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Abstract
Time-delay reservoir computing uses a nonlinear node associated with a feedback loop to construct a large number of virtual neurons in the neural network. The clock cycle of the computing network is usually synchronous with the delay time of the feedback loop, which substantially constrains the flexibility of hardware implementations. This work shows an asynchronous reservoir computing network based on a semiconductor laser with an optical feedback loop, where the clock cycle (20 ns) is considerably different to the delay time (77 ns). The performance of this asynchronous network is experimentally investigated under various operation conditions. It is proved that the asynchronous reservoir computing shows highly competitive performance on the prediction task of Santa Fe chaotic time series, in comparison with the synchronous counterparts.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Recurrent neural networks (RNNs) have memory effect owing to the feedback connections between neurons. In comparison with forward neural network, RNNs are good at processing time-dependent or sequential data, such as speech recognition, natural language processing, and video games. Long-short term memory and gated recurrent unit are two representative types of RNNs. However, training RNNs requires high computational cost, and is even impossible due to the vanishing or exploding gradient problem [1]. In order to reduce the training cost of common RNNs, Jaeger and Hass proposed the echo state neural network [2], while Maass et al. independently proposed the liquid state machine [3] at the beginning of 2000s. Both architectures are now classified into the framework of reservoir computing (RC) [4]. In comparison with common RNNs, RCs only require training the weights at the output layer, while the weights at the input layer and at the hidden layers are left untrained [5]. Owing to the lightweight architecture, RCs become very popular for various hardware implementations including memristors, spintronics, quantum devices, etc [6–9]. In the field of optics, Vandoorne et al. demonstrated a passive RC on silicon based on the combination of optical splitters and couplers [10]. Nakajima et al. showed an on-chip RC based on a photonic linear processer consisting of delay lines and coherent cavities [11]. In addition, the concepts of optical RCs based on semiconductor optical amplifiers and micro-rings have been theoretically proposed [12,13]. However, it remains challenging to achieve a large-scale RC consisting of a huge number of hardware neurons.
In order to simplify the hardware requirement in large-scale RCs, Appeltant et al. proposed the architecture of time-delay reservoir computing (TDRC), which consists of a single nonlinear node and a feedback loop [14]. When the nonlinear system is operated in the transient state, a large amount of virtual neurons are produced, which play similar role as physical neurons [15]. The interval between virtual neurons θ is defined by a random mask, which is multiplied with the input signal before injecting into the reservoir. In order to maintain the transient state, the interval is set to be smaller enough than the characteristic time of the system. The total number of virtual neurons N is determined by the clock cycle Tc of the input signal divided by the neuron interval (N = Tc/θ). In most cases, the clock cycle is synchronous with the delay time τd of the feedback loop (Tc=τd) [15]. However, a slight mismatch between the clock cycle and the delay time (τd =Tc +θ) has been found to be helpful to enrich the dynamics of the reservoir [16]. Photonic TDRCs are usually implemented by using a semiconductor laser with an optical feedback loop [17] or by using an optical modulator with an optoelectronic feedback loop [16,18]. In comparison with the optoelectronic scheme, the first all-optical scheme does not require optical-to-electrical conversion and radio-frequency amplification in the feedback loop, and hence is more power efficient [19]. In addition, parallel TDRCs can be achieved by taking advantage of the multiple multiplexing dimensions of semiconductor lasers, including the wavelength in multimode lasers [20] and the polarization in VCSELs [21,22,23].
It is worthwhile to point out that the above definite relationship between the clock cycle of the input signal and the delay time of the feedback loop significantly constrains the flexibility of the hardware implementation of TDRCs. For a fixed clock cycle, it is challenging to construct a precisely matched or slightly mismatched optical or optoelectronic feedback delay line in practice. Very recently, Hülser et al. theoretically pointed out that TDRCs did not require a predefined relationship between the clock cycle and the delay time [24]. The independent relationship gained a degree of freedom to tune the memory capacity of the delay-coupled oscillators. This work experimentally shows an asynchronous TDRC based on a semiconductor laser with an optical feedback loop, where the clock cycle (20 ns) of the input signal is much different to the delay time (about 77 ns) of the feedback loop. The performance of the asynchronous TDRC is tested under various operation conditions, including the optical injection condition, the optical feedback condition, the pump current of the laser, as well as the modulation format of the input signal. It is proved that the asynchronous TDRC shows competitive performance on the prediction task of Santa Fe chaotic time series, in comparison with the synchronous counterparts reported in literatures. The minimum normalized mean square error of the chaos prediction reaches as small as 0.023.
2. Experimental method
Figure 1 illustrates the experimental setup for the asynchronous TDRC based on a semiconductor laser with an optical feedback loop. The slave laser is a commercial Fabry-Perot laser. The optical feedback loop is provided by an optical circulator in combination with several optical fibers. The optical feedback strength is changed by tuning the optical attenuator. The input data is supplied to the reservoir through the optical injection locking technique [25]. The master laser is a tunable single-mode external cavity laser (Santec TSL-710), which governs the detuning frequency with the slave laser. The laser power is amplified by an EDFA (Amonics AEDFA-23), which tunes the optical injection strength. The input data is multiplied by a random binary mask with {1, 0} [20]. This preprocessed data is sent to an arbitrary waveform generator (AWG, Keysight 8195A, 25 GHz bandwidth) to generate the corresponding voltage sequence. After amplification, the preprocessed signal is superimposed onto the carrier wave of the master laser through a Mach-Zehnder intensity modulator (IM, EOSPACE, 40 GHz bandwidth). The modulated light is injected into the slave laser through the optical circulator. The polarization of the light is aligned with the IM and the slave laser by using two polarization controllers, respectively. The optical spectrum is measured by an optical spectrum analyzer (OSA, Yokogawa, 0.02 nm resolution). The output optical signal is detected by a high-speed photodetector (Finisar, 50 GHz bandwidth). Finally, the states of the virtual neurons are tracked on the digital oscilloscope (Keysight DSAZ594A, 59 GHz bandwidth). The average signal-to-noise ratio (SNR) of the measurements in the experimental setup is around 15 dB. The finite SNR limited reservoir performance can be further improved by using a multi-level mask instead of a binary one [26].
Fig. 1. Experimental setup for the asynchronous TDRC. AWG: arbitrary waveform generator; IM: intensity modulator; OSA: optical spectrum analyzer.
In the experiment, the interval between the virtual neurons of the asynchronous TDRC is fixed at θ=0.05 ns by modulating the IM at 20 GHz. The sampling rate of the AWG is set at 60 GSa/s, and hence three sampling points represent one-bit masked signal. The clock cycle of the input data is set at Tc = 20 ns, corresponding to a data processing speed of 50 MHz. Within each clock cycle, the total number of virtual neurons is N = 400 (N = Tc/θ). The sampling rate of the oscilloscope is set at 80 GSa/s, and thereby we record four sampling points for each neuron duration. The neuron state is obtained by averaging the four sampling points. The delay time of the feedback loop is around τd = 77 ns, which is determined by measuring the external cavity modes induced by optical feedback. In contrast to common synchronous TDRCs, the delay time τd of the asynchronous TDRC in the experimental setup is much larger than the clock cycle Tc.
3. Experimental results
The slave laser exhibits a lasing threshold of Ith = 8 mA at the operation temperature of 20 °C. Throughout the experiment, the laser is biased at 1.88×Ith with an output power of 1.0 mW, unless stated otherwise. The relaxation resonance frequency of the laser is about 4.7 GHz, corresponding to a characteristic time of 0.21 ns. Figure 2(a) shows that the free-running laser emits multiple longitudinal modes around 1543 nm, and the mode spacing is about 155 GHz. When applying optical injection to the laser, only the mode subject to the injection keeps lasing, while other modes are well suppressed with a side mode suppression ratio of about 52 dB. The nonlinear dynamics of the semiconductor laser with optical injection is mainly determined by the detuning frequency and the injection ratio [25]. The detuning frequency is defined as the lasing frequency difference between the master laser and the slave laser. The injection ratio is defined as the power ratio of master laser to the slave laser, which is measured at port 2 of the circulator in Fig. 1. As shown in Fig. 2(b), the stable locking regime is bounded by the Hopf bifurcation and the saddle-node bifurcation. It is shown that the stable locking range broadens with increasing injection ratio, from 6 GHz at the ratio of -10 dB up to 105 GHz at the ratio of 10 dB. The data injection of TDRCs requires that the optical injection is operated within the stable locking regime. Outside the stable regime, the laser becomes unstable and produces nonlinear pulses, including periodic oscillations, aperiodic oscillations, and chaotic oscillations [25]. These pulse oscillations can significantly disturb the data injection from the IM in Fig. 1. For the optical feedback, there is a critical feedback level separating the stable regime and the unstable regime [25]. The critical feedback level of the slave laser (without optical injection) is measured to be about -21 dB. Similar as optical injection, TDRCs are usually operated in the stable regime of optical feedback as well.
Fig. 2. (a) Optical spectra of the slave laser with and without optical injection. (b) Stable locking regime bounded by the Hopf bifurcation and saddle-node bifurcation.
In order to test the performance of the asynchronous TDRC, we employ the benchmark prediction task of the Santa Fe chaotic time series [27]. The aim is to predict the point one-step ahead in the chaos sequence through using the states of virtual neurons triggered by the past points of the sequence. The optimal weights for approaching the target value are obtained through the algorithm of ridge regression [20]. The first 3000 points in the time series are used for the training, while the second 1000 points are used for the test. All the data points are normalized to the range of [0,1] before the multiplication with the random binary mask. The performance of the asynchronous TDRC is quantified by the normalized mean square error (NMSE):
Fig. 3. Santa Fe chaotic time series and the best prediction result using the asynchronous TDRC. The corresponding NMSE is as small as 0.019.
3.1 Effects of optical injection
The dynamics of optical injection is primarily determined by the frequency detuning Δfinj and the injection ratio Rinj. For Rinj = 0 dB, Fig. 4(a) shows the measured performance of the asynchronous TDRC as a function of the detuning frequency. For each detuning frequency, the measurement is repeated by four times, and the error bar is displayed in the figure. It is shown that the NMSE of the chaos prediction generally declines, as the detuning frequency is tuned from the side of the saddle-node bifurcation to the side of the Hopf bifurcation. The NMSE decreases from 0.099 at Δfinj = -26.8 GHz down to 0.063 at Δfinj = -4.1 GHz. Similar behavior has been observed in synchronous TDRCs as well [28,29]. The reason is likely due to the fact that the slave laser becomes less damped when the optical injection is operated in the vicinity of the Hopf bifurcation [30–32]. As a result, the dynamics of the virtual neurons becomes richer when the slave laser is perturbed by the input data. Various neuron states are helpful to improve the performance of TDRCs. Therefore, we always operate the frequency detuning close to the Hopf bifurcation in Fig. 4(b), for all the measured injection strengths. It is clearly shown that the NMSE first declines significantly with increasing injection ratio, from 0.19 at Rinj = -6.0 dB down to 0.032 at Rinj = 3.0 dB. This is because the increased power of the master laser raises the signal-to-noise ratio of the TDRC system. This suggests a moderate strong injection strength is beneficial to improve the performance of the asynchronous TDRC. However, the NMSE saturates around 0.03 for stronger injection strengths (Rinj > 3.0 dB), which is due to the noise limit of the system. The noise of the TDRC system mainly comes from the noise of both the slave laser and the master laser, as well as the noise of the photodetector. The noise sources of the lasers originate from the intrinsic spontaneous emission noise, the intrinsic carrier noise, and the technical noise (including current source noise, temperature fluctuation, and mechanical vibration) [33].
Fig. 4. Effects of (a) frequency detuning and (b) injection ratio on the performance of the asynchronous TDRC. The injection ratio for (a) is fixed at 0 dB. The detuning frequency for (b) is operated in the vicinity of the Hopf bifurcation. The feedback ratio is fixed at -26 dB. The pump current is fixed at 1.88×Ith.
3.2 Effects of optical feedback
The dynamics of optical feedback is mainly determined by the feedback strength, the feedback delay time, as well as the feedback phase. It has been theoretically shown that the performance of TDRCs are sensitive to the feedback phase [34]. In practice, nevertheless, it is challenging to accurately control the feedback phase, because a minor variation of the feedback length on one-wavelength scale results in a phase change of 2×π [35,36]. In addition, the feedback delay time was theoretically proved to affect the memory capacity of the TDRCs [24,37]. Figure 5 focuses on the impact of the feedback strength on the performance of the asynchronous TDRC. The NMSE does not show clear dependence on the feedback ratio, but slightly varies around the average value of 0.028 (dashed line). This tendency suggests the chaos prediction is insensitive to the feedback strength, as long as the optical feedback is operated below the critical feedback level of -21 dB. Similarly, synchronous TDRCs reported in literatures have shown insensitivity to the feedback strength as well [38,39]. When the optical feedback is operated above the critical feedback level, the NMSE of the chaos prediction still has little variation, which is out of expectation and the details refer to Supplement 1.
Fig. 5. Effects of the feedback ratio on the performance of the asynchronous TDRC. The injection ratio is fixed at 7.0 dB. The detuning frequency is near the Hopf bifurcation. The pump current is fixed at 1.88×Ith. The dashed line indicates the average value of 0.028.
3.3 Effects of pump current
Raising the pump current of the slave laser not only increases the optical power, but also reduces the relative intensity noise [40,41]. Therefore, a high pump current is beneficial to improve the signal-to-noise ratio of the TDRC system. As expectation, the NMSE of the chaos prediction in Fig. 6 first declines substantially from 0.125 at 1.16×Ith down to 0.023 at 2.71×Ith. This NMSE of 0.023 is the best performance of the asynchronous TDRC on the prediction of chaos throughout this work. For higher pump currents, the NMSE slightly increases and saturates around 0.03, which can be attributed to the thermal effect of the laser and the noise limit of the TDRC system. Indeed, this saturated value of NMSE is similar to the one of the optical injection effect shown in Fig. 4(b). Consequently, a moderate high pump current of the slave laser is helpful to improve the performance of the asynchronous TDRC on the task of chaos prediction. This conclusion is in agreement with the observation of the synchronous TDRC in [29]. However, this is different to the situation in [17], which exhibited optimal performance when the laser is operated close to the lasing threshold.
Fig. 6. Effects of the pump current on the performance of the asynchronous TDRC. The injection ratio is fixed at 3.0 dB. The detuning frequency is near the Hopf bifurcation. The feedback ratio is fixed at -24 dB.
3.4 Effects of modulation format
Most reports of laser-based TDRCs use the intensity modulation to inject the input signal to the reservoir, which means the injection ratio of the slave laser is modulated during the signal input [17,21,28,29,42,43]. On the other hand, several reports employ the phase modulation to inject the input signal, which modulates the injection phase instead of the injection ratio [38,39]. This section investigates the performance of the asynchronous TDRC using the phase modulation, through replacing the IM in Fig. 1 with a phase modulator. The phase modulator has a bandwidth of 40 GHz (EOSPACE), which is the same as the IM. The modulation depth is fixed at 2.0×π without any optimization. Figure 7(a) shows that the NMSE generally decreases from 0.223 down to 0.069, when the detuning frequency is tuned from the side of saddle-node bifurcation to the side of Hopf bifurcation. In addition, the NMSE in Fig. 7(b) declines from 0.151 for the injection ratio of Rinj = -6.0 dB down to 0.085 for Rinj = 7.0 dB. Therefore, the effect of the optical injection on the asynchronous TDRC with phase modulation is similar to the case with intensity modulation discussed in Fig. 4. Figure 7(c) presents that the optical feedback strength has little impact on the value of NMSE, which slightly varies around the average value of 0.120. This trend is in agreement with the case of the intensity modulation in Fig. 5 as well. As for the effects of pump current, Fig. 7(d) illustrates that the NMSE goes down from 0.200 at 1.16×Ith down to the minimum value of 0.052 at 4.38×Ith. The declining behavior with increasing pump current is also similar to the case of the intensity modulation in Fig. 6. However, the overall performance of the asynchronous TDRC with phase modulation is not as good as the one with intensity modulation. This is likely due to the fact that the modulation depth of the phase modulator is not optimized. A very recent work by Bauwens et al. theoretically reported that the phase modulation had optimal performance for the modulation depths in the range of 0.5×π to 1.0×π [44]. In addition, a TDRC with combined intensity and phase modulation exhibited similar performance as the one with only phase modulation.
Fig. 7. Performance of the asynchronous TDRC with phase modulation. Effects of (a) detuning frequency, (b) injection ratio, (c) feedback ratio, and (d) pump current. The dashed line in (c) indicates the average value of 0.120.
4. Discussion
The above section has proved that the asynchronous TDRC generally exhibits a smaller NMSE on the prediction task of Santa Fe chaos, for the operation conditions of a larger detuning frequency near the Hopf bifurcation, and/or a larger injection ratio, and/or a higher pump current. However, the performance is insensitive to the feedback strength, as long as the feedback is operated in the stable regime (that is, below the critical feedback level). Both intensity modulation and phase modulation of the input signal work well for the asynchronous TDRC. The minimum NMSE for the intensity modulation scheme is 0.023, while it is 0.052 for the phase modulation scheme. Table 1 compares the performance of this asynchronous TDRC with the synchronous ones reported in literatures. It is shown that the NMSEs of the synchronous TDRCs are usually larger than 0.1, except the one using VCSEL with rotated optical feedback. Therefore, the asynchronous TDRC presented in this work exhibits superior performance on the prediction task of Santa Fe chaos.
Table 1. Comparison of the TDRC performances on chaos prediction
For the asynchronous architecture of TDRCs, the clock cycle and the delay time of the feedback loop become independent parameters. It is expected that varying either parameter can affect the performance of the TDRC. For a fixed delay time of 77 ns, Fig. 8 proves that the NMSE of the chaos prediction generally increases with the clock cycle, from 0.046 at Tc = 20 ns up to 0.15 at Tc = 200 ns. This is because increasing the clock cycle also raises the neuron interval (θ=Tc/N), when the neuron number is fixed at N = 400. However, a sufficiently short neuron interval is helpful to enrich the transient dynamics and hence to improve the TDRC performance [44,45]. While it is challenging to continuously tune the feedback delay time in experiment, future work will investigate its effects theoretically.
Fig. 8. Prediction performance of the asynchronous TDRC versus the clock cycle. The delay time is fixed at 77 ns, and the neuron number is fixed at 400.
5. Conclusion
We have presented an asynchronous TDRC based on a semiconductor laser with an optical feedback loop, where the delay time is much longer than the clock cycle of the computing system. It is proved that a moderate high injection strength and/or a detuning frequency in the vicinity of the Hopf bifurcation are helpful to improve the performance of the TDRC. In addition, a moderate large pump current of the slave laser is favorable. However, the TDRC performance is insensitive to the feedback strength, as long as the optical feedback is operated in the stable regime. Both intensity modulation and phase modulation of the input signal are available for the asynchronous TDRC. The optimal NMSE for the prediction task of Santa Fe chaos reaches as small as 0.023, which is strongly competitive against common synchronous ones. These results suggest the TDRCs do not need to constrain the feedback delay time to be identical as the clock cycle of the system. This independent relationship between the two time scales earns high flexibility for hardware implementations. Finally, it is worthwhile to point out that the conclusion of the asynchronous TDRC merits is not limited to the chaos prediction task, but can be generalized to other tasks, such as the spoken digit recognition and the nonlinear channel equalization. This is because the asynchronous scheme improves the memory capacity of the system, which is independent on the testing task.
Funding
Natural Science Foundation of Shanghai (20ZR1436500).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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