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Abstract
Photonic computing is widely used to accelerate the computational performance in machine learning. Photonic decision making is a promising approach utilizing photonic computing technologies to solve the multi-armed bandit problems based on reinforcement learning. Photonic decision making using chaotic mode-competition dynamics has been proposed. However, the experimental conditions for achieving a superior decision-making performance have not yet been established. Herein, we experimentally investigate mode-competition dynamics in a chaotic multimode semiconductor laser in the presence of optical feedback and injection. We control the chaotic mode-competition dynamics via optical injection and observe that positive wavelength detuning results in an efficient mode concentration to one of the longitudinal modes with a small optical injection power. We experimentally investigate two-dimensional bifurcation diagram of the total intensity of the laser dynamics. Complex mixed dynamics are observed in the presence of optical feedback and injection. We experimentally conduct decision making to solve the bandit problem using chaotic mode-competition dynamics. A fast mode-concentration property is observed at positive wavelength detunings, resulting in fast convergence of the correct decision rate. Our findings could be useful in accelerating the decision-making performance in adaptive optical networks using reinforcement learning.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The demand for large computational resources to support the rapid evolution of artificial-intelligence technologies has increased in recent years. However, the performance of electronic general-purpose computers cannot satisfy the requirements of many calculations, owing to the limitations of semiconductor integration technologies, known as the end of Moore's law. To overcome this problem, photonic computing has been widely applied, including photonic artificial neural networks [1], photonic reservoir computing [2–4], coherent Ising machines [5], and nonlinear classification based on silicon photonic circuits [6]. These studies are expected to lead to the development of photonic accelerators that can enhance specific computational tasks using photonic technologies [7,8].
Photonic decision making is a promising photonic computing technology that can solve multi-armed bandit problems (MABs) based on reinforcement learning [9–14]. In MAB, a player maximizes the total reward from multiple slot machines with unknown hit probabilities (an example of multiple choice) [15,16]. Two actions are important for maximizing the total reward. One action, called exploration, searches for the best choice (e.g., the slot machine with the highest hit probability). The second action is exploitation, which involves finding a choice based on information obtained from exploration. However, a tradeoff between exploration and exploitation has been reported, in which the total reward is reduced if either exploration or exploitation is biased [16]. The tradeoff between exploration and exploitation is an important issue that must be resolved in MAB.
Photonic decision making for solving MAB with two slot machines has been reported using chaotic temporal waveforms of a semiconductor laser and a threshold value [9], spontaneous mode dynamics in a semiconductor laser with a ring cavity [11], and coupled semiconductor lasers [12]. Several schemes have been proposed to increase the number of slot machines, such as a comparison of two slot machines in a hierarchical architecture [10], a unidirectionally coupled laser network [13], and the assignment of random bits based on a chaotic semiconductor laser [14]. However, this approach has several limitations. For example, the performance is degraded based on the slot machine arrangement in a hierarchical architecture [10], and decision making cannot be achieved when the number of slot machines is increased using a coupled laser network [13]. Digital processing is required for random number generation and slot-machine selection [14]. Thus, it is important to propose a method to increase the number of slot machines required for photonic decision making.
Recently, decision making using the dynamics of a multimode semiconductor laser has been proposed by exploring slot machines based on chaotic itinerancy (chaotic mode-competition dynamics) [17]. Numerical simulations show that the proposed method requires a smaller number of plays than existing software-based algorithms when the number of slot machines is increased, which is useful for solving large-scale MAB. A proof-of-concept experiment has also been reported [17]; however, a detailed experimental investigation is still lacking, and the use of proper dynamics is important for accelerating decision-making performance.
In the dynamics of multimode semiconductor lasers with optical feedback, low-frequency fluctuations (LFFs) have been observed numerically and experimentally [18–22]. A bifurcation diagram of the total intensity is experimentally investigated using a multimode semiconductor laser when the feedback power and injection current are changed [23]. In addition, several studies have reported experimental observations of chaotic antiphase dynamics [24] and adaptive mode selection of the maximum modal intensity (called the dominant mode) [25].
Semiconductor lasers with both optical feedback and injection have been utilized for photonic reservoir computing [26,27], bandwidth enhancement of chaotic outputs [28], and control of chaotic mode-competition dynamics in multimode semiconductor lasers [29]. However, these studies are primarily limited to numerical simulations. It has been reported that semiconductor lasers with optical injection exhibit rich nonlinear dynamics by changing the initial wavelength detuning and the optical injection power [30–33]. It is important to experimentally investigate the conditions for controlling the mode-competition dynamics in a chaotic multimode semiconductor laser because the use of proper dynamics can improve the decision-making performance.
In this study, we experimentally investigate the control of mode competition dynamics in a chaotic multimode semiconductor laser in the presence of both optical feedback and injection. We investigate the bifurcation diagram when the optical injection power and initial wavelength detuning are varied. We experimentally evaluate decision-making performance under different conditions of initial wavelength detuning and nonlinear dynamics.
2. Dynamics of a chaotic multimode semiconductor laser with optical feedback and injection
2.1 Experimental setup
We investigate the dynamics of optical injection into one of the longitudinal modes of a multimode semiconductor laser in the presence of optical feedback. Figure 1 shows the experimental setup for the multimode semiconductor laser with optical feedback and injection. We used a Fabry–Perot multimode semiconductor laser (Anritsu, GF5B5003DLL). The injection current and temperature of the multimode laser were set to I = 44.73 mA (= 2.0 Ith,multi) and T = 25.00 °C, respectively, where Ith,multi was the lasing threshold of the multimode laser (Ith,multi = 22.4 mA). The output from the multimode laser was reflected by a reflector and reinjected as optical feedback. The round-trip time of the feedback light between the multimode laser and the reflector was set to τ = 25.3 ns. The dynamics of the multimode laser were controlled via optical injection from an external single-mode semiconductor laser (NTT Electronics, KELD1C5GAAA). The output of the single-mode laser was stable, and the wavelength of the single-mode laser was controlled by changing the temperature of the single-mode laser. The injection current of the single-mode laser was set to I = 57.2 mA (= 5.3 Ith,single), where Ith,single was the lasing threshold of the single-mode semiconductor laser (Ith,single = 10.7 mA). The output of the single-mode laser was injected into the multimode laser. The output of the multimode laser was divided into two outputs to observe the total and modal dynamics. Four wavelength filters (WL Photonics, IW-WLTF-NE-S-1550-50/0.25-PM-0.9/1.0-FC/APC-USB, FWHM = 0.25 nm) were used to extract the four longitudinal modes. The laser output was converted into an electric signal using a photodetector (Newport, 1544-B, 12 GHz bandwidth), and the temporal waveform was observed using a digital oscilloscope (Tektronix, DPO72304SX, 23 GHz bandwidth, 50 GSample/s for four channels). In the experiment, the direct current (DC) component of the photodetectors was not eliminated in order to compare the powers of the modal intensities. The RF spectrum of the converted laser output was observed using an RF spectrum analyzer (Keysight, N9010B, 26.5 GHz bandwidth). Total output in the multimode semiconductor laser (i.e., total intensity) was observed at the fiber indicated by “Total dynamics” in Fig. 1. The optical spectrum of the multimode semiconductor laser was obtained using an optical spectrum analyzer (Yokogawa, AQ6370D). The output power of the multimode laser without optical feedback or injection was 6.9 mW.
Fig. 1. Experimental setup for multimode semiconductor laser with optical feedback and injection. ISO: isolator, VA: variable attenuator, FC: fiber coupler, Ref: reflector, Filter: wavelength filter, PD: photodetector.
2.2 Modal intensities and optical spectra
Figure 2 shows the dynamics of a multimode semiconductor laser under optical feedback (without optical injection). Optical feedback power to the multimode laser is estimated as 53 µW, and the total output power of the multimode laser is 6.9 mW. Thus, the power ratio of the feedback light to the total output of the multimode laser is κf = 0.0077. Figure 2(a) shows the temporal waveform of the total intensity, where chaotic oscillations are observed. Figure 2(b) shows the RF spectrum corresponding to Fig. 2(a). The RF spectrum has smooth and broad frequency components, with a maximum peak at a relaxation oscillation frequency of approximately 4 GHz, which is the characteristic frequency of semiconductor lasers [32]. Figure 2(c) shows the optical spectrum of the multimode semiconductor laser with optical feedback. Multiple longitudinal modes are observed in the optical spectra. The longitudinal mode spacing is 0.29 nm (36 GHz in frequency). Four neighboring high-power longitudinal modes are selected and defined as modes 1, 2, 3, and 4, as shown in Fig. 2(c). These modes have almost the same power and are located near the highest peak of the optical spectrum with optical feedback. The wavelengths of modes 1, 2, 3, and 4 are 1547.536, 1547.244, 1546.951, and 1546.661 nm, respectively. The center wavelength of each wavelength filter is set to the wavelength of each longitudinal mode to extract the modal intensities. Figure 2(d) shows the temporal waveforms of the modal intensities for modes 1, 2, 3, and 4 extracted using the four-wavelength filters. Each mode oscillates chaotically, and the temporal waveforms are different. Chaotic mode-competition dynamics are observed; that is, one mode oscillates with a large amplitude, and the other modes are suppressed (also known as chaotic antiphase dynamics [24]).
Fig. 2. Dynamics of the multimode semiconductor laser under optical feedback without optical injection. (a) Temporal waveform of total intensity. (b) RF spectrum of total intensity. (c) Optical spectrum of the multimode semiconductor laser. (d) Temporal waveforms of modal intensities for four modes (modes 1, 2, 3, and 4 are defined in Fig. 2(c)).
We control the chaotic mode-competition dynamics using optical injection. Optical injection from a single-mode semiconductor laser is applied for one of the longitudinal modes (e.g., mode 1) in the multimode laser under the optical feedback of 53 µW (κf = 0.0077). We consider the wavelength of the optical spectral peak of the single-mode semiconductor laser as λsingle. The wavelength of the spectral peak of mode 1 in the multimode laser is also defined as λmulti,1 under optical feedback. The initial wavelength detuning is defined as Δλini = λmulti,1 − λsingle, and we investigate the dynamics for different Δλini. The wavelength detuning is defined to match the sign of the frequency detuning used in the literature [32,33], where the frequency detuning is defined as Δfini = fsingle − fmulti,1.
The initial wavelength detuning is tuned by varying λsingle, which is controlled by the temperature of the single-mode laser. For example, the wavelength detuning of Δλini = −0.060 nm is obtained by setting λsingle = 1547.596 nm at the temperature of 23.00 °C when λmulti,1 is fixed at 1547.536 nm. The wavelength detuning of Δλini = 0.060 nm is obtained by setting λsingle of 1547.476 nm at the temperature of 21.91 °C.
We investigate the optical spectra and modal intensities at different initial wavelength detunings under the optical injection of 350 µW into the multimode laser with optical feedback. The power ratio of the optical injection light of the single-mode laser to the total output of the multimode laser (6.9 mW) is κinj = 0.051. In addition, the power ratio of the optical injection light to the optical feedback light in the multimode laser is κinj / κf = 6.6.
Figure 3(a) shows the optical spectrum of the multimode laser at Δλini = −0.060 nm. The power of mode 1 increases under optical injection, whereas the powers of the other modes are slightly suppressed. The mode around the wavelength of 1543.426 nm has a higher power than the other modes, which corresponds to the original peak wavelength of the multimode laser without optical feedback and injection. Figure 3(b) shows the optical spectrum at Δλini = 0.00 nm. Mode 1 has high power, and the other modes are suppressed by more than 40 dB, which indicates that the laser power is concentrated in one mode. Figure 3(c) shows the optical spectrum at Δλini = + 0.060 nm. Mode 1 has a higher power, and the other modes are suppressed. However, some side modes with high power exist, compared to Fig. 3(b). Thus, the power of the mode is enhanced via optical injection, and the suppression of the other modes is less effective, even at the same optical injection power.
Fig. 3. (a), (b), (c) Optical spectra and (d), (e), (f) temporal waveforms of modal intensities in the multimode semiconductor laser with optical feedback and injection for different initial wavelength detunings. Optical injection power to the multimode laser is estimated to 350 µW (κinj = 0.051). (a), (d) Δλini = −0.060 nm, (b), (e) Δλini = 0.00 nm, (c), (f) Δλini = + 0.060 nm.
Figure 3(d) shows temporal waveform of the modal intensities for the four modes at Δλini = −0.060 nm. The four modes compete with a small amplitude, and mode 1 has a slightly higher intensity than the other modes. Figure 3(e) shows the temporal waveform of the modal intensities at Δλini = 0.00 nm. Mode 1 is enhanced, and periodic oscillations with large amplitudes are observed. Figure 3(f) shows the temporal waveform of the modal intensities at Δλini = + 0.060 nm. Mode 1 is enhanced and chaotic oscillations are observed. Therefore, different temporal dynamics are obtained for the three different Δλini, even though the injection power is the same.
Next, we increase the optical injection power to 1030 µW (κinj = 0.15 and κinj / κf = 19.4) and observe the optical spectra and modal intensities at different initial wavelength detunings. Figure 4(a) shows the optical spectrum at Δλini = −0.060 nm. The power of mode 1 increases and the powers of the other modes are suppressed. A sharp peak is observed, unlike the weak optical injection of 350 µW in Fig. 3(a). Figures 4(b) and 4(c) show the optical spectra at Δλini = 0.00 nm and Δλini = + 0.060 nm, respectively. Mode 1 has high power, and the other modes are suppressed, as well as in Figs. 3(b) and 3(c).
Fig. 4. (a), (b), (c) Optical spectra and (d), (e), (f) temporal waveforms of modal intensities in the multimode semiconductor laser with optical feedback and injection for different initial wavelength detunings. Optical injection power to the multimode laser is estimated to 1030 µW (κinj = 0.15). (a), (d) Δλini = −0.060 nm, (b), (e) Δλini = 0.00 nm, (c), (f) Δλini = + 0.060 nm.
Figure 4(d) shows the temporal waveform of the modal intensities for four modes at Δλini = −0.060 nm. Mode 1 is stabilized and always has the maximum intensity. Figure 4(e) shows the temporal waveform of the modal intensities at Δλini = 0.00 nm. Mode 1 is enhanced and periodic oscillations with large amplitudes are observed. Figure 4(f) shows the temporal waveform of the modal intensities at Δλini = + 0.060 nm. Mode 1 is enhanced and oscillates chaotically. Therefore, different nonlinear dynamics are observed by varying the initial wavelength detuning.
Figure 5 shows the enlarged view of optical spectra of the multimode laser without and with optical injection for the three different Δλini. Figures 5(a), 5(b), and 5(c) correspond to Figs. 3(a), 3(b), and 3(c), and Figs. 5(d), 5(e), and 5(f) correspond to Figs. 4(a), 4(b), and 4(c), respectively. Figures 5(a) and 5(d) show the case of Δλini = −0.060 nm. In Fig. 5(a), the peak wavelength of mode 1 is shifted to match the wavelength of the injected light of the single-mode laser. In Fig. 5(d), one peak of the optical spectrum is obtained with the suppression of the other side modes. The peak of the multimode laser is shifted to a longer wavelength (i.e., red-shifted) by optical injection and exactly matches that of the single-mode laser, indicating injection locking (optical frequency matching). The other side modes are also shifted by the optical injection, although the peaks of the side modes are very small. The amount of the wavelength shift is 0.060 nm (7.5 GHz in frequency), corresponding to the absolute value of the initial wavelength detuning. This stable locking results in a stable laser output, as shown in Fig. 4(d).
Fig. 5. Optical spectra of the multimode semiconductor laser without (black) and with (red) optical injection. (a), (b), (c) Enlarged view of Figs. 3(a), 3(b), and 3(c). (d), (e), (f) Enlarged view of Figs. 4(a), 4(b), and 4(c). Optical injection power to the multimode laser is estimated to (a), (b), (c) 350 µW (κinj = 0.051) and (d), (e), (f) 1030 µW (κinj = 0.15). The optical wavelength detuning is set to (a), (d) Δλini = −0.060 nm, (b), (e) Δλini = 0.00 nm, (c), (f) Δλini = + 0.060 nm.
Figures 5(b) and 5(e) show the case of Δλini = 0.00 nm. The optical spectra appear similar even though the peak corresponding to the injected light is enhanced by increasing the injection power, as shown in Fig. 5(e). Two main peaks are observed in the optical spectra. The left peak corresponds to the wavelength of the injected single-mode laser, and the right peak corresponds to the wavelength of the multimode laser, which is red-shifted by 0.052 nm (6.5 GHz) by optical injection. The two peaks do not match, and injection locking is not achieved at Δλini = 0.00 nm. Other small peaks generated by four-wave mixing are observed at intervals of 0.052 nm, as shown in Fig. 5(e). In addition, the other side modes are red-shifted. The existence of these two main peaks induces a periodic temporal waveform, as shown in Figs. 3(e) and 4(e).
Figures 5(c) and 5(f) show the case of Δλini = + 0.060 nm, and the results are similar to the case of Figs. 5(b) and 5(e). Two main peaks are obtained: the left and right peaks correspond to the wavelengths of the single-mode and multimode lasers, respectively. In Fig. 5(f), the wavelength shift of the multimode laser is 0.024 nm (3.0 GHz), and the mode spacing between the two peaks is 0.084 nm (10.5 GHz). Injection locking is not achieved, and the interaction between the two peaks results in four-wave mixing and chaotic oscillations of the laser intensity, as shown in Figs. 3(f) and 4(f).
From these results, different optical spectra and temporal dynamics are observed for different wavelength detunings. For negative wavelength detuning, injection locking is achieved at a longer wavelength (red-shifted), and a stable temporal output is obtained when the optical injection power is sufficiently large. For zero and positive wavelength detuning, injection locking is not achieved, and periodic or chaotic oscillations are observed. The peak wavelength of the multimode laser cannot be shifted to a shorter wavelength (blue-shifted) by optical injection, and multiple peaks appear in the optical spectrum. This asymmetric behavior results from the linewidth enhancement factor (known as the α parameter) of the semiconductor laser [32,33].
2.3 Mode power ratio and dominant mode ratio
We measure the interaction between the longitudinal modes using two quantities when the optical injection power for mode 1 is changed at different initial wavelength detunings. The first quantity is the mode power ratio (MPR), which is defined as the difference in the peak power in the optical spectrum as follows:
where Pmain is the maximum peak power of mode 1 and Pside is the maximum peak power of the side modes, except for the main mode in the optical spectra. The power of the main mode is defined in the range of 1547.260–1547.810 nm to obtain Pmain; therefore, multiple peaks within mode 1, as shown in Figs. 5(e) and 5(f), are not considered as side modes. For example, in Fig. 5(f), MPR of 39.80 dB is obtained from Pmain = −10.12 dB at 1547.476 nm and Pside = −49.92 dB at 1548.432 nm.Figure 6(a) shows MPR for the three initial wavelength detunings (Δλini = −0.060, 0.00, and +0.060 nm) when the optical injection power for mode 1 is increased. Optical spectra are acquired three times at each wavelength detuning and injection power. The average MPR is estimated from the optical spectra obtained by varying the injection power. The MPR increases with the injection power. However, the curves are different between the positive and negative detunings (i.e., Δλini = −0.060 and +0.060 nm) even though the absolute value of the initial wavelength detuning is the same, which indicates that there are asymmetric characteristics regarding the initial wavelength detuning. The MPR gradually increases with injection power for positive wavelength detuning. One of the longitudinal modes can be excited more easily by supplying power from the optical injection light to the longitudinal mode for positive wavelength detuning. However, in the case of negative wavelength detuning, the MPR increases rapidly with the large injection power. Mode concentration does not occur under a small injection power. When the injection power reaches the threshold for injection locking, a rapid mode concentration is observed, and a large MPR is obtained. Thus, higher power is required to achieve a large MPR for negative wavelength detuning because of injection locking. For a zero-detuning case (Δλini = 0.00 nm), rapid transition of MPR is observed with a small injection power without injection locking.
Fig. 6. (a) Mode power ratio (MPR) and (b) dominant mode ratio (DMR) of mode 1 for the three different initial wavelength detunings of −0.060 nm (black), 0.00 nm (red), and +0.060 nm (blue) when optical injection power is varied. Average value is plotted as a dot, and maximum and minimum values are indicated as error bar. The injected power is adjusted using a variable attenuator, so that the minimum injection power of 3 µW is added even at the lowest injection power (leftmost point in the figure).
When we focus on the value of the maximum MPR, it reaches more than 50 dB for the negative wavelength detuning (Δλini = −0.060 nm), which indicates that the modal power is efficiently concentrated on one of the longitudinal modes. This is because all laser powers are concentrated on a sharper spectrum by injection locking, as shown in Fig. 5(d). However, for the positive wavelength detuning (Δλini = + 0.060 nm), the maximum MPR is limited to approximately 40 dB. This is because the suppression level of the other modes is lower than that of the negative detuning, and the laser power is distributed to the coexistence of the two peaks of the injected light and the multimode laser. Thus, the asymmetric characteristics appear in the process of mode concentration in the optical spectra, because of the nonzero α parameter of the semiconductor laser [32,33].
The second quantity is the dominant mode ratio (DMR), which represents the ratio of the dominant mode (i.e., the mode with the maximum modal intensity) by comparing the temporal waveforms of the modal intensities [17,25,29]. The DMR for mode m (DMRm) is defined as:
where S denotes the total number of sampled points. Dm(j) is 1 if mode m is the dominant mode at the j-th sampled point and 0 otherwise.Figure 6(b) shows the DMR of mode 1 at three wavelength detunings when the optical injection power of mode 1 is changed. Temporal waveforms of S = 5 × 105 points (10 µs in time at the sampling interval of 20 ps) for each mode are used to calculate DMR. Temporal waveforms are acquired ten times at each wavelength detuning and injection power, and the average of the DMRs is shown. The DMR increases with the injection power. However, there are asymmetric characteristics in the DMR between the positive and negative detunings. For the positive detuning (Δλini = + 0.060 nm), DMR is large at small injection power, and DMR reaches 1 by slightly increasing the injection power. However, for the negative detuning (Δλini = −0.060 nm), DMR increases gradually as the injection power increases. This characteristic agrees with the MPR results shown in Fig. 6(a).
We investigate the MPR and DMR when the initial wavelength detuning and optical injection power are changed continuously. Figure 7(a) shows the two-dimensional map of MPR. The MPR increases with the injection power for all initial wavelength detunings. However, a higher injection power is required to increase the MPR for negative wavelength detuning. In addition, high MPR values are achieved for negative wavelength detuning under strong optical injection. By contrast, for positive wavelength detuning, the MPR rapidly increases with a small injection power. However, the MPR is limited to less than 50 dB.
Fig. 7. Two-dimensional maps for (a) MPR and (b) DMR of mode 1 when initial wavelength detuning and optical injection power are changed. The injection power of 1000 µW corresponds to κinj = 0.145. The initial wavelength detuning of 0.1 nm in wavelength corresponds to 12.5 GHz in frequency.
Figure 7(b) shows a two-dimensional map of the DMR. For negative wavelength detuning, a high optical injection power is required to increase the DMR. By contrast, for positive wavelength detuning, a large DMR is achieved, even for a small injection power.
From these results, the asymmetric properties of MPR and DMR are evident in terms of optical wavelength detuning. More injection power is required for negative wavelength detuning because of the injection-locking condition, whereas a small injection power results in effective mode concentration for positive wavelength detuning without using injection locking. However, larger MPR and DMR values are obtained for negative detuning than for positive detuning. The use of injection locking at negative detunings is not always required to improve the performance of certain applications, such as decision making, as described in Section 3.
2.4 Two-dimensional bifurcation diagram
We experimentally investigate a two-dimensional bifurcation diagram of the total intensity dynamics when the initial wavelength detuning and optical injection power are changed continuously. Temporal waveforms and RF spectra of the total intensity are observed when the injection power is increased under a fixed initial wavelength detuning, and the bifurcation points of the laser dynamics are recorded. A two-dimensional bifurcation map is created by repeating this procedure and changing the initial wavelength detuning.
These dynamics can be classified based on the characteristics of the temporal waveforms and RF spectra [32]. Stable outputs can be identified using temporal waveforms with noise-level fluctuations and RF spectra with no peaks at the noise level. Periodic oscillations can be determined from the temporal waveforms in which sinusoidal oscillations are observed, and the RF spectra exhibit sharp peaks. Relaxation oscillations (RO) can be found using temporal waveforms with a smaller amplitude than the periodic oscillation and RF spectra with a non-sharp small peak. Quasi-periodic oscillations can be identified by temporal waveforms with oscillations with multiple irrational frequencies and RF spectra with multiple discrete peaks and irregular frequency intervals. Chaotic oscillations can be determined using temporal waveforms with irregular oscillations and RF spectra with smooth and broadband frequencies.
Figure 8 shows a two-dimensional bifurcation diagram of a multimode semiconductor laser without optical feedback under optical injection. The horizontal axis represents the injection power and the vertical axis represents the initial wavelength detuning. The injection power is changed from 0 to 1000 µW (0.0 to 0.145 in κinj). λmulti,1 is defined as the wavelength of mode 1 in the absence of optical feedback. The total intensity of the multimode laser exhibits a stable output when the injection power is increased for negative wavelength detuning. However, for positive wavelength detuning, period-1 oscillations are observed over a wide range of positive wavelength detunings, and period-2 oscillations are observed inside the region of period-1 oscillations. Relaxation oscillations are observed between the regions of period-1 oscillations and stable outputs. Period-doubling bifurcation and chaotic oscillations have been observed in a single-mode semiconductor laser with optical injection when the injection power and initial wavelength detuning is changed [30,31,33]. We observe a very small region of chaotic oscillations within the region of period-2 oscillations in Fig. 8 (not shown).
Fig. 8. Two-dimensional bifurcation diagram of the intensity dynamics when the initial wavelength detuning (vertical axis) and optical injection power (horizontal axis) are continuously changed without optical feedback. The injection power of 1000 µW corresponds to κinj = 0.145. The initial wavelength detuning of 0.1 nm in wavelength corresponds to 12.5 GHz in frequency.
We also investigate a two-dimensional bifurcation diagram of the total intensity dynamics under weak optical feedback power. Figure 9 shows the two-dimensional bifurcation diagram when the optical feedback power is set to 14 µW (κf = 0.0020). λmulti,1 is defined as the wavelength of mode 1 under optical feedback and without optical injection. Stable outputs (blue) are observed for a wide range of negative wavelength detunings. The stabilization of the chaotic oscillations of the multimode semiconductor laser results from injection locking, where the wavelength of the multimode laser matches that of the injected light of the stable single-mode semiconductor laser. The injection-locking region is observed for negative wavelength detuning, similar to the case of single-mode semiconductor lasers [32]. Period-1 oscillations (light green) are observed over a wide range of positive wavelength detunings. Period-2 and quasi-periodic oscillations are also observed. Chaotic oscillations (red) are observed for large absolute values of wavelength detuning under weak optical feedback.
Fig. 9. Two-dimensional bifurcation diagram of the total intensity dynamics when the initial wavelength detuning (vertical axis) and optical injection power (horizontal axis) are continuously changed under optical feedback of 14 µW (κf = 0.0020). The injection power of 1000 µW corresponds to κinj = 0.145. The initial wavelength detuning of 0.1 nm in wavelength corresponds to 12.5 GHz in frequency.
Figure 10 shows the two-dimensional bifurcation diagram under the strong optical feedback power of 53 µW (κf = 0.0077) and with optical injection. Bifurcation from chaos (red) to other dynamics occur when the optical injection power is increased. The dynamics are stabilized (blue) in a narrower range of negative wavelength detunings and a larger injection power compared to those in Fig. 9. By contrast, period-1 oscillations (light-green) are observed near zero with positive wavelength detuning. This is because of the coexistence of the peaks of the injected light and the mode of the multimode laser without injection locking, as observed in Fig. 5(b), where the beat frequency of the two-peak spectra appears as periodic oscillations. In addition, mixed-dynamics regions are widely distributed between the chaotic, stable, and period-1 regions. The bifurcation diagram is similar to that shown in Fig. 9, although the injection powers at the bifurcation points are different. It is worth noting that mixed dynamics are more likely to be observed under strong optical feedback.
Fig. 10. Two-dimensional bifurcation diagram of the total intensity dynamics when the initial wavelength detuning (vertical axis) and optical injection power (horizontal axis) are continuously changed under optical feedback power of 53 µW (κf = 0.0077). The injection power of 1000 µW corresponds to κinj = 0.145. The initial wavelength detuning of 0.1 nm in wavelength corresponds to 12.5 GHz in frequency.
Figure 11 summarizes examples of mixed dynamics in the two-dimensional bifurcation diagram shown in Fig. 10. In the mixed dynamics, the temporal waveforms and RF spectra change over time at a slower time scale (seconds) than in the original dynamics. Figures 11(a), 11(b), and 11(c) show examples of Chaos + Stable (C + S in Fig. 10). The temporal waveforms of the chaotic oscillations and stable output are observed at different times in Figs. 11(a) and Fig. 11(b), respectively. In Fig. 11(c), the RF spectrum suddenly changes at 3.74 GHz from a constant (stable output) to a broad (chaotic oscillations) spectrum, which shows the change in the two dynamics. Figures 11(d), 11(e), and 11(f) show examples of Period-1 + Relaxation Oscillation (P1 + RO). The temporal waveforms in Figs. 11(d) and 11(e) exhibit period-1 and relaxation oscillations with smaller amplitudes, respectively. In this case, the amplitudes changes at different times. The RF spectrum in Fig. 11(f) exhibits a sharp peak (periodic oscillations) with a small broad spectrum (relaxation oscillations). This sharp peak in the RF spectrum appears and disappears over time. Figures 11(g), 11(h), and 11(i) show examples of Chaos + Period (C + P). Chaotic and periodic oscillations are shown in Figs. 11(g) and 11(h), respectively. In Fig. 11(i), the RF spectrum has broad components (chaotic oscillations) with a very sharp peak (period-1 oscillations).
Fig. 11. Examples of the mixed dynamics in the multimode semiconductor laser with optical feedback and injection. Optical feedback power is fixed at 53 µW. (a), (b), (c) Chaos + Stable (C + S). Optical injection power is 210 µW and Δλini = −0.030 nm. (d), (e), (f) Period-1 + Relaxation Oscillation (P1 + RO). Optical injection power is 690 µW and Δλini = −0.030 nm. (g), (h), (i) Chaos + Period (C + P). Optical injection power is 240 µW and Δλini = + 0.050 nm. (a), (b), (d), (e), (g), (h) Temporal waveforms at different times. (c), (f), (i) RF spectra.
We speculate that the mixed dynamics in Figs. 9 and 10 result from the phase fluctuations between the optical injection and feedback light. The coexistence of the optical injection and feedback light results in interference within the laser cavity, and phase fluctuations would change the dynamics over time. In fact, we have found numerically that the temporal dynamics are strongly influenced by changes in the phase fluctuations and small wavelength detuning corresponding to the inverse of the round-trip time in the external cavity of the optical feedback [29]. The fluctuations in the optical linewidth in single-mode semiconductor lasers and the phase fluctuations of the optical feedback in multimode semiconductor lasers may result in the mixed dynamics observed in the experiment shown in Fig. 10.
3. Experiment for decision making
3.1 Experimental setup for decision making
In this section, we experimentally investigate the decision-making process for solving the MAB using a multimode semiconductor laser with optical feedback and injection. Figure 12 shows the experimental setup for decision making using a multimode semiconductor laser. Each longitudinal mode was assigned to a slot machine, and the slot machine corresponding to the dominant mode was selected for decision making [17]. Four single-mode lasers were used to control the modal intensity dynamics of the four modes. The dominant mode changed spontaneously with chaotic mode-competition dynamics, which were used for the exploration of the slot machine with maximum hit probability. The dynamics were controlled via optical injection, which was used for exploitation in MAB. Four slot machines were emulated in a computer, and online decision making was performed experimentally. The temporal waveform of each mode was obtained for 1000 points using an oscilloscope and transferred to a computer. The modal intensities at the sampled points in the temporal waveforms were compared and the mode with the maximum intensity was determined to be the dominant mode. Subsequently, a slot machine corresponding to the dominant mode was selected. The result of the slot-machine selection (hit or miss) was obtained according to the hit probability of the selected slot machine in the computer, and the optical injection power from the single-mode lasers was controlled according to the result (see the Appendix for the algorithm). The injection power from a single-mode semiconductor laser could be adjusted by controlling the voltage of a variable electric attenuator. The computer, oscilloscope, and voltage controller were used to control the injection power and connected by LAN cables for online control. The single-mode semiconductor laser m was used to excite mode m in the multimode semiconductor laser, and its peak wavelength was defined as λsingle,m. Mode m was excited by adjusting λsingle,m close to the peak wavelength of mode m (λmulti,m) in the presence of optical feedback. The initial wavelength detuning for mode m was defined as Δλini,m = λmulti,m − λsingle,m, and the initial wavelength detunings for four modes were set to the same value for decision making. We conducted decision making under the optical feedback power of 53 µW (κf = 0.0077) in the multimode semiconductor laser.
Fig. 12. Experimental setup for decision making using the multimode semiconductor laser with optical feedback and injection. ISO: isolator, VA: variable attenuator, FC: fiber coupler, Ref: reflector, Filter: wavelength filter, PD: photodetector. Four modes in the multimode semiconductor laser are assigned to four slot machines. Result of slot machine selection is emulated in a computer. Optical injection power from four single mode lasers is controlled based on the result of slot machine selection.
3.2 Dominant mode ratios and decision-making performance
In this section, we focus on three different wavelength detunings of Δλini,m = −0.060, 0.00, and +0.060 nm, which show stable output, periodic oscillation, and chaotic oscillation when optical injection power is increased (see Figs. 3 and 4). We conduct decision making at the three wavelength detunings when we set the same initial wavelength detunings for all modes. We investigate the DMR when the injection power for mode 1 is increased in the presence of optical injection.
Figure 13 shows the DMRs for the four modes when the injection power for mode 1 is changed. The injection power for mode 2, 3, and 4 are fixed at 9 µW (κinj,2,3,4 = 0.001). DMR is obtained from the temporal waveforms of 5 × 105 points (10 µs in time) for each mode. The average of ten DMRs is shown, and their maximum and minimum values are indicated by error bars. Figures 13(a), 13(b), and 13(c) show DMRs for the wavelength detunings of Δλini,m = −0.060, 0.00, and +0.060 nm, respectively. In all cases, the DMR of mode 1 increases as the injection power increases, and the dominant mode can be controlled. Comparing Figs. 13(a) with 13(b), the average DMRs appear to be similar. However, the large error bars at low injection powers in Fig. 13(b) show that the DMRs fluctuate between 0 and 1 and the DMR changes significantly over time. In fact, one of the modes has the maximum intensity for more than 10 µs with periodic oscillations, and the dominant mode changes slowly. This indicates that mode competition occurs at a time scale slower than the GHz chaotic oscillations. In the case of Δλini,m = + 0.060 nm in Fig. 13(c), the DMR of mode 1 increases rapidly with less injection power than the cases in Figs. 13(a) and 13(b). In addition, the DMR of mode 1 is larger than that of the other modes under a small injection power, as shown in Fig. 13(c). This rapid convergence of the DMR of mode 1 may result from the coexistence of the injected and original modes without injection locking at positive wavelength detunings.
Fig. 13. DMRs for different wavelength detunings when optical injection power for mode 1 is increased in the presence of optical injection for four modes. Optical injection power for mode 2,3,4 is fixed at 9 µW (κinj,2,3,4 = 0.001). (a) Δλini,m = −0.060 nm, (b) Δλini,m = 0.00 nm, (c) Δλini,m = + 0.060 nm.
Next, we perform decision making under different wavelength detunings when the hit probability of slot machine 1 is set to 0.7, and the hit probabilities of slot machines 2, 3, and 4 are set to 0.3. Figure 14 shows the modal intensities for the four modes (Figs. 14(a), 14(b), and 14(c)) and the corresponding slot machine selection (Fig. 14(d), 14(e), and 14(f)) as a function of the number of plays. Figures 14(a) and 14(d) show modal intensities and slot machine selection for each play at Δλini,m = −0.060 nm. The dominant mode among the four modes changes up to approximately the 10th play, as shown in Fig. 14(a), and the slot machines are randomly selected, as shown in Fig. 14(d). By contrast, the modal intensity of mode 1 has the highest intensity and is stabilized at a value of 0.08 in Fig. 14(a). As the number of plays increases, slot machine 1 is continuously selected in Fig. 14(d) after around the 10th play. Figures 14(b) and 14(e) show the modal intensities and slot machine selection for each play at Δλini,m = 0.00 nm, and Figs. 14(c) and 14(f) show the case at Δλini,m = + 0.060 nm. The four modes compete when the number of plays is small and the slot machines are selected randomly. Mode 1 is enhanced as the number of plays increases; however, mode 1 oscillates irregularly, unlike that shown in Fig. 14(a). Mode 1 has the largest modal intensity compared with the other modes after around the 20th play, which allows slot machine 1 to be continuously selected.
Fig. 14. Results of decision making when the hit probability of slot machine 1 is set to 0.7 and hit probabilities of slot machines 2, 3, and 4 are set to 0.3. (a), (b), (c) modal intensities, (d), (e), (f) slot machine selection as a function of the number of plays. (a), (d) Δλini,m = −0.060 nm, (b), (e) Δλini,m = 0.00 nm, (c), (f) Δλini,m = + 0.060 nm.
To evaluate the decision-making performance, we perform multiple cycles of decision making and evaluate the correct decision rate. Correct decision rate is defined as the rate at which the slot machine with the highest hit probability is selected for each play when multiple decision-making cycles are performed. The correct decision rate for the t-th play is described as follows [9,10].
where N is the total number of decision-making cycles (N = 100), C(i, t) is a function that takes the value of 1 if the slot machine with the highest hit probability is selected at the t-th play in the i-th cycle, and 0 otherwise.Figure 15(a) shows the correct decision rates at the three different wavelength detunings of Δλini,m = −0.060, 0.00, and +0.060 nm. In all cases, the correct decision rate increases as the number of plays increases, and the slot machine with the highest hit probability can be selected correctly as a result of the exploration. The curves of the correct decision rates for Δλini,m = −0.060 nm and Δλini,m = 0.00 nm look similar. However, the curve of the correct decision rate for Δλini,m = + 0.060 nm saturates faster than those for the other wavelength detunings.
Fig. 15. (a) Correct decision rates for the three different wavelength detunings of −0.060 nm (red), 0.00 nm (blue), and +0.060 nm (green) when the hit probability of slot machine 1 is set to 0.7 and hit probabilities of slot machine 2, 3, and 4 are set to 0.3. (b) DMRs of mode 1 when optical injection power for mode 1 is increased in the presence of optical injection for four modes. Average values of DMRs of mode 1 in Fig. 13 are replotted.
We also compare the DMRs when the optical injection power for mode 1 increases during the decision-making process. Figure 15(b) shows the DMRs of mode 1 for different wavelength detunings (the average DMRs of mode 1 in Fig. 13 are replotted). The DMR for Δλini,m = + 0.060 nm increases with less injection power, and the DMRs for Δλini,m = −0.060 nm and Δλini,m = 0.00 nm are close to each other. Comparing Figs. 15(a) with 15(b), the fast convergence of the DMR corresponds to the fast convergence of the correct decision rate. Correct decision rate increases rapidly at Δλini,m = + 0.060 nm in Fig. 15(a), because DMR saturates rapidly in Fig. 15(b), which accelerates the selection of the best slot machine. Therefore, the fast mode-concentration property results in fast decision making.
4. Discussions
In single-mode semiconductor lasers, injection locking can be achieved via negative wavelength detuning and is used to match the optical wavelength [32]. We found that fast mode concentration can be achieved without injection locking in positive wavelength detunings. Therefore, positive wavelength detuning plays an important role in achieving an effective mode concentration in a multimode semiconductor laser even without injection locking. We also observe complex mixed dynamics in the presence of optical injection and feedback light. We speculate that these mixed dynamics result from phase fluctuations between the optical injection and feedback light.
One play of the slot machine selection requires 0.3 s in this decision-making experiment because of the feedback configuration using a computer, voltage controller, and digital oscilloscope. The main bottleneck in the processing speed of decision making is the acquisition time of temporal waveforms in the oscilloscope and communication time using LAN cables among the measurement equipment and computer. The maximum modulation speed of the variable electric attenuator is 1 kHz, which is another bottleneck in improving the processing speed of decision making. The use of a photonic integrated circuit with an FPGA would help reduce the processing time for decision making. The potential of high-speed laser dynamics can be utilized to improve decision-making performance in ultrafast photonic computing and will be addressed in future work.
DMR has an impact on decision making, and no significant difference in correct decision rates appears for Δλini,m = −0.060 nm and Δλini,m = 0.00 nm. The numerical simulations in Ref. [29] show that the DMR has a mode dependence because it is sensitive to the change in the optical feedback phase. It is required to carefully adjust the optical frequency (wavelength) detuning within the frequency corresponding to 1/τ to obtain a similar curve of DMR for each mode. However, we have experimentally confirmed that the curves of the DMRs for the other modes are similar to those shown in Fig. 13. In our experiment, the inverse of the feedback delay time 1/τ is 39.6 MHz, which is smaller than the resolution of the wavelength controller (0.001 nm in wavelength and 125 MHz in frequency). We consider that fluctuations in the optical linewidth and the optical feedback phase exist in the semiconductor lasers in the experiment, which induce the occurrence of mixed dynamics, as shown in Figs. 9 and 10. Therefore, the curve of the DMRs is averaged, and similar DMRs are obtained for the four modes in the experiment.
5. Conclusions
We experimentally investigated the control of the mode-competition dynamics in a chaotic multimode semiconductor laser. The dynamics of the optical injection for one of the longitudinal modes in the multimode laser were investigated, and it was found that the laser power was focused on one longitudinal mode with a smaller optical injection power when the initial wavelength detuning was positive. We experimentally investigated two-dimensional bifurcation diagrams of the total intensity dynamics when the initial wavelength detuning and injection power were continuously changed. We observed complex dynamics in the bifurcation diagram, including mixed dynamics. The region of mixed dynamics was enhanced, and the regions of period-1 oscillation and stable output were diminished by increasing the optical feedback power.
We experimentally conducted decision making for four slot machines using four longitudinal modes. A slot machine corresponding to the dominant mode was selected, and the chaotic mode-competition dynamics were controlled based on the result of the slot machine selection. Decision making was conducted at different wavelength detunings (negative, zero, and positive), which exhibited stable output, periodic oscillation, and chaotic oscillation as the injection power increased. Correct decision making was achieved, and positive wavelength detuning resulted in faster convergence of the correct decision rate because the power was concentrated in one of the longitudinal modes with a smaller injection power. The fast mode concentration property at positive wavelength detuning could be useful for accelerating effective decision making.
Appendix
We explain the algorithm of the tug-of-war method for solving the MAB, which is inspired by the behavior of amoebas [34,35]. To control the optical injection power, we change the m-th evaluation value Xm(t) at the t-th play in the tug-of-war method, which corresponds to the evaluation of slot machine m at the t-th play [17,34,35]. The initial value is set as Xm(0) = 0. If slot machine s is selected and a hit is obtained at the t-th play, Xm(t) of each slot machine m is changed as follows:
Funding
Japan Society for the Promotion of Science (JP19H00868, JP20K15185, JP22H05195).
Disclosures
The authors declare no conflicts of interests.
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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