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Abstract
The minimum requirements for an optical reservoir computer, a recent paradigm for computation using simple algorithms, are nonlinearity and internal interactions. A promising optical system satisfying these requirements is a platform based on coupled degenerate optical parametric oscillators (DOPOs) in a fiber ring cavity. We can expect advantages using DOPOs for reservoir computing with respect to scalability and reduction of excess noise; however, the continuous stabilization required for reservoir computing has not yet been demonstrated. Here, we report the continuous and long-term stabilization of an optical system by introducing periodical phase modulation patterns for DOPOs and a local oscillator. We observed that the Allan variance of the optical phase up to 100 ms was suppressed and that the homodyne measurement signal had a relative standard deviation of 1.4% over 62,500 round trips. The proposed methods represent important technical bases for realizing stable computation on large-scale optical hybrid computers.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The physical reservoir computer is a recent paradigm for computation on a physical system, which originally used a randomly connected recurrent neural network [1,2]. When we feed an input signal to a physical system, the system outputs complex responses via internal nonlinear interactions. The relationship between the input and response signals can be considered as fixed but unknown information processing. The output of the physical reservoir computer is obtained as a linear combination of these complex nonlinear responses, where only the output weights are optimized to perform a computation. Thanks to this simple learning rule, physical reservoir computers have been realized in a wide range of physical systems [3–6]. Optics is the largest emerging field seeking the development of physical reservoir computers, which have been demonstrated using all-optical systems [7–10] as well as hybrid ones with analog electrical circuits [11,12] and digital circuits [13,14]. Owing to the simple learning rule, an optical reservoir computer can, for example, predict an output value of a chaotic laser, equalize a nonlinear wireless communications channel, and classify spoken digits even with a low learning cost. An important problem is the limitation on the number of nodes in the physical reservoir layer—with a larger number of nodes, a wider variety of functions can be expressed. Another problem is excess noise in the physical system, which directly affects the accuracy of the computation results.
Recently, a coherent Ising machine (CIM) using degenerate optical parametric oscillators (DOPOs) has been attracting research interest as a different type of optical computers [15–19]. It has been demonstrated that a CIM can solve combinatorial optimization problems on a large graph within a short computation time [15,16] and simulate the Ising model at low temperatures [17,18]. Although a CIM was developed for simulating the Ising model, its optical system is similar to that of an optical reservoir computer using a time-delay system [20], where a nonlinear device is implemented in a ring structure and interactions occur between temporally separated nodes inside the ring. Therefore, we can use complex DOPO responses against external injection signals for reservoir computing. Using an optical system of a CIM has several advantages for reservoir computing. Time multiplexing of the nodes and a measurement-feedback scheme for a CIM can improve the scalability of the reservoir nodes. These approaches have used to achieve the interaction of 2,000 nodes arbitrary [15,19] and generate more than one million independent nodes [21]. Moreover, optimizing the interaction weights in the measurement-feedback scheme enables us to extend optical reservoir computers to a more general recurrent neural network. In addition, we can reduce the excess noise in the physical reservoir layer thanks to the ultimately low noise figure of optical phase sensitive amplification (PSA) [22].
In constructing a large and low-noise optical reservoir computer using an optical system consisting of DOPOs, stabilization of the optical phases is important because the system consists of phase-sensitive components, including a long fiber ring cavity, a phase-sensitive amplifier, a homodyne detector, and a coherent feedback injection part. Furthermore, a reservoir computer inherently requires continuous stability because reservoir computing is a sequential computation for many tasks. For instance, in the case of prediction of a chaotic laser output, the task is to predict its future output from past outputs, and the computation can continue as long as the chaotic laser outputs a new signal. Because it is necessary to evaluate the accuracy of the computation results within a finite time, it has been evaluated by performing over several thousand computation steps in typical experiments on physical reservoir computers. To evaluate the computation performance as a reservoir computer, we therefore need to stabilize an optical system with over 10,000 steps, which corresponds to 50 ms in the optical system of a CIM. In addition, a long computation time has the benefit of improving the computation accuracy for a CIM when we increase the size of combinatorial optimization problems [19].
Here, toward the development of large-scale optical hybrid computers, we report continuous and long-term phase stabilization of an optical system comprising 4,500 DOPOs using a 1-km fiber ring cavity and homodyne detection. We stabilized the optical system by introducing periodic modulation patterns for the DOPO pulses and a local oscillator for homodyne detection. We evaluated the stability of the optical phase using the power spectrum density and Allan variance [23], which had not been observable with simple homodyne detection. From these measurements, we confirmed the long-term stability of the DOPOs up to 100 ms, which is required to evaluate a sequential computation of a physical reservoir computer. The stabilized homodyne detection signal had a relative standard deviation of $\pm$ 1.4% that was not related to the optical phase noise.
2. Stabilization of DOPOs and homodyne detection
First, we describe the optical reservoir computer using DOPOs. Then, we focus on our methods to stabilize the optical system. Finally, we explain the methods to evaluate the phase noise for the proposed stabilization.
2.1 Optical hybrid system using DOPOs for reservoir computing
As shown in Fig. 1(a), a reservoir computer consists of three layers: the input layer, reservoir layer and output layer. In the input layer, an external signal is input into nodes in the reservoir layer using random weights called the input mask. The nodes in the reservoir layer are randomly connected to each other and have a nonlinear response against input signals. By using a linear combination of these nonlinear responses with optimized weights, the reservoir computer can perform a computation for a target task.
Fig. 1. (a) Schematic diagram of a general reservoir computer. (b) Schematic diagram of the reservoir computer using DOPOs and the stabilization methods. The dotted lines outline the equipment used to stabilize the cavity (blue), pump pulses (orange) and local oscillator (green).
Figure 1(b) schematically shows our setup for the reservoir computer and stabilization methods described later. We generate nodes in the reservoir layer as follows. The fiber ring cavity at the center of the setup includes a long optical fiber and a periodically poled LiNbO$_3$ (PPLN) waveguide. By launching pump pulses into the PPLN waveguide, we can generate independent DOPO pulses in time division via spontaneous parametric down conversion and PSA in the following round trips [15,16,21]. Because PSA only amplifies the in-phase amplitude relative to the phase of the pump pulses, the in-phase amplitude dominates the optical power and the quadrature amplitude becomes negligible. Several DOPO pulses are used for computation while the other pulses are used to stabilize the optical system, where the number of computation pulses depends on the size of the optical system. We hereafter refer to the pulses for the stabilization as dummy pulses. Each computation pulse represents a node in the reservoir. The nonlinearity required for the reservoir computer is introduced by the saturation of the output amplitude from the phase-sensitive amplifier, whose implementation is similar to that in the optical reservoir computer using a semiconductor optical amplifier [7].
To connect these independent nodes, we use a measurement-feedback scheme implemented in field programmable gate arrays (FPGAs) [15,16]. Portions of the DOPOs inside the cavity are split by an optical coupler and measured by a balanced homodyne detector (BHD). The observed signal from the BHD, which is the in-phase amplitude of the DOPO denoted by $c_i$ for the $i$th computation pulse, is fed into FPGAs. The FPGAs calculate $\tilde {c_i} = \sum _j J_{ij} c_j$ and modulate amplitudes of other optical pulses according to the value of $\tilde {c_i}$. Here, $J_{ij}$ is a random matrix representing the connection between nodes. We inject these modulated pulses into the cavity to implement the interaction between computation pulses.
The input layer is implemented by adding an electrical signal on the feedback signal from the FPGAs. Here, the input signal for every computation step is added for every round trip. On the other hand, the output is retrieved as $\sum _i m_i c_i$ with an output mask $m_i$ for every round trip, where $m_i$ is optimized so that the output $\sum _i m_i c_i$ is close to the desired output depending on a target task.
2.2 Stabilization of the optical system
First, we describe the stabilization of the fiber ring cavity. We employ the Pound–Drever–Hall (PDH) method [24], which is widely used to stabilize cavity lengths and laser frequencies [25]. To use the PDH method for our setup, we set the amplitudes of the dummy pulses large enough so that the phase of the dummy DOPO pulses inherits that of injected dummy pulses. Otherwise, the DOPOs randomly take a phase of 0 or $\pi$, and the PDH method fails because the interference between DOPOs and injection pulses is inconsistent in different trials. We apply a small dithering signal on the phase of injection pulses using a fiber stretcher [FS 1 in Fig. 1(b)]. A signal from a photodetector (PD 1) is input into a lock-in amplifier (not shown in the figure). The output from the lock-in amplifier is used as an error signal to stabilize the cavity. Here, the lock-in phase is chosen to minimize the optical power monitored by PD 1. This error signal is fed into a PID controller, which adjusts the cavity length by using another fiber stretcher, FS 2, inside the cavity.
Likewise, we stabilize the relative phase of the pump pulses by applying a dithering signal on FS 3. The error signal is obtained from PD2, and a lock-in phase is chosen to maximize the optical power monitored by it. Here, a PID control is applied to FS 3.
Although we separately described stabilization for the cavity length and relative phase of the pump pulses, the former cannot be stabilized unless the latter is, and vice versa. For example, let us assume that the cavity is stabilized but the phase of the pump pulses is unstable. Even though the PDH method is robust against amplitude fluctuation, in this case, the amplitudes of the DOPOs largely fluctuate and the fluctuation destabilizes the PID control for the cavity length. Thus, a stable DOPO oscillation implies successful stabilization of both the cavity length and relative phase of the pump pulses. Note that we use injection pulses as references for the cavity length. Therefore, we can also stabilize the relative phase between the injection pulses and DOPO pulses at the same time.
Next, we describe the stabilization of the local oscillator. It is possible to stabilize the phase of the local oscillator by using a scheme similar to the PDH method. However, the BHD signal is zero on average regardless of the relative phase because DOPO pulses take both 0 and $\pi$ phase. Therefore, a square-law detector following the BHD is required to obtain an error signal, where PID parameters are adjusted to maximize the squared BHD signal. Unfortunately, this method locks the optical phase at the inverted sign with a 50% probability due to the square operation, and we need to implement an additional filter to detect and avoid the phase flip [18,19]. In addition, it imposes a dithering signal on the computation pulses and degrades the accuracy of the computation results. It is also possible to use a phase estimation technique in a digital coherent receiver and compensate for the phase fluctuation in digital processing [26,27]. This method has advantages in that it requires neither the dithering signal nor analog feedback to stabilize the phase and all processing can be implemented on the FPGA used to introduce the interactions between DOPOs. However, using phase-diversity homodyne detection instead of single homodyne detection introduces a 3-dB degradation of the signal-to-noise ratio, which affects the computation accuracy of the reservoir computer. To overcome these drawbacks, we introduce a stabilization method using a binary feature of the DOPO phase and phase modulation for the local oscillator so that we can stabilize the phase at the correct polarity using single homodyne detection.
To introduce a reference for the optical phase, we employ a periodical pattern of phase 0 and $\pi$ for the dummy injection pulses as shown in Fig. 1(b). The complex amplitude of the $n$th dummy injection pulse in the $k$th round trip $E_{\mathrm {d}}^{n,k}$ is given by
Here, rather than a sinusoidal wave, we use rectangular waveforms for the phase-modulation signal and reference signal. Because the pulse width of the DOPOs is much shorter than that of these waveforms, we can expect that the detection of the error signal and elimination of the computation pulses are robust against timing jitter and misalignment of the phase of electrical signals. In addition, the error signal is detected as a radio frequency (RF) signal at the BHD to avoid the instability of the low-frequency component often observed in a baseband measurement.
Note that the proposed stabilization method locks the phase of the local oscillator at the correct polarity and, unlike the PDH method, does not impose a dithering signal on the computation pulses. Furthermore, it can avoid the 3-dB degradation of the signal-to-noise ratio because it uses single homodyne detection.
2.3 Evaluation of the stability of the optical system
We evaluate the phase stability from the BHD signal as follows. The BHD signal for the $n$th dummy pulse is given by Eq. (3). Therefore, we can estimate $\delta$ for the $k$th round trip in the cavity by
where $\tilde {\delta }^k$ is the estimated value of $\delta$, and $N$ is the number of pulses accumulated in each formula.Furthermore, we measure the optical phase noise for two other setups: In one, we use a continuous-wave (CW) light to estimate the noise floor of the measurement system; in the other, we use DOPOs without the stabilization of the local oscillator to separately analyze the stability of the cavity and the homodyne detection. Here, we employ a phase-diversity homodyne detection with a 90-degree optical hybrid to measure these phase noises [26].
Figure 2(a) shows the setup for measuring the noise floor of the measurement system. A CW light is split by an optical coupler, and a portion of it is launched into a phase modulator followed by a fiber stretcher and an erbium-doped fiber amplifier (EDFA). Since the purpose is to measure the noise floor, we do not apply any signals to the phase modulator or fiber stretcher. We input the CW light from the EDFA to a 90-degree optical hybrid as a local oscillator while we directly input the other portion of the CW light to the 90-degree optical hybrid as a signal light. From the measurement signals from the two BHDs, we can estimate the phase of the signal light relative to the local oscillator, $\varphi _{\mathrm s} - \delta$, where $\varphi _{\mathrm s}$ is the phase of the signal light. In this setup, the difference between the two optical path lengths is short enough so that the phase noise of the original light source does not mainly contribute to the observed phase noise. Therefore, the observed noise can be considered to consist of noises from the inserted devices, a small fluctuations in the fiber length due to a temperature drift and acoustic noises, and electrical noises from the equipment following the two BHDs.
Fig. 2. Measurement setup to evaluate (a) the noise floor of the measurement system and (b) relative phase noise of DOPOs without local oscillator stabilization. Insets on the right side are schematics of output signals in the in-phase and quadrature phase plane for the different setups.
In the setup in Fig. 2(b), the signal light is replaced by the output of the DOPOs using the long fiber ring cavity. Here, the cavity is stabilized, but the phase of the local oscillator is not locked so that its phase noise can be compared with the noise using local oscillator stabilization estimated by Eq. (4). Because the DOPOs oscillate as pulses, the measured signals in the in-phase and quadrature phase plane vary between two points, which are peak values of the pulses for the 0 and $\pi$ phase [see the inset in Fig. 2(b)]. Note that the phase fluctuates only a little within a time shorter than the round trip time. Therefore, we estimate the optical phase corresponding to the $k$th round trip by linearly fitting the two BHD signals.
3. Experimental setup
Figure 3 shows the experimental setup. A CW light from a laser with a wavelength of 1535.8 nm was split by a 90:10 optical coupler (OC 1). Ninety percent of the CW light was modulated into a pulse train by an intensity modulator (IM 1), whose pulse width was 62 ps and repetition frequency was about 1 GHz adjusted to the resonance condition of the 1-km fiber cavity. The optical pulses were amplified by an EDFA followed by an optical band-pass filter to suppress amplified spontaneous emission (ASE) noise. We split the optical pulses with a 50:50 optical coupler (OC 2) to prepare pump pulses and injection pulses.
Fig. 3. The experimental setup. OC: Optical coupler. IM: Intensity modulator. PC: Polarization controller. EDFA: Erbium doped fiber amplifier. BPF: Optical band-pass filter. FS: Fiber stretcher. PPLN: Periodically poled LiNbO$_3$ waveguide. PM DSF: Polarization maintaining dispersion shifted fiber. PD: Photodetector. DL: Optical delay line. BHD: Balanced homodyne detector. LPF: Low-pass filter. All devices used polarization maintaining fibers except for the EDFAs used for the pulse train. The PC was used to compensate for the polarization in these EDFAs. The dotted lines indicate the paths which we manually switch. Optical isolators are inserted where we suppress reflections from PDs.
The pump pulses were divided into two temporal blocks by suppressing a fraction of them with an intensity modulator (IM 2). The odd-numbered pulses in the second block were used as computation pulses, and remaining unsuppressed pulses, including those in the first block, were used as dummy pulses for the stabilization. The first and second blocks had 2,500 and 2,000 pulses, respectively, and they were separated by 50 and 489 vacant time slots. These pulses were input into a fiber stretcher (FS 3) to stabilize the relative phase of the pump pulses, and they were amplified by an EDFA. The amplified pulses were launched into a PPLN waveguide to generate pulses with a wavelength of 768 nm via second-harmonic generation (SHG). We then launched the 768-nm pump pulses into another PPLN waveguide through a dichroic mirror placed in a fiber ring cavity, which also contains an optical band-pass filter, two optical couplers, a 1-km polarization maintaining dispersion shifted fiber (PM DSF), and a fiber stretcher (FS 2). As a result, 4,500 DOPO pulses oscillated in this cavity while 539 pulses remain in a vacuum state as a marker of the boundary for each round trip. A portion of the DOPO pulses were split by a 90:10 optical coupler (OC 3) followed by an optical isolator and another 99:1 optical coupler (OC 4). The output from the 99% port was measured by using stabilized single homodyne detection or phase-diversity homodyne detection described later. We split the DOPO pulses from the other 1% port with a 50:50 optical coupler (OC 5) and measured the optical power with a photodetector (PD 2). The measured signal of the optical power was used to stabilize the relative phase of the pump pulses via FS 3. We used a PID controller accompanied with a dithering signal generator and a lock-in amplifier for the stabilization. We set the frequency of the dithering signal at 23 kHz and the cut-off frequency of the lock-in detection at 2 kHz.
We used an intensity modulator (IM 3) to generate injection pules from the other portion of the pulse train from OC 2. We modulated the amplitudes of the dummy pulses according to Eq. (1). On the other hand, we randomly set the phase of computation pulses at 0 or $\pi$ to confirm that we could observe any combination of the phases stably. We did not inject optical pulses for the 50 and 489 vacant time slots. In this experiment, we used fixed amplitudes for injection pulses and pump pulses to focus on the evaluation of the stability rather than on that of the computation performance as a reservoir computer or CIM. These pulses were injected into the cavity by a 99:1 optical coupler (OC 6) after their temporal position had been adjusted by an optical delay line and a dithering signal had been added by a fiber stretcher (FS 1). A photodetector (PD 1) measured the optical power from an output port of OC 6, where an error signal was obtained to stabilize the cavity length. We set the frequency of the dithering signal at 42 kHz and the cut-off frequency of the lock-in detection at 7 kHz.
To prepare the local oscillator and CW signal light shown in Fig. 2(a), we split the portion of CW light from OC 1 with a 90:10 optical coupler (OC 7). We modulated the phase of the local oscillator according to Eq. (2) with a phase modulator only when we performed the proposed local oscillator stabilization. The phase-modulated CW light was amplified by an EDFA following a fiber stretcher (FS 4) to stabilize the relative phase of the local oscillator. An optical band-pass filter and optical isolator were inserted after the EDFA to reduce the ASE noise and reflection from the BHDs, respectively. On the other hand, the other portion of the CW light from OC 7 was used as the signal light shown in Fig. 2(a).
For the stabilized single homodyne detection, the DOPO pulses were combined with the local oscillator by a 50:50 optical coupler (OC 8) and measured by a BHD. We input the signal from the BHD to an RF amplifier and a low-pass filter with a cut-off frequency of 2.95 GHz. The signal from the low-pass filter was divided by a power splitter. We used the divided signals to stabilize the local oscillator and record the measurement results. To obtain an error signal described in the section 2.2, we generated a reference signal $S_{\mathrm {ref}}^{n,k}$ with an arbitrary waveform generator whose sampling rate was synchronized to ten times the pulse repetition frequency. The BHD signal was multiplied by the reference signal by using an RF mixer. Then, we extracted the baseband signal of the error signal through a low-pass filter with a cut-off frequency of 20 kHz. After an additional digital low-pass filter with a cut-off frequency of 5 kHz had been applied, the error signal was fed back to FS 4 using a PID controller. The other BHD signal divided by the power splitter was input into a low-pass filter with a cut-off frequency of 1.87 GHz and measured by a real-time oscilloscope. The oscilloscope was triggered at the beginning of each round trip using the arbitrary waveform generator, and it recorded 1,000 data points per trigger for 62,500 round trips. The sampling rate was 6.25 GSps, which was up-converted to 32 samples per pulse in a postprocessing to extract the peak value of the pulses with better precision. We extracted only 152 dummy pulses per round trip from this data [$N=38$ in Eqs. (5)–(8)] so that we could maximize the measurement time under the limitation of memory size.
For the phase-diversity homodyne detection, the signal light, which was either the DOPO pulses or the CW light, were combined with the local oscillator by a 90-degree optical hybrid. Four optical outputs from the 90-degree optical hybrid were input into the two BHDs. We used 1.87-GHz low-pass filters when we measured the DOPOs, while, to avoid an aliasing effect due to a low sampling rate, we added 2.5-MHz low-pass filters when using CW light. The sampling rates were 6.25 GSps and 62.5 MSps when we measured the DOPO pulses and CW light, respectively. In contrast to the stabilized single homodyne detection, we did not up-convert the sampling rate because the peak value of the pulses were unnecessary. Note that the sampling rate of the phase was much lower than that of the BHD signals for the DOPOs because we estimated the phase as an average over the first $4N$ dummy pulses for each round trip. In addition, the total measurement time for the DOPOs became shorter than that for the CW light because the averaging operation consumed larger memory in an oscilloscope.
4. Results
Figure 4(a) shows the measured signal from the BHD when we employed all the stabilization methods. The elapsed times of 0.5 and 2.5 ns correspond to the peak values of dummy pulses projected onto the in-phase amplitude. We observed that these peaks stably take the same values over 300 round trips [see also Fig. 4(b)]. We evaluated the relative standard deviation (RSD) of the peak value for each pulse by
Fig. 4. (a) Measured signal from the BHD when we used all the stabilization methods. This figure only shows the first four out of 152 dummy pulses and the first 300 out of 62,500 round trips. (b) Peak values of the BHD signals for the first four dummy pulses, i.e., $n=0, 1, 2$ and $3$ in Eq. (3).
On the other hand, the elapsed times of 1.5 and 3.5 ns in Fig. 4(a) correspond to the peak values of dummy pulses projected onto the quadrature amplitude. Because $\sin \delta \approx \delta$ is more sensitive to a fluctuation of $\delta$ than $\cos \delta \approx 1 - \delta ^2/2$ is, the fluctuation of these peak values was larger than that of the in-phase amplitudes [see Fig. 4(b)]. By combining the in-phase and quadrature amplitudes, we evaluated the phase noise using Eq. (4).
Figure 5 shows the power spectrum densities of the relative phase noises for different experimental setups. The phase noise due to the measurement system dominated a frequency of $<\!1$ kHz (blue line). In particular, we observed peaks around 100–200 Hz, which are considered as noise from devices such as the fiber stretcher, phase modulator, and EDFA.
Fig. 5. Single-sided power spectrum densities of the phase noises of the measurement system (blue), DOPOs without the local oscillator stabilization (orange), and DOPOs with the local oscillator stabilization (green), observed by the measurement setups shown in Fig. 2(a) and (b) and Fig. 1(b), respectively. The black dashed line is the fitted curve of the orange line from 10 Hz to 10 kHz. The power spectrum densities were estimated by using Welch’s method [28] with parameters so that the frequency resolution became 10 Hz. The power spectrum densities for CW light and DOPOs were averaged over three and ten trials to compensate for the different total measurement time of 1 and 0.32 s, respectively.
The DOPOs without the local oscillator stabilization had a larger phase noise than the noise floor over the whole frequency range (orange line). We also observed the noise from the measurement devices around 100–200 Hz, residual dithering signal for the pump pulses at 23 kHz, and the resonance peak of FS 3 around 10–20 kHz. Although the DOPOs showed noises from these devices, they showed $1/f^2$ dependence up to 10 kHz, where $f$ is frequency. If we assume white frequency noise whose single-sided spectral density is given by $S_F$ [26], laser linewidth and single-sided spectral density of phase noise are given by $\pi S_F$ and $S_F / f^2$, respectively. By fitting the phase noise of the DOPOs by $S_F / f^2$ from 10 Hz to 10 kHz, $\pi S_F$ was evaluated to be 1.8 Hz (black dashed line). This result indicates that the DOPOs stably oscillated relative to the measurement system even without the local oscillator stabilization.
By stabilizing the local oscillator, the phase noise was significantly suppressed (green line). The power spectrum density reached the noise floor for a frequency of $<\!1$ kHz, and the peaks around 100–200 Hz were also suppressed. Furthermore, the proposed method rarely affected the phase noise for a frequency of $>\!1$ kHz because it didn’t introduce additional dithering signal.
As we discussed in the introduction, an important property for a physical reservoir computer is long-term stability. To evaluate the time interval of the stability, we observed the Allan variance of the relative phases of the DOPOs [23]. The Allan variance is defined as
Figure 6 shows the Allan variance of the relative phase for the DOPOs. The Allan variance began to increase at around 5 to 10 ms without the local oscillator stabilization (blue line). This result indicates that the drift and random walk of the optical phase were not negligible compared to the other noise sources for $\tau$ > 10 ms. On the other hand, the Allan variance measured using the local oscillator stabilization continued to decease with increasing $\tau$ and became flat at $\tau \approx$ 100 ms (orange line). Thus, the proposed methods clearly achieved long-term stability, at least, up to 100 ms. Note that $\tau$ was limited to < 0.16 s because the total measurement time was 0.32 s due to the memory size of the oscilloscope. However, the measured time interval of the stability was long enough for the current purpose. As we discussed in the introduction, the number of computation steps used to evaluate a reservoir computer has been several thousands in typical experiments on physical reservoir computers. Even if we perform physical reservoir computing with over 10,000 steps, it takes only 50 ms for the total computation with the proposed methods. Moreover, we observed a small $\mathrm {RSD}$ of the BHD signals over 62,500 round trips. Therefore, the proposed stabilization methods successfully achieved the long-term stability required to evaluate a physical reservoir computer.
Fig. 6. Allan variance of relative phase for the DOPOs without local oscillator stabilization (blue) and with local oscillator stabilization (orange). Shaded regions indicate the envelope of the minimum and maximum values over ten trials, and the solid lines indicate average values. The Allan variance was estimated according to the overlapping Allan variance [29].
5. Discussion
An important question is whether we can further reduce the RSD of the BHD signal by improving the stability of the optical phase. Because we measured the fluctuation of the phase $\delta$, we were able to estimate the contribution of $\delta$ to the fluctuation of the BHD signal. The RSD of $\cos \delta$ was estimated to be $\pm$ 0.5%. If we assume that the phase noise is independent from other noise sources and subtract it from the total variance, the RSD becomes $\pm$ 1.3%. Thus, the phase noise was negligible compared to the RSD of the BHD signal. Although the phase noise was significantly suppressed, we can use a phase modulator rather than a fiber stretcher to eliminate the dithering signal imposed on the injection and pump pulses as we did for the local oscillator. In addition, the phase noise of the light source could be suppressed by using a laser with a narrower linewidth.
Recently, the effect of optical phase noise was investigated for a coherent spatially parallel optical reservoir computer, which used a Fabry-Perot cavity and a spatial light modulator [30]. It was numerically shown that a phase noise of a few milliradians can drastically degrade the computation performance if we use the simplest learning approach. However, the interesting learning methods introduced in [30] can strengthen the phase noise robustness up to an order of 1 rad if we can estimate the optical phase. Note that our stabilization method can estimate the phase while we perform a computation. Therefore, it could be possible to use those learning methods although this requires further investigation during an actual computation.
In this experiment, we excluded the computation steps that are performed in a reservoir computer or CIM. If we include the computation steps, the amplitudes of the computation pulses largely change, and the variation of the amplitudes can affect the stability of the proposed methods. The stabilization of the local oscillator can still work even if we include the computation steps because we do not use the computation pulses to obtain the error signal $S_{\mathrm {err}}^{k}$. Unfortunately, the computation steps can affect the stabilization of the cavity and pump pulses with the current implementation. We can mitigate this effect by introducing additional intensity modulators in front of the photodetectors to eliminate the computation pulses. In addition, it is known that a reservoir computer requires the echo state property [20], with which the initial information in the reservoir computer is asymptotically eliminated. This property can be satisfied if we set the pump amplitude below the oscillation threshold for the computation pulses, because a DOPO decays and becomes a vacuum state without injection pulses. In this case, the average optical power for the computation pulses is smaller than the power of the dummy pulses oscillating above the threshold. Because an off-the-shelf intensity modulator has a high extinction ratio ($>\!20$ dB) and the PDH method is robust against amplitude noises, we expect that a simple modification using additional intensity modulators will mitigate amplitude noises enough to achieve the same stability.
6. Conclusion
We proposed stabilization methods for the optical phases for optical hybrid computers using DOPOs. The proposed methods employed the PDH method and periodical modulation pattern for the injection pulses and local oscillator. We also achieved, for the first time, the measurement of the relative phase for an optical computing system including a CIM. We evaluated the phase stability of DOPOs using the power spectrum density and Allan variance, which clearly showed long-term stability up to 100 ms. We also observed a stable BHD signal that had the RSD of $\pm$ 1.4% over 62,500 round trips. The proposed stabilization methods constitute important bases of stable computation for large-scale optical hybrid computers.
Acknowledgments
The authors thank T. Akatsuka and K. Oguri for fruitful discussions, and H. Tamura for his support during this research.
Disclosures
TIkuta,TInagaki,KI,TH,HT: NTT Corporation (P,R)
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