1. Introduction

With the current dramatic expansion of artificial neural network (ANN) usage, photonics is gaining increasing attention as a physical medium for implementing power-efficient, high-speed analog ANNs, or neuromorphic hardware. Among various ANN classes, such as deep neural networks [1] and spiking neural networks [2], reservoir computing (RC) [3] is outstanding from the aspects of its implementation feasibility and low learning cost. These features have led to intensive studies of photonic RC [4–16], which have demonstrated its applicability to computationally challenging problems despite its relatively simple implementation. Among the notable advantages of photonic implementation are (1) ultra-high-speed linear operations by use of light interference [10,16], (2) the availability of low-loss delay lines to enable node formation in the time domain [5–9,11–16], and (3) sophisticated wavelength-division-multiplexing (WDM) technologies [17–20] that may be exploited to enlarge the network structure toward the wavelength domain [16]. Furthermore, recent advances in the fabrication of large-scale photonic integrated circuits (PICs), especially of silicon photonics (SiPh) [21,22], are expected to enable spatial enlargement of photonic ANNs on a compact chip.

In contrast to these promising advantages, the execution of nonlinear operations remains an important issue for photonic implementation. It is widely recognized that nonlinear transformation (i.e., nonlinear activation) of processed signals is essential for ANNs to achieve high processing performance [23–25]. In the photonic ANNs demonstrated so far, nonlinear operations have been implemented through various phenomena. These phenomena include gain saturation in semiconductor optical amplifiers (SOAs) [4,5,14], sigmoidal nonlinearity via an SOA-Mach-Zehnder-interferometer setup with a WDM scheme [26], absorption saturation in saturable absorbers [9], the power-dependent phase nonlinearity of the optical Kerr effect [15], chaotic behavior in injection-locked laser diodes [13], and the opto-electronic sinusoidal nonlinearity of Mach-Zehnder modulators (MZMs) [6,7]. Among these phenomena, the nonlinearity in SOAs is particularly promising for three reasons: (1) it is all-optical and thus avoids costly O/E/O conversions; (2) SOAs rather provide optical gain that can compensate various kinds of optical losses; and (3) SOAs have wide gain spectra (typically several tens of nm) and do not rely on optical resonators, which makes them compatible with WDM-extended implementations.

However, conventional SOAs typically consume large electrical power on the scale of 10² mW to achieve sufficient gain and saturable nonlinearity [4,5,26]. This large power requirement is considered to be an important issue for SOA-based implementation, and it would induce a severe problem of power inefficiency when implementing large-scale ANN circuits with multiple densely integrated SOAs. Accordingly, reduction of the power requirement is of particular importance to make SOAs a more attractive candidate and to make such dense integration practically feasible. In addition, the use of SOAs’ nonlinearity for RC has so far been limited to only the saturable nonlinearity itself [4,5,14], even though SOAs themselves offer richer nonlinear functionalities. Among the most typical of these functionalities is cross-gain modulation (XGM), in which multiple input signals to a saturated SOA interact with each other via its shared gain medium. In the field of optical communication, XGM has found notable applications for all-optical signal transfer, such as wavelength conversion [27,28] and logic gate implementation [29,30]. Here, the capability of XGM to nonlinearly correlate different optical signals could be harnessed for RC as a kind of nonlinear operator; that is, it could act as a nonlinear function with multiple optical input signals as arguments. However, such a potential application of XGM in the field of photonic RC has not yet been explored.

Hence, in this study, we apply our recently developed membrane SOA on a Si platform [31] to photonic RC to demonstrate its usability as a low-power, all-optical, nonlinear operator. Specifically, we implement a delay-based, all-optical reservoir using the membrane SOA chip coupled to a fiber delay line, which employs XGM in the SOA for the delayed information feedback. The reservoir exhibits XGM-based nonlinear operation with a low power consumption of the SOA on the scale of 10¹ mW and a low optical input power to the reservoir on the scale of 10² µW. Such low-power nonlinear operation is enabled by the membrane SOA’s features of small active volume and strong optical confinement. In the proposed XGM-based configuration, the feedback light (i.e., a past signal) is injected into the SOA from a direction opposite to the input light (i.e., a new signal). This unique feedback scheme enables the reservoir to operate robustly, because it is insensitive to the phase of feedback light and free from the loop oscillation no matter how large the SOA’s gain is.

In a preliminary report [32], we showed that the XGM-based reservoir operated at a low power scale and that the RC system could solve a nonlinear benchmark task with high accuracy. However, we neither analyzed nor discussed how the information was nonlinearly transformed, and the study only used a single reservoir-driving condition. The preliminary report also did not characterize the membrane SOA itself, which left the key device characteristics unclear.

Hence, in this extended study, we systematically perform both the device characterization and the RC performance evaluation. The membrane SOA chip is characterized to be a low-power-scale nonlinear amplifier, with 10¹-mW-scale electrical power consumption and 10²-µW-scale optical input power. Within these low-power ranges, the reservoir exhibits the XGM-based nonlinear operation. The power consumption is about an order of magnitude lower than that of conventional SOAs with significant saturable nonlinearity [4,5,26]. Under various reservoir-driving conditions, the RC performances are evaluated in terms of the information processing capacity (IPC) and a nonlinear benchmark task. The IPC evaluation reveals that the XGM-based reservoir performs strong nonlinear transformation of input time-series signals. As the series of results are reasonable and consistent throughout, they clearly demonstrate that the membrane SOA performs RC-applicable nonlinear operations through XGM at a low power scale. The results highlight the significant potential of membrane SOAs as a low-power-scale, all-optical, nonlinear operator, and they suggest new XGM application schemes for photonic RC.

2. Experimental setup and device characterization

2.1 XGM-based reservoir

Figure 1(a) illustrates the XGM-based all-optical reservoir that we implemented in this study. Before explaining the details of the implemented system, we first briefly describe the reservoir's operation principle. An optical input signal is first fed to an SOA from an input port (left side in the figure), and the SOA's output signal is then launched into a fiber feedback loop (right side). After a round trip through a fiber delay line, the resulting feedback signal with delay time $\tau $ is fed to the SOA from the output port. Accordingly, the input and delayed feedback signals are fed to the same SOA from opposite directions, which causes XGM between them to occur when the SOA is in its saturation regime. When such XGM occurs, the intensity information of the delayed feedback signal (i.e., the past signal) is transferred to the input signal, and the output signal thus incorporates this past information. This makes the reservoir dependent on past information and thus gives the reservoir a memory. At the same time, the SOA also functions as a nonlinear operator through the saturable nonlinearity and the XGM: it generates nonlinear correlations of the input and feedback signals. In other words, it nonlinearly transforms input time-series signals. Thus, we expect the proposed configuration to behave as a reservoir that is capable of both memorizing information and nonlinearly transforming it, which are the key functionalities that RC requires [3,24].

 

Fig. 1. (a) Schematic illustration of the XGM-based all-optical RC system that was implemented in this study. Optical/electrical components and lines are shown in orange/blue, respectively. Propagating optical signals are drawn as red arrows. (b) The equivalent ANN structure. The ring-shaped recurrent neural network is configured along the delay line. The masking of the input signal and weighting of the output signal are performed offline.

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Regarding the specifics of our implementation, first, to generate optical input signals, we use a tunable laser diode (TLD, santec TSL-550) as a continuous-wave (CW) light source and a Mach-Zehnder modulator (MZM, SHF 46210C) as an E/O converter. A time-series voltage signal ${V_{\textrm{pp}}}{m_i}u(n )$, where $u(n )$ is the input time-series signal and ${m_i}$ is a mask function, is applied to the MZM by an arbitrary waveform generator (AWG, Keysight M8195A). For the MZM, we set the bias point close to the quadrature point and ${V_{\textrm{pp}}}$ smaller than ${V_\pi }$ so that its E/O characteristics can be regarded as roughly linear (i.e., ${f_{\textrm{MZM}}}(x )\sim x$). This makes the RF component of the optical input signal roughly identical to the voltage input signal itself, ${m_i}u(n )$. The SOA chip is coupled to optical fibers for signal input and feedback, and the optical input signal from the MZM is fed to the chip through an optical circulator (CIR, Thorlabs 6015-3) and a polarization controller (PC, Oyokoden Lab MPCA-1550). Then, the output signal is launched into the fiber feedback loop, which comprises another CIR, a 10-dB coupler (CPL, Thorlabs TW1550R2F1), a delay line (DL), and a PC. The delayed feedback signal is fed to the SOA chip via the CIR, and finally, the amplified feedback signal is removed from the reservoir circuit through the PC and CIR on the input side. The removed feedback signal is guided to an optical spectrum analyzer (OSA, Advantest Q8384) to monitor and ensure the circuit's stability. The 10-dB coupler in the feedback loop taps the SOA's output signal, which is regarded as the reservoir response ${x_i}(n )$. The tapped optical signal is O/E-converted by an erbium-doped fiber amplifier (EDFA, Alnair Labs LNA-220-C), a band-pass filter (BPF, santec OTF-920), and a photodiode with a trans-impedance amplifier (PD/TIA, Hewlett-Packard 11982A); then, the signal is recorded by a real-time digital storage oscilloscope (DSO, Keysight DSO-Z 634A).

Figure 1(b) shows the equivalent ANN structure. Here, n denotes the time step of the time-series signals ($n = 1,2, \cdots ,L$), and i denotes the index for the virtual nodes ($i = 1,2, \cdots ,N$), which are defined along the delay line in the time domain. As is performed elsewhere [5–9,11–13,15], this virtual-node formation is enabled through offline multiplication of a mask function ${m_i}$. By setting the node interval $\theta = \tau /({N + 1} )$, where $\tau $ is the round-trip delay time, we can make connections between adjacent nodes and obtain the ring-shaped recurrent network illustrated in Fig. 1(b). Note that the node index for the feedback signal in Fig. 1(a) is denoted as “$i - 1$” because of the ring-shaped network. In the offline processing, the time signal recorded by the DSO is demultiplexed with the node interval $\theta $, which allocates each time slot to the corresponding reservoir response ${x_i}(n )$. The calculation of the output signal $o(n )= {\boldsymbol w} \cdot {\boldsymbol x}(n )$ and learning of the output weights are also performed offline. Here, ${\cdot} $ denotes the vector dot product, and ${\boldsymbol w} = [{{w_1},{w_2}, \cdots ,{w_N}} ]$ and ${\boldsymbol x}(n )= [{{x_1}(n ),{x_2}(n ), \cdots ,{x_N}(n )} ]$ are the vectors of the output weights and the reservoir responses, respectively. The learning is executed by a linear regression method such as least squares method or ridge regression, which are widely used, low-cost learning methods.

In terms of system stability, this XGM-based configuration provides the feedback circuit's robustness because of two features: the phase insensitivity and the elimination of loop oscillation risk. First, because the feedback light and input light propagate in opposite directions, they do not interfere, and only the intensity information is fed back via XGM. This makes the reservoir circuit insensitive to the phase of feedback light, which enables easy, simple implementation of the feedback loop by a free-standing fiber without any phase-locked loop (PLL) system. In contrast, conventional fiber-based coherent reservoirs that use optical couplers for information feedback [11] require a costly PLL system for precise control of the feedback phase. Second, because the feedback signal is removed from the reservoir after the information transfer via XGM, the feedback circuit is open and does not form an optical cavity. Hence, provided that any unwanted reflection from the input port is small enough, the SOA will not oscillate as a laser no matter how large the gain is. Accordingly, this feedback circuit is free from the risk of loop oscillation, and ideally we can set the SOA gain to an arbitrary value. In contrast, for conventional reservoirs that have SOAs inserted in closed feedback loops [4,5], the net feedback gain must be precisely controlled to prevent the SOAs from lasing. These two features make us easily, successfully implement and operate the XGM-based reservoir circuits.

2.2 Device design and characteristics of membrane SOA on Si

Figure 2(a) shows a cross-sectional schematic of the membrane SOA on Si that was used in this study. As described in [31], the III-V membrane comprises a 230-nm-thick InP slab and a 600-nm-wide buried active core which consists of an InGaAsP-based six-period multi-quantum well (MQW). A Si waveguide is heterogeneously integrated with the III-V membrane, which will enable fabrication of III-V/Si monolithically integrated RC circuits in the future. Because the effective refractive index of the III-V membrane has the same range as that of the Si waveguide, it is easy to control the mode coupling and the optical confinement factor in the III-V core.

 

Fig. 2. (a) Cross-sectional schematic of the membrane SOA on Si. (b) Its cross-sectional mode field, which was calculated in [31]. (c) Top-view schematic of the SOA chip coupled to high-numerical-aperture (HNA) optical fibers for signal input and output.

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Here, we briefly explain the device design's key points for achieving low-power nonlinear operation. The unsaturated modal gain of an SOA can be expressed by a well-known equation:

Here, ${\tau _\textrm{c}}$ is the carrier lifetime which is assumed to be constant, ${\eta _\textrm{i}}$ is the injection efficiency, ${I_{\textrm{SOA}}}$ is the SOA injection current, q is the elementary charge, and ${A_\textrm{a}}$ is the active region's cross-sectional area.

These simple expressions indicate a key measure to achieve low power consumption in SOAs: reducing the active area while maintaining strong optical confinement. That is to say, a small active area enables inversion of the gain medium with a small injection current per length (Eqs. (2) and (3)), and strong optical confinement enables amplification with a short active length (Eq. (1)). Both of these contribute to a reduction in the injection current required for amplification. On the other hand, regarding the optical power scale, the input power to induce 3-dB gain saturation under a given unsaturated modal gain, ${P_{\textrm{in, - 3dB}}}$, is known to satisfy the following proportional relation [34]:

The factor $\Gamma /{A_\textrm{a}}$ expresses how strongly the optical power is localized in the active region on a per-area basis, which corresponds to the interaction strength between the gain medium and the optical field. This means that strong interaction results in saturation by a low-power input. Accordingly, the small active area with strong optical confinement, which is a unique feature of the membrane structure, has the benefit of lowering both the electrical and optical power requirements to achieve nonlinear amplification operation.

Accordingly, we designed the size of the Si waveguide in our device to be 220 nm × 440 nm so that the confinement factor in the active region would be large, while modal overlap with the p-InP region was suppressed [31]. Figure 2(b) shows the calculated mode profile [31], which indicates that the optical mode is strongly localized in the active region. The fill factor in the MQW is about 9.4%. Figure 2(c) shows a top-view schematic of the SOA chip coupled to high-numerical-aperture (HNA) optical fibers for signal input and output. The SOA's active region is 300 µm long, and its facets are butt-coupled to 40-µm-long InP tapers that efficiently convert the optical mode between the SOA and the Si waveguide with a low loss [31]. The SOA is connected to SiO_x spot-size converters (SSCs) at the chip facets via the waveguide, which enables low-loss coupling with the HNA fibers.

Prior to our RC experiments, we measured the basic amplification characteristics of the SOA chip in a fiber-to-fiber setup to characterize its power scale. The SOA injection current was varied from 5 to 25 mA, within which a fiber-to-fiber gain larger than 0 dB would be expected according to our previous study [31]. The corresponding applied voltage ${V_{\textrm{SOA}}}$ was measured, and we calculated the power consumption of the SOA by ${I_{\textrm{SOA}}}{V_{\textrm{SOA}}}$. A TLD with the lasing wavelength set to 1530.0 nm was used as a light source, and a variable optical attenuator (VOA, Anritsu MN9605C) was used to vary the input power. The SOA chip was mounted on a temperature-controlled stage that was kept at 25°C. Figures 3(a) and (b) show the measurement results: an input-gain plot (logarithmic scale) and an input-output plot (linear scale), respectively. Remarkably, an injection current of only 15 mA (23-mW power consumption) was enough to compensate the fiber-to-fiber loss and provide a net fiber-to-fiber gain of about 2 dB at a fiber input power of −10 dBm. At the same time, the input power to induce 3-dB gain saturation (${P_{\textrm{in, - 3dB}}}$) measured from the minimum point in the graph was as low as about −9 dBm. This means that strong saturable nonlinearity occurred within an input power range of 10² µW, as can be clearly seen in Fig. 3(b), as well. Accordingly, these characteristics confirm that the membrane SOA chip works as a nonlinear amplifier with 10¹-mW-scale electrical power consumption and 10²-µW-scale optical input power. The device is thus characterized as a low-power nonlinear amplifier, as compared to conventional SOAs [4,5,26], which require 10²-mA-scale current injection (10²-mW-scale power consumption) to exhibit significant saturable nonlinearity. To reiterate, this notable low-power characteristic is enabled by the above-mentioned features of the membrane structure.

 

Fig. 3. Fiber-to-fiber amplification characteristics of the SOA chip with various SOA injection current values: (a) an input-gain plot on a logarithmic scale, and (b) an input-output plot on a linear scale.

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3. Demonstration of RC operation

In the following RC experiments, we again set the TLD wavelength to 1530.0 nm and the stage temperature to 25°C. The constant bias optical power at the input fiber, which corresponded to the input signal's zero level (i.e., ${m_i}u(n )= 0$), was set to be about 160 µW. The RF voltage signal from the AWG, ${V_{\textrm{pp}}}{m_i}u(n )$, was applied to the MZM, whose positive and negative peaks corresponded to the maximum and minimum levels (i.e., ${m_i}u(n )={\pm} 1$), respectively. As described above, to ensure good linearity, the MZM's bias point was set close to the quadrature point and ${V_{\textrm{pp}}}$ was set smaller than ${V_\pi }$. The mask function ${m_i}$ was set to take arbitrary random values ranging from −1 to 1. The total node number N was set to 200. The total length of the feedback delay loop was about 11 m, which corresponded to a round-trip delay time $\tau $ of about 54 ns. As a result, the node interval $\theta = \tau /({N + 1} )$ was about 270 ps. In practice, we determined the node interval from the actual delay time $\tau $ which was precisely evaluated by impulse response measurement. The DSO's sampling rate was set to 20 GSa/s, which resulted in an average of ∼5.4 sampling points per node. We averaged all the sampling points in each node interval to obtain its reservoir response ${x_i}(n )$.

To examine the dependence on the SOA injection current, we performed a series of measurements at 15, 20, and 25 mA, which corresponded to power consumptions of 23, 33, and 43 mW, respectively. In addition, to clearly understand the effects of the optical feedback, we also evaluated a case without optical feedback at an injection current of 25 mA, by disconnecting and terminating the fiber connectors between the PC and the CIR in the delay loop.

3.1 Impulse response

First, we measured the reservoir's impulse response. An input optical pulse was generated by modulating the MZM with a positive pulse voltage, whose amplitude was ${V_{\textrm{pp}}}/2$. The pulse duration was set to 1 ns. Figures 4(a-d) show the measurement results for the cases of 25, 20, and 15 mA with feedback, and 25 mA without feedback, respectively. The former three cases all exhibited a periodic response, whose pulse polarity was inverted for each period, up to the third pulse. The period was measured to be about 54 ns, which exactly corresponds to the round-trip delay time $\tau $ expected from the delay loop length of ∼11 m. As is well known, the inversion of the pulse polarity is a distinctive feature of XGM. Accordingly, this result indicates that XGM actually occurred in the membrane SOA even with such low electrical power consumption and low optical power input. Comparison with Fig. 4(d), in which only the input pulse itself can be seen, more clearly confirms that the periodic response originated from the optical feedback and the accompanying XGM.

 

Fig. 4. Measured impulse response of the reservoir under various reservoir-driving conditions. Specifically, the SOA injection current and the presence of optical feedback were varied: (a) 25 mA, (b) 20 mA, and (c) 15 mA, all with feedback, and (d) 25 mA without feedback. The insets show close-ups of the time slots corresponding to the periodic response peaks.

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By comparing Fig. 4(c) (15 mA) with Figs. 4(a) (25 mA) and 4(b) (20 mA), we can see that the shapes of the second and third pulses in the first graph are more obscure than those in the latter two graphs. This trend can be reasonably attributed to the difference in the net feedback gain. As Fig. 3 indicated, a larger injection current provides a larger gain and a larger output power. This implies that when the injection current is larger, the feedback has a larger optical power, which more strongly modulates the CW input through the XGM and thus results in response pulses with larger amplitudes.

3.2 Information processing capacity

Next, we measured and evaluated the information processing capacity (IPC), which was introduced by [35] and is widely used for task-independent specification of an RC’s processing performance [15,36–38]. In IPC evaluation, the input signal $u(n )$ is set to a random sequence taking arbitrary values from −1 to 1, and the corresponding reservoir responses ${x_i}(n )$ are experimentally measured. The target signal is given by the following equation:

In this IPC evaluation experiment, we set the data length L of the random input signal to 2,000 samples. As described in [35], when the data length is finite, the MC is plagued by a positive bias because of the finite nature of the statistics. It is thus necessary to apply a cutoff capacity ${\textrm{C}_{\textrm{CO}}}$ [15,36]: if the capacity $\textrm{C}$ is below ${\textrm{C}_{\textrm{CO}}}$, then it is assumed to be zero in the evaluation of each MC. ${\textrm{C}_{\textrm{CO}}}$ is an arbitrarily set value but has to be high enough. According to [35], the positive bias follows a Chi-squared distribution with a mean of $N/L$ and a variance of $2N/{L^2}$, which were equal here to 0.1 and 0.0001, respectively. Accordingly, we set ${\textrm{C}_{\textrm{CO}}}$ to 0.15, which was high enough to eliminate the positive bias's contribution.

Figure 5 shows the IPC evaluation results under the reservoir-driving conditions described above. We evaluated the series of $D$-th order MCs from $D = 1$ to $D = 5$, which are plotted in different colors in the graph and stacked to give the TMC.

 

Fig. 5. Information processing capacities (IPCs) evaluated under the same reservoir-driving conditions as in Fig. 4: 25, 20, and 15 mA, all with feedback, and 25 mA without feedback. The memory capacities for the $D$-th order polynomial degree, $\textrm{M}{\textrm{C}^D}$, are plotted in different colors with respect to D and are stacked to give the TMC.

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First, it is evident that the cases with optical feedback had significant MCs for various values of D, with TMCs ranging from 10.18 (at 15 mA) to 18.60 (at 25 mA). On the other hand, the case without feedback had an MC only for $D = 1$, and the MC had a small value of 1.06. This clearly indicates that the memory in this reservoir is provided by the optical feedback and the accompanying XGM. Ideally, the case without feedback should not have had any memory, giving an MC of zero, but there was actually a slight linear MC. We consider it to have originated from the finite response speeds of the instruments in the E/O/E route from the AWG to the DSO, as reported in other delay-based RC experiments [12]. Because this residual memory was significantly smaller than the MCs in the cases with feedback, it should be practically irrelevant in terms of the processing performance.

Second, there was a clear trend with regard to the dependence on the injection current: the larger the injection current was, the larger the TMC became. We can reasonably attribute this trend to the differences in the feedback signal power, as was the case for the impulse responses. That is to say, the feedback signal has a larger power when the injection current is larger, which more strongly modulates the next input signal via XGM. This should cause the feedback signal to have a stronger influence, which results in a stronger dependence on the past information, and thus, a larger TMC.

Third, it is remarkable that the reservoir was highly nonlinear with every injection current: the contribution of the nonlinear MCs ($D \ge 2$) to the TMC was as high as 78.4%, 72.9%, and 72.1% in the cases of 25, 20, and 15 mA, respectively. This means that the input signals here were strongly nonlinearly transformed. Although the whole system from the AWG to the DSO could contain some residual nonlinearities (e.g., deviation of the MZM’s characteristics from ideal linearity), the fact that the SOA operated in a regime with significant nonlinearity (i.e., gain saturation and XGM) strongly suggests that the SOA itself should provide the dominant contribution to the nonlinear transformation.

For comparison, we refer to conventional delay-based RC circuits that use optical couplers for feedback [5,11]. In [5] and [11], the ratio of $\textrm{M}{\textrm{C}^1}$ (the linear MC) to the TMC was at least 41.6% and 42.3%, respectively (i.e., the nonlinear MCs’ contribution was at most 58.4% and 57.7%), when the reservoir-driving conditions were set so as to maximize $\textrm{M}{\textrm{C}^1}$. Note that this calculation of a lower bound on $\textrm{M}{\textrm{C}^1}$’s contribution is based on the theoretical notion that the maximum TMC is bounded by the total number of nodes, N [35]. In [5], the saturable nonlinearity of the SOA was simply used for the nonlinear transformation; in [11], where the reservoir was coherently driven in the amplitude domain, square-law detection by a readout photodiode was used for quadratic nonlinear transformation. On the other hand, in the XGM-based reservoir here, the input signal is first nonlinearly transformed by the saturable nonlinearity, and then the feedback signal is nonlinearly fed back via the XGM, which is a significant nonlinear phenomenon, in contrast to the linear feedback by optical couplers. As a result of this nonlinear feedback scheme, nonlinear correlations between the input and feedback signals should be generated with a variety of polynomial order combinations, which leads to the high contribution of the nonlinear MCs. This strong nonlinear transformation functionality is a notable feature of the proposed XGM-based configuration.

3.3 Santa-Fe time-series prediction

Finally, we measured and evaluated the RC performance on the Santa-Fe time-series prediction task [6,13,15,24], which is a widely adopted nonlinear benchmark task for RC. In this task, discrete time-series data of a laser output in a chaotic oscillation state are taken as the input signal $u(n )$. The target signal to predict is set to a version of the input itself shifted one step ahead, i.e., $y(n )= u({n + 1} )$. Thus, the task requires one-step-ahead prediction of a chaotic time series, which is believed to require significant nonlinearity to achieve high accuracy [15,24]. The time-series data used here comprised 4,000 samples in total. We allocated 3,000 samples for training (i.e., determining the output weights ${\boldsymbol w}$) and the remaining 1,000 samples for testing (i.e., evaluating the prediction performance). We adopted ridge regression as the training method. The prediction performance was quantified in terms of the normalized mean-square error (NMSE):

Figures 6(a-d) show the testing results obtained under the same reservoir-driving conditions as in Figs. 4(a-d), respectively. The NMSE values are shown in the figures. The best performance was obtained with an injection current of 25 mA (Fig. 6(a)), giving an NMSE as low as 0.112. This value is comparable to those obtained by other experimental nonlinear photonic RC systems, such as an MZM opto-electronic type [6] and an injection-locked laser type [13]. On the other hand, in the 25-mA case without optical feedback (Fig. 6(d)), the NMSE was as high as 0.480: indeed, it is apparent that the prediction signal hardly traced the target signal. This poor performance is a reasonable result, because ideally no past information is available without optical feedback, and the RC system thus has to do its best with only the current information. This clear difference between the results shown in Figs. 6(a) and (d) again confirms that the optical feedback and XGM play an essential role in this reservoir.

 

Fig. 6. Test results for the Santa-Fe time-series prediction task under the same reservoir-driving conditions that were used for Fig. 4, in the same order. The target signals are shown in gray, and the prediction signals are overlain in the same colors as in Fig. 4. The NMSE values are shown in each graph.

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As for the injection current dependence (Figs. 6(a-c)), the performance gradually deteriorated as the injection current decreased. We can consistently explain this trend as having the same reason as for the IPC results: a larger injection current provides a larger feedback power and stronger XGM, thus providing more abundant past information that helps the prediction in higher accuracy. Because the IPC is a generalized measure of processing performance, the correspondence in the injection current dependence between the TMC and the Santa-Fe task performance is reasonable. In the cases of 20 and 15 mA, the performance was not as good as in the 25-mA case. However, it was still significantly better than in the case without feedback, which indicates that the reservoir can provide decent processing performance even in those cases.

To summarize, the experimental results demonstrate that the XGM-based reservoir implemented by the membrane SOA chip performs RC-applicable nonlinear operations with a low power consumption on the 10¹-mW scale and a low input optical power of ∼160 µW on average.

Before concluding this paper, we discuss how we could further enhance the processing performance of the proposed XGM-based reservoir. While the XGM-based configuration is uniquely advantageous in terms of its feedback robustness and strong nonlinear transformation, the current TMC value is significantly restricted: according to [35], the TMC value (i.e., 18.60 at 25 mA) can ideally reach the total node number N (i.e., 200) at maximum, which indicates that the reservoir implemented here did not retain past information very efficiently. Given the SOA injection current dependencies discussed above, it is highly likely that the TMC can be increased by boosting the feedback power to induce stronger XGM.

One simple way to boost the feedback power would be to reduce the excess optical losses: the delay loop illustrated in Fig. 1(a) includes a series of optical losses by the CIR, the 10-dB CPL, the PC, and the CIR again, which accumulate to a total of about 3 dB. Otherwise, a more active approach would be to insert another SOA into the delay loop to pre-amplify the feedback signal. Since an SOA with low-power nonlinearity cannot exhibit a large output power due to the gain saturation, a single SOA alone cannot simultaneously achieve both the low-power XGM-based operation and the large-power feedback by large-gain amplification. Thus, insertion of another SOA for an amplifying purpose would be effective. Even if this were done, the feedback robustness would be maintained, because the pre-amplifying SOA's output would not circulate in the delay loop and thus would not form an oscillator. For such an amplifying purpose, the optical confinement of the SOA should be set adequately weak to mitigate the saturation (i.e., nonlinearity) and achieve the large output power. Membrane SOAs on Si can easily realize such an intentional reduction in the optical confinement by control of the Si core width [31,39]. This feature would enable us to integrate an XGM-exhibiting SOA with strong confinement and a pre-amplifying SOA with weak confinement on a same chip. Boosting of the feedback power by these means could improve the memory and enhance the processing performance of the XGM-based reservoir.

4. Conclusion

In this study, we have demonstrated XGM-based RC by using a low-power-consumption membrane SOA on Si. We show that the membrane SOA works as a nonlinear amplifier and enables low-power nonlinear operation of the reservoir with 10¹-mW-scale power consumption and 10²-µW-scale input optical power. This power consumption is about an order of magnitude lower than that of conventional SOAs exhibiting saturable nonlinearity. Our proposed implementation uses XGM to facilitate a delay-based reservoir, where past information is nonlinearly fed back via XGM. This XGM-based configuration provides the feedback circuit's robustness because of its phase insensitivity and elimination of loop oscillation risk.

To demonstrate RC operation, the impulse response, IPC, and performance on the Santa-Fe time-series prediction task have been systematically measured and evaluated. The IPC evaluation reveals that the XGM-based reservoir performs strong nonlinear transformation of input time-series signals. The series of results are reasonable and consistent throughout, thus clearly showing that the membrane SOA performs RC-applicable nonlinear operations through XGM at a low power scale. The results demonstrate the significant potential of the membrane SOAs as a low-power-scale, all-optical, nonlinear operator. They also suggest new ways to effectively apply XGM to photonic RC.

In the future, we expect that membrane SOAs will be used not only for delay-based configurations but also for spatial parallelization on SiPh chips. The SOAs used in this study are heterogeneously integrated with Si waveguides and are thus fully compatible with linear reservoir circuits implemented via SiPh. This feature opens up the possibility of using these SOAs as on-chip, densely integrable, all-optical, nonlinear operators to complement SiPh linear circuits. There, the membrane SOAs would be expected to play a crucial role in enabling high-performance processing while maintaining the required power cost low. Toward the development of such advanced reservoir circuits, the RC architecture, device design, and device fabrication process will have to be further explored in the future.