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Abstract
Photonic reservoir computing is a new paradigm for performing high-speed prediction and classification tasks in an efficient manner. The major challenge for the miniaturization of photonic reservoir computing is the need for the use of photonic integrated circuits. Herein, we experimentally demonstrate reservoir computing using a photonic integrated circuit with a semiconductor laser and a short external cavity. We propose a method to increase the number of virtual nodes in delayed feedback using short node intervals and outputs from multiple delay times. We perform time-series prediction and nonlinear channel equalization tasks using reservoir computing with the photonic integrated circuit. We show that the photonic integrated circuit with optical feedback outperforms the photonic integrated circuit without optical feedback for prediction tasks. To enhance the memory effect we feed past input signals in the current input data and demonstrate successful performance in an n-step-ahead prediction task.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Artificial intelligence using neural networks has been attracting increasing interest for use in a variety of information processing tasks. Recurrent neural networks consisting of self-feedback nodes have been used for time-dependent information processing, such as time-series prediction and speech recognition. However, recurrent neural networks require complex algorithms for training the connection weights owing to the existence of self-feedback nodes. Reservoir computing is a simplified form of computing using recurrent neural networks in which the connection weights between the input and network nodes are fixed. Furthermore, the weights of the links of the network nodes are also fixed, and only the connection weights between the output and the network nodes are trained using machine learning [1,2]. Reservoir computing has an advantage compared to other computational techniques in that it can be implemented using simple algorithms for calculating the output weights. Correspondingly, it has been successfully demonstrated experimentally and numerically [3–19].
The use of optical and laser devices in reservoir computing achieve processing speeds in the gigahertz range [3]. A scheme of delay-based reservoir computing has already been proposed [4]. In this scheme, a single nonlinear device with a time-delayed feedback loop is used as a reservoir in which the existence of virtual nodes is assumed by sampling the temporal waveform of the output of the nonlinear device in the feedback loop. Time-multiplexing of the network nodes has also been used in delay-based reservoir computing. Delay-based reservoir computing has been implemented experimentally using electronic circuits [4], optoelectronic devices [5–8], nonlinear optical devices [9–13], and laser systems [3,14–19]. In general, delay-based reservoir computing requires a large number of virtual nodes in order to achieve high performance. Long feedback delays are required and achieved with optical fibers lengths of a few meters.
Photonic integrated circuits (PICs) are good candidates for implementing reservoir computing at a millimeter scale. Moreover, reservoir computing has been implemented using integrated passive silicon photonic circuits [12,13]. However, in these implementations, the number of nodes is limited owing to optical losses in the silicon photonic circuit, and nonlinearity has to be introduced in the detection parts. Recently, PICs that generate chaotic outputs have been proposed that consist of a distributed-feedback semiconductor laser, an optical amplifier, a phase modulator, and an external cavity for optical feedback [20]. Different versions of PICs have been fabricated experimentally and have been used for investigating novel nonlinear dynamics in PICs with short external cavities [21–24]. These PICs are robust against external fluctuations and turbulence, and they are practical for applications in secure optical communication [25], random number generation [26–30], and secure key distribution [31].
These PICs with delayed feedback loops could be used for delay-based reservoir computing. However, the number of virtual nodes in the PICs is limited owing to the short external cavity at the order of millimeters. The number of virtual nodes N is dependent on the feedback delay time τ and the node interval θ (e.g., N ≈ τ/θ). The feedback delay time is of the order of hundreds of picoseconds with these PICs, and the node interval is of the order of tens of picoseconds. Therefore, only ~10 virtual nodes can be obtained. The lack of techniques for realizing a large number of virtual nodes makes the use of these PICs difficult for delay-based reservoir computing.
In this study, we propose a method for implementing delay-based reservoir computing using a PIC with a semiconductor laser and a short external cavity. We introduce short node intervals and multiple delay times to obtain a large number of virtual nodes. We experimentally demonstrate a time-series prediction task and evaluate the prediction errors for different conditions of laser parameter values. We evaluate the memory capacity of the reservoir, and introduce a method for enhancing the performance of the time-series prediction task using the past input signals. Finally, we also use a nonlinear channel equalization task as a different measure of evaluation.
2. Reservoir computing scheme with a photonic integrated circuit
2.1 Experimental setup
We implement delay-based photonic reservoir computing with a PIC, as shown in Fig. 1(a). The scheme consists of three stages: the input layer, the reservoir, and the output layer. In the input layer, an analog input signal (time series) is expanded for a time duration T, and a temporal mask signal with a length T is applied to each input signal. The expanded input signal is multiplied with the mask signal and used as the modulation signal (see [4] for details). A binary mask signal with a scaling factor of γ = 1.0 is used in this experiment [17].
Fig. 1 (a) Experimental setup for reservoir computing using a photonic integrated circuit. (b) Structure of photonic integrated circuit. Amp, electric amplifier; ATT, optical attenuator; DFB laser, distributed-feedback semiconductor laser; FC, fiber coupler; ISO, optical isolator; LD, semiconductor laser diode; PD, photodetector; PIC, photonic integrated circuit; PM, phase modulator; SOA, semiconductor optical amplifier.
We used a semiconductor laser (NTT Electronics, NLK1C5GAAA) to feed the modulation signal into the reservoir, as shown in Fig. 1(a). The injection current of the semiconductor laser was set to 30.0 mA (2.83 Ith, where Ith is the lasing threshold), and had a relaxation oscillation frequency of 5.6 GHz. The phase of the semiconductor laser was modulated by a phase modulator (EO Space, AX–0MSS–20–LV, 20 GHz bandwidth) with the modulation signal. The modulation signal was generated using an arbitrary function generator (Tektronix, AWG70002A, 25 Giga samples/s). We used a PIC with time-delayed optical feedback (delay time τ) as a reservoir, and the phase-modulated laser signal was injected into the PIC. The temporal waveforms of the PIC were detected using a photodetector (New Port, 1554–B, 12 GHz bandwidth) and amplified by an electric amplifier (New Port, 1422–LF, 20 GHz bandwidth). This electronic signal was sampled using a digital oscilloscope (Tektronix, DPO72304DX, 100 Giga sample/s, 23 GHz bandwidth) at N nodes at every node interval θ, in accordance to the condition τ ≈ Nθ. The sampled outputs were treated as virtual node states, and were used to calculate the output signal. The output signal was obtained from the sum of the weighted values of all the virtual node states. The learning of the weight values was carried out using the linear least-squares method with the use of training data.
The structure of the PIC is shown in Fig. 1(b). The PIC consisted of a distributed-feedback (DFB) semiconductor laser, a semiconductor optical amplifier (SOA), a phase modulator (PM), a passive waveguide, and an external mirror for optical feedback [31]. The lengths of the DFB laser, SOA, and PM were 0.3, 0.44, and 0.3 mm, respectively. The structures of the DFB laser, SOA, and PM were based on ridge-waveguide type structures. The optical output port was placed on the left side of the DFB laser and was coated with anti-reflection (AR). It was connected to an optical fiber through a lens that avoided the implementation of a photodetector on the chip [27–30]. This configuration allowed the external detection of light that was generated by the PIC, and facilitated the optical injection into the DFB laser. The modulation signal was injected into the optical output port on the left side of the DFB laser. The output signal from the DFB laser (right side) was reflected with the use of an external mirror. The mirror was constructed with high-reflection (HR) coating on the right edge of the PIC. The external cavity length between the right facet of the DFB laser and the external mirror was 10.6 mm. This length corresponded to a round-trip delay time of τ = 254 ps in the external cavity, and the external cavity frequency was 3.9 GHz (The refractive index was 3.6). The strength of the optical feedback light was adjusted by the SOA injection current.
We set the injection current of the DFB lasers in the PIC at the lasing threshold Ith of 13.3 mA. At this condition, the relaxation oscillation frequency of the DFB laser in the PIC was 1.0 GHz. The PICs operated in the so-called short cavity regime, where the external cavity frequency was higher than the relaxation oscillation frequency [32]. The wavelengths of the semiconductor laser for signal injection and the DFB laser in the PIC were set by using a temperature controller to 1553.502 and 1553 462 nm, respectively. The injection power from the semiconductor laser to the DFB laser in the PIC was 62.3 μW. Injection locking was achieved between the two lasers at this experimental condition.
2.2 Method for reservoir computing with a photonic integrated circuit
It is difficult to directly apply the previous method [3,4] for defining the virtual nodes for the PIC, because the delay time of the feedback loop was very small and of the order of hundreds of picoseconds. For example, only six virtual nodes can be obtained from the PIC with a feedback delay time of τ = 254 ps when sampled at 40 ps, which is the minimum sampling time of an arbitrary waveform generator used for the input signal modulation in our experiment (i.e., corresponds to a sampling frequency of 25 Giga samples/s). Some techniques have been proposed to allow changes in the number of virtual nodes by dividing the feedback loop [33] and by adding another feedback loop [34].
We propose herein two methods to increase the number of virtual nodes. The first method is the reduction of the node interval. In the previous method of delay-based reservoir computing, the mask interval θM and the node interval θ were matched [3,4]. However, the node interval θ can be decreased because an analog temporally varying waveform with continuous variation is used to represent the virtual node states. The fundamental oscillation period of laser dynamics is approximately 100 ps (10 GHz in frequency) and different node states can be obtained during the mask interval θM = 40 ps. The mask interval is limited to 40 ps in our experiment owing to the sampling time of the arbitrary waveform generator. However, the node interval can be reduced to 10 ps by using the digital oscilloscope with the minimum sampling time of 10 ps (i.e., at the sampling frequency of 100 Giga samples/s).
The second method uses the signal output based on the use of multiple delay times. Specifically, a temporally varying waveform with multiple delay times kτ (k is a positive integer) of the feedback loop can be used as the set of virtual node states for a single input signal. We do not change the configuration of the PIC with a single delay loop with τ, and we construct a virtual network from a temporal waveform with the length of kτ. We use the same input signal for the mask length of T = kτ, and the mask pattern changes within the period of T with piecewise constant over the mask interval θM (random binary mask consisting of a sequence {−1, 1}). The number of virtual nodes N can be increased with increases in k (see Eq. (1) in Sec. 3.1).
3. Experimental results
3.1 Time-series prediction task
We used the Santa Fe time-series prediction task [35] to evaluate the performance of reservoir computing. We performed single-point prediction of chaotic data generated from a far-infrared laser. In this task, the input signal corresponded to the chaotic waveform at the i-th sampling point, and reservoir computing aimed to the prediction of the chaotic data at the (i + 1)-th sampling point. We used 3000 steps for training and 1000 steps for testing.
We use the output of multiple delay times in the feedback loop to increase the number of virtual nodes. We define the total number of virtual nodes N as follows.
where (int) indicates the conversion to an integer. The multiple delay times kτ for a single input is divided by the mask interval θM and is rounded down to calculate the number of mask patterns used for a single input signal. This integer is multiplied by the ratio of the mask interval θM to the node interval θ, and is rounded down to obtain the total number of virtual nodes N.Figures 2(a) and 2(b) shows the results of the time-series prediction task when the node interval is changed. The node intervals θ are set to 40 and 10 ps, respectively, and a single delay time is used for virtual nodes (k = 1). These parameter settings correspond to the number of virtual nodes of N = 6 and 24, respectively, and are obtained from Eq. (1). Prediction is not successful with a node interval of 40 ps [Fig. 2(a)], and the normalized mean-square error (NMSE) [3,4] is 0.423. When the node interval is decreased to 10 ps [Fig. 2(b)], the NMSE decreases to 0.320, and the prediction error improves slightly. However, large errors still exist in the predicted reservoir output.
Fig. 2 (a) Time-series prediction task for different node intervals θ and different number of delay times k used as virtual nodes. (a) θ = 40 ps, k = 1, (b) θ = 10 ps, k = 1, (c) θ = 40 ps, k = 5, and (d) θ = 10 ps, k = 5. The numbers of the virtual nodes are (a) N = 6, (b) N = 24, (c) N = 31, and (d) N = 124, respectively. The mask interval is set to θM = 40 ps.
Figures 2(c) and 2(d) show the results of the time-series prediction task when the outputs of five delay times (k = 5) are used as virtual nodes for node intervals of θ = 40 and 10 ps. The numbers of virtual nodes are N = 31 and 124 for Figs. 2(c) and 2(d), respectively, as obtained from Eq. (1). A significant improvement in the prediction errors is exhibited in Figs. 2(c) and 2(d). In addition, a small NMSE value that equals 0.086 is obtained for the shorter node interval of 10 ps and the multiple delay times of 5τ [Fig. 2(d)]. Thus, the reservoir output closely matched the original signal.
We investigated the prediction errors when the number of delay times used for the virtual nodes is changed. Figure 3(a) shows the prediction error when the number of delay times k is changed. The prediction error decreases with increases in k, and saturates at k = 5. The increase in k is an effective method for improving the prediction error. We also use different node intervals at θ = 10, 20, and 40 ps that yield similar curves. Smaller errors are elicited for smaller node intervals. The best NMSE of 0.109 is obtained with k = 5 and θ = 10 ps. This result is comparable to the results in the literature (e.g., the NMSE of 0.106 in [3]). We can further improve the prediction error by increasing the signal-to-noise ratio (SNR). In this experiment, an SNR of ~10 dB is obtained. We speculate that this low SNR may result from the low-injection current of the DFB laser in the PIC. The injection current of the DFB laser is set at the lasing threshold to obtain consistent outcomes with the reservoir output. The amplitude of the response laser is not large enough and detection noise is included in the laser output.
Fig. 3 Prediction errors (NMSEs) as a function of (a) the number of the delay times k and (b) the number of virtual nodes N. The node intervals θ are set to 10 ps (black circles), 20 ps (red squares), and 40 ps (blue diamonds). The mean NMSE values for ten different repetitions of the experiments are plotted, and the maximum and minimum values of the ten repetitions are shown as error bars. The injection power is 62.3 μW.
In addition, we compared NMSEs for the number of virtual nodes N, as shown in Fig. 3(b). Larger node intervals result in smaller NMSEs, even for the same number of nodes. We speculate that this characteristic is a result of the correlations among the virtual nodes that are being reduced to generate larger node intervals. Furthermore, a variety of node states can be obtained for larger node intervals. This result indicates that for improving the prediction errors, increasing the interval between two adjacent virtual nodes is effective if the number of virtual nodes does not change.
Figure 3(a) shows that a smaller value of θ is better when the number of delay times k is fixed. This result suggests that a smaller value of θ is better because it allows the implementation of more nodes with a small value of k. Additionally, Fig. 3(b) shows that a larger value of θ is better when the number of nodes N is fixed. A larger value of θ is more appropriate to reduce correlations between adjacent virtual nodes and to obtain the variability of node states.
We investigated the prediction errors of PICs both with and without optical feedback. Figure 4 shows the comparison of the variations of NMSEs with and without optical feedback in the PIC for different SOA injection currents as a function of the number of delay times k. The node interval is set to θ = 10 ps. Smaller NMSE values are obtained in the presence of optical feedback for most of the cases associated with different k, as indicated in Fig. 4. We consider that the optical feedback can enhance the memory effect (the ability to hold past results in the reservoir), and it plays a crucial role in improving the errors of the time-series prediction task. Saturation in the values of NMSEs occurs at k = 5 with optical feedback, as shown in Fig. 4. We interpret that the number of nodes is large enough to perform the time-series prediction task for k ≥ 5.
Fig. 4 NMSEs with (red curve) and without (black curve) optical feedback in the PIC as a function of the delay times k. The SOA injection currents with and without the optical feedback are respectively set to 4 and 0 mA. The mean NMSE values for ten different repetitions of the experiments are plotted, and the maximum and minimum values of the ten repetitions are shown as error bars.
Next, we investigated the parameter dependence of reservoir computing with the PIC. We continually changed the feedback strength of the delay loop in the PIC (i.e., the injection current of the SOA) and measured the NMSE for the time-series prediction task. Figure 5(a) shows the NMSE as the feedback strength is changed. The NMSE decreases slightly with an increase in feedback strength, and the minimum NMSE (0.098) is observed for an SOA injection current of 4.0 mA. Subsequently the NMSE increases further. We also investigated the consistency of the PIC outputs with respect to the modulation signal, including the input data. Consistency can be measured using the cross-correlation of the two temporal waveforms of the laser output in the PIC, driven by the same input signal [32,36]. Small NMSEs areobtained for high values of cross-correlation. Therefore, the condition of consistency corresponds to the parameter regions where small prediction errors are obtained. The correlation value decreases for SOA currents larger than 5 mA, and the NMSE starts to increase. In fact, this region corresponds to the appearance of chaotic outputs in the reservoir laser without optical injection. Therefore, the lowest NMSE is obtained before the appearance of chaotic oscillations of the reservoir laser (known as the edge of chaos).
Fig. 5 (a) Variations of NMSE values (black curve) and cross-correlation (red curve) as a function of SOA injection current (feedback strength). (b) Variations of NMSE values as a function of normalized injection current of the PIC with (red curve) and without (black curve) optical feedback. The SOA injection currents with and without the optical feedback are respectively set to 4 and 0 mA. The number of delay times is set to k = 5. The mean NMSE values for ten different repetitions of the experiments are plotted, and the maximum and minimum values of the ten repetitions are shown as error bars.
We also changed the injection current of the PIC. Figure 5(b) shows the NMSEs when the injection current of the PIC is changed, with and without optical feedback. The minimum NMSE values are obtained at 0.98Ith and 1.06Ith with and without optical feedback, respectively. The injection current needs to be close to the lasing threshold (1.00Ith) to reduce prediction errors [3]. It is also found that smaller NMSE values are observed in the case of the PIC in the presence of optical feedback.
3.2 Memory capacity
We investigated the memory capacity of the PIC with optical feedback. The memory function m(i) has been introduced as follows [37,38].
Herein, y(n) is a random input signal in the range [-1, 1], oi(n) is the reservoir output at time n when the output weights are trained with the i-th past input signal y(n-i), σ2 is the variance, and < > denotes the time average. The memory capacity MC has been introduced as the sum of m(i) as follows [37,38].The memory capacity MC quantifies how much information of the past input signals can be reproduced by reservoir computing.Figure 6(a) shows the memory capacity for different numbers of delay times k, with and without optical feedback. The memory capacity changes slightly as k increases, and it ranges from 0 to 1.5. Note that the memory capacity for the PIC with optical feedback is larger than the case without optical feedback. Therefore, optical feedback is important for maintaining the memory of past information. However, the memory capacity is approximately equal to one. This is much smaller than the memory capacities of previous reservoir computing systems that used a semiconductor laser with a long external cavity (a few meter length), where the elicited memory capacities were in the range of 2–8 [18]. We consider that this low-memory capacity of the PIC reservoir may be attributed to the low SNR of the reservoir outputs in our experiment. We also speculate that this low memory capacity may result from the state of full injection locking between the semiconductor laser for input signal injection and the DFB laser in the PIC because the memory capacity can be enhanced with the use of a partial injection-locking state [18].
Fig. 6 (a) Memory capacity MC as a function of the number of delay times k with (red curve) and without (black curve) optical feedback. The SOA injection currents are respectively set to 4 and 0 mA with and without the optical feedback. (b) Memory capacity as a function of the SOA injection current (the feedback strength) in the cases where three (black curve), four (red), and five (blue) delay times are used.
Figure 6(b) shows the memory capacity as the SOA injection current (feedback strength) of the delayed feedback loop is changed for different k. The memory capacity is almost constant (~0.4) for SOA currents in the range of 0–2.5 mA, and it increases monotonically up to ~1 for SOA currents in the range of 2.5–5 mA. However, it starts decreasing for currents exceeding 5 mA, because of the lack of consistent outputs in the PIC reservoir.
3.3 Use of past input data for n-step-ahead prediction task
In this section, we propose a method for enhancing the memory effect in the PIC reservoir for the time-series prediction task. We introduce past input signals to correct the lack of memory. We use a weighted sum of previous input signals to denote the current input signal U(t) as follows,
where gi is the weight for the i-th past input signal of u(n-i), and mi(t) is the mask signal for u(n-i). Additionally, u(n) is defined using the same expression listed in [6], i.e., u(n) is an input signal evolving in discrete time n and u(t) is a piecewise constant function of continuous time t: u(t) = u(n), nT ≤ t ≤ (n + 1)T, where T = kτ is the mask length. The input mask mi(t) = mi(t + T) is a periodic function of period T, and is piecewise constant over the mask interval θM, i.e., mi(t) = mj when nT + jθM ≤ t ≤ nT + (j + 1)θM for j = 0, 1, …, M-1, and M = T/θM. The values of mj are randomly chosen from −1 or 1. We construct the current input signal U(t) so that the effect of the past input signal u(n-i) decreases linearly with an increase in i, thus indicating the fading past input signals. We change the number of past inputs P and evaluate the NMSE of the time-series prediction task. A different scheme has been introduced to assist the lack of the memory effect in the network in the framework of extreme learning machines, in which several past input signals are sent in parallel to the neural network without internal feedback [37].The number of delay times for the virtual nodes is set to k = 5 and the node interval is set to θ = 10 ps in the following experiments. Figure 7 shows the NMSE values as a function of P with and without optical feedback. The NMSE decreases as P increases in the case of the PIC without optical feedback (the black curve in Fig. 7), and the minimum value is obtained with P = 8. However, the NMSE increases slightly as P increases in the case of the PIC with optical feedback (the red curve in Fig. 7). This result indicates that past input signals are more effective at improving the NMSEs for a PIC without optical feedback.
Fig. 7 NMSEs as a function of the number of past inputs with (red curve) and without (black curve) optical feedback. The SOA injection currents with and without optical feedback are respectively set to 4 and 0 mA. The number of delay times is set to k = 5. The node interval is set to θ = 10 ps.
We consider a more difficult task with n-step-ahead prediction (n > 1) with past input signals instead of the single-step-ahead prediction considered in the previous results. Figure 8 shows the NMSEs as a function of P for the n-step-ahead prediction task using PICs with and without optical feedback. In the case where there is no optical feedback [Fig. 8(a)], the prediction errors decrease as P increases for n-step-ahead prediction. In fact, the NMSEs start to decrease for P values in the range of 0 to 8, and saturate for larger values of P. As n increases, there is no significant change in the values of NMSEs, and large NMSEs are obtained [curve plotted in blue color in Fig. 8(a)]. In the case where optical feedback is used [Fig. 8(b)] similar characteristics are found, however, smaller NMSEs are obtained for small values of n and P. This result indicates that optical feedback strongly affects NMSEs when both n and P are small. While the prediction task becomes more difficult for larger values of n, the inclusion of several past input data improves the NMSEs, and the effect of optical feedback is not significant.
Fig. 8 NMSEs as a function of the number of past input signals P (a) without and (b) with optical feedback for different prediction steps n. The SOA injection currents in the cases with and without the optical feedback are respectively set to 4 and 0 mA. The prediction steps include n = 1 (black circles), n = 2 (red squares), n = 3 (green diamonds), and n = 7 (blue triangles).
We also continually changed the prediction step n for different numbers of the past inputs P and measured the NMSEs. Figure 9 shows the NMSEs as a function of n for different values of P with and without optical feedback. It is worth noting that the NMSE values fluctuate periodically in Figs. 9(a) and 9(b). This periodic oscillation becomes small when P is large, and the prediction error increases as n increases. In fact, this periodic curve corresponds to the auto-correlation function of the original chaotic input data, as shown in Fig. 10. Larger positive correlations in Fig. 10 correspond to smaller NMSEs in Figs. 9(a) and 9(b). It is found that n-step-ahead prediction with a large value of n is more difficult because of the sensitive dependence on the initial conditions of chaotic input data. However, we succeeded in performing ten-step predictions with an NMSE less than 0.4. The inclusion of past input signals is effective in assisting the memory effect of the PIC with a short external cavity, and n-step-ahead prediction can thus be achieved.
Fig. 9 NMSEs as a function of the prediction step n (a) without and (b) with optical feedback for different past input signals P. The SOA injection currents with and without the optical feedback are respectively set to 4 and 0 mA. The numbers of past input signals include P = 0 (black circles), P = 2 (red squares), P = 6 (green diamonds), and P = 15 (blue triangles).
Fig. 10 Auto-correlation function of the Santa Fe time series of the laser chaos [35] used for the prediction task.
3.4 Nonlinear channel equalization task
To test the effectiveness of our PIC for reservoir computing, we used different types of tasks, e.g., the nonlinear channel equalization task [6] for classification. This task requires the classification of four digital signals {-3, −1, 1, 3} transmitted through a communication channel with nonlinear distortion. The nonlinear transformation of the communication channel is described by the following model equations [6].
Herein, d(n) is the input signal consisting of a random sequence with values from {−3, −1, + 1, + 3}, q(n) is the linear channel output, u(n) is the noisy nonlinear channel output, and v(n) is the white Gaussian noise with a zero mean adjusted in power to yield SNRs that range from 11 to 31 dB. The term u(n) is used to determine d(n) in this classification task for reservoir computing.We set the same parameter values used for the time series prediction task, except for the scaling factor of the input signal (γ = 0.4) [17]. In particular, five delay times (k = 5) and a node interval of θ = 10 ps are used for the PIC reservoir. Figure 11(a) shows the signal error rate (SER) as a function of the SNR of the nonlinear channel signal. The SER decreases monotonically as the SNR increases. The best SER of ~0.03 is obtained in Fig. 11(a), and is worse compared to previous reports [6]. We speculate that this SER may be attributed to the low SNR of the reservoir outputs in our experiment. In addition, there is no significant difference between the PICs in the cases with and without optical feedback. This indicates that the lack of memory is not the limiting factor for the system’s performance in this classification task.
Fig. 11 Results of nonlinear channel equalization task. (a) Signal error rate (SER) as a function of the signal-to-noise ratio (SNR) of the nonlinear channel signal. (b) SER as a function of the SOA injection current (the feedback strength) for different past input signals. The numbers of past input signals are P = 0 (black), P = 1 (red), and P = 7 (blue).
Figure 11(b) shows the SER as a function of the optical feedback strength of the PIC (see the curve plotted in black). The consistency region corresponds to SOA injection currents in the range of 0–5.2 mA. We found that the SER decreases slightly with increases in optical feedback strength, and the minimum SER of 0.022 is obtained at 4.5 mA. We also plotted the cases when one and seven past input signals are included (P = 1 and 7) in the input modulation signal (see the red and blue curves in Fig. 11(b)). The SER improved when only one past input signal is used (e.g., the minimum SER of 0.012 at 4.75 mA). However, it becomes worse when seven past input signals are included for SOA currents larger than 3 mA. Therefore, a finite memory effect is required for the nonlinear channel equalization task.
4. Conclusions
We experimentally demonstrated delay-based reservoir computing using a PIC with a semiconductor laser and a short external cavity. We increased the number of virtual nodes in the delayed feedback by using the short node interval and outputs in multiple delay times. We performed time-series prediction tasks and evaluated the prediction errors. We found that the PIC with optical feedback outperformed the PIC without optical feedback for the prediction task owing to the presence of past memory. To enhance the memory effect, we included several past input data in the modulation signal. When additional past data were used, there was a smaller difference between the PICs in the cases with and without optical feedback. We succeeded in performing an n-step-ahead prediction task by using several past input data. We also demonstrated the nonlinear channel equalization task with the PIC reservoir.
We confirmed that PICs with lasers and feedback loops have a promising potential for miniaturizing delay-based reservoir computing. The lack of memory capacity can be compensated for including past input signals to the current input data. Moreover, parallel implementation of a large number of PICs on a single chip could be possible to further enhance the performance of reservoir computing. In this scheme, we could measure multiple temporal waveforms from multiple PICs with optical feedback, and the concept of delay-based reservoir computing can be applied to each temporal waveform. The total number of nodes can be determined by the number of PICs and the number of virtual nodes in the delay loop for each PIC. This hybrid scheme using the spatiotemporal approach of reservoir computing could be very promising for realizing reservoir computing that solves more difficult tasks in more efficient manners [39]. Compact reservoir computing with PICs could become a new paradigm of photonic artificial intelligence.
Funding
Grants-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (JSPS KAKENHI Grant Number JP16H03878), and JST CREST Grant Number JPMJCR17N2, Japan.
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