1. Introduction

The physical implementation of machine-learning schemes, such as photonic neural networks, coherent Ising machines, and reservoir computing, has attracted significant attention, greatly reducing learning costs and power consumption [1–4]. Reservoir computing is a promising machine learning paradigm inspired by brain information processing [5,6]. Originally, reservoir computing utilized recurrent neural networks with a large number of nodes [7,8]. Various types of physical reservoir computing have been demonstrated using photonics, electronic circuits, spintronics, and soft robotics [9–12]. Recently, simple implementations of reservoir computing, known as delay-based reservoir computing [13,14], have been studied intensively. In this scheme, a single dynamical system with a time-delayed loop is used rather than recurrent neural networks. Virtual nodes are considered by sampling the temporal waveforms of the reservoir outputs in the delayed feedback loop. When an input signal is injected into a time-delayed dynamical system, the system produces transient responses, and virtual node states can be obtained from the response outputs. The reservoir computing output is given by a weighted linear sum of the virtual node states, where the weights are determined using a training procedure to ensure consistency between the input and output signals.

Delay-based reservoir computing using photonic systems has been intensively studied, where the reservoir is based on a semiconductor laser with time-delayed optical feedback [3,15–17]. The laser system consists of a semiconductor laser and a time-delayed optical feedback loop. The semiconductor laser produces response outputs at high frequencies (up to a few gigahertz), resulting in high-speed information processing. Applications of optical fiber communications using photonic reservoir computing have been studied, such as modulation format identification [18] and modulation encoding [19]. In this photonic reservoir computing scheme, an input signal is injected into the reservoir laser via optical injection from another semiconductor laser. This optical injection scheme requires injection locking (wavelength matching) of the reservoir laser to the injected light [20,21]. This requirement restricts the magnitude of the injection current because an excessive injection current in the reservoir laser may prevent injection locking. The best performance was obtained when the injection current was set close to the lasing threshold (or 2.5 times the threshold at the maximum [22]) of the optical injection system [15,20,23]. The output power of the reservoir laser became relatively low at a small injection current, and the signal-to-noise ratio (SNR) was relatively low.

Recently, reservoir computing based on an erbium-doped microchip solid-state laser with an input signal by modulating the intensity of the optical feedback has been proposed [24]. This scheme does not require an extra laser for the signal injection, and SNR can be improved by increasing the pumping power. However, the speed of information processing is relatively low (several microseconds per sample) owing to the low relaxation oscillation frequency of the microchip laser (a few megahertz). Semiconductor lasers can be more suitable reservoirs for faster information processing because the relaxation oscillation frequency can reach gigahertz.

The process of converting the input information into photonic reservoir computing is not well understood. In most photonic reservoir computing procedures, the intensity of the laser output or injection current is modulated by the input signal, and the reservoir output is obtained as a virtual node state. However, optical phase modulation has not been used for signal injection in photonic reservoir computing. Therefore, understanding how the optical feedback intensity and phase modulations are converted into photonic reservoir computing output in terms of the change in the structure of the external cavity modes is crucial.

In this study, we numerically and experimentally investigate reservoir computing using a semiconductor laser with optical feedback modulation. An input signal is injected into the laser by modulating either the optical intensity or the phase of the feedback signal. We perform a chaotic time-series prediction task using the reservoir computing system and evaluate the prediction performance. We compare the reservoir computing performance of the intensity and phase modulation schemes. We further analyze the intensity and phase modulation schemes using the external cavity modes obtained from the steady-state analysis.

2. Reservoir computing based on semiconductor laser with time-delayed optical feedback

This section describes the proposed reservoir computing scheme based on an external-cavity semiconductor laser with optical feedback modulation. A schematic of the reservoir computing system is shown in Fig. 1. It consists of three parts: an input layer, a reservoir, and an output layer. The input signal is preprocessed in the input layer (see Appendix for details). In the reservoir layer, the input signal is injected into a reservoir laser consisting of an external cavity semiconductor laser with feedback modulation. The intensity or phase of the feedback signal is modulated by the input signal in the feedback loop (delay time $\tau$) to obtain the transient dynamics of the reservoir laser. In the output layer, postprocessing of the output from the reservoir laser is performed. The delay-based reservoir computing scheme is described in detail in the Appendix.

 

Fig. 1. Schematic of the proposed reservoir computing system based on external-cavity semiconductor laser with optical feedback modulation. LD is the laser diode, CIRC is the optical circulator that realizes optical feedback, and MOD is the intensity or phase modulator. The delay time of optical feedback is represented by $\tau$.

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The dynamics of the external-cavity semiconductor laser with feedback phase modulation for the reservoir are described by the Lang–Kobayashi equations as follows [25]:

In Eq. (1), $E_{\textrm {fb}}(t,\tau )$ represents the optical feedback signal with a modulation signal. The optical feedback signals for the intensity and phase modulation are as follows:

Intensity modulation:

Phase modulation:

The feedback strength $\kappa$ is given by $\kappa = (1-r^{2}_2)r_3/(\tau _{\textrm {in}}r_2)$, where $r_3$ is the feedback ratio (intensity reflectivity of the external mirror), $r_2$ is the intensity reflectivity of the laser facet, and $\tau _{\textrm {in}}$ is the round-trip time in the internal laser cavity. The feedback strength is adjusted using the feedback ratio $r_3$. Here, $\tau$ denotes feedback delay time, $\omega \tau + \pi s(t)$ represents the phase shift owing to the delay and phase modulation, and $\omega$ is the angular optical frequency of the laser and is given by $\omega = 2 \pi / \lambda$, where $\lambda$ is the optical wavelength of the laser. The term $\xi (t)$ in Eq. (1) is white Gaussian noise with properties $\left\langle \xi (t) \right\rangle= 0$ and $\left\langle \xi (t) \xi (t_0) \right \rangle = \delta (t-t_0)$, where $\left\langle \cdot \right\rangle$ denotes the ensemble average and $\delta (t)$ is Dirac’s delta function. The parameter values are summarized in Table 1.

Table 1. Parameter values used in the numerical simulation.

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In Eqs. (3) and (4), $s(t)$ is the input signal generated by the masking procedure in the delay-based reservoir computing scheme. We consider discrete-time data $s_n$ ($n = 1, 2, \ldots$ as the discrete-time) as the input data. Each input data point is injected into the reservoir for a duration of $N\theta$. A mask signal $m(t)$ is multiplied by the input data to induce transient dynamics in the laser output. The mask signal is considered as the input weight to the virtual nodes. To implement the same input weight for all input data, the period of the input mask $T$ is set as the product of the number of nodes $N$ and node interval $\theta$ (i.e., $T=N\theta$). The mask used in this study is a piecewise step function with a step interval $\theta$, and the value of the mask is randomly selected from the set $\{-1, -0.3, 0.3, 1\}$. The input signal multiplied by the mask is given by

We fix the number of virtual nodes in the numerical simulation at $N=200$. The node interval $\theta$ is adjusted. A smaller value than the characteristic time scale of a reservoir was used for the node interval $\theta$ in a previous study [13]. The characteristic time scale of the reservoir is related to the relaxation oscillation frequency, which depends on the injection current. We thus change the injection current in the numerical simulation. We adjust the node interval for the different injection current values. Subsequently, the feedback delay time is given by $\tau =N\theta$.

We use the fourth-order Runge-Kutta method for the numerical integration of Eqs. (1) and (2). The step size for the numerical integration is 1 ps. We numerically integrated the variables using rate equations for a transient time of 10000 ns. The transient time of the dynamics is sufficiently large to converge. A stable laser output converges to the external cavity mode nearest to the solitary laser mode, although the laser is subject to optical feedback. Therefore, the reservoir computing performance does not depend strongly on the initial conditions.

3. Numerical results of the reservoir computing performance

3.1 Comparison between intensity and phase modulation schemes

To evaluate the performance of the reservoir computing scheme, we use the Santa Fe time-series prediction task [26]. This task aims to perform single-point predictions of a chaotic time series. The time series is generated using a far-infrared laser. We use the first 3000 points in the time-series for training ($N_{tr}=3000$) and the last 1000 points for testing ($N_t=1000$). The signal level of the time series is in the $[0, 255]$ range. The time series is normalized between $[0, 1]$ by dividing the original values by 255. In addition, the input scaling $\gamma$ is adjusted for the intensity and phase modulation after investigating the dependence of the prediction performance on $\gamma$. For the intensity modulation, we select $\gamma =1.0$, which produces the maximum amplitude of the input signal $s(t)$. For phase modulation, a low prediction error is obtained around $\gamma =0.60$, which is used in the numerical simulation. From Eqs. (4) and (5), the setting $\gamma =0.60$ produces phase modulations with a range of $\left [ -0.60 \pi, 0.60 \pi \right ]$. A more detailed investigation of the dependence of performance on $\gamma$ is described in Section 3.2.

The performance of the prediction task is quantitatively evaluated using the normalized mean-square error (NMSE) as follows:

We first use the node interval $\theta =0.1$ ns, and the feedback delay time is given as $\tau = N \theta = 20.0$ ns. The node interval $\theta$ is selected to be smaller than the characteristic time scale of the semiconductor laser with a normalized injection current of $j=2.0$. For $j=2.0$, the relaxation oscillation frequency is 4.5 GHz, corresponding to a characteristic time of 0.22 ns.

Figure 2 shows the response intensity signal of the laser with the input signal modulation. Figures 2(a), 2(b), and 2(c) show the input signal $s(t)$, response signal for the intensity modulation scheme, and response signal for the phase modulation scheme, respectively. The amplitude of the response signal to phase modulation is larger than that of the intensity modulation scheme. The difference in the amplitudes of the response signal is expected to affect the performance of the time-series prediction task because a larger amplitude leads to a better SNR.

 

Fig. 2. (a) Input signal for chaotic time-series prediction task after the input mask is applied. (b), (c) Temporal waveforms of the reservoir laser output when the input signal is injected by (b) intensity and (c) phase modulation in the delayed feedback loop. The feedback ratios for the intensity and phase modulation are set to (b) $r_3 = 0.025$ and (c) $r_3 = 0.016$ to optimize the NMSE.

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Figures 3(a) and 3(b) show the prediction results for the intensity and phase-modulation schemes, respectively. The black, red, and blue curves represent the target signal, predicted signal, and prediction error, respectively. The predicted signal for phase modulation is similar to the target signal, whereas the predicted signal for intensity modulation is dissimilar. NMSE values of 0.238 and 0.031 are obtained for the intensity and phase modulation schemes, respectively. Therefore, the prediction error in phase modulation is smaller than that in intensity modulation.

 

Fig. 3. Numerical results of the chaotic time-series prediction task for the (a) intensity and (b) phase modulation schemes. The black, red, and blue curves represent the target signal, prediction result, and error between them. The feedback ratios are $r_3 = 0.025$ and $0.016$ for the intensity and phase modulation schemes, respectively.

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3.2 Dependence on the input scaling coefficient

We optimize the input scaling coefficient $\gamma$ to obtain a high-performance reservoir computing. Figure 4 shows the dependence of the prediction error (i.e., NMSE) on input scaling coefficient $\gamma$. The green solid curve represents the result of feedback intensity modulation. The prediction error gradually decreases as $\gamma$ increases, and the minimum error is obtained at $\gamma =1.0$. From Eq. (3), input signal $s(t)$ must be larger than $-1.0$. From Eq. (5), the range of the input signal $s(t)$ depends on $\gamma$, mask $m(t)$, and input data $s_n$. The value of the mask is sampled from the set $\{ -1.0, -0.3, 0.3, 1.0 \}$, and the input data $s_n$ are scaled to $\left [ 0, 1 \right ]$. Therefore, the signal $s(t)$ has a range $\left [ -1, 1 \right ]$ at $\gamma =1.0$, and we use $\gamma =1.0$ for feedback intensity modulation. The dashed blue curve in Fig. 4 represents the result for the feedback phase modulation. The minimum error is obtained at $\gamma =0.60$. Therefore, we use $\gamma =0.60$ for the phase modulation scheme. From Eqs. (4) and (5), a phase-modulation signal with a range of $\left [ -0.60\pi, 0.60\pi \right ]$ can be obtained for $\gamma =0.60$. The maximum peak-to-peak modulation width is $1.2\pi$, which is smaller than the phase variation corresponding to one period ($2.0\pi$). If the modulation amplitude is larger than $2.0\pi$, different input data cannot be correctly identified. Therefore, the modulation amplitudes must be smaller than $2.0\pi$ for the phase modulation.

 

Fig. 4. Dependence of the prediction error (i.e., NMSE) on the input scaling factor $\gamma$. The signal injection schemes are intensity and phase modulations for the green solid and blue dashed curves, respectively. The reflectivity of for optical feedback is $r_3 = 0.025$ for the intensity modulation and $r_3 = 0.016$ for the phase modulation. The values of $r_3$ are selected to obtain a low prediction error in Fig. 5. The other parameter values are the same as in Table 1.

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3.3 Dependence on laser parameters

We investigate the dependence of the NMSE on the reflectivity $r_3$ and compare the performance of the intensity and phase modulation schemes. Figure 5 shows the NMSE variations as the feedback ratio $r_3$ is changed. The green solid and blue dashed curves represent the NMSEs of the intensity and phase modulation schemes, respectively. For $r_3 = 0$, the NMSE is nearly equal to 1 because no signal injection occurs via feedback modulation. The minimum value of the NMSE is observed at an intermediate feedback strength for both the intensity and phase modulation schemes. A larger value of reflectivity $r_3$ results in an increase in the NMSE in both cases. From the comparison between the intensity and phase modulation schemes, the NMSE for the phase modulation scheme is smaller than that for the intensity modulation scheme for all values of reflectivities.

 

Fig. 5. Normalized mean–square error (NMSE) as a function of the feedback ratio (reflectivity of the external mirror $r_3$). The green solid and blue dashed curves represent the intensity and phase modulations, respectively.

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The differences between the NMSE values for the intensity and phase modulations are related to the consistency and SNR. Consistency can be defined as the ability of a dynamic system to produce an identical response output when the system is driven by a repeated drive signal [27,28]. In reservoir computing, consistency is one of the criteria for information processing [20]. We quantitatively evaluate the degree of consistency by using the cross-correlation between two temporal waveforms generated by repeatedly driving the reservoir laser with an identical signal as follows:

Figure 6(a) shows the cross-correlation value as a function of the feedback ratio $r_3$ for the intensity modulation (green solid curve) and phase modulation (blue dashed curve). In both modulation cases, the cross-correlation increases with an increase in $r_3$ until $r_3$ exceeds 0.025. When $r_3$ exceeds $0.025$, the correlation value decreases to zero. The cross-correlation value for phase modulation is larger than that for intensity modulation. Therefore, the phase modulation scheme is superior to the intensity modulation scheme in terms of consistency of the response output for an injection signal.

 

Fig. 6. (a) Consistency and (b) SNR as a function of the feedback ratio (reflectivity of the external mirror $r_3$). The green solid and blue dashed curves represent the intensity and phase modulations, respectively. (c) A bifurcation diagram is created from the probability distribution of the laser intensity. The probability of the laser intensity is represented with the grayscale. The black color indicates a large probability.

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Figure 6(b) shows the SNR as a function of feedback ratio $r_3$. The SNR is defined as $\textrm {SNR} = 10\log _{10}(\sigma _I^{2}/\sigma _N^{2})$, where $\sigma _I$ and $\sigma _N$ are the standard deviations of the laser intensity with and without input signal injection, respectively. The SNR increases as the feedback strength increases for both modulations for small feedback strengths and decreases suddenly for larger values of the feedback strength. This result indicates that the phase modulation in optical feedback produces a large amplitude in the temporal response (Fig. 2). Thus, we show that phase modulation is superior to intensity modulation in the feedback modulation scheme.

The sudden decrease in the cross-correlation value and SNR at $r_3=0.025$ results from the dynamical bifurcation of the reservoir system. Figure 6(c) shows a bifurcation diagram of the reservoir without feedback modulation. We created a diagram of the probability distributions of the laser intensity. The horizontal axis represents the reflectivity $r_3$ and the vertical axis is the laser intensity. In the bifurcation diagram, we plot the probability distribution of the laser intensity using a grayscale for each reflectivity. The black color represents a high probability. For $r_3 \leq 0.025$, the probability converges to almost one, indicating that the laser dynamics exhibit a stable output. The probability distribution spreads slightly as reflectivity increases. When the reflectivity exceeds $r_3 = 0.025$, a wide probability distribution of the laser intensity is obtained, indicating the appearance of periodic or chaotic dynamics. When the dynamics of the reservoir are unstable, the consistency property is degraded. Therefore, for reservoir computing, it is necessary to adjust the feedback strength to maintain a stable output without an input signal.

Then, we investigate the dependence of the reservoir computing performance on the normalized injection current $j$. When the injection current is varied, the feedback ratio $r_3$ and node interval $\theta$ are appropriately adjusted to enhance the reservoir computing performance. Figures 7(a) and 7(b) show two-dimensional NMSE maps on the parameter space of $r_3$ and $\theta$ at $j = 1.50$ and $3.00$, respectively. In Fig. 7(a), the lowest value of the NMSE is obtained at $r_3 = 0.013$ and $\theta = 0.14$ ns. By contrast, as shown in Fig. 7(b), the lowest value of NMSE is obtained at $r_3 = 0.045$ and $\theta = 0.04$ ns. These results indicate that large $r_3$ and small $\theta$ values are required for large injection currents to achieve high performance. A large $r_3$ value is required for a larger value of $j$ because the prediction performance can be improved near the dynamical transition point [29]. In addition, the node interval $\theta$ must be less than the inverse of the relaxation oscillation frequency for a high reservoir computing performance [13]. The relaxation oscillation frequency of the semiconductor laser is enhanced with an increase in $j$. Therefore, smaller values of $\theta$ are required for larger $j$ values and faster relaxation oscillation frequencies.

 

Fig. 7. Two-dimensional maps of the NMSE as functions of the feedback ratio (reflectivity of external mirror $r_3$) and the node interval $\theta$. The gray scale map shows the NMSE values. The normalized injection currents are (a) $j = 1.5$ and (b) $j = 3.0$. The feedback phase modulation is used.

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Figure 8(a) shows the dependence of the NMSE on injection current $j$ when feedback ratio $r_3$ and node interval $\theta$ are optimized. The dependence of SNR on $j$ is also shown. We found that the NMSE decreases as $j$ increased. In addition, the NMSE decreases as the SNR improves, as depicted by the dashed red curve. Figure 8(b) shows the optimal values of $r_3$ and $\theta$ used in Fig. 8(a) for different injection currents. The black curve with circles indicates the optimal node interval. The optimal value of $\theta$ decreases as the injection current increases, because faster oscillations are obtained for larger injection currents. The red dashed curve with diamonds indicates the optimal $r_3$. The optimal value of $r_3$ increases as the injection current increases because a higher power is observed for larger injection currents.

 

Fig. 8. (a) NMSE (black solid curve) and SNR (red dashed curve) as a function of the normalized injection current $j$. The feedback ratio $r_3$ and the node interval $\theta$ are optimized at each $j$. (b) Optimal values of the node interval $\theta$ and the feedback ratio $r_3$ for each injection current in (a). The feedback phase modulation scheme is used.

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We also investigate the memory capacity of our scheme and compare the performances of the intensity and phase modulation schemes. The memory capacity of the phase modulation scheme is superior to that of the intensity modulation scheme, as described in the Appendix.

4. Experimental results of the reservoir computing performance

4.1 Experimental setup

In this section, we experimentally demonstrate reservoir computing based on a semiconductor laser with optical feedback modulation to confirm the numerical results discussed in previous sections. We investigate the performance of reservoir computing by using intensity- and phase-modulated optical feedback signals.

Figure 9 shows the experimental setup for reservoir computing using a semiconductor laser with feedback modulation. We used a distributed-feedback semiconductor laser (NTT Electronics, KELD1C5GAAA, wavelength: 1547 nm) with a fiber pigtail. The optical isolator is removed from the semiconductor laser to provide optical feedback to the laser. A fiber circulator was placed in front of the laser for optical feedback. A fiber loop configuration, which is considered as a reservoir, was used to introduce optical feedback. A fiber attenuator was inserted into the fiber loop to adjust the optical feedback strength. An intensity modulator (iXblue, MXAN-LN-20, 20 GHz bandwidth) or phase modulator (EO Space, AX-DMSS-20-LV, 20 GHz bandwidth) was inserted in the fiber loop to realize the intensity or phase modulation of the feedback light. A bias controller (iXblue, MBC-AN-LAB-A1) was used for the intensity modulation to stabilize the bias point of the intensity modulator. An input signal was generated by an arbitrary waveform generator (Tektronix, AWG70002A, 25 GigaSample/s) and injected into the modulator. The modulated feedback light is reinjected into the laser as an input signal. For the detection of the reservoir output, part of the light in the fiber loop was separated by a fiber coupler. The output of the reservoir laser was detected using a photodiode (Newport, 1554-B, 12 GHz bandwidth) and converted into an electric signal. The electrical signal was measured using a digital oscilloscope (Tektronix, DPO72304DX, 23 GHz bandwidth, 100 GigaSample/s), and the virtual node states were obtained by sampling the temporal waveform of the reservoir output.

 

Fig. 9. Experimental setup for reservoir computing using a semiconductor laser with feedback modulation. Either intensity or phase modulator is used to produce feedback modulation light.

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Note that the phase modulation scheme does not require any components to stabilize the operating point of the phase modulator. We numerically confirmed that the feedback phase does not significantly affect the performance of reservoir computing. However, a bias controller is required for the intensity modulation scheme in the experiment because the operating point of the intensity modulator fluctuates. This is an advantage of the phase-modulation scheme because no additional component is required to stabilize the operating point of the phase modulator.

The experimental parameter values were set as follows. The injection current of the semiconductor laser was set to 66.0 mA (6.0 $I_{th}$, $I_{th}$ = 11.0 mA was the lasing threshold). The optical fiber lengths of the delayed feedback loops were set to 13.5 and 9.4 m for the intensity and phase modulation schemes, respectively, because a longer fiber was required for the intensity modulation to support power monitoring for the bias controller. Thus, the corresponding feedback delay times $\tau$ were 67.9 and 47.4 ns for the intensity and phase modulation schemes, respectively. The node interval was set to 0.04 ns for both the intensity and phase modulation schemes. The period of the input mask was set to $T$ = 47.4 ns for both modulation schemes, and the number of nodes was $N$ = 1185 in both cases for a fair comparison. The rest of the virtual node states within the feedback delay time is discarded for the intensity modulation scheme.

4.2 Experimental results

Figure 10 shows the experimental results of the chaotic time-series prediction task using intensity and phase modulation of the feedback light. We use 3000 points for training and 1000 points for testing. For the intensity modulation shown in Fig. 10(a), the prediction signal is dissimilar to the target signal. A large NMSE of 0.322 is obtained for the intensity modulation. In contrast, for the phase modulation in Fig. 10(b), the prediction signal resembles the target signal. A small NMSE of 0.070 is obtained for the phase modulation. Therefore, we confirm that the phase modulation of the feedback light outperforms intensity modulation. This experimental result is consistent with the numerical results described in the previous section.

 

Fig. 10. Experimental results of chaotic time-series prediction task using (a) intensity modulation and (b) phase modulation of the feedback light. Black: target signal, red: prediction signal, and blue: error between them. NMSEs are (a) 0.322 and (b) 0.070, respectively.

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Figure 11(a) shows the changes in the prediction error (NMSE) as the feedback strength changes for the intensity and phase modulation schemes. The values of the NMSE for phase modulation are smaller than those for intensity modulation for different feedback strengths. The minimum NMSE value of 0.056 is obtained at a feedback strength of 2.6 $\mu$W for the phase modulation scheme. Therefore, the minimum NMSE value is obtained for an intermediate feedback strength.

 

Fig. 11. Experimental results of (a) prediction error (NMSE) and (b) cross-correlation for consistency measure as the feedback strength is changed for the intensity (green solid curve) and phase modulation (blue dashed curve) schemes.

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We measure the consistency of reservoir outputs generated from the same input signal. Figure 11(b) shows the cross-correlation between two reservoir outputs generated from the same input signal for the intensity and phase modulation as the feedback strength is changed. The phase modulation shows higher correlation values than the intensity modulation. The maximum correlation value of 0.75 is obtained for the phase modulation scheme, whereas a maximum correlation of 0.36 is observed for the intensity modulation scheme. The region over the feedback strength of 3 $\mu$W corresponds to the loss of consistency owing to the bifurcation of the laser output. This result indicates that the phase modulation of the feedback provides higher consistency than the intensity modulation.

We change the parameter values of the reservoir laser and investigate the performance of the chaotic time-series prediction. Here, we use the feedback phase modulation scheme. Figure 12(a) shows the prediction error (NMSE) for different values of the node interval $\theta$, mask interval $\theta _m$, and number of virtual nodes $N$ as the feedback strength is changed. The NMSE is reduced when $\theta$ and $\theta _m$ are decreased from 0.2 to 0.08 ns (the black and blue curves in Fig. 12(a)). Therefore, smaller $\theta$ and $\theta _m$ values lead to a better performance of the time-series prediction task. In addition, the NMSE is further reduced when $N$ is increased (blue and red curves). Therefore, a larger $N$ is preferable for improving the performance of reservoir computing.

 

Fig. 12. Experimental results of prediction error (NMSE) as the feedback strength is changed with respect to feedback phase modulation. (a) Node interval $\theta$, mask interval $\theta _m$, and the number of virtual nodes $N$ are changed. (b) Normalized injection current $J/J_{th}$ is changed. The feedback strength is normalized by the maximum feedback power for each injection current. The maximum feedback power is as follows: $102$ $\mu$W (at $J/J_{th} = 2.0$), $190$ $\mu$W (at $J/J_{th} = 3.0$), $278$ $\mu$W (at $J/J_{th} = 4.0$), $367$ $\mu$W (at $J/J_{th} = 5.0$), and $451$ $\mu$W (at $J/J_{th} = 6.0$).

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We change the injection current of the semiconductor laser and investigate the dependence of the performance on the injection current. Figure 12(b) shows the changes in the prediction error (NMSE) as the feedback strength is changed for different injection currents. We optimize $\theta$ and $\theta _m$ for different injection currents because the relaxation oscillation frequency of the laser strongly depends on the injection current. For Fig. 12(b), the values of NMSE decrease as the injection current increases. Moreover, the minimum value of the NMSE is observed at a large feedback strength as the injection current increases. Therefore, a larger injection current ($J/J_{th} = 6.0$ in Fig. 12) is preferable in the feedback light modulation scheme, unlike the case of semiconductor lasers with optical injection modulation [15], where the injection current must be set close to the lasing threshold to optimize the reservoir performance. This characteristic is preferable for obtaining a high SNR to improve the performance of reservoir computing. These experimental results are in good agreement with the numerical results.

In this experiment, we used different delay times for the intensity and phase modulations because of the limitations of the experimental equipment. We consider that this difference does not strongly affect our experimental results because we use the same number of nodes for both the modulation schemes. Additionally, we match the mask period and feedback delay time for each modulation scheme [30]. The superior performance of the phase modulation results from its high consistency property, as shown in Fig. 11, but not the shorter delay time for the phase modulation scheme. We checked the dependence of the prediction performance on the feedback delay time in the numerical simulation. We confirmed that the performance does not change significantly for different feedback delay times, and the phase modulation scheme is superior to the intensity modulation scheme for a wide range of delay times, as described in detail in the Appendix.

5. Steady-state analysis for external cavity modes

In the previous section, we showed that the phase modulation scheme produces larger amplitude response signals than the intensity modulation schemes, resulting in higher performance of the chaotic time-series prediction task. In this section, we provide a detailed analytical investigation of the physical origin of the large response during phase modulation. We first derive the external cavity modes, which are steady-state solutions to the Lang–Kobayashi equations. Steady-state solutions have been used to explain various dynamical phenomena such as low-frequency fluctuations [31,32] and chaos synchronization [33,34]. We investigate the dependence of the external cavity modes on the feedback strength and phase, and clarify the effect of the dynamical response of the semiconductor laser on feedback modulation. The parameter values used for the analysis are the same as those in Table 1 except for the reflectivity $r_3$. The derivation of the steady-state solutions is described in the Appendix.

First, we examine the external cavity modes in the phase space of the optical frequency and intensity ($\omega$–$I$ plane) for different parameter values of the feedback strength and phase. Figure 13(a) shows the external cavity modes calculated using Eqs. (14)–(16) in the Appendix. The horizontal axis is denoted by $(\omega _s - \omega ) / (2 \pi )$. For Fig. 13(a), the black and red dots represent the external cavity modes at the feedback strengths of $r_3 = 0.02$ and $0.01$, respectively. The feedback strength is varied to observe the changes in the external cavity modes caused by intensity modulation. The external cavity modes are shifted in the vertical direction (intensity), and the size of the ellipse is changed. However, the external cavity modes do not change in the horizontal direction (optical frequency), as shown in the enlarged view of the phase space in Fig. 13(b). Therefore, the frequency and phase of the external cavity modes remain unchanged during the intensity modulation.

 

Fig. 13. External cavity modes of semiconductor laser with optical feedback in the phase space of the optical frequency and laser intensity ($\omega$-$I$ plane). (a) The feedback ratios (reflectivity of external mirror $r_3$) are set to $r_3 = 0.02$ (black dots) and $0.01$ (red dots) to observe the changes in the external cavity modes by intensity modulation. (b) Enlarged view of (a). (c) Feedback phases are set to $\phi _0=0$ (black dots) and $\pi$ (red dots) to observe the changes in the external cavity modes by phase modulation. (d) Enlarged view of (c).

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Figure 13(c) shows the external cavity modes when the feedback phase changes. The black and red dots shown in Fig. 13(c) represent the external cavity modes at the feedback phases of $\phi _0=0$ and $\pi$, respectively. The feedback phase is varied to observe the changes in the external cavity modes caused by phase modulation. The external cavity modes are shifted towards the direction of the ellipse. An enlarged view of the external cavity modes shown in Fig. 13(d), and the external cavity modes are shifted toward the optical frequency. Therefore, a change in the feedback phase results in a change in the optical frequency of the external cavity modes.

Figure 14(a) shows the trajectory of the response outputs under intensity modulation with a four-level mask signal (see Fig. 2(a)) and the corresponding external cavity modes. The attractor is located near the external cavity modes, and its size is very small. The enlarged view of the attractor shown in Fig. 14(b), the external cavity modes corresponding to the four-level mask signal are plotted as color dots. The external cavity modes are shifted vertically and the changes in the external cavity modes are small. The trajectory of the response output is located around these external cavity modes.

 

Fig. 14. Trajectories of the laser dynamics with feedback modulation in the phase space of the optical frequency and laser intensity ($\omega$-$I$ plane). The input signal is injected via (a),(b) intensity and (c) phase modulations. (b) is the enlarged view of (a). The parameter values are the same as in Fig. 2. The trajectories of (a) and (c) are in the same ranges of the vertical and horizontal axes for comparison. The color dots indicate the external cavity modes when the four-level input mask is used for the input signal.

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For comparison, Fig. 14(c) shows the trajectory of the response outputs under the phase modulation with a four-level mask signal and the corresponding external cavity modes. The external cavity modes are shifted in the horizontal direction. The size of the attractor is shown in Fig. 14(c) is larger than the case of the intensity modulation in Fig. 14(a) when using the same range of the vertical and horizontal axes for comparison. Notably, changes in the optical frequency of the external cavity modes result in a two-dimensional transformation of the attractor, as shown in Fig. 14(c). Therefore, we consider that phase modulation is more effective than intensity modulation for the nonlinear transformation of a low-dimensional input signal to a high-dimensional trajectory in phase space.

We consider that the changes in the optical feedback phase significantly change the external cavity modes in the optical frequency, inducing variations in laser intensity. This indicates that a semiconductor laser with feedback phase modulation produces a mapping of a one-dimensional signal into a two-dimensional phase space of intensity and frequency. We speculate that higher-dimensional mapping, which is an important requirement for reservoir computing, can be achieved using phase modulation in a time-delayed reservoir computing system. Our steady-state analysis provides qualitative insights into the trajectory changes in the phase space for intensity and phase modulation. More quantitative analysis, such as dimensionality measurement, could be performed to clarify the mechanism for the superior performance of phase modulation compared to intensity modulation.

6. Conclusion

We numerically and experimentally demonstrate reservoir computing using a single-semiconductor laser with optical feedback modulation. An input signal was injected into the semiconductor laser via the intensity or phase modulation of the optical feedback signal. We performed a chaotic time-series prediction task using reservoir computing and compared the reservoir computing performance of the intensity and phase modulation schemes. The results indicate that the feedback phase modulation scheme outperformed the intensity modulation scheme. Furthermore, the prediction error was improved for large injection currents, unlike the results for a semiconductor laser modulated by an optical input signal from an external laser. We analyzed the physical origin of the superior performance of the phase modulation using external cavity modes obtained from steady-state analysis in the phase space. A high-dimensional mapping from the input signal to the trajectory of the reservoir laser output can be achieved using feedback phase modulation.

The phase modulation of the feedback light is more suitable than the intensity modulation to facilitate nonlinear high-dimensional mapping for reservoir computing. Complex-field reservoir computing using intensity and phase information can be easily realized using the proposed phase-modulation scheme to enhance its performance. In addition, the differential phase-shift keying scheme can be used to convert phase information to intensity information to enhance the SNR. These findings have a potential to aid in the implementation of complex-field reservoir computing with a single semiconductor laser.

Appendix A: Delay-based reservoir computing scheme

In the delay-based reservoir computing scheme, a network with several connected nodes is emulated using delayed nonlinear nodes, and the nodes are virtually implemented within a delay line via time multiplexing (virtual nodes) [13]. The virtual nodes are connected to neighboring nodes and included self-feedback. Therefore, the configuration of a virtual network is considered a ring topology. Our reservoir has optical feedback with delay time $\tau$, and virtual nodes are implemented by dividing $\tau$ by a small time interval $\theta$, known as the node interval. Therefore, the number of virtual nodes $N$ is given by $N = \tau / \theta$. The states of the virtual nodes are extracted from the temporal waveform of the laser output.

In the input layer, the input data are preprocessed, as discussed in the main text, where the mask signal $m(t)$ with mask period $T$ is multiplied with the input data $s_n$ to induce transient dynamics in the laser output. In previous studies, the delay time $\tau$ did not match the product of the number of virtual nodes $N$ and the node interval $\theta$ ($\tau \neq N\theta$) [30,35], and the mask period $T$ differed slightly from the feedback delay time $\tau$. This setting is often called a desynchronized scheme, which can enhance the performance of reservoir computing [36,37]. However, for simplicity, we match the delay time $\tau$ with the mask period $T$ in this study [15].

A weighted linear combination of the virtual node states is calculated in the output layer, and the result is used as the reservoir-computing output. The output $y(n)$ for the $n$-th input data is given by the following equation:

(8)$$y(n) = \sum^{N}_{j = 1}w_j x_j(n),$$

where $x_j$ is the virtual node state and $w_j$ is the weight of the $j$-th node state. The node state $x_j$ is obtained from the temporal waveform of the laser output. The weight $w_j$ is trained by minimizing the mean-square error between the target function $\bar {y}(n)$ and the reservoir computing output $y(n)$ as follows:

(9)$$\frac{1}{N_{tr}} \sum^{N_{tr}}_{n = 1}(y(n) - \bar{y}(n))^{2} \rightarrow \textrm{min},$$

where $N_{tr}$ is the number of input data points for training. We use a standard linear regression technique, the least-squares method, to minimize Eq. (9).

Appendix B: Comparison of memory capacity

We investigate the linear memory capacity [38] of our reservoir for the intensity and phase modulation schemes. To evaluate memory capacity, we consider a task whose goal is to recall past input data. We consider input data $s_n$ randomly sampled from a uniform distribution of $\left [ -1, 1 \right ]$. When the input data $s_n$ are injected into the reservoir, the target signal $\hat {y}_n$ is the previous input data $s_{n-k}$, where $k \geq 1$ is the number of memory steps. To calculate the similarity between the target $\hat {y}_n$ and reservoir output $y_n$, we use the following cross-correlation function [39]:

(10)$$m(k) = \frac{ \left\langle s_{n-k} y_n \right\rangle}{ \sigma(s_n) \sigma(y_n) },$$

where $\sigma (s_n)$ and $\sigma (y_n)$ are the standard deviations of $s_n$ and $y_n,$, respectively. $\left\langle \cdot \right\rangle$ represents the time averaging. The number of input data points for calculating $m(k)$ is 10000. The number of input data points for the training is $N_{tr}=5000$. Memory capacity is defined as the square sum of the correlation function $m(k)$:

(11)$$MC = \sum^{\infty}_{k=1}m^{2}(k),$$

where the maximum step for $k$ is theoretically infinite. In reality, we stop the calculation at $k=100$, where the correlation value $m(k)$ approaches zero.

Figure 15(a) shows $m^{2}(k)$ as a function of the memory step $k$. The solid green and dashed blue curves represent the intensity and phase modulation schemes, respectively. For $1 \leq k \leq 18$, the phase-modulation scheme produces a higher value of $m^{2}(k)$ than the intensity-modulation scheme. However, for $k \geq 19$, $m^{2}(k)$ for the intensity modulation scheme is larger than that for the phase modulation scheme. Figure 15(b) shows the dependence of memory capacity on the reflectivity $r_3$ of the external mirror for optical feedback. The solid green and dashed blue curves represent the intensity and phase modulation schemes, respectively. In the intensity modulation scheme, the maximum memory capacity, $MC=15.0$, is obtained at $r_3=0.025$. By contrast, in the phase modulation scheme, the maximum memory capacity $MC=16.3$ is obtained at $r_3=0.023$. Therefore, the phase modulation scheme produces a slightly higher value than the intensity modulation scheme.

 

Fig. 15. (a) Cross-correlation function $m(k)$. The intensity (green solid curve) and phase (blue dashed curve) modulations are used. The reflectivities are $r_3 = 0.025$ for the intensity modulation and $r_3 = 0.023$ for the phase modulation. (b) Memory capacity as a function of the reflectivity $r_3$. Input scalings are $\gamma = 0.70$ and $0.25$ for the intensity and phase modulation schemes, respectively.

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The memory capacity of the intensity modulation scheme is sensitive to changes in reflectivity $r_3$. In Fig. 15(b), the memory capacity for the intensity modulation scheme decreases suddenly with an increase in the reflectivity from $r_3=0.025$. In contrast, sudden changes in memory capacity are not observed in the phase-modulation scheme. Therefore, the phase-modulation scheme is more robust than the intensity-modulation scheme against changes in reflectivity $r_3$.

When the reflectivity exceeds $r_3 = 0.025$, the semiconductor laser exhibits a bifurcation from a stable output to a periodic state in the temporal dynamics. When the temporal dynamics of the laser are unstable, the consistency in the laser is reduced (that is, the cross-correlation value defined in Eq. (7) decreases). In the intensity modulation scheme, the cross-correlation value immediately approaches zero when the reflectivity exceeds $r_3 = 0.025$ as indicated by the solid green curve in Fig. 6(a). In contrast, the cross-correlation value for the phase modulation scheme (dashed blue curve in Fig. 6(a) decreases slowly. The difference between the two modulation schemes in the variation of the cross-correlation value results from the amplitude of the laser output to that of the input modulation. As shown in Fig. 2, the amplitude of the laser output for the intensity modulation scheme is small, whereas a large amplitude is obtained for the phase modulation.

Appendix C: Comparison between numerical and experimental results

Different values of feedback delay times were used for the intensity and phase modulation schemes in the experiment. We numerically confirm that the difference in delay time does not affect our conclusions. Further, the prediction performance of the phase modulation scheme is superior to that of the intensity modulation schemes. In our numerical simulations based on the experimental parameter values, we use a normalized injection current of $j=6.0$, corresponding to the value at which the best performance is achieved in the experiment (Fig. 12(b)). We also match the strength of the spontaneous emission noise between the experiment and numerical simulations (the optimal value of the SNR is 12 dB). The feedback delay times are set to 67.9 ns and 47.4 ns for the intensity and phase modulation schemes, respectively. In addition, we use the node interval $\theta =0.04$ ns and number of nodes N = 1185. The conditions for $\tau$, $\theta$, and $N$ are the same as those used in the experiment.

Figure 16(a) shows the numerical results of the prediction error (i.e., the NMSE) as a function of reflectivity $r_3$. The green solid and blue dashed curves represent the results of the intensity and phase modulation schemes, respectively. The prediction error for the phase modulation scheme is smaller than that of the intensity modulation schemes for all values of $r_3$. This result is in good agreement with the experimental results (Fig. 11(a)).

 

Fig. 16. Numerical results of the prediction error (NMSE) as a function of (a) the reflectivity $r_3$ and (b) the delay time $\tau$. The feedback delay times are 67.9 ns and 47.4 ns for the intensity and phase modulation schemes in (a), respectively. The reflectivity is fixed at $r_3=0.10$ in (b). The injection current for the laser is set to six times the threshold ($j=6.0$). The green solid and blue dashed curves represent the NMSEs of the intensity and phase modulation schemes, respectively.

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We also investigate the dependence of performance on the feedback delay time in the numerical simulation. Figure 16(b) shows the prediction error (i.e., NMSE) as a function of feedback delay time $\tau$. We fix the node interval at $\theta =0.04$ ns and the number of nodes at $N=1185$, which provides the mask period $T = N \theta = 47.4$ ns. In Fig. 16(b), the green solid and blue dashed curves represent the NMSEs of the intensity and phase modulation schemes, respectively. The NMSE does not change significantly for different feedback delay times, and the phase modulation scheme is superior to the intensity modulation scheme for a wide range of delay times. Therefore, we consider that the performance is not strongly affected by the difference in delay times between the intensity and phase modulation schemes in our experiment.

Appendix D: Steady state analysis for external cavity modes

Here, we describe the derivation of the steady-state solutions of the Lang-Kobayashi equations. We consider rotating wave solutions of the forms $E(t) = A_s e^{i (\omega _s - \omega ) t}$ and $N(t) = N_s$ as the external cavity modes, where $A_s$, $\omega _s - \omega$, and $N_s$ are the constant electric field amplitude, optical frequency shift from the solitary laser mode, and carrier density, respectively. We obtain the following equations by inserting these solutions into the Lang–Kobayashi equations without signal injection [29,31,40],

(12)$$i\omega_s A_s e^{i(\omega_s-\omega)t} = \frac{1 + i\alpha}{2} \left[ \frac{G_N (N_s - N_0)}{1 + \epsilon A_s^{2}} - \frac{1}{\tau_p} \right] A_s e^{i(\omega_s-\omega)t} + \kappa A_s e^{i[(\omega_s-\omega) (t-\tau) - \omega \tau - \phi_0]},$$

(13)$$0 = J - \frac{N_s}{\tau_s} - \frac{G_N (N_s - N_0)A_s^{2}}{1 + \epsilon A_s^{2}},$$

where $\phi _0$ denotes feedback phase shift. From Eqs. (12) and (13), we obtain the following steady-state solutions:

(14)$$A_s^{2} = \frac{jN_{th}-N_s}{\tau_s G_N (N_s - N_0) + \varepsilon (N_s - jN_{th})},$$

(15)$$\omega_s - \omega ={-} \kappa \sqrt{1+\alpha^{2}} \sin(\omega_s \tau + \tan^{{-}1}\alpha + \phi_0),$$

(16)$$N_s - N_{th} = \frac{\varepsilon N_{th}(j-1) - 2 \kappa \tau_s \cos(\omega_s \tau + \phi_0) }{\tau_s G_N + \varepsilon}.$$

The solutions of these equations are the external cavity modes. These modes are placed on an ellipse in the phase space of $\omega$ and $N$ in [29,31,40]. In this study, we observe the external cavity modes in the phase space of the optical frequency and intensity ($\omega$ and $I=|E|^{2}$) because we address the relationship between the variations in the external cavity modes and the evolution of the laser intensity on the attractor. The optical frequency is given by $\Delta f(t) = ( \phi (t) - \phi (t-\tau ) ) / (2 \pi \tau )$, where $\phi (t)$ is the optical phase calculated using the complex electric field amplitude $E(t)$. The optical phase is given by $\phi (t) = \tan ^{-1}(E_{\textrm {im}}(t)/E_{\textrm {re}}(t)) + 2 \pi k$, where $E_{\textrm {re}}(t)$ and $E_{\textrm {im}}(t)$ are the real and imaginary parts of $E(t)$, respectively. Coefficient $k$ is the number of rotations of the laser trajectory in the complex plane of the electric-field amplitude. The optical frequency corresponds to a frequency shift from the initial optical frequency $\omega / (2\pi )$.

Funding

Telecommunications Advancement Foundation; Japan Science and Technology Agency (JST) CREST (JPMJCR17N2); Japan Society for the Promotion of Science (JSPS) KAKENHI (JP19H00868, JP20K15185).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.

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