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Performance-enhanced time-delayed photonic reservoir computing system using a reflective semiconductor optical amplifier

Xiaoyu Li, Ning Jiang, Qiang Zhang, Chuanjie Tang, Yiqun Zhang, Gang Hu, Yongsheng Cao, and Kun Qiu

Open Access Open Access

Abstract

We propose a time-delayed photonic reservoir computing (RC) architecture utilizing a reflective semiconductor optical amplifier (RSOA) as an active mirror. The performance of the proposed RC structure is investigated by two benchmark tasks, namely the Santa Fe time-series prediction task and the nonlinear channel equalization task. The simulation results show that both the prediction and equalization performance of the proposed system are significantly improved with the contribution of RSOA, with respect to the traditional RC system using a mirror. By increasing the drive current of the RSOA, the greater nonlinearity of the RSOA gain saturation is achieved, as such the prediction and equalization performance are enhanced. It is also shown that the proposed RC architecture shows a wider consistency interval and superior robustness than the traditional RC structure for most of the measured parameters such as coupling strength, injection strength, and frequency detuning. This work provides a performance-enhanced time-delayed RC structure by making use of the nonlinear transformation of the RSOA feedback.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Reservoir computing (RC), which was first coined by Verstraeten [1], is derived from two closely related recurrent neural network (RNN) architectures, namely the echo state network [2] and the liquid state machine [3]. It has emerged in analog neuromorphic computing for processing time-dependent information [4]. The main advantage of RC is that it simplifies the training process of the RNN structure and reduces the training cost. Specifically, only the connection weights between the reservoir and the readout layer need to be trained, while the input connection weights of the injected information and the internal connection weights of the reservoir do not participate in the training process. In addition, the photonic RC structure has many advantages over the traditional electronic RC structure, including extremely low power consumption, higher operating speed, greater scalability, and greater robustness. Consequently, there has been a resurgence of interest in neuromorphic computing using photonics in recent years. So far, the photonic RC has been extensively applied in various fields such as optical chaos prediction [5,6], optical signal-to-noise ratio monitoring [7], modulation format identification [7,8], stock market forecasting [9], adaptive parameter-variant channel equalization [10], and other areas.

In the last few years, the all-optical delayed-based RC structures [11–16] have been widely researched due to the huge advantage of their minimal hardware requirements over other RC systems [17–20]. Up to the present, numerous RC architectures based on various all-optical devices have been explored. Xiang and Xia proposed the use of vertical cavity surface emitting lasers to construct photonic RC systems [21–24]. Donati et al. demonstrated a time-delayed RC structure utilizing microring resonators with external optical feedback [25]. Xiang and colleagues investigated a performance-enhanced RC system by introducing a semiconductor nanolaser with double phase conjugate feedbacks [26]. Wang et al. constructed a deep time-delay RC architecture using cascading injection-locked lasers [27]. Nguimdo and coworkers realized the time-delayed RC schemes using a single delayed quantum cascade laser with optical feedback [28] and a semiconductor ring laser with optical feedback [29], respectively. Phang demonstrated a photonic RC structure based on the acousto-optic effect with a stimulated Brillouin scattering fiber kernel [30]. It has also been reported that a semiconductor optical amplifier (SOA) can be used in an ultrafast photonic RC system as a nonlinear operator at low power scales [11,31–33]. Although the reflective semiconductor optical amplifier (RSOA) can achieve low noise figure and high optical gain at low driving current compared to the conventional SOA [34], its application in photonic RC systems remains unexplored.

In this paper, we present a time-delayed photonic RC architecture with enhanced performance in which RSOA is adopted as a nonlinear mirror. The performance of the proposed RC structure is evaluated by the Santa Fe time-series prediction task and the nonlinear channel equalization task. We introduce the traditional RC structure for comparison and analyze the effects of several key parameters on the performance of both RC structures. It is shown that by exploiting the nonlinearity of RSOA, the proposed RC system can achieve excellent prediction and equalization performance over the traditional RC system using a mirror.

2. Principles and theoretical model

The schematic of the proposed time-delayed photonic RC structure, abbreviated as RSOA-RC, is illustrated in Fig. 1. Different from the typical linear feedback RC structure that employs a mirror, the RSOA-RC structure utilizes an RSOA as an active mirror. The RSOA-RC system consists of three components: the input layer, the reservoir, and the output layer. In the input layer, the injection signal is sampled as a discrete signal u(n), and each sampling point is held for a time period T. Then, the masked input signal s(t) is obtained by multiplying u(n) by a temporal masking signal m(t) of length T (i.e., s(t) = u(n)×m(t)×γ, where γ is a scaling factor). Here, the mask signal m(t) acts as a random weight connection from the input layer to the reservoir, which can lead to a highly diverse reservoir response. The amplitudes of u(n) vary between 0 and 1. m(t) is a random sequence with a mean value of 0 and a standard deviation of 1, derived by rescaling a chaotic signal with an effective bandwidth of 7.45 GHz generated by an external cavity semiconductor laser reported in [35]. Following the masking process, s(t) is injected into the reservoir by modulating the optical signal from the drive laser with a phase modulator (PM). The modulated injection signal can be expressed as:

$${E_d}(t) = \sqrt {{I_d}} \exp ({i\pi s(t)} ),$$
where Ed(t) represents the electric field injected into the reservoir and Id refers to the steady-state optical intensity of drive laser. The drive laser is also used to achieve the consistency of the response laser, which measures the ability to produce similar responses to similar inputs.

 figure: Fig. 1.

Fig. 1. The schematic of the RSOA-RC architecture.

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In the reservoir, the nonlinear dynamics of the response laser are simultaneously influenced by both the phase-modulated masked injection signal from the input layer and the time-delayed optical feedback signal from the RSOA. The RSOA acts as a nonlinear operator owing to its saturable nonlinearity. In addition, the reservoir is given a memory by converting the intensity feedback information into the injection signal. In the reservoir, the states of the virtual nodes are determined from the transient responses of the response laser with interval θ, where N virtual nodes xi(n) (i = 1, 2, …, N) are available within the delay time τ. It is worth mentioning that similar results can be obtained by using the RSOA transient responses to determine the virtual node states in the proposed RSOA-RC system. For the sake of simplicity, only the case utilizing the transient responses of the response laser is discussed in this work. The state of each virtual node is a high-dimensional mapping of the input signal and the roles of the virtual nodes are analogous to the nodes of traditional reservoirs. By applying the de-synchronization scheme, the delay time τ is set to τ = T+θ, where T = N×θ. In the output layer, the optimal readout weights are obtained by minimizing the deviation between the predicted values and the target values during the training process. By applying the ridge regression algorithm to compute the values of the optimal readout weight wi, the reservoir output y(n) is computed by taking a linear combination of virtual nodes xi(n) with readout weights wi for each time period T as follows:

$$y(n) = \sum\limits_{i = 1}^N {{w_i}{x_i}(n)} .$$

By modifying the feedback term in the Lang-Kobayashi rate equations of the traditional RC structure, the nonlinear dynamics of the response laser in the proposed RSOA-RC system can be illustrated as follows [10,16]:

$$\begin{array}{c} \frac{{dE(t)}}{{dt}} = \frac{{1 + i{\alpha _r}}}{2}\left\{ {\frac{{G({{N_r}(t) - {N_0}} )}}{{1 + \varepsilon {{|{E(t)} |}^2}}} - \frac{1}{{{\tau_p}}}} \right\}E(t) + {k_{inj}}{E_d}(t)\exp (i\Delta \omega t)\\ + {k_c}{E_R}(t - \tau )\exp ( - i\omega \tau ) + \sqrt {2{\beta _r}{N_r}(t)} \chi (t), \end{array}$$
$$\frac{{d{N_r}(t)}}{{dt}} = {J_r} - \frac{{{N_r}(t)}}{{{\tau _{c,r}}}} - \frac{{G({{N_r}(t) - {N_0}} )}}{{1 + \varepsilon {{|{E(t)} |}^2}}}{|{E(t)} |^2},$$
where E(t) and Nr(t) are the slowly varying complex electric field amplitude and carrier density of response laser, respectively. ER(t-τ) is the feedback signal from the RSOA with the delay time τ. αr is the linewidth enhancement factor of response laser, G is the gain coefficient, N0 is the carrier density at transparent and ε is the saturation coefficient. Additionally, τp and τc,r are the photon and carrier lifetimes of response laser, respectively. kinj is the injection strength from drive laser to response laser and kc is the mutual coupling strength between response laser and RSOA. ω is the optical angular frequency of response laser and Δω = 2πΔv is the angular frequency detuning, where Δv is the frequency detuning between drive laser and response laser. χ(t) is the white Gaussian noise added to simulate the spontaneous emission noise, where βr is the spontaneous emission rate. Finally, Jr is the injection current of the response laser.

During the operation of the RSOA, the signal is injected and collected from the facet that has an anti-reflective coating, which is similar to the reflection mode of vertical-cavity semiconductor optical amplifiers (VCSOA). In this work, we model the interaction of photons and carriers in the amplifier cavity of the RSOA with reference to the rate equations of the VCSOA and taking the difference of RSOA and VCSOA into consideration. The RSOA rate equations are used to model the interaction between photons and carriers in the amplifier cavity. It is obtained by modifying the well-known rate equations that are commonly used to analyze lasers, with the consideration of external input signal. The rate equations for the carrier density N(t) (which is related to the active medium of RSOA) and the total photon density S(t) (which is related to the electric field inside RSOA) take the following forms [36]:

$$\frac{{dS(t)}}{{dt}} = \frac{{{P_{in}}(t)}}{{hvV}} + \Gamma {\beta _R}({{B_{rad}} + {B_{nrad}}} ){N^2}(t) + \Gamma {v_g}{g_m}S(t) - {\alpha _i}{v_g}S(t),$$
$$\frac{{dN(t)}}{{dt}} = \frac{I}{{qV}} - [{({{A_{rad}} + {A_{nrad}}} )N(t) + } {({{B_{rad}} + {B_{nrad}}} ){N^2}(t) + {C_{aug}}{N^3}(t)} ]- {v_g}{g_m}S(t),$$
where the total photon density S(t) in the active cavity of RSOA is determined by the photon density of the external injection signal, the photon density produced by the spontaneous and stimulated radiation recombination, and the loss of photon density due to internal loss. Pin(t) refers to the power of the injected optical signal, while h denotes the Planck constant. v = c0/λ is the frequency of the optical signal, where c0 is the speed of light in vacuum and λ is the wavelength of the injected optical signal. V = WHL is the active region volume of the RSOA, where W, H, and L represent the width, height, and length of the active region respectively. Γ is the confinement factor and βR is the spontaneous emission coefficient. vg = c0/ng represents the group velocity, where ng is the refractive index of the active region. αi = K0+ΓK1N(t) is the internal loss of the active region, where K0 and K1 stand for the carrier-independent and carrier-dependent loss coefficients respectively. In particular, the cavity mirror losses are ignored for the sake of model complexity reduction [37,38]. The ROSA reflective facet is assumed to be an ideal facet with a reflectivity of 1. I is the injection current and q is the elementary charge. Arad and Anrad are the linear radiative and linear non-radiative recombination coefficients respectively. Brad and Bnrad are the bimolecular radiative and bimolecular non-radiative recombination coefficients respectively. Caug is the Auger recombination coefficient. gm = g/(1 + Pout/Psat) denotes the RSOA material gain considering gain saturation, where Pout is the total output power and Psat is the saturated output power. The RSOA material gain model considered in this paper relies on density-matrix and perturbation theory formalisms, and takes into account the bandgap shrinkage due to carrier injection, as well as the light-hole and heavy-hole band transitions. The expression for the material gain coefficient g is given by [38,39]:
$$\begin{aligned} g(v,N) &= \frac{{c_0^2}}{{4\sqrt 2 {\pi ^{3/2}}n_g^2{\tau _{c,R}}{v^2}}}{\left( {\frac{{2{m_e}{m_{hh}}}}{{h/(2\pi )({m_e} + {m_{hh}})}}} \right)^{3/2}}\\ &\times \sqrt {v - \frac{{{E_{g0}} - \Delta {E_g}(N)}}{h}} ({{f_c}(v) - {f_v}(v)} ), \end{aligned}$$
where τc,R is the carrier lifetime of RSOA, me is the electron effective mass in the conduction band and mhh is the heavy hole effective mass in the valence band. Eg0 = q(a + by + cy2) is the bandgap energy for no carrier injection, where y represents the molar fraction of arsenide in the active region, and a, b, and c are the bandgap energy coefficients. ΔEg(N) = qKgN3/2 describes the bandgap shrinkage for injection, where Kg is the bandgap shrinkage coefficient. fc(v) and fv(v) are Fermi-Dirac probability distributions for the conduction and valence bands, separately.

Specifically, the complex electric field amplitude E(t) can be related to the optical power P(t) by P(t) = |E(t)|2hv, and P(t) can be related to the photon density S(t) by S(t) = ΓP(t)/(hvvgWH) [40]. Therefore, the dynamics of the proposed RSOA-RC system can be characterized by Eqs. (3)-(6), where the feedback signal ER(t) from RSOA to response laser and the injection optical power Pin(t) from response laser to RSOA can be described as follows:

$${E_R}(t) = \sqrt {S(t)WH{v_g}/\Gamma } \exp (i\phi ),$$
$${P_{in}}(t) = {|{{k_c}E(t - \tau )\exp ( - i\omega \tau )} |^2}hv,$$
where the phase shift ϕ of the injection signal in the RSOA during a single round trip can be determined by ϕ = π-αRΓgmL, where αR is the linewidth enhancement factor of the RSOA [41].

In the following simulations, the traditional linear feedback RC structure using a mirror [10,16], abbreviated as M-RC, is considered for comparison. The performance of the RSOA-RC and M-RC structures are quantitatively investigated. In order to unify the parameters of both systems, the feedback strength kf in [10] is considered as the square of the mutual coupling strength kc between the response laser and the mirror. The fourth-order Runge-Kutta algorithm with a 1 ps step is employed to solve the rate equations of the RSOA-RC system. In the simulations, the time duration T is fixed at 1 ns, which corresponds to an information processing rate of 1 Gsample/s. The parameter values of response laser and RSOA are chosen as the typical values reported in [16,38,39,41], which are summarized in Table 1.

Tables Icon

Table 1. Parameter values of RSOA and response laser used in numerical simulations.

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3. Results and discussion

3.1 Performance of Santa Fe chaotic time-series prediction

In this section, the Santa Fe chaotic time-series prediction task is employed to evaluate the prediction performance of the RSOA-RC and M-RC architectures. The objective of this task is to predict the next step of the Santa Fe chaotic data generated by a far-infrared laser. In this task, 3000 steps and 1000 steps are used for training and testing, respectively. The discrete chaotic time series is taken as the input signal u(n) and the amplitudes of the data are adjusted between 0 and 1. Moreover, the scaling factor γ is set to 0.6, the virtual node interval θ is set to 0.01 ns and the number of virtual nodes N is set to 100. The prediction performance is measured by the normalized mean-square error (NMSE), which is defined as follows:

$$NMSE = \frac{1}{L}\frac{{\sum\limits_{n = 1}^L {{{({\bar{y}(n) - y(n)} )}^2}} }}{{{\mathop{\rm var}} (\bar{y})}},$$
where L is the total number of input data and n is the index of the data. $\bar{y}$(n) denotes the target value, y(n) denotes the prediction result of the system and var represents the variance. In the following simulations, the prediction results are the average of 5 tests.

The effect of the RSOA drive current in the RSOA-RC structure on the prediction performance is first analyzed, and the dependence of the NMSE on the RSOA drive current for kinj = 7.5 × 109 and kc = 2.5 × 105 is shown in Fig. 2. As depicted in the figure, the NMSE decreases as the RSOA drive current increases in the range from 50 mA to 200 mA, which is because increasing the RSOA drive current not only increases the optical gain of the RSOA [42], but also increases the phase shift variation of the injection signal in the RSOA [41]. Therefore, the greater nonlinearity of gain saturation is achieved in the proposed RSOA-RC architecture, as such the prediction accuracy is enhanced. In further analysis, the drive current of the RSOA is set to 80 mA.

 figure: Fig. 2.

Fig. 2. NMSE as a function of the RSOA drive current I for kinj = 7.5 × 109, kc = 2.5 × 105.

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Then, the values of the injection strengths kinj are fixed at 7.7 × 109 and the effects of the coupling strengths kc on the prediction performance in both systems are investigated. The bifurcation diagrams for the optical output intensity of the response lasers versus the coupling strengths in the M-RC and RSOA-RC structures are shown in Figs. 3(a) and 3(b), respectively. It can be seen that each response laser transitions from a relative steady state to a chaotic state as kc increases. However, the response laser of the RSOA-RC system has a wider range of nonchaotic states than that of the M-RC system. Figure 3(c) shows the dependence of the NMSE on the coupling strengths in both RC structures. When the response laser is in a nonchaotic state for each RC system, the NMSE decreases and then increases as the coupling strength increases, which is consistent with the results reported in [16]. When the coupling strength is weak, the transfer function of RSOA is linear, as such, the RSOA-RC system can hardly perform the tasks. As the increase of coupling strength, the gain saturation would gradually occur, and the nonlinearity of RSOA is enhanced. Under such a scenario, the features extracted in the periodic regime adequately reflect the original data, and results in lower NMSE values. Since the gain saturation occurs at the periodic regime, the optimum operation range is slightly away from the edge of stability. However, if the coupling strength is further increased, the consistency feature of RC would be degraded, due to the larger and larger frequency detuning between the drive laser and the response laser, the reservoir would gradually become unstable and even more complex, and consequently, the NMSE values would increase rapidly. Therefore, in this work, the RSOA-RC system prefers to operate at periodic regime. It is also shown that, the RSOA-RC system has superior prediction performance and a wider consistency interval than the M-RC system, and the optimum consistency intervals of the two structures are different. With the injection strengths kinj set to 7.7 × 109, the M-RC system achieves the minimum NMSE at kc = 1 × 105 while the minimum NMSE of the RSOA-RC system occurs at kc = 2 × 105.

 figure: Fig. 3.

Fig. 3. Bifurcation diagram of the optical output intensity of the response laser in the M-RC structure (a) and in the RSOA-RC structure (b) versus the coupling strength kc for kinj = 7.7 × 109. (c) Dependence of the NMSE on the coupling strength kc in the M-RC structure (blue) and in the RSOA-RC structure (red) for kinj = 7.7 × 109.

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In the next step, the influence of the injection strength kinj on the prediction performance is investigated, where the optimal values of coupling strengths in the M-RC and RSOA-RC systems are fixed at kc = 1 × 105 and kc = 2 × 105 respectively. Figures 4(a) and 4(b) present the bifurcation diagrams for the optical output intensity of the response lasers versus the injection strength in both structures. For small values of injection strength, the response lasers in the M-RC and RSOA-RC systems exhibit chaotic behavior. As the injection strength increases, both response lasers transition from the chaotic states to the relative steady states, with the response laser in the RSOA-RC structure reaching the periodic state earlier. Figure 4(c) depicts the dependence of the NMSE on the injection strength in both RC structures. As the injection strength increases, the NMSE values initially decrease in both systems. This is because the increased power of the drive laser raises the signal-to-noise ratio of the RC system and enriches the dynamics of the virtual neurons, as such the prediction performance of the system is improved [43]. As kinj continues to increase, the NMSE saturates and fluctuates slightly due to the noise limit of the RC structure. It is shown that the prediction performance of the RSOA-RC system is better than that of the M-RC system for most of the injection strength values tested. Since the M-RC system performs better at the edge of the stability point, the NMSE values of the RSOA-RC system are higher in the vicinity of the point (kinj = 4 × 109). For larger values of injection strength, as that shown in the inset of Fig. 4, the proposed RSOA-RC structure behaves better robustness due to its smaller NMSE fluctuations.

 figure: Fig. 4.

Fig. 4. Bifurcation diagram of the optical output intensity of the response laser in the M-RC structure for kc = 1 × 105 (a) and in the RSOA-RC structure for kc = 2 × 105 (b) versus the injection strength kinj. (c) Dependence of the NMSE on the injection strength kinj in the M-RC structure (blue) for kc = 1 × 105 and in the RSOA-RC structure (red) for kc = 2 × 105.

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Furthermore, the effect of the frequency detuning Δv between the drive laser and the response laser on the prediction performance is additionally analyzed, as well as the prediction performance for different prediction steps. For the M-RC and RSOA-RC architectures, the injection strengths kinj are fixed at 7.7 × 109 and the optimum coupling strength kc of each system is chosen. As depicted in Fig. 5(a), as the frequency detuning is varied from -20 GHz to 10 GHz, the NMSE values of the M-RC and RSOA-RC systems first decrease, reaching minimum values at about -4 GHz, and then increase. Although the prediction performance of the RSOA-RC structure is not significantly better than that of the M-RC structure in the optimal consistency region, the RSOA-RC system exhibits greater robustness and performs excellently for other measured frequency detuning values. Figure 5(b) presents the NMSE values of both structures at different prediction steps, which become larger as the prediction steps increase. It can also be seen that the prediction performance of the RSOA-RC system is consistently better than that of the M-RC system. When the prediction steps are varied in the range of 1 to 5, the NMSE values of the proposed RSOA-RC structure are all below 0.1, showing excellent prediction performance.

 figure: Fig. 5.

Fig. 5. Dependence of the NMSE on the frequency detuning Δv between drive laser and response laser (a) and on the prediction steps (b), where blue and red indicate the dependence of the prediction NMSE in the M-RC structure for kc = 1 × 105 and kinj = 7.7 × 109 and in the RSOA-RC structure for kc = 2 × 105 and kinj = 7.7 × 109, respectively.

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3.2 Performance of nonlinear channel equalization

In this section, the nonlinear channel equalization task is applied to evaluate the equalization performance of the RSOA-RC and M-RC architectures. Nonlinear channel equalization is an essential task in wireless communications, aiming to recover the transmitted signal as accurately as possible and improve the overall performance of the communications system. The original signal sequence d(n) is randomly chosen from the set {-3, -1, 1, 3}. The signal sequence is then propagated through a linear channel with memory and transformed into q(n). Next, q(n) passes through a nonlinear channel of Gaussian white noise to become the final transmitted signal m(n). The above transmission procedure can be illustrated as follows [44]:

$$\begin{array}{c} q(n) = 0.08d(n + 2) - 0.12d(n + 1) + d(n) + 0.18d(n - 1)\\ - 0.1d(n - 2) + 0.091d(n - 3) - 0.05d(n - 4)\\ + 0.04d(n - 5) + 0.03d(n - 6) + 0.01d({n - 7} ), \end{array}$$
$$m(n) = q(n) + 0.036{q^2}(n) - 0.011{q^3}(n) + \xi (n),$$
where ξ(n) is the Gaussian white noise and its amplitude can be adjusted according to the signal-to-noise ratio (SNR). In this task, the original signal sequence of 105 symbols is generated, with 3000 symbols and the remainder used for training and testing respectively. In the following simulation, the number of virtual nodes N is set to 50, the virtual node interval θ is set to 20 ps, the scaling factor γ is set to 0.3 and the SNR is set to 20 dB. The equalization performance of the RC system is measured by the symbol error rate (SER), which is the proportion of error symbols to total symbols. The following equalization results represent the average of 5 tests.

Figure 6 presents the effect of the RSOA drive current on the equalization performance of the proposed RSOA-RC structure for kinj = 5 × 109 and kc = 1.5 × 105. It shows the same trend as Fig. 2 that the SER decreases as the drive current of RSOA increases from 50 mA to 200 mA. Since the increasing drive current introduces rich amplitude and phase variations in the RSOA signal, the nonlinear dynamics of the RSOA-RC system caused by the gain saturation is enhanced. As a result, the equalization performance of the proposed RSOA-RC structure is improved. However, for larger values of RSOA drive current, the RSOA-RC system becomes more complicated, resulting in larger SER variations. In the following investigations, the drive current of the RSOA is fixed at 80 mA.

 figure: Fig. 6.

Fig. 6. SER as a function of the RSOA drive current I for kinj = 5 × 109, kc = 1.5 × 105.

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In Fig. 7, we illustrate the bifurcation diagrams for the optical output intensity of the response lasers in the M-RC and RSOA-RC structures versus the coupling strengths, and the dependence of the SER on the coupling strengths in both structures for kinj = 5 × 109. As shown in Figs. 7(a) and 7(b), the response laser in the proposed RSOA-RC system requires a larger kc than that in the M-RC system to enter the chaotic state, indicating that the response laser of the former system has a broader relative steady-state range than that of the latter system. It can be seen in Fig. 7(c) that as the coupling strength increases, the SER values in both structures first decrease and then increase. In addition, the proposed RSOA-RC system shows a wider consistency interval and superior performance compared to the M-RC system. It shows a similar trend to Fig. 3(c), but the RSOA-RC structure performs better for all measured coupling strengths, even in the consistency range of the M-RC structure. The minimum SER value of the M-RC structure can be obtained when kc is set to 0.9 × 105, and the optimum equalization performance of the RSOA-RC structure can be achieved when kc = 1.3 × 105.

 figure: Fig. 7.

Fig. 7. Bifurcation diagram of the optical output intensity of the response laser in the M-RC structure (a) and in the RSOA-RC structure (b) versus the coupling strength kc for kinj = 5 × 109. (c) Dependence of the NMSE on the coupling strength kc in the M-RC structure (blue) and in the RSOA-RC structure (red) for kinj = 5 × 109.

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Figure 8 depicts the bifurcation diagrams for the optical output intensity of the response lasers in the M-RC and RSOA-RC structures versus the injection strengths, along with the dependence of the SER on the injection strengths in both structures. The optimum coupling strength values of the M-RC and RSOA-RC systems are set to kc = 0.9 × 105 and kc = 1.3 × 105, respectively. As shown in Figs. 8(a) and 8(b), the response lasers in both structures transition from chaotic to relative stable states almost simultaneously as the injection strength increases. It can be seen in Fig. 8(c) that the SER of the proposed RSOA-RC system decreases and saturates with increasing kinj, which exhibits a similar trend to Fig. 4(c). However, as the injection strength increases, the SER variation is more dramatic in the M-RC system than in the RSOA-RC system due to the effect of system noise, suggesting that the RSOA-RC structure is more robust.

 figure: Fig. 8.

Fig. 8. Bifurcation diagram of the optical output intensity of the response laser in the M-RC structure for kc = 0.9 × 105 (a) and in the RSOA-RC structure for kc = 1.3 × 105 (b) versus the injection strength kinj. (c) Dependence of the NMSE on the injection strength kinj in the M-RC structure (blue) for kc = 0.9 × 105 and in the RSOA-RC structure (red) for kc = 1.3 × 105.

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Furthermore, the effect of frequency detuning Δv between drive laser and response laser on the equalization performance is investigated in Fig. 9, as well as the equalization performance at different SNRs. As illustrated in Fig. 9(a), the SER values in the M-RC and RSOA-RC structures first decrease and then increase with the increase of Δv, and the minimum SER values are obtained at about Δv = -5 GHz and Δv = -2 GHz, respectively. Moreover, the RSOA-RC system outperforms the M-RC system for most of the measured frequency detuning values. However, the consistency intervals of both structures are different. This is because there are various stable and unstable oscillation states of the response laser for the variations in frequency detuning [45], while the introduction of RSOA expands the relative steady-state region of response laser in the proposed RSOA-RC structure and also changes the optimal equalization performance intervals of the system. The results in Fig. 9(b) show that both the RSOA-RC and M-RC architectures can achieve the nonlinear channel equalization task. It can be seen that the SER values of both structures decrease as the SNR increases, and the equalization performance of the RSOA-RC system is always better than that of the M-RC system. There is no significant difference in equalization performance between the two systems when the SNR is less than 16 dB. However, as the SNR continues to increase, the difference becomes more pronounced. The equalization performance of the M-RC structure is only marginally improved when the SNR is above 28 dB, whereas the SER of the RSOA-RC structure is further reduced and better equalization performance is achieved. As can be seen, the RSOA-RC system displays superior equalization performance in comparison to the M-RC system.

 figure: Fig. 9.

Fig. 9. Dependence of the SER on the frequency detuning Δv between drive laser and response laser (a) and on the SNRs (b), where blue and red indicate the dependence of the equalization SER in the M-RC structure for kc = 0.9 × 105 and kinj = 5 × 109 and in the RSOA-RC structure for kc = 1.3 × 105 and kinj = 5 × 109, respectively. The black curve represents the SER before equalization.

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4. Conclusion

In conclusion, we propose and demonstrate a time-delayed photonic RC architecture with improved performance by using RSOA as a nonlinear mirror. Two benchmark tasks, namely the Santa Fe time-series prediction and nonlinear channel equalization, are carried out to evaluate the performance of the proposed RSOA-RC system. Furthermore, we discuss the traditional RC system for comparison and analyze the effects of several key parameters on the performance of both systems. The simulation results demonstrate that the proposed RSOA-RC structure outperforms the traditional M-RC structure due to the richer nonlinear dynamics caused by the RSOA gain saturation. Although the nonlinear effects such as cross-phase modulation and spatial hole burning may occur, the performance of the RSOA-RC system is almost unaffected. The results also show that the nonlinear transformation of the feedback signal from RSOA to response laser can enhance the performance of the RC system.

Funding

National Natural Science Foundation of China (62171087); Sichuan Science and Technology Program (2021JDJQ0023); Fundamental Research Funds for the Central Universities (ZYGX2019J003).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (9)
Fig. 1.
Fig. 1. The schematic of the RSOA-RC architecture.

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Fig. 2.
Fig. 2. NMSE as a function of the RSOA drive current I for kinj = 7.5 × 109, kc = 2.5 × 105.

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Fig. 3.
Fig. 3. Bifurcation diagram of the optical output intensity of the response laser in the M-RC structure (a) and in the RSOA-RC structure (b) versus the coupling strength kc for kinj = 7.7 × 109. (c) Dependence of the NMSE on the coupling strength kc in the M-RC structure (blue) and in the RSOA-RC structure (red) for kinj = 7.7 × 109.

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Fig. 4.
Fig. 4. Bifurcation diagram of the optical output intensity of the response laser in the M-RC structure for kc = 1 × 105 (a) and in the RSOA-RC structure for kc = 2 × 105 (b) versus the injection strength kinj. (c) Dependence of the NMSE on the injection strength kinj in the M-RC structure (blue) for kc = 1 × 105 and in the RSOA-RC structure (red) for kc = 2 × 105.

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Fig. 5.
Fig. 5. Dependence of the NMSE on the frequency detuning Δv between drive laser and response laser (a) and on the prediction steps (b), where blue and red indicate the dependence of the prediction NMSE in the M-RC structure for kc = 1 × 105 and kinj = 7.7 × 109 and in the RSOA-RC structure for kc = 2 × 105 and kinj = 7.7 × 109, respectively.

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Fig. 6.
Fig. 6. SER as a function of the RSOA drive current I for kinj = 5 × 109, kc = 1.5 × 105.

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Fig. 7.
Fig. 7. Bifurcation diagram of the optical output intensity of the response laser in the M-RC structure (a) and in the RSOA-RC structure (b) versus the coupling strength kc for kinj = 5 × 109. (c) Dependence of the NMSE on the coupling strength kc in the M-RC structure (blue) and in the RSOA-RC structure (red) for kinj = 5 × 109.

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Fig. 8.
Fig. 8. Bifurcation diagram of the optical output intensity of the response laser in the M-RC structure for kc = 0.9 × 105 (a) and in the RSOA-RC structure for kc = 1.3 × 105 (b) versus the injection strength kinj. (c) Dependence of the NMSE on the injection strength kinj in the M-RC structure (blue) for kc = 0.9 × 105 and in the RSOA-RC structure (red) for kc = 1.3 × 105.

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Fig. 9.
Fig. 9. Dependence of the SER on the frequency detuning Δv between drive laser and response laser (a) and on the SNRs (b), where blue and red indicate the dependence of the equalization SER in the M-RC structure for kc = 0.9 × 105 and kinj = 5 × 109 and in the RSOA-RC structure for kc = 1.3 × 105 and kinj = 5 × 109, respectively. The black curve represents the SER before equalization.

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Tables (1)

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Table 1. Parameter values of RSOA and response laser used in numerical simulations.

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Equations (12)

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E d ( t ) = I d exp ⁡ ( i π s ( t ) ) ,
y ( n ) = ∑ i = 1 N w i x i ( n ) .
d E ( t ) d t = 1 + i α r 2 { G ( N r ( t ) − N 0 ) 1 + ε | E ( t ) | 2 − 1 τ p } E ( t ) + k i n j E d ( t ) exp ⁡ ( i Δ ω t ) + k c E R ( t − τ ) exp ⁡ ( − i ω τ ) + 2 β r N r ( t ) χ ( t ) ,
d N r ( t ) d t = J r − N r ( t ) τ c , r − G ( N r ( t ) − N 0 ) 1 + ε | E ( t ) | 2 | E ( t ) | 2 ,
d S ( t ) d t = P i n ( t ) h v V + Γ β R ( B r a d + B n r a d ) N 2 ( t ) + Γ v g g m S ( t ) − α i v g S ( t ) ,
d N ( t ) d t = I q V − [ ( A r a d + A n r a d ) N ( t ) + ( B r a d + B n r a d ) N 2 ( t ) + C a u g N 3 ( t ) ] − v g g m S ( t ) ,
g ( v , N ) = c 0 2 4 2 π 3 / 2 n g 2 τ c , R v 2 ( 2 m e m h h h / ( 2 π ) ( m e + m h h ) ) 3 / 2 × v − E g 0 − Δ E g ( N ) h ( f c ( v ) − f v ( v ) ) ,
E R ( t ) = S ( t ) W H v g / Γ exp ⁡ ( i ϕ ) ,
P i n ( t ) = | k c E ( t − τ ) exp ⁡ ( − i ω τ ) | 2 h v ,
N M S E = 1 L ∑ n = 1 L ( y ¯ ( n ) − y ( n ) ) 2 var ( y ¯ ) ,
q ( n ) = 0.08 d ( n + 2 ) − 0.12 d ( n + 1 ) + d ( n ) + 0.18 d ( n − 1 ) − 0.1 d ( n − 2 ) + 0.091 d ( n − 3 ) − 0.05 d ( n − 4 ) + 0.04 d ( n − 5 ) + 0.03 d ( n − 6 ) + 0.01 d ( n − 7 ) ,
m ( n ) = q ( n ) + 0.036 q 2 ( n ) − 0.011 q 3 ( n ) + ξ ( n ) ,
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