1. Introduction

Photonic accelerators, which accelerate specific information processing using light, have been widely studied as novel computational technologies in the post-Moore era [1]. Photonic technology has been used for artificial intelligence (AI) such as photonic neural networks [2], coherent Ising machines [3], photonic reservoir computing [4–6], and photonic decision making [7–10]. The use of semiconductor laser dynamics for the development of photonic AI technologies is promising.

The technique of controlling chaos has been applied to many nonlinear dynamical systems that stabilize chaotic outputs into a steady state or periodic output [11,12]. The concept of controlling chaos, referred to as the OGY method, was first proposed by Ott et al. [11] and numerous studies on controlling chaos have been reported [12,13]. Controlling chaos in lasers has been experimentally achieved [14], and chaotic oscillations in lasers can be stabilized into steady state or periodic outputs. The techniques of controlling chaos in lasers have been applied to the stabilization of chaos into high periodic oscillations [15], suppression of relative intensity noise [16], and dynamic associative memory [17]. The diversity of controlling chaos could be useful for achieving advanced information processing in AI, because its complex behavior and controllability allow spontaneous exploration and exploitation functions in reinforcement learning [18]. Thus, the evaluation of controlling chaos in lasers is of highly significant.

Numerous studies have been conducted on the dynamics of semiconductor lasers in the presence of optical feedback [19–22] or injection [23–25]. Various nonlinear dynamics and bifurcation phenomena have been reported in the literature. However, studies on the dynamics of semiconductor lasers in the presence of both optical feedback and injection are limited compared with those on semiconductor lasers with either of them. Particularly, the bandwidth enhancement of chaotic oscillations has been studied in semiconductor lasers with optical feedback and injection [26]. Moreover, semiconductor lasers with optical feedback and injection have been used for photonic reservoir computing [27–29]. The performance of photonic reservoir computing in these systems is sensitive to the phase of the feedback and injection light because a change in the optical phase results in different dynamics [27,28]. The optical feedback phase strongly affects the dynamics of a semiconductor laser in the short external cavity regime [30].

Various studies on nonlinear dynamics have been reported for multimode (Fabry–Perot) semiconductor lasers with multiple longitudinal modes. Spontaneous mode hopping is induced by optical injection in a multimode semiconductor laser [31,32]. Modal and total intensity dynamics in the low-frequency fluctuation (LFF) regime have been studied in multimode semiconductor lasers with optical feedback [33–38]. The interaction of the longitudinal modes plays an important role in the LFF of multimode semiconductor lasers, which indicates the existence of an anti-correlated interaction among the modal intensities (called anti-phase dynamics [12,39]). Chaotic antiphase dynamics have been experimentally observed in multimode semiconductor lasers with optical feedback [39]. Moreover, the mode with the maximum intensity (i.e., the dominant mode) competes chaotically, and can be adaptively selected from the chaotic mode-competition dynamics by changing the optical feedback strength or injection current [40].

However, control of chaotic mode-competition dynamics via optical injection has not yet been studied. A control technique for chaotic mode-competition dynamics in a multimode semiconductor laser can be applied to photonic information processing to solve the multi-armed bandit problem in reinforcement learning [18]. Additionally, multimode semiconductor lasers have been used for photonic reservoir computing to process multiple tasks in parallel using multimode dynamics [29]. Therefore, they are expected to have a high potential as photonic accelerators for computing applications.

In this study, we numerically evaluate the chaotic mode-competition dynamics in a multimode semiconductor laser with optical feedback and injection. We introduce a technique for controlling the dominant mode in a multimode laser by injecting optical signals from a stable single-mode semiconductor laser. We also evaluate the relationship between the parameter regions of the large dominant mode ratios and injection-locking range.

2. Numerical model

Figure 1 shows the schematic of our numerical model for a multimode semiconductor laser with optical feedback and injection. We consider a single-mode semiconductor laser for optical injection. The mode-competition dynamics in a multimode semiconductor laser with optical feedback are controlled by exciting the longitudinal mode m with modal frequency ν_m via optical injection from a single-mode semiconductor laser. The optical frequency of the single-mode semiconductor laser is defined as f_m, whose frequency is near ν_m. The initial optical frequency detuning between the injection light and injected mode is defined as Δf_m = f_m − ν_m.

 

Fig. 1. Schematic of our numerical model for the multimode semiconductor laser with optical feedback and injection. Light from a single-mode semiconductor laser is injected into a multimode semiconductor laser with optical feedback to control one of the longitudinal modes in the multimode laser.

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We use a numerical model described by the Lang–Kobayashi equations [12,41,42], which are well-known rate equations for a single-mode semiconductor laser with optical feedback. The Lang–Kobayashi equation can be extended to multiple longitudinal modes [37,40,42]. The effect of optical injection from a single-mode semiconductor laser is added [12,26,42]. The numerical model of a multimode semiconductor laser with M longitudinal modes under optical injection is described as follows

The initial conditions of the modal amplitude and carrier density are set to 1.3 × 10¹⁰ and 1.9 × 10²⁴, respectively, which are of the same order as the steady-state solutions. We numerically integrate the rate equations using the fourth-order Runge–Kutta method. Temporal waveforms are obtained after calculating 5000 ns to remove the transient response. The modal gain is assumed to have a parabolic gain profile, as described in Eq. (3), where the peak value of the gain coefficient is set to G_N = 8.40 × 10⁻¹³. In our numerical simulations, we do not include Langevin noise to evaluate the deterministic mechanism of mode-competition dynamics.

Table 1. Parameter values used in numerical simulations

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We consider the mode coupling of the electric-field amplitudes via the total carrier density (the term with Σ in Eq. (2)). Additionally, we use different gain coefficients for the different modes, as described in Eq. (3). We also include the self-saturation effect of the electric field amplitude (terms with ε) in Eqs. (1) and (2). However, there is no mode coupling term via the electric field amplitude, such as the cross saturation in Eq. (1). This is a simpler model than the existing models in the literature, considering the cross-saturation effect of carrier density gratings [43] and intra-band nonlinearities related to spectral hole burning [44]. In this study, we generate mode-competition dynamics with a minimum mode-coupling effect using a simple model.

The physical origin of chaotic mode coupling in our model is the energy consumption of each modal intensity from the common carrier density. Chaotic dynamics occur in a semiconductor laser with optical feedback owing to the nonlinear interaction between the electric field amplitude and carrier density. The longitudinal modes compete with each other via the carrier density to obtain the energy for lasing. This lasing energy competition is perturbed by optical feedback, and chaotic mode-competition dynamics occur. Moreover, a larger mode spacing results in a weaker coupling effect for the edge modes, because the coupling strength among the modes depends on the gain profile.

The electric field for each mode in Eq. (1) is expanded from the central frequency of each mode, and the envelope approximation is based on the angular frequency ω_m of the m-th mode. The optical phase is obtained from the real and imaginary parts of the electric field, i.e., ϕ_m(t) = tan^-1(E_m,im(t) / E_m,re(t)), where E_m,re(t) and E_m,im(t) are the real and imaginary parts of the complex electric field amplitude of the laser output, respectively. To obtain the phase difference between the central mode and phase of the m-th mode, we can add the term exp(–i(ω_m – ω_c)) to ϕ_m(t) after the numerical integration of the rate equations is carried out. The optical phase of the total intensity ϕ_total(t) can be obtained from the coherent sum of the electric field for each mode E_total(t) = ΣE_m(t), i.e., ϕ_total(t) = tan^-1(E_total,im(t) / E_total,re(t)). The optical frequency detuning of the total intensity can be calculated from ϕ_total(t), and it moves between the different modes, reported as chaotic itinerancy in [18].

3. Numerical results

3.1 Mode-competition dynamics with respect to optical injection

We evaluate the temporal waveforms with and without optical injection under optical feedback. Figure 2 shows the numerical results of the temporal waveforms of the multimode semiconductor laser. Figure 2(a) shows the five modal intensities when only optical feedback is applied (i.e., no optical injection). The optical feedback strength is set to κ = 4.411 ns^-1. In Fig. 2(a), each modal intensity exhibits chaotic oscillation; however, the oscillation is different for each mode. We define the mode with the maximum intensity as the dominant mode, which changes over time and chaotic mode-competition dynamics occur. Figure 2(b) shows the total intensity corresponding to Fig. 2(a). The total intensity exhibits chaotic oscillation.

 

Fig. 2. Temporal waveforms of multimode semiconductor laser with optical feedback and injection. (a) Modal intensities when optical feedback is only applied without optical injection. (b) Total intensity corresponding to (a). (c) Modal intensities when optical injection is applied to mode 3 at κ_inj,3 = 6.0 ns^-1 under optical feedback. (d) Modal intensities when optical injection is applied to mode 1 at κ_inj,1 = 6.0 ns^-1 under optical feedback. Initial optical frequency detuning is fixed at Δf_m = –4.0 GHz in (c) and (d).

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Figure 2(c) shows the modal intensities when the optical injection is applied to mode 3 with κ_inj,3 = 6.0 ns^-1 under optical feedback. Here, the initial optical frequency detuning is fixed at Δf_m = –4.0 GHz. Mode 3 (green curve) oscillates with a large amplitude owing to the optical injection into mode 3, and the duration of the dominant mode for mode 3 is longer than that for the other modes. In other words, mode 3 is excited by optical injection. Figure 2(d) shows the modal intensities when the optical injection is applied to mode 1 with κ_inj,1 = 6.0 ns^-1 under optical feedback. In Fig. 2(d), the oscillation of mode 1 (red curve) is suppressed compared to that of the other modes, and the duration of the dominant mode for mode 1 is shorter than that of the other modes. From Figs. 2(c) and 2(d), the behaviors of the mode-competition dynamics are different among the injected modes, even though the injection strength is set to the same value.

We determine the change in the dominant mode to evaluate the mode-competition dynamics quantitatively when the optical injection strength is changed. The dominant mode ratio is defined as the ratio of the dominant mode of mode m over a long period [40]. The dominant mode ratio DMR_m for mode m is expressed as follows:

Figure 3 shows the dominant mode ratio of mode m when optical injection is applied only to mode m and the optical injection strength for mode m (κ_inj,m) is changed. The initial optical frequency detuning between the injection light and injected mode are set to Δf_m = –4.0 GHz for all the modes. In Fig. 3, the dominant mode ratio changes differently for each mode; however, the characteristics of the dominant modes for modes 1 and 5 are similar. Particularly, the dominant mode ratio for mode 3 increases as the optical injection strength increases, whereas those of modes 1, 2, and 5 decrease to nearly zero for a small κ_inj,m. The dominant mode ratio eventually reached 1 for all the modes when the optical injection strength is sufficiently large. A large injection strength results in the growth of the dominant mode; however, the characteristics of the dominant mode ratio in terms of the optical injection strength differ among the modes, as shown in Fig. 3.

 

Fig. 3. Comparison of dominant mode ratios of mode m as a function of optical injection strength for mode m when optical injection is applied for only mode m under optical feedback. Initial optical frequency detuning for each mode is Δf_m = –4.0 GHz.

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3.2 Control of dominant mode ratio

In this subsection, we evaluate the effect of the chaotic mode-competition dynamics by changing the optical feedback and injection phases. We replace the term for the optical feedback phase of mode 3 in Eq. (1) from ω₃τ to ω₃τ + Φ₃, where Φ₃ is the phase shift and an optical injection signal to mode 3 is applied. Figure 4(a) shows the dominant mode ratios of mode 3 as a function of Φ₃ at different optical injection strengths for mode 3 (κ_inj,3). The initial optical frequency detuning between the injection light and mode 3 is fixed at Δf₃ = –4.0 GHz. The dominant mode ratio of mode 3 changes as Φ₃ changes for different κ_inj,3. Particularly, the maximum dominant mode ratio is observed near Φ₃ = 0 and 2π, whereas the minimum value is obtained near Φ₃ = π. Thus, the dominant mode ratio strongly depends on the optical feedback phase Φ₃. From these results, we consider that the difference in the characteristics of the dominant mode ratio in Fig. 3 is owing to the difference in the optical feedback phases between the modes. The optical feedback phase for each longitudinal mode is not matched in a multimode semiconductor laser owing to the different optical frequencies (wavelengths).

 

Fig. 4. Comparison of dominant mode ratios of mode 3 when optical injection is applied for mode 3 at different optical injection strengths under optical feedback. Each color corresponds to the optical injection strength to mode 3 (κ_inj,3). The optical injection strength is fixed at κ_inj,3 = 1.0 ns^-1, 2.0 ns^-1, 3.0 ns^-1, 4.0 ns^-1, and 5.0 ns^-1. (a) Dominant mode ratio of mode 3 at different injection strengths as a function of Φ₃ when we replace the optical feedback phase ω₃τ of mode 3 by ω₃τ + Φ₃ in Eq. (1). (b) Dominant mode ratio of mode 3 at different injection strengths as a function of Δf₃.

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We propose a method to compensate for the differences in the optical feedback phases among the modes by adjusting the initial optical frequency detuning, because it is difficult to precisely adjust the optical feedback phase for each mode in the experiment. Both the optical feedback and injection phases affect the dynamics of a semiconductor laser with optical feedback and injection [27,28]. We fix the optical feedback phase of mode 3 (i.e., Φ₃ = 0) and change the initial optical frequency detuning of mode 3 (Δf₃). Figure 4(b) shows the dominant mode ratio of mode 3 as a function of Δf₃ at different κ_inj,3. Comparing Fig. 4(b) with Fig. 4(a), very similar characteristics of the dominant mode ratio are obtained within the range of 0.1 GHz of Δf₃. Here, 0.1 GHz corresponds to the inverse of the round-trip time τ = 10.01 ns in the external cavity of the optical feedback (i.e., the inverse of the feedback delay time, 1/τ). Therefore, similar characteristics of the dominant mode ratio can be obtained by changing Δf₃ instead of Φ₃.

From these results, the difference in the characteristics of the dominant mode ratio can be compensated for by adjusting Δf₃. The difference in the optical feedback phase between neighboring modes can be described as 2πΔντ, where Δν is the longitudinal mode spacing. We can compensate for the optical frequency detuning of mode m to match the characteristics of the central mode m_c as follows:

The initial optical frequency detuning of the central mode (mode 3) is set to Δf₃ = –4.0 GHz, whereas the initial optical frequency detuning Δf_m of mode m is adjusted using Eqs. (5) and (6), respectively. Particularly, Δf_m of mode 1, 2, 3, 4, and 5 are set to –3.951, –3.975, –4.000, –4.025, and –4.049 GHz, respectively. Figure 5 shows the dominant mode ratio for mode m as the optical injection strength of mode m (κ_inj,m) is changed under the adjustment of Δf_m. The characteristics of the change in the dominant mode ratio are almost the same for all the modes. Therefore, the differences in the characteristics of the dominant mode ratio in terms of κ_inj,m can be compensated for by adjusting Δf_m using Eqs. (5) and (6), respectively.

 

Fig. 5. Comparison of dominant mode ratios of mode m as a function of optical injection strength for mode m when optical injection is applied for only mode m under optical feedback. Initial optical frequency detuning for each mode is adjusted using Eqs. (5) and (6).

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We have performed numerical simulations with a large number of modes (up to 129) and compared the results. We have confirmed that the dominant mode ratio can reach one by increasing the optical injection strength although many modes are used. The optical frequency required for phase compensation depends on 1/τ. The value of 1/τ can be increased if τ is decreased, and the optical frequency detuning can be adjusted even in the experiment.

The difference in the characteristics of the dominant mode ratio among the modes can be explained by the distribution of the steady-state solutions in the phase space (see the Appendix for details).

4. Parameter dependence

4.1 Parameter dependence of dominant mode ratio

In the previous section, we showed that the change in the dominant mode strongly depends on the optical injection strength and initial optical frequency detuning. In this section, we evaluate the characteristics of the dominant mode ratio when the optical injection strength and initial optical frequency detuning are systematically changed. Figure 6(a) shows two-dimensional (2D) maps of the dominant mode ratio when the optical injection strength (κ_inj,3) and initial optical frequency detuning (Δf₃) for mode 3 are changed simultaneously. The value of κ_inj,3 required for a large dominant mode ratio increases when Δf₃ is far from 0 GHz, which indicates that external light with an optical frequency closer to the longitudinal mode can easily excite the dominant mode. For negative Δf₃, the dominant mode ratio is reduced in the region where κ_inj,3 is small (blue region), whereas this region is smaller for positive Δf₃. Thus, the asymmetric characteristics with respect to Δf₃ = 0 GHz result from the characteristics of semiconductor lasers with nonzero α parameters [12].

 

Fig. 6. Two-dimensional maps of (a) dominant mode ratio for mode 3 and (b) its enlarged view. Horizontal axis represents optical injection strength κ_inj,3 for mode 3 and vertical axis represents initial optical frequency detuning Δf₃ for mode 3.

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Figure 6(b) shows an enlarged view of Fig. 6(a) in the range of Δf₃ from –4.0 to –3.0 GHz to observe the fine structure of Fig. 6(a). As shown in Fig. 6(b), the dominant mode is changed periodically by changing Δf₃ with a frequency interval of 0.1 GHz, which corresponds to the inverse of the feedback delay time 1/τ. The dominant mode ratio is significantly changed for small changes in Δf₃ even if κ_inj,3 has the same value. This periodic structure results from the optical phase shift between the optical feedback and injection owing to the change in Δf₃, as described in the previous section.

4.2 Relationship between dominant mode ratio and injection locking

In this subsection, we evaluate the relationship between the characteristics of the dominant mode ratio and injection locking. Figure 7(a) shows the 2D map of the dominant mode ratio of mode 3 as κ_inj,3 and Δf₃ are changed in wider ranges than in Fig. 6(a). The region of the large dominant mode ratio of 1 (red region) increases as κ_inj,3 increases. The characteristics of the dominant mode ratio are asymmetric for Δf₃. In wide regions of positive Δf₃, the dominant mode ratio ranges between 0.4 and 0.8. In contrast, the dominant mode ratio is close to zero, and mode 3 is perfectly suppressed in wide regions of negative Δf₃.

 

Fig. 7. (a) Two-dimensional map of dominant mode ratio for mode 3 (expanded view of Fig. 6(a)). (b) Absolute value of actual optical frequency detuning |Δf_inj,3| for mode 3. The horizontal axis represents the optical injection strength κ_inj,3 for mode 3 and the vertical axis is the initial optical frequency detuning Δf₃ for mode 3.

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We calculate the actual optical frequency detuning Δf_inj,3 for mode 3 under optical injection. Δf_inj,3 can be described using the initial optical frequency detuning Δf₃ and the difference in the phase changes between the injected light and mode 3, as follows [12]:

Figure 7(b) shows the absolute value of the actual optical frequency detuning of mode 3 |Δf_inj,3| under optical injection. In wide regions of positive Δf₃, |Δf_inj,3| increases as Δf₃ increases from 0 GHz and injection locking does not occur. In contrast, |Δf_inj,3| approaches 0 GHz, and injection locking occurs in wide regions of negative Δf₃ as κ_inj,3 increases (blue region). However, the injection locking range (blue region) in Fig. 7(b) does not directly correspond to the region with a large dominant mode ratio (red region) in Fig. 7(a).

Figure 8 summarizes the relationship between the injection locking range and the region of the large dominant mode ratio by comparing Figs. 7(a) and 7(b). We define |Δf_inj,3| ≤ 0.001 GHz as the injection-locking range, as shown in Fig. 7(b). Figure 8 is categorized into four regions: (a) a dominant mode ratio of 1 and injection locking is achieved (blue region in Fig. 8), (b) a dominant mode ratio of 1 and injection locking is not achieved (red region), (c) the dominant mode ratio is not 1 and injection locking is achieved (light green region), and (d) the dominant mode ratio is not 1 and injection locking is not achieved (purple region).

 

Fig. 8. Two-dimensional map of the dominant mode ratio of 1 and injection locking range. (Blue) dominant mode ratio (DMR) is 1 and injection locking is achieved; (red) dominant mode ratio is 1 and injection locking is not achieved; (light green) dominant mode ratio is not 1 and injection locking is achieved; and (purple) dominant mode ratio is not 1 and injection locking is not achieved. (a)-(d) correspond to the temporal dynamics shown in Fig. 9.

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To show the dynamics in the four regions of Fig. 8, we evaluate the temporal waveforms of the modal intensities and short-term optical frequency detuning under optical injection for mode 3. The dynamics of short-term optical frequency detuning for mode 3 (Δf_inj,3(t)) are described as follows:

Figure 9 shows the temporal waveforms of the five modal intensities and short-term optical-frequency detuning for mode 3 Δf_inj,3(t) in the presence of optical injection (κ_inj,3 = 30.0 ns^-1) for different Δf₃. The dynamics in Figs. 9(a)–(d) correspond to examples of the four regions indicated by (a)–(d) in Fig. 8.

 

Fig. 9. Temporal waveforms for different initial optical frequency detuning Δf₃ for mode 3 at κ_inj,3 = 30.0 ns^-1. (upper) Five modal intensities and (lower) actual optical frequency detuning of mode 3. (a) Δf₃ = –5.0 GHz, (b) Δf₃ = 3.5 GHz, (c) Δf₃ = –15.0 GHz, and (d) Δf₃ = 15.0 GHz. (a)-(d) correspond to the regions shown in Fig. 8.

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Figure 9(a) shows the temporal waveforms of the five modal intensities and Δf_inj,3(t) for Δf₃ = –5.0 GHz, which corresponds to the blue region in Fig. 8. The temporal oscillations of all the modes are stabilized in steady states. Mode 3 has the maximum intensity, whereas the other modes have zero intensity. Therefore, mode 3 becomes the dominant mode. Moreover, Δf_inj,3(t) is stabilized at 0 GHz, and perfect injection locking is achieved.

Figure 9(b) shows the temporal waveforms of the five modal intensities and Δf_inj,3(t) for Δf₃ = 3.5 GHz, which corresponds to the red region in Fig. 8. The temporal waveform of mode 3 oscillates quasi-periodically, and the other modes are perfectly stabilized with zero intensity (no oscillations). Therefore, mode 3 is always the dominant mode, although it exhibits quasi-periodic oscillations. Additionally, Δf_inj,3(t) oscillates between –2 and 8 GHz, and injection locking does not occur, even on an average. Therefore, a large dominant mode ratio can be achieved even without injection locking in the red region of Fig. 8.

Figure 9(c) shows the temporal waveforms of the five modal intensities and Δf_inj,3(t) for Δf₃ = –15.0 GHz, which corresponds to the light green region in Fig. 8. Only the temporal waveform of mode 3 is stabilized with small fluctuations by optical injection, and the other modes exhibit large chaotic oscillations. Therefore, mode 3 is not the dominant mode. Moreover, Δf_inj,3(t) fluctuates chaotically around 0 GHz within the range of ±0.1 GHz; however, the mean of Δf_inj,3(t) is close to 0 GHz, where injection locking seems to be achieved on average. In fact, Δf_inj,3(t) fluctuates slightly because of the chaotic mode-competition dynamics of the other modes, and injection locking is achieved on average. However, the dominant mode ratio of mode 3 does not become 1 because the temporal dynamics of mode 3 are almost stabilized, whereas the other modes fluctuate chaotically.

Figure 9(d) shows the temporal waveforms of the five modal intensities and Δf_inj,3(t) for Δf₃ = 15.0 GHz, which corresponds to the purple region in Fig. 8. All modes oscillate chaotically, and the dominant mode ratio is not 1. Mode 3 oscillates with faster frequencies than the other modes owing to the beat frequency of Δf₃ (= 15.0 GHz). Additionally, Δf_inj,3(t) remains approximately equal to 17 GHz and occasionally moves to 0 GHz, indicating that injection locking is not achieved on average.

From these results, we conclude that the region of the dominant mode ratio of 1 does not perfectly match the injection-locking range. A dominant mode ratio of 1 under injection locking can only be obtained in the blue region of Fig. 8. However, a dominant mode ratio of 1 can still be achieved without injection locking in the red region of Fig. 8, where the intensity and optical frequency of mode 3 oscillate quasi-periodically (or chaotically), and the other modes are perfectly stabilized. In contrast, the dominant mode ratio cannot reach 1 even though injection locking is achieved on average in the green-light region of Fig. 8, where only mode 3 is suppressed, and the other modes fluctuate chaotically. The optical frequency of mode 3 fluctuates slightly at approximately 0 GHz, which indicates that injection locking is achieved, on average. Finally, a dominant mode ratio of 1 is not obtained, and injection locking is not achieved in the purple region in Fig. 8. Therefore, the relationship between the region of the dominant mode ratio of 1 and the injection locking range is not straightforward in the multimode semiconductor laser with optical feedback and injection, unlike in the case of the single-mode semiconductor laser [12].

Finally, we evaluate the temporal dynamics of different values of κ_inj,3 and Δf₃. We distinguish four different nonlinear dynamics of stable, periodic, quasi-periodic, and chaotic oscillations using the standard deviation of the amplitude of the temporal waveforms and the peak value of the autocorrelation functions. We determine a steady state if the standard deviation σ of the amplitude of the temporal waveform is less than a threshold, i.e., σ < T_s (T_s = 0.02). In other cases, we calculate the autocorrelation function of the temporal waveform to classify the dynamics. The autocorrelation function is calculated by changing the time shift t′, and the maximum correlation value C_max except at t′ = 0 is used to determine the dynamics. For example, we define that a periodic oscillation is observed in the case of C_max > T_p (T_p = 0.9999995), a quasi-periodic oscillation is obtained in the case of T_p ≥ C_max > T_qp (T_qp = 0.95), and a chaotic oscillation is observed in the case of C_max ≤ T_qp.

Figure 10(a) shows a 2D map of the temporal dynamics of the total intensity in the multimode semiconductor laser as κ_inj,3 and Δf₃ change simultaneously. A steady state is observed in the blue triangle region for a wide range of negative Δf₃ and large κ_inj,3. The periodic (green region) and quasi-periodic (orange region) oscillations are located around the upper side of the blue triangular region of the steady state (near zero Δf₃ and large κ_inj,3). The other region corresponds to chaotic oscillations, indicated by the red region in Fig. 10(a), owing to the optical feedback. The regions of steady state and periodic oscillations in Fig. 10(a) correspond to the blue region in Fig. 8, where the dominant mode ratio is 1, and injection locking is achieved.

 

Fig. 10. Two-dimensional maps of the temporal dynamics of (a) total intensity and (b) modal intensity of mode 3 as κ_inj,3 and Δf₃ change simultaneously. (a)-(d) correspond to the temporal dynamics shown in Fig. 9. (Blue) stable outputs, (green) periodic oscillations, (orange) quasi-periodic oscillations, and (red) chaotic oscillations.

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Figure 10(b) shows a 2D map of the temporal dynamics of mode 3 as κ_inj,3 and Δf₃ change simultaneously. A new region of the steady state (blue triangle region) appears at a small negative Δf₃ and large κ_inj,3 which is different from the temporal dynamics of the total intensity in Fig. 10(a). This region is included in the light-green region of Fig. 8, where the dominant mode ratio is not 1, even though injection locking is achieved. The dynamics of mode 3 are almost stabilized under injection locking; however, the dominant mode ratio is not 1 owing to the appearance of chaotic oscillations of the other modes. Therefore, the difference in the dynamics between the total and modal intensities results in a mismatch between the region of the large dominant mode ratio and the injection-locking range, as shown in Fig. 8.

5. Conclusions

In this study, we numerically evaluated the chaotic mode-competition dynamics in a multimode semiconductor laser with optical feedback and injection. Chaotic mode-competition dynamics were observed in the longitudinal modes, and one of them was enhanced by injecting an optical signal with a wavelength similar to that of a single-mode semiconductor laser. The dominant mode was defined as the mode with the maximum intensity, and the dominant mode ratio for the injected mode increased as the optical injection strength increased. However, the characteristics of the dominant mode ratio in terms of the optical injection strength differed among the modes owing to the difference in the optical feedback phase. We proposed a control method to match these characteristics by adjusting the initial optical frequency detuning between the injection signal and injected mode in the multimode laser. Moreover, we evaluated the relationship between the region with a large dominant mode ratio and the injection-locking range. The large dominant mode ratio region did not match the injection-locking range. The dominant mode ratio could reach one when the injected mode was larger than the other modes, even though the oscillation of the injected mode was not stabilized. Therefore, there was a region where the dominant mode ratio became one when the oscillation was not stabilized by injection locking.

It is easy to enhance one of the longitudinal modes in a multimode laser via optical injection from a single-mode laser with a similar frequency. The optical injection method for controlling mode-competition dynamics can be applied to reinforcement learning, that is, the best choice is selected from multiple choices [18]. The control technique of chaotic mode-competition dynamics in multimode semiconductor lasers could be useful for photonic computing applications in reinforcement learning and reservoir computing as a novel photonic AI.

6. Appendix

6.1 Steady state analysis

In the Appendix, we explain the influence of the optical feedback phase on the dominant mode ratio using steady-state analysis. An initial optical frequency detuning of 0.1 GHz (= 1/τ) corresponds to an optical feedback phase of 2π, as shown in Figs. 4(a) and 4(b). Generally, multiple steady-state solutions appear in the frequency interval corresponding to 1/τ in a single-mode semiconductor laser with optical feedback [12,42]. Thus, the distribution of steady-state solutions may be related to the change in the dominant mode ratio.

For simplicity, we consider steady-state solutions for only one longitudinal mode in a multimode semiconductor laser with optical feedback. In this case, the steady-state solutions are almost identical to those of a single-mode semiconductor laser with optical feedback, except for the steady-state solution of the carrier density, which depends on the gain coefficient of each mode. The steady-state solutions with optical feedback for the carrier density N_s and angular frequency ω_s,m of the longitudinal mode m are expressed as follows:

(9)$${\Delta {\omega _{s,m}} ={-} \kappa \sqrt {1 + {\alpha ^2}} \sin ({\mathrm{\Delta }{\omega_{s,m}}\tau + {\omega_m}\tau + {{\tan }^{ - 1}}\alpha } )}$$

(10)$${{N_s} = \frac{{{\tau _s}{G_m}{N_0} + {\tau _s}/{\tau _p} + \varepsilon {\; }{N_{th}}{\; }J/{J_{th}}{\; } - 2\kappa {\tau _s}\cos ({{\omega_{s,m}}\tau } )}}{{{\tau _s}{G_m} + \varepsilon }}}$$

where Δω_s,m indicates Δω_s,m = ω_s,m – ω_m, ω_s,m is the steady-state solution for the angular frequency of the longitudinal mode m with optical feedback, and ω_m is the angular frequency of mode m without optical feedback. G_m indicates the gain coefficient of mode m. The steady-state solutions for the angular frequency (ω_s,m) are converted to those of the frequency, that is, ν_s,m = ω_s,m / 2π.

In this analysis, steady-state solutions are obtained without optical injection. The steady-state solutions may be changed by optical injection; however, those with only optical feedback can be used to describe the original dynamics because the optical injection is interpreted as a small perturbation to the chaotic multimode semiconductor laser with optical feedback.

Figure 11 shows the steady-state solutions for mode m obtained using Eqs. (9) and (10), respectively. Figure 11(a) shows the distribution of steady-state solutions for each longitudinal mode with respect to the central mode (mode 3). Distributions are obtained by adding the mode spacing ν_m – ν₃ to the steady-state solution. Multiple steady-state solutions are elliptically distributed around each of the five longitudinal modes. Figure 11(b) shows the distribution of the steady-state solutions for each longitudinal mode with respect to the modal frequency ν_m of the longitudinal mode m. The steady-state solutions of all the modes are overlapped and elliptically distributed. Figure 11(c) shows an enlarged view of Fig. 11(b). The steady-state solutions for each longitudinal mode are distributed at intervals of 0.1 GHz (corresponding to 1/τ); however, they are slightly shifted for each longitudinal mode and are placed at different frequencies.

 

Fig. 11. Steady-state solutions obtained using Eqs. (9) and (10). (a) Distribution of steady-state solutions for five longitudinal modes with respect to mode 3. (b) Distribution of steady-state solutions with respect to each longitudinal mode. Steady-state solutions are overlapped for the five modes. (c) Enlarged view of (b). The steady-state solutions for the five modes are indicated by different colors.

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Here, steady-state solutions can be obtained by determining the intersection of Δω_s,m (the left-hand-side term in Eq. (9)), and sinusoidal functions (the right-hand-side term in Eq. (9)) [12,42]. The initial phase of the sinusoidal function on the right side of Eq. (9) is determined by ω_mτ and the position of the intersection, and the difference in the optical feedback phase for each mode (ω_mτ) affects the positions of steady-state solutions. Particularly, the steady-state solutions of modes 1 and 5 almost overlap in Fig. 11(c), because the feedback phases ω_mτ for modes 1 and 5 are very similar. Thus, the characteristics of the dominant mode ratios of modes 1 and 5 in terms of the optical injection strength are very similar, as shown in Fig. 3.

Figure 12(a) shows the dominant mode ratio of mode 3 (black curve) when the initial optical frequency detuning for mode 3 (Δf₃) is changed, and the optical injection strength for mode 3 is fixed at κ_inj,3 = 5.0 ns^-1. The dominant mode ratio periodically changes with a frequency interval of 0.1 GHz (= 1/τ) for a wide range of over ±5.0 GHz. In other words, the dominant mode ratio strongly depends on the change in the optical feedback phase.

 

Fig. 12. (a) Dominant mode ratio of mode 3 (black curve) when initial optical frequency detuning for mode 3 is changed and optical injection strength for mode 3 is fixed at κ_inj,3 = 5.0 ns^-1. Distributions of steady-state solutions for mode 3 are also shown (red and blue dots correspond to modes and anti-modes, respectively). (b), (c) Enlarged views of (a) at different frequency ranges.

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Figure 12(a) also shows the distribution of the steady-state solutions of mode 3 obtained from Eqs. (9) and (10) to understand the relationships between the dominant mode ratio and distribution of the steady-state solutions. The lower half of the ellipse (red dots) is known as the external-cavity mode (or mode), which is a stable solution, whereas the upper half of the ellipse (blue dots) is known as the antimode, which is an unstable solution [12,42].

Figures 12(b) and 12(c) show the enlarged views of Fig. 12(a) for different frequency ranges. The change in the dominant mode ratio occurs in the frequency interval of 0.1 GHz (= 1/τ). The periodic change in the dominant mode ratio corresponds to the periodic distribution of the modes and antimodes.

We discuss the relationship between the initial optical frequency detuning and the steady-state solutions. The relative positions of the modes and antimodes on the horizontal axis in Fig. 12 (optical frequency detuning) are related to the dominant mode ratio. In Fig. 4, the increase in the optical feedback phase (corresponding to the shift in the steady-state solutions in the negative direction) is equivalent to the shift in the initial optical frequency detuning in the positive direction. Therefore, a change in the initial optical frequency detuning is effective in changing the position of the steady-state solution, which results in a change in the dominant mode ratio.

It has also been reported that the distribution of the modes and antimodes can be used to control chaos in a semiconductor laser with optical feedback [45]. The dominant mode ratio can be controlled by changing the distribution of the modes and antimodes in the proposed method, as described in Section 3.2.

Funding

Telecommunications Advancement Foundation; Core Research for Evolutional Science and Technology (JPMJCR17N2); 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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