the error between the two is no more than 0.2. Therefore, the system exhibits unique advantages in high-speed information processing.

3.2. NARMA-10 Task

To further analyze the performance of the proposed RC system, we tested it using the NARMA-10 task, which was widely used to evaluate the machine learning performance [29]. Due to the complexity of this task, the processing requires higher nonlinearity and memory ability. We generated the sequences of 4000 points, choosing 3000 and 1000 for training and testing, respectively. NARMA-10 is defined as follows:

$$y_{k+1}=0.3y_k+0.05y_k{\displaystyle \sum _{i=0}^9{y}_{k-i}}+1.5u_k{u}_{k-9}+0.1$$

(6)

where y_k is the output result of the system at time k, and u_k is obtained from the uniform distribution in the interval [0, 0.5].

We use normalized root mean square error (NRMSE) to measure the system’s implementation of NARMA-10 tasks. The definition of NRMSE is as follows:

$$\mathrm{NRMSE}=\sqrt{\frac{1}{m}\frac{{\displaystyle {\sum }_{k=0}^m{\left({\widehat{y}}_k-y_k\right)}^{{}^2}}}{{\sigma }^2\left(y_k\right)}}$$

(7)

where y_k is the target value, ŷ_k is the predicted value, and m represents the total number of test data. σ represents the standard deviation of the target value.

As in the above text, we analyze the dependence of the performance of the RC system on the κ_inj and ∆ν. Figure 8a,b illustrate the distributions of NRMSE under different κ_inj and ∆ν within the range of κ_f from 0 to 2.5% (interval is 0.1%), respectively.

In Figure 8a, Δν is −25 GHz and κ_inj is 0.2, 0.4, 0.6 and 0.8, respectively. In the case of κ_inj = 0.2, NRMSE first decreased and then increased in the considered κ_f and was mainly distributed in the interval of 0.3–0.4. When κ_inj increased to 0.4, the NRMSE decreased significantly, and all were below 0.2. The NRMSE oscillates around 0.1 over the entire κ_f range when κ_inj is 0.6. In addition, an NRMSE of less than 0.1 can be obtained in a large range of κ_f with κ_inj = 0.8, which means that the system has better predictability, and the lowest NRMSE can reach 4.1 × 10⁻². With the increase in κ_inj, the overall NRMSE showed a decreasing trend.

Figure 8b describe the effects when Δν is −10 GHz, 0 GHz, 10 GHz and 20 GHz in the same κ_f range under κ_inj is 0.8. The NRMSE is basically below 0.2 when κ_f is lower than 1.8% with Δν = −10 GHz, and the prediction performance significantly degrades with the increase in κ_f. When Δν = 0 GHz, NRMSE almost never exceeds 0.1 in the range κ_f ≤ 1.5%, and NRMSE expands nearly two times as κ_f increases. The NRMSE is averaged over the entire κ_f range, mostly hovering around 0.1 at Δν = 10 GHz. In the case of Δν = 20 GHz, the prediction performance is slightly reduced compared to the case where Δν is 10 GHz. It is basically balanced in the overall κ_f range. From Figure 8b, under different Δν, the NRMSE distribution also shows obvious asymmetry.

3.3. NCE Task

The NCE task refers to an anti-fading measure taken to improve the transmission performance in a fading channel during the signal transmission process of a communication system [30]. When performing this task, the random sequence d(n) from {−3, −1, +1, +3} is fed into the linear channel to obtain q(n), which is defined as follows [31]:

$$\begin{array}{cc}\hfill q\left(n\right)& =0.08d\left(n+2\right)-0.12d\left(n+1\right)+d\left(n\right)+0.18d\left(n-1\right)\hfill \\ & -0.1d\left(n-2\right)+0.091d\left(n-3\right)-0.05d\left(n-4\right)\hfill \\ & +0.04d\left(n-5\right)+0.03d\left(n-6\right)+0.01d\left(n-7\right)\hfill \end{array}$$

(8)

The q(n) is transmitted through a nonlinear channel and then superimposed with noise to generate u(n). It is defined as follows:

$$\begin{array}{l}u\left(n\right)=q\left(n\right)+0.36q{\left(n\right)}^2-0.011q{\left(n\right)}^3+\xi \left(n\right)\\ \end{array}$$

(9)

where ξ(n) is a Gaussian white noise with mean 0 and variance determined by the desired output signal-to-noise ratio (SNR). We generated 9000 symbols, from which 3000 were selected as a training set and 1000 as a test set. Taking u(n) as the input signal, the original signal d(n) was expected to reconstruct after being processed by the RC system.

The performance of the system for this task is usually evaluated using symbol error rate (SER), which is defined as follows:

$$\mathrm{SER}=\frac{N_{error}}{N_{all}}$$

(10)

where N_error represents the sum of the output error data after passing through the system, and N_all is the sum of the input data. When SER ≤ 0.01, the classification performance of the system is considered to be good [32].

Figure 9a shows the change in the system classification performance when the SNR is between 12 dB and 32 dB (with an interval of 2 dB). We set M to 50; the effect will be degraded somewhat unexpectedly if M = 25 and, when increasing M, although the system performance is improved, the speed problem also needs to be considered. Other parameters are considered, such as I = 12 I_th, κ_inj = 0.8, κ_f = 1.3%, and Δν = −25 GHz, according to the above study. SER shows a downward trend with an increase in SNR, which indicates that the classification performance is improved.

The analysis of the influence of κ_f on the system classification performance is shown in Figure 9b. Here, we set the SNR to 32 dB according to the results in Figure 9a. It can be seen from this figure that with the increase in κ_f, the SER shows a trend of first decreasing and then increasing. When κ_f is 1.3% near the Hopf bifurcation point (κ_f = 1.2%), a minimum SER of 1.05 × 10⁻³ can be obtained. After that, the performance of the system degrades as κ_f increases. This is due to the gradual chaotic state of the laser, which is very sensitive to initial conditions. In addition, compared with no feedback, the SER significantly decreases with feedback, which indicates that the delay-based RC system can effectively improve the classification accuracy of information.

4. Conclusions

In summary, a delay-based RC using a CSHR microlaser with optical feedback and injection is proposed. The performance of this RC system is measured using three typical benchmark tasks: Santa-Fe chaotic time series prediction task, NARMA-10 task and NCE task. Meanwhile, the optimal working area of the system could be found by adjusting key parameters such as κ_inj, κ_f, Δν and M. In this system, when M is small (25), a better performance can still be obtained. This means that it is easier for the system to achieve a small τ and realize high-speed information processing.

In the test of the Santa-Fe chaotic time series prediction task, the system can achieve the information processing rate of 25 Gbps (NMSE is 6.1 × 10⁻³); under a strong injection, high-accuracy predictions are almost always satisfied with κ_f ≤ 2% and −40 GHz ≤ Δν ≤ 40 GHz, where the lowest NMSE can reach 2.7 × 10⁻³; the NMSE is essentially less than 0.01 when Δν = −25 GHz, κ_f ≤ 2.5% and 0.32 ≤ κ_inj ≤ 1.2.

We analyzed the effects of different κ_inj and Δν on NRMSE within the same κ_f range for the NARMA-10 task. The results show that a larger κ_inj helps to obtain a smaller NRMSE, and the minimum NRMSE is 4.1 × 10⁻²; a different Δν shows obvious asymmetry. Moreover, the predictive power of the system remains undiminished for tasks with a higher memory capacity and nonlinearity requirements, indicating the strong applicability of the system. In addition, the system can also perform well on the classification task with good generalizability. Boosting the SNR has a positive effect on the system’s classification performance for the NCE task. The smallest SER (1.05 × 10⁻³) is obtained near the Hopf bifurcation point. In general, when R_inf is at 10 Gbps, the system can show low-error prediction over a wide range of parameters (e.g., Δν in the interval 80 GHz, κ_inj in the range of 0.9 variation and 2% range for κ_f). This is of great significance for RC to achieve high-speed information processing.

Furthermore, we have considered using the system in practical applications (e.g., signal recovery in high-speed optical communications and image processing) with an integrated design in possible future work.