Frequency-multiplexed optical reservoir computing using a microcomb

3.1 Numerical demonstration

Numerical simulations are conducted using the Lugiato–Lefever equation [24], [25]:

Here ψ denotes the intracavity electric field. The variables t and θ represent slow time and fast time. Slow time describes how the field evolves over successive round trips inside the resonator, whereas fast time labels the angular position inside the cavity in a co-moving reference frame. The parameters δ, D₂, and F correspond to the normalized detuning, normalized second-order dispersion, and normalized pump amplitude. The normalized pump power (F²) and second-order dispersion are fixed at 32 and 0.048, values that match the experimental microresonator. On the physical time scale, we assume a free spectral range of 100 GHz and a loaded resonance linewidth of 28 MHz, again consistent with the device used in the experiments. When input signals are applied, δ becomes a function of the slow time.

First, we examine the static behavior of a soliton comb while varying the detuning. With the fixed pump power F² = 32 a soliton exists for detuning values between 12 and 39, as illustrated in Figure 2(a). Figure 2(b) shows the optical spectra obtained at detunings of 13 (red), 23 (green), and 33 (blue). These spectra are symmetric because higher-order dispersion and the Raman effect are omitted in the simulation. For positive comb mode numbers, we observe that the power of comb modes below 7 decreases as the detuning increases, whereas the power of comb modes above 7 grows when the detuning is reduced. Figure 2(c) plots the normalized power of the +1st (red), +10th (green), +20th (blue), and +30th (purple) modes versus detuning, scaled to one at the initial detuning of 12. The power change is more pronounced when the detuning is smaller (that is, when the pump frequency lies closer to the cavity resonance). The distinct and nonlinear power responses of the different modes indicate rich dynamics that are promising for RC. We next analyze the dynamic response. The detuning is abruptly stepped from 12 to 14 for 2 ns, a duration shorter than the photon lifetime ( ≈ 1 l i n e w i d t h ) . Figure 3 reveals that the power of the −1st comb mode continues to evolve for more than 100 ns and exhibits oscillations with a period of 6.8 ns. The prolonged response results from photon storage in the microresonator, and the oscillatory behavior arises from the interplay between the Kerr nonlinearity and the changing intracavity power.

Figure 2:

Static response of the simulated soliton microcomb to pump-laser detuning. (a) Total comb power versus detuning. (b) Optical spectra of the soliton comb at the detuning of 13 (red), 23 (green), and 33 (blue). (c) Normalized powers of selected comb modes as a function of detuning. The comb mode numbers are +1st (red), +10th (green), +20th (blue), and +30th (purple).

Figure 3:

Transient response of the power of the −1st comb mode (red) when the detuning is stepped from 12 to 14 for 2 ns (black).

As a first benchmark we employ the one-step-ahead Santa Fe time-series prediction task. This chaotic benchmark was introduced during the Santa Fe Institute competition, where the data come from intensity fluctuations of a far-infrared NH₃ laser. After a soliton comb is generated at a detuning of 12, the detuning is swept between 12 and 18 in response to the input signal. The detection sampling rate is 5 GSa/s. Up to 60 comb modes are used, and the intensity of each mode is normalized to one in the absence of modulation. Only one side of the comb is taken because the opposite side shows the same response when the Raman effect and third-order dispersion are neglected. This is not true for the experiment, and both sides of the comb modes are used in the experiment. The reservoir outputs, that is the intensities of the individual comb modes, are trained on 3,000 samples with ridge regression to obtain the optimal weights, and a further 1,000 samples are used for test. To determine the best modulation rate, we evaluate normalized mean-square errors (NMSEs) with N_comb = 60 while varying the modulation rate, as shown in Figure 4(a):

Figure 4:

Optimization of NMSE for numerical Santa Fe prediction. (a) NMSE when modulation rate is varied for N_comb = 60. Normalized modulation rate is defined as f m l i n e w i d t h . (b) NMSE when N_comb is varied with the normalized modulation rate fixed at ≈2. (c) Predicted (red) and target (blue) Santa Fe time-series waveforms, and the residual (green) between them, when N_comb and normalized modulation rate are 60 and around 2, respectively.

Here N is the number of samples, and d and y are the target and predicted signals. σ²(∗) denotes the variance of *. The minimum NMSE occurs when the normalized modulation rate f m l i n e w i d t h is about 2, corresponding to 50 MSa/s for a resonance linewidth of 28 MHz, which shows that the short-term memory is used effectively. NMSE increases gradually up to a normalized rate of 10 and then rises sharply for higher rates, behavior that matches earlier observations that excessive memory degrades performance on the Santa Fe task. Figure 4(b) plots NMSE as a function of N_comb with the normalized modulation rate fixed near 2. Increasing N_comb markedly improves performance, reaching an NMSE as low as 0.015. The best predicted Santa Fe waveform (red) is compared with the actual data (blue) in Figure 4(c). The prediction follows the target closely, and most residuals lie between −0.1 and +0.1.

The results in Figure 4 are effectively noise-free except for a small amount of random noise added during each round trip. In a real soliton comb, however, amplified spontaneous emission (ASE) from the optical amplifiers placed before and after the microresonator raises the noise floor. To mimic these conditions, we add Gaussian noise with standard deviation σ _G = 0.02 to the comb-mode intensities, as shown by the red trace in Figure 5(a). This level corresponds to a signal-to-noise ratio (SNR) of 34 dB at 0.02 nm resolution bandwidth (RBW) in the telecom band. With this noise present the NMSE increases by about 0.4, as indicated by trace (2) in Figure 5(b). We explore two mitigation strategies. First, we insert random delays between comb modes, with the maximum delay set to 1 2 f m . These delays reduce the NMSE to 0.163, trace (3) in Figure 5(b), and the error bar gives the standard deviation across ten independent delay patterns. The random delays enhance the temporal complexity of the comb-mode power waveforms, similar to time-division multiplexing, without increasing the number of nodes. Second, we apply a low-pass filter (LPF) with a 75 MHz cutoff to the comb-mode powers. The filter attenuates the noise according to

Figure 5:

NMSE for numerical Santa Fe prediction under four post-processing schemes. (a) Normalized power of the −30th comb mode when the detuning is driven by the Santa Fe signal under four conditions: noise-free, no low-pass filter (LPF) (black); noisy, no LPF (red); noise-free, LPF applied (blue); noisy, LPF applied (green). (b) Corresponding NMSE values: (1) no noise/no delay/no LPF; (2) noise/no delay/no LPF; (3) noise/delay/no LPF; (4) noise/delay/LPF. σ _G = 0.02, N_comb = 60, and normalized modulation rate ≈2.

Here σ_LPF is the noise standard deviation after filtering, and f_LPF and f _s are the LPF cutoff frequency and the ADC sampling rate. Because of the LPF, the impact of noise is less pronounced, as indicated by the blue and green curves in Figure 5(a). Combining the random delays with LPF processing lowers the NMSE of the noise-corrupted comb-mode waveforms to 0.0293, as shown by trace (4) in Figure 5(b).

Next we apply our ORC to a nonlinear equalization (NLEQ) task. The NLEQ problem is common in digital communications, where the goal is to remove nonlinear distortion and inter-symbol interference produced by the channel and to recover the transmitted symbols. In the model the original bit sequence d(i), which takes the values −3, −1, +1, and +3, is mixed by inter-symbol interference according to

The resulting sequence q(i) is then distorted nonlinearly and corrupted by Gaussian noise v(i), giving

We use u(i) as the input signal to the RC. In this task the detuning is swept between 12 and 16, the normalized modulation rate is set to about 4 (which corresponds to 100 MSa/s for a 28 MHz linewidth), and N_comb = 60. We prepare 8,000 samples, half for training and half for testing, and determine the output weights with ridge regression. Without the ORC the symbol-error rate (SER) shown by the red curve in Figure 6 stays near 0.1 even at high SNR because nonlinear distortion and inter-symbol interference cannot be removed. Using the ORC lowers the SER. When no additional noise is added to the comb-mode intensities the SER falls by more than one order of magnitude at SNR values above 24, as shown by the blue curve in Figure 6. The improvement saturates at high SNR, which suggests that the available nonlinearity in the present ORC is already fully exploited since increasing the modulation rate, i.e. enhancing the memory effect, does not yield further gains. When noise is added to the comb-mode intensities the improvement is smaller, as indicated by the green curve. A LPF cannot be used for this task because the signal is broadly distributed in frequency and the filter would remove both signal and noise.

Figure 6:

Symbol-error-rate (SER) versus SNR for the NLEQ task. (Red) baseline without the ORC. (Blue) ORC with random inter-mode delays, no added comb-intensity noise. (Green) ORC with both random delays and comb-intensity noise (σ _G = 0.02). Normalized modulation rate and N_comb are around 4 and 60, respectively. Error bars denote one standard deviation across ten delay patterns.

3.2 Experimental demonstration

Figure 7(a) shows the proof-of-concept experimental setup. In this demonstration, the output weights are applied electronically instead of with microring arrays. A single-frequency CW laser operating at 1,536 nm provides the pump light. The pump CW laser, with an average power of 200 mW, is coupled into a Si₃N₄ microresonator that has a loaded Q of 7 × 10⁶ [17]. To reach the soliton state the pump frequency is rapidly swept by a dual-parallel Mach–Zehnder modulator (DP-MZM) driven in carrier-suppressed single-sideband mode [26], [27]. When the input signal is applied, a voltage-controlled oscillator driven by an arbitrary waveform generator (SDG6052X, SIGLENT) adds a modulation to the DP-MZM in addition to the static bias that sets the soliton detuning. The modulation waveform is strictly positive, so the detuning only increases from its initial value. After the soliton comb is generated, a programmable bandpass filter (WaveShaper 1000A, FINISAR) selects four comb modes. These modes are amplified by an erbium-doped fiber amplifier (EDFA) and then split by a programmable wavelength-division multiplexer (WSS-2000, santec) so that each selected mode carries 100 μW of optical power. Each mode is detected by a separate photodiode (PDA05CF2, Thorlabs), and the electrical outputs are sampled by an oscilloscope (DHO4804, RIGOL), yielding four reservoir nodes at once. A computer then retunes the filter and multiplexer to additional sets of modes, ultimately covering 37 modes in total, from +6 to +1 and from −1 to −31. The reservoir outputs are multiplied by the trained weights offline. For stable long-term operation the static detuning is maintained by a feedback loop. One comb mode is monitored, and its power is held constant by adjusting the pump frequency, utilizing the one-to-one relationship between comb-mode power and detuning [28], [29]. The feedback bandwidth is less than 100 Hz, far below the modulation rate, so the loop stabilizes only the static detuning and does not affect the dynamic detuning modulation. Figure 7(b) displays the optical spectrum of the soliton comb. The soliton exists over a detuning range of about 550 MHz, which corresponds to a normalized soliton existence range of about 20 and is 25 % narrower than predicted numerically. Figure 7(c) shows the transient response of a single comb mode when the detuning is increased for 5 ns. The power decays slowly because of the photon lifetime in the resonator and exhibits oscillations with a period of roughly 6 ns, consistent with the numerical results in Figure 3. These oscillations arise from the interplay among the Kerr effect, detuning, and thermal dynamics. The responses of all comb modes are then measured while a Santa Fe chaotic signal drives the detuning at a sampling rate of 50 MSa/s, corresponding to a normalized modulation rate of 1.8. Figure 7(d) presents the resulting waveforms after photodetection. The traces vary gradually with the comb index, and representative examples at the −2nd, −15th, and −29th modes are plotted in Figure 7(e) in blue, green, and red. The distinct responses of different modes supply the high-dimensional and nonlinear mapping required for RC. All traces are normalized to compensate for variations in comb-mode power, and these normalized signals are used in the subsequent benchmark tasks.

Figure 7:

Setup and results of experimental the proposed ORC. (a) Schematic of the experimental setup. AWG, arbitrary waveform generator; DP-MZM, dual-parallel Mach–Zehnder modulator; EDFA, erbium-doped fiber amplifier; BPF, bandpass filter; WDM, wavelength-division multiplexer. (b) Optical spectrum of the soliton comb. RBW is 0.2 nm. (c) AC-coupled photodetected signal of a single comb mode when the detuning is stepped for 5 ns. (d) DC-coupled photodetected signal of the used comb modes when a Santa Fe chaotic signal drives the detuning at 50 MSa/s. The signals are low-pass filtered with the cutoff frequency of 8.5 MHz by post-processing. (e) Traces from (d) extracted at the −2nd (blue), −15th (green), and −29th (red), vertically offset for clarity.

The one-step-ahead Santa Fe prediction task is also evaluated experimentally. To obtain reliable statistics, the measurement is repeated five times. As in the numerical study, the NMSE reaches about 0.8 when neither random delay between comb modes nor a LPF is used (trace 1, red, in Figure 8). Introducing a random delay between the comb modes during post-processing lowers the NMSE to 0.21 ± 0.025 (trace 2, blue, in Figure 8). The same delay pattern is applied to all five data sets. Adding a LPF with a 13.5 MHz cut-off further reduces the NMSE to 0.081 ± 0.0056 (trace 3, green). By choosing a more suitable random delay through trial and error we obtain an NMSE as low as 0.061. The experimental errors remain higher than the numerical ones because the measured soliton comb contains more noise than its simulated counterpart. Later we describe how this noise can be reduced to the level assumed in the simulations.

Figure 8:

Experimental NMSE for Santa Fe prediction under three post-processing schemes. Red bar (1): no random inter-mode delay, no LPF. Blue bar (2): random delay only. Green bar (3): random delay plus 13.5 MHz LPF. Error bars represent one standard deviation over five independent measurement runs.

To minimize NMSE the static detuning must be optimized. Figure 9(a) plots the photodetected signal from the −2nd comb mode while the initial detuning is swept, with the modulation range held constant. From red to green to blue to purple traces the initial setting is shifted toward longer wavelengths, and the response amplitude decreases. This behavior occurs because the comb-mode power is more sensitive to detuning changes at the shorter-wavelength side of the soliton-existence window, as indicated in Figure 2(c). As a result, the NMSE increases, as shown in Figure 9(b). When the initial detuning lies near the blue edge (red data in Figure 9(a) and (b)) the NMSE is 0.064 ± 0.0025. In contrast, when the initial detuning is near the red edge (purple data) the NMSE rises to 0.17 ± 0.0031.

Figure 9:

Effect of the initial detuning on reservoir performance. (a) Photodetected signal of the −2nd comb mode for four initial detuning settings. The red trace lies near the blue edge of the soliton existence window; successive shifts toward longer wavelength produce the green, blue and purple traces. (b) Corresponding NMSE for each initial detuning. Colors match (a). Error bars indicate one standard deviation over five data sets. NMSE is evaluated with a fixed random inter-mode delay and a 13.5 MHz LPF.

The nonlinear equalization task is also verified experimentally. The SNR of the distorted input is fixed at 40, and the modulation sampling rate is 50 MSa/s. A total of 4,000 samples are used for training and another 4,000 for testing. Figure 10(a) shows the photodetected traces of every comb mode. The responses vary from mode to mode, confirming that the reservoir maps the one-dimensional input into a high-dimensional space. This contrast is highlighted in Figure 10(b), which plots representative modes at +6 (blue), −12 (green), and −19 (red). In this task, a random delay can be inserted between comb modes, but a LPF cannot be applied because the signal power is widely spread in frequency, so filtering removes useful information. Nonetheless, the measured SER is down to 0.0635 ± 0.0036. The numerical study in Figure 6 indicates that further suppression of intensity noise would lower the SER.

Figure 10:

Photodetected comb-mode responses in the NLEQ experiment. (a) Time-domain traces of all measured comb modes when the distorted input defined by Eq. (5) drives the detuning. The modulation sampling rate is 50 MSa/s. (b) Representative modes taken from (a) at the +6th (blue), −12th (green), and −19th (red), vertically offset for clarity.