Open Access
Add to BibSonomy
Abstract
We experimentally investigate the complex dynamics of a multi-mode quantum-dot semiconductor laser with time-delayed optical feedback. We examine a two-dimensional bifurcation diagram of the quantum-dot laser as a comprehensive dynamical map by changing the injection current and feedback strength. We found that the bifurcation diagram contains two different parameter regions of low-frequency fluctuations. The power-dropout dynamics of the low-frequency fluctuations are observed in the sub-GHz region, which is considerably faster than the conventional low-frequency fluctuations in the MHz region. Comparing the dynamics of quantum-dot laser with those of single- and multi-mode quantum-well semiconductor lasers reveals that the fast low-frequency fluctuation dynamics are unique characteristics of quantum-dot lasers with time-delayed optical feedback.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum-dot (QD) semiconductor lasers have been intensively investigated for several decades, where QDs of nanometer size are introduced to the active layer in a semiconductor laser cavity [1–5]. Low loss and high energy efficiency can be achieved owing to the confinement of electrons in QDs, which provides a low lasing threshold. In addition, the superior temperature insensitivity and high modulation bandwidth of QD lasers provide reliable light sources for optical communications. Recently, QD lasers with shorter wavelengths in the range of 1.0–1.3 µm (i.e., T- and O-bands) have been developed using silicon photonics technologies to achieve ultrawide bandwidths in novel optical communications [4,5]. Multi-stacked QD structures of the active layer have been fabricated to obtain high-power laser output for these communication applications [5].
QD lasers are less sensitive to optical feedback than quantum-well (QW) semiconductor lasers [6,7]. Nevertheless, rich nonlinear dynamics of QD lasers with optical feedback or injection have been reported [6–19]. Theoretical analysis of bifurcation toward chaotic dynamics has been significantly investigated in QD lasers with optical feedback or injection [6–13]. Compared with intensive theoretical and numerical examinations, experimental investigations of the nonlinear dynamics of QD lasers with optical feedback are noticeably limited [14–18]. For example, different chaotic dynamics of the ground-state (GS) and excited-state (ES) emissions in QD lasers have been observed experimentally [14–16]. QD lasers exhibit chaotic dynamics when large optical feedback is introduced [14,15]. In addition, the linewidth of the optical spectrum is enhanced in the ES emission [16].
Low-frequency fluctuations (LFF) are typical dynamics observed in QW semiconductor lasers with optical feedback [19–23], where abrupt power dropouts and the subsequent gradual power recovery are observed at MHz frequencies, which are significantly slower than primal chaotic oscillations at GHz frequencies determined by the relaxation oscillation frequency. Slow power dropouts come from the spontaneous irregular mode switching among the external-cavity modes, which is determined by the propagation delay time of light from the laser to the external mirror for optical feedback, also known as chaotic itinerancy [20,21]. However, LFF dynamics in QD lasers have not been investigated experimentally thoroughly.
QD lasers are excellent temperature-independent light sources based on nonlinear laser dynamics. Hence, it is important to clarify the dynamical regimes of QD lasers with optical feedback for promising engineering applications, such as random number generation [24,25] and photonic artificial intelligence including decision making [26–28]. For example, LFF dynamics can be applied for random number generation when irregular intervals of power dropouts are used (i.e., intermittency) because the statistical characteristics of the intermittent dynamics intervals follow a power-law distribution, which is appropriate as a physical entropy source [25]. In addition, LFF dynamics can be applied to decision making for solving multi-armed bandit problems using lag synchronization of chaos [27,28]. Therefore, photonic implementation based on LFF dynamics of semiconductor lasers can be realized in ultrafast decision-making systems.
In this study, we experimentally investigate nonlinear dynamics and bifurcation phenomena in a QD laser with time-delayed optical feedback. We focus on the LFF dynamics of the laser output from a QD laser. We compare the LFF dynamics between the QD and QW lasers to identify the novel dynamics of QD lasers.
2. Dynamics of QD lasers
2.1 Experimental setup
Figure 1 shows a schematic of the experimental setup for a QD laser with optical feedback [29]. We used a multi-mode Fabry-Perot QD laser (PQLD1190, fabricated by Pioneer based on our developed technology). We fabricated a seven-stack InAs/InGaAs QD structure on an n-type GaAs substrate [5]. The photoluminescence peak wavelength of the QD structure is observed at approximately 1.25 µm for novel optical communications. The Fabry-Perot cavity is introduced as the internal cavity of the QD laser, and multi-longitudinal-mode oscillations are observed (see [5] for more details of the QD structure). The internal cavity length of the QD laser is 2 mm.
Fig. 1. Experimental setup for QD laser with optical feedback. ATT, optical attenuator; FC, fiber coupler; ISO, optical isolator; PC, polarization controller; PD, photodetector.
We introduce optical feedback to the QD laser to observe the chaotic temporal dynamics. The output light from the QD laser is sent to a polarization controller to adjust the polarization direction because the QD laser has a single-mode fiber pigtail. Polarization-maintaining fiber components for a wavelength of 1250 nm are used in the experiment, except for the single-mode fiber pigtail of the QD laser. The light is sent to a fiber coupler to be split into two directions, whereas one of the directed lights is sent to a fiber circulator to generate optical feedback. We introduce a loop configuration for optical feedback using a fiber circulator, an optical isolator, and an optical attenuator. The feedback strength can be adjusted using an optical attenuator in the feedback loop. The other directed light in the fiber coupler is sent to another optical isolator, attenuator, and photodetector (Newport, 1554-B, 12-GHz bandwidth) to observe the dynamics of the QD laser output. The optical output is converted into an electric signal using a photodetector. The temporal waveforms of the laser output are measured using a digital oscilloscope (Tektronix, DPO72304DX, 23-GHz bandwidth, 100 GigaSamples/s). Radio-frequency (RF) spectra of the laser output are obtained using an RF spectrum analyzer (Agilent, N9010A-544, 44-GHz bandwidth). The optical spectra of the laser output are measured using an optical spectrum analyzer (Yokogawa, AQ6370C, 0.02-nm resolution).
The laser operation parameter values are set as follows: The lasing threshold of the QD laser is Ith = 65.0 mA. The injection current of the QD laser is changed to observe different temporal dynamics. For example, the optical power of the QD laser is 6.66 mW at an injection current of I = 162.5 mA ( = 2.5 Ith). Under this condition, the maximum feedback strength is κmax = 2.12 mW in the experiment. The feedback strength κ is normalized by κmax at each injection current in the following sections. The delay time of optical feedback in the feedback loop and corresponding external-cavity frequency are τ = 75.6 ns (roundtrip) and 13.2 MHz, respectively.
2.2 Experimental results of QD laser dynamics
We experimentally observe the temporal QD laser dynamics for different parameter values. First, we fix the injection current at I = 162.5 mA ( = 2.5 Ith) and change the relative feedback strength κ normalized by κmax.
Figure 2 shows the optical spectra of the QD lasers with and without optical feedback at the maximum feedback strength (κ = 1.0). Without optical feedback, as shown in Fig. 2(a), the QD laser exhibits multi-longitudinal oscillations. In the enlarged view of Fig. 2(b), a longitudinal-mode interval of 0.102 nm (20.0 GHz frequency) is observed, which is determined by the internal cavity length. In the presence of optical feedback, as shown in Fig. 2(c), the entire shape of the optical spectrum does not change. However, each longitudinal mode is observed more clearly, and there is a modulation of the optical spectrum at an interval of 2.0 nm (390 GHz), as shown in the enlarged view of Fig. 2(d). The center wavelength of the QD laser is 1235.1 nm in the presence of optical feedback, as shown in Fig. 2(c). We only observe the spectral peaks shown in Fig. 2(c) when the injection current increases further. Consequently, only GS emission is observed, and no ES emission exists in the QD laser with optical feedback.
Fig. 2. Optical spectra of the QD laser output (a), (b) without and (c), (d) with optical feedback (κ = 1.0). (b), (d) Enlarged views of (a), (c). The injection current is fixed at I = 2.5 Ith.
Figure 3 shows the temporal waveforms of the QD laser output for different normalized feedback strengths κ. A steady state with noisy oscillation is observed without optical feedback (κ = 0.0), as shown in Fig. 3(a). Quasiperiodic oscillations appear as the feedback strength increases, as shown in Figs. 3(b) and 3(c). Chaotic oscillation is observed for the intermediate feedback strength (κ = 0.151), as shown in Fig. 3(d). More interestingly, LFF dynamics (i.e., sudden power dropouts and gradual power recovery) are observed in the chaotic temporal waveforms as the feedback strength increases further, as shown in Figs. 3(e) and 3(f). Such power dropouts frequently appear in the temporal waveforms; hence, the frequency of power dropouts is extremely faster than the ordinary LFF dynamics at MHz frequencies [22,23]. In addition, the feedback strengths in the regime of these LFF dynamics are relatively large (κ = 0.2 ∼ 0.3) compared with the ordinary LFF dynamics.
Fig. 3. Temporal waveforms of the QD laser output for different normalized feedback strength κ: (a) κ = 0.0, (b) 0.060, (c) 0.106, (d) 0.151, (e) 0.225, and (f) 0.301. The injection current is fixed at I = 2.5 Ith. The vertical axis corresponds to the electric signal voltage of the laser intensity converted at the photodetector.
Figure 4 summarizes the RF spectra for different feedback strengths corresponding to the temporal waveforms in Fig. 3. A flat spectrum appears without optical feedback, as shown in Fig. 4(a). Many discrete spectral peaks are observed in the case of quasiperiodic dynamics in Fig. 4(b), and the widths of these peaks broaden as the feedback strength increases, as shown in Fig. 4(c). Figure 4(b) shows that quasiperiodic oscillations are generated, and the peak heights of RF spectrum are distributed irregularly. Figure 4(c) illustrates other quasiperiodic oscillations, whereas the RF spectrum shows a smoother distribution of the peak heights. The frequency peaks in Fig. 4(c) correspond to the relaxation oscillation frequency, its harmonics, and sub-harmonics. In addition, many spectral peaks are observed in the enlarged view of the RF spectra, as shown in the insets of Figs. 4(b) and 4(c). This frequency interval corresponds to the external cavity frequency of 13.2 MHz.
Fig. 4. RF spectra of the QD laser output for different feedback strengths κ: (a) κ = 0.0, (b) 0.060, (c) 0.106, (d) 0.151, (e) 0.225, and (f) 0.301. The injection current is fixed at I = 2.5 Ith. These figures correspond to the temporal waveforms in Fig. 3. The insets show the enlarged view of the RF spectra at around the peak frequency.
In the chaotic regime, broad and continuous spectral components are observed, as shown in Fig. 4(d). The peak frequency of the chaotic spectrum is 1.745 GHz, which corresponds to the relaxation oscillation frequency of the QD laser. The inset shows that many peaks appear and their frequency interval corresponds to the external cavity frequency of 13.2 MHz, similar to that shown in Figs. 4(b) and 4(c). Figure 4(e) shows that another peak appears at the lower frequency side (0.227 GHz) as the feedback strength increases further. This frequency component corresponds to the LFF dynamics observed in the temporal waveform shown in Fig. 3(e). The frequency of power dropouts (0.227 GHz) is one order of magnitude smaller than the main peak frequency of the chaotic oscillations (2.102 GHz). Figure 4(f) shows that a comparable RF spectrum is observed for a large feedback strength.
To quantitatively define the LFF dynamics regions, we compare the spectral peak heights between the main frequency of the chaotic oscillation Pmain(f) and the frequency of LFF dynamics PLFF(f). We define LFF dynamics when the following condition is satisfied:
The unit of dB is used in Eq. (1) because the ratio of the power of the two spectral peaks is used to define the occurrence of LFF dynamics. The value of −15 dB is determined from the appearance of power dropouts in the low-pass-filtered temporal waveforms (see Subsection 2.3). The condition of Eq. (1) is satisfied in Figs. 4(e) and 4(f); i.e., LFF dynamics are observed under these conditions.We change the parameter values of the feedback strength κ and injection current I to systematically examine the QD laser dynamics. We first fix the injection current at I = 1.0Ith and increase the feedback strength from zero (κ = 0.0) to the maximum value (κ = 0.3) to generate a one-dimensional (1D) bifurcation diagram. Subsequently, we slightly increase the injection current with a step size of 0.1Ith and change the feedback strength to generate another 1D bifurcation diagram. This procedure is repeated up to an injection current of I = 3.0Ith, and these 1D bifurcation diagrams are combined to generate a two-dimensional (2D) bifurcation diagram.
Figure 5 shows the 2D bifurcation diagram of the temporal waveforms of the QD laser output. The feedback strength is normalized by the maximum feedback strength at each injection current. Different colors indicate the regions for different dynamic regimes. The dynamics change from steady state (S), quasiperiodic oscillations (QP), and chaotic oscillations (C) as the feedback strength increases for intermediate injection currents. This is a typical bifurcation scenario for a QW semiconductor laser with optical feedback [23]. However, in contrast to conventional lasers, two regions of LFF dynamics are observed for small and large injection currents in this QD laser. LFF dynamics appear at an injection current of 1.1Ith close to the lasing threshold, which is a similar parameter region for the LFF dynamics in QW semiconductor lasers [22,23]. However, different LFF dynamics are also observed at injection currents over 2.0Ith for large feedback strength, as shown in Fig. 5. Such LFF dynamics have not been observed in QW lasers because LFF dynamics appear only for small injection currents close to the lasing threshold [22,23]. Therefore, the LFF dynamics for a large injection current may be unique characteristics of QD lasers with optical feedback.
Fig. 5. Two-dimensional bifurcation diagram of the temporal waveforms of the QD laser output. C, chaotic oscillations; LFF, low-frequency fluctuations; QP, quasiperiodic oscillations; S, steady state.
2.3 Characteristics of LFF dynamics
We examine the difference in the LFF dynamics of the two-parameter regions. Figure 6 shows the temporal waveforms and RF spectra of the LFF dynamics for a small injection current of I = 1.0 Ith. The feedback strength is set at κ = 0.150. Figure 6(a) shows the original temporal waveform and low-pass filtered temporal waveform. The cut-off frequency of the low-pass filter is set at 0.534 MHz in order to show the LFF dynamics clearly. LFF dynamics are observed in the low-pass filtered temporal waveform in Fig. 6(a). Figure 6(b) shows the RF spectra of the original temporal waveform in the GHz range. A large peak appears in the low-frequency component in Fig. 6(b). Figure 6(c) shows an enlarged view of the RF spectrum in the low-frequency region of a few MHz. A spectral peak is observed at a frequency of 0.356 MHz, which is an indication of the appearance of LFF dynamics in the order of MHz. These LFF dynamics are comparable to those observed in the ordinary QW semiconductor lasers with optical feedback [22,23].
Fig. 6. Temporal waveforms and the RF spectra of the QD laser for the small injection current of I = 1.0 Ith. The feedback strength is fixed at κ = 0.150. (a) Temporal waveform (black) and low-pass filtered temporal waveform (red) in the microsecond time scale. The cut-off frequency of the low-pass filter is set at 0.534 MHz. (b) RF spectrum in the GHz range. (c) Enlarged view of RF spectrum in the MHz range.
Figure 7 shows the temporal waveforms and the RF spectra of the LFF dynamics for a large injection current of I = 2.5 Ith. The feedback strength is set at κ = 0.301. The black and red curves in Fig. 7(a) show the original and low-pass filtered temporal waveforms, respectively. The cut-off frequency of the low-pass filter is 1.0 GHz. LFF dynamics are observed in the low-pass filtered temporal waveform; however, these power dropouts are extremely faster than those shown in Fig. 6(a). Figure 7(b) shows the RF spectrum of the original temporal waveform in the GHz range. Two spectral peaks appear, which correspond to the main frequency of chaotic oscillations (the relaxation oscillation frequency) at 2.425 GHz and LFF frequency at 0.240 GHz. The LFF frequency in Fig. 7(b) is considerably faster than that in Fig. 6(b). Figure 7(c) shows an enlarged view of the RF spectrum in the MHz range. No clear peaks are observed in the MHz frequency region, as shown in Fig. 7(c). Therefore, fast LFF dynamics at sub-GHz frequencies are observed for large injection currents in QD lasers. We consider that these dynamics differ from the conventional LFF dynamics observed at MHz frequency region. This will be discussed further in Sections 3 and 4.
Fig. 7. Temporal waveforms and the RF spectra of the QD laser for the small injection current of I = 2.5 Ith. The feedback strength is fixed at κ = 0.301. (a) Temporal waveform (black) and low-pass filtered temporal waveform (red) in the nanosecond time scale. The cut-off frequency of the low-pass filter is set at 1.0 GHz. (b) RF spectrum in the GHz range. (c) Enlarged view of RF spectrum in the MHz range.
Additionally, we investigate the parameter dependence of the spectral power at the peak frequencies of the main chaotic dynamics Pmain(f) and LFF dynamics PLFF(f) in the RF spectra. Figure 8 shows the spectral power at the peak frequencies in the RF spectra for the cases of small (I = 1.0 Ith) and large (I = 2.5 Ith) injection currents as the feedback strength is changed. For a small injection current of I = 1.0 Ith in Fig. 8(a), the peak spectral power of the LFF dynamics in the MHz range (denoted as PLFF1) is always larger than that of the main frequency of chaotic oscillations in the GHz range (denoted as Pmain). Therefore, LFF dynamics always exist at different feedback strengths. For a large injection current of I = 2.5 Ith in Fig. 8(b), the peak spectral power of LFF dynamics in the MHz range PLFF1 is extremely smaller than that of the main frequency of chaotic oscillation in the GHz range Pmain. However, the peak spectral power of LFF dynamics in the sub-GHz range (denoted as PLFF2) is close to that of Pmain. Therefore, fast LFF dynamics in the sub-GHz range are observed for large injection currents and large feedback strengths.
Fig. 8. Spectral power at peak frequencies in the RF spectra for the cases of (a) small (I = 1.0 Ith) and (b) large (I = 2.5 Ith) injection currents as the feedback strength is changed. PLFF1, spectral height of LFF dynamics at the MHz range; PLFF2, spectral height of LFF dynamics at the GHz range; Pmain, spectral height of main frequency of chaotic oscillations at the GHz range. See also Figs. 6 and 7 for PLFF1, PLFF2, and Pmain.
3. Comparison of LFF dynamics
3.1 Bifurcation diagrams
We compare the multi-mode QD laser dynamics with a single- and multi-mode QW semiconductor laser to identify the unique characteristics of fast LFF dynamics in the sub-GHz range. We replace the QD laser with a single-mode QW laser (NTT Electronics, KELD1C5GAAA, center wavelength of 1547.6 nm, Ith = 10.2 mA) or a multi-mode QW laser (Anritsu, GF5B5003DLL, center wavelength of 1542.9 nm, Ith = 19.2 mA) in the experimental setup shown in Fig. 1, and investigate the dynamics for different parameter values of the injection current and feedback strength. Polarization-maintaining fiber components for a wavelength of 1550 nm are used in this experiment. We use a variable fiber reflector to produce optical feedback instead of using an optical-feedback loop with a fiber circulator. The delay times of optical feedback in the feedback loop are τ = 22.9 and 25.7 ns (roundtrip) for the single- and multi-mode QW lasers, respectively. The corresponding external-cavity frequencies are 43.7 and 38.8 MHz for the single- and multi-mode QW lasers, respectively.
Figure 9 shows the 2D bifurcation diagrams for single- and multi-mode QW lasers. The feedback strength is normalized by the maximum feedback strength at each injection current. The colors in Fig. 9 indicate types of dynamics, similar to Fig. 5. For the single-mode QW laser in Fig. 9(a), a similar bifurcation is observed from the steady states (S), quasiperiodic oscillations (QP), to chaotic oscillations (C). LFF dynamics are observed only in the region with small injection currents. In contrast to that of QD lasers shown in Fig. 5, no LFF dynamics are observed in the region of large injection currents.
Fig. 9. Two-dimensional bifurcation diagrams for (a) single-mode and (b) multi-mode QW lasers. C, chaotic oscillations; IP, irregular pulsations; LFF, low-frequency fluctuations; QP, quasiperiodic oscillations; S, steady state.
For the multi-mode QW laser in Fig. 9(b), similar bifurcations are observed from the steady states (S), quasiperiodic oscillations (QP), to chaotic oscillations (C). However, irregular pulsations (IP) appear in the region of small injection currents instead of LFF dynamics. LFF dynamics are also observed in a wide parameter region of the injection current and feedback strength. The LFF dynamics region is continuously distributed from small to large injection currents in Fig. 9(b), whereas the two regions of the LFF dynamics are clearly separated for the QD laser, as shown in Fig. 5.
The relaxation oscillation frequency of the QD laser is relatively lower than that of the QW laser in our experiment. For example, the relaxation oscillation frequency is less than 3 GHz for the QD laser at a large injection current of I = 3.0Ith, whereas it is less than 7 GHz for the QW laser under the same conditions. In addition, the peak value of RF spectrum for the QD laser is relatively smaller than that of the QW laser, and it is more difficult to observe the peak of the relaxation oscillation frequency in the RF spectra for the QD laser.
3.2 Characteristics of LFF dynamics in QW lasers
We examine the LFF dynamics in single- and multi-mode QW lasers comprehensively. Figure 10 shows the temporal waveforms and RF spectra of the LFF dynamics for the single-mode QW laser. The black and red curves in Fig. 10(a) show the temporal waveforms of LFF dynamics and its low-pass-filtered temporal waveform (the cut-off frequency of the low-pass filter is 2.467 MHz), respectively, when the injection current is set at I = 1.0 Ith and the normalized feedback strength is set at κ = 0.012. LFF dynamics are observed in the low-pass filtered temporal waveform, as shown in Fig. 10(a). Figures 10(b) and 10(c) show the RF spectra of LFF dynamics in the GHz and MHz ranges, respectively. In Fig. 10(b), the main peak appears at 1.985 GHz (the relaxation oscillation frequency), and a lower frequency peak is also observed. In Fig. 10(c), the low-frequency peak appears at 1.645 MHz, which is the frequency range of typical LFF dynamics [22,23].
Fig. 10. Temporal waveforms and RF spectra of the single-mode QW laser for the small injection current of I = 1.0 Ith. The feedback strength is fixed at κ = 0.012. (a) Temporal waveform (black) and low-pass filtered temporal waveform (red) in the microsecond time scale. The cut-off frequency of the low-pass filter is set at 2.467 MHz. (b) RF spectrum at the GHz range. (c) Enlarged view of RF spectrum at the MHz range.
Figure 11 shows the temporal waveforms and RF spectra of the LFF dynamics in the multi-mode QW laser when the injection current is set at I = 2.0 Ith and the feedback strength is set at κ = 0.055. The black and red curves in Fig. 11(a) show the temporal waveforms of the LFF dynamics and their low-pass-filtered temporal waveforms (the cut-off frequency of the low-pass filter is 2.229 MHz), respectively. The low-pass filtered temporal waveform shows typical LFF dynamics, as shown in Fig. 11(a). In the GHz range RF spectrum in Fig. 11(b), the peak frequency appears at 5.023 GHz (the relaxation oscillation frequency). In the MHz range RF spectrum in Fig. 11(c), the peak frequency is also observed at 1.486 MHz. Therefore, the two frequency peaks at the GHz and MHz ranges appear; that is, typical LFF dynamics are confirmed in the multi-mode QW laser. Moreover, no peaks appear in the sub-GHz range, unlike the QD laser with large injection currents, as shown in Fig. 7.
Fig. 11. Temporal waveforms and RF spectra of the multi-mode QW laser for the small injection current of I = 2.0 Ith. The feedback strength is fixed at κ = 0.055. (a) Temporal waveform (black) and low-pass filtered temporal waveform (red) in the microsecond time scale. The cut-off frequency of the low-pass filter is set at 2.229 MHz. (b) RF spectrum at the GHz range. (c) Enlarged view of RF spectrum at the MHz range
Figure 12 shows a comprehensive comparison of 2D bifurcation diagrams for the multi-mode QD, single-mode QW, and multi-mode QW lasers. We change the horizontal-axes ranges (the feedback strengths) of the three bifurcation diagrams (Figs. 5 and 9) for comparison so that the steady-state and quasiperiodic-state regions can roughly match among the three figures. Similar bifurcations are observed from the steady state (S), quasiperiodic oscillation (QP), to chaotic oscillations (C) for the three types of lasers. However, the feedback strengths required for observing chaotic dynamics are of different orders of magnitude. For the injection current of I = 2.5 Ith, a feedback strength of ∼0.1, normalized by the maximum feedback strength, is required to generate chaotic dynamics in the multi-mode QD laser, whereas feedback strengths of only ∼0.001 or ∼0.002 are required for generating chaotic dynamics in the single- and multi-mode QW lasers, respectively. Therefore, stronger optical feedback of two orders of magnitude is necessary to induce chaotic oscillations in the QD laser. More importantly, the LFF dynamics regions are different in the bifurcation diagrams, and LFF dynamics appear for large injection currents and large feedback strengths only for the QD lasers. Fast oscillations of the LFF dynamics at sub-GHz frequencies are observed in this LFF region, whereas no similar dynamics are observed in the single- and multi-mode QW lasers. Consequently, the fast LFF dynamics at sub-GHz frequencies are unique characteristics of QD lasers with optical feedback.
Fig. 12. Two-dimensional bifurcation diagram of the temporal waveforms of the outputs of (a) multi-mode QD laser, (b) single-mode QW laser, and (c) multi-mode QW laser. The ranges of the horizontal axes (feedback strengths) are adjusted to roughly match the dynamic regions of steady state and quasiperiodic oscillations. C, chaotic oscillations; IP, irregular pulsations; LFF, low-frequency fluctuations; QP, quasiperiodic oscillations; S, steady state.
4. Discussion
We found that fast LFF dynamics are observed at sub-GHz frequencies in the multi-mode QD laser with optical feedback, which is not the case for single- and multi-mode QW lasers. Indeed, this fast LFF dynamics looks comparable to the dynamics of regular pulse packages (RPP) [30], where the envelope of fast chaotic oscillations is modulated with a lower frequency component, as observed in QW lasers with a short external cavity (known as the short cavity regime [30]). This lower frequency corresponds to the external cavity frequency, and RPP is observed when the external cavity frequency is close to the relaxation oscillation frequency owing to the short external-cavity length. The existence of two frequency components induces chaotic oscillations with slow modulation. Moreover, similar dynamics have been observed experimentally in a QW laser with a short external cavity implemented on a photonic integrated circuit at large injection currents [31]. In addition, pulse-like oscillations and quasiperiodic dynamics related to the relaxation oscillation frequency and external cavity frequency have been found in a QD laser with optical feedback [17,18]. However, the peak frequency at sub-GHz region for the fast LFF dynamics in the QD laser does not originate from the external cavity frequency of 13.2 MHz in our experiment.
We speculate the physical origin of the sub-GHz-peak appearance for the fast LFF dynamics in Fig. 7(b) as follows: The longitudinal-mode interval of our QD laser is 0.102 nm (20.0 GHz), as shown in Fig. 2, which is two orders of magnitude larger than the fast LFF frequency at 0.240 GHz shown in Fig. 7. In addition, the peak frequency of fast LFF dynamics is approximately constant between 0.2 and 0.3 GHz as the injection current and feedback strength are changed. The spectral peak height at 0.2–0.3 GHz becomes larger as the feedback strength increases. Therefore, we speculate that the appearance of this sub-GHz frequency might originate from a larger relaxation lifetime owing to the phonon bottleneck effect [32–34], where the relaxation between the discrete eigenstates of QD lasers is slower than the recombination processes because of the lack of appropriate final states. The energy distribution of carriers may change in the presence of large optical feedback, and the carriers cannot be directly relaxed to the ground state, which may cause slow sub-GHz spectral components in the QD laser dynamics. Self-phase modulation in the QD laser cavity might be another possibility. The beat frequency between the original and self-phase-modulated optical signals may cause the appearance of sub-GHz frequency component.
Numerical simulations are required to identify the origin of the sub-GHz peak of fast LFF dynamics. Indeed, RPP dynamics have been observed in numerical simulations presented in [7]. This frequency corresponds to the difference between the external cavity frequency and the relaxation oscillation frequency. However, the external cavity frequency of 13.2 MHz is considerably smaller than the relaxation oscillation frequency (2–3 GHz) in our experiment. We will develop a theoretical model and perform numerical simulations to clarify the physical origin of the peak-frequency appearance at the sub-GHz region in our future work.
5. Conclusions
We experimentally investigated the nonlinear dynamics of a multi-mode QD semiconductor laser with time-delayed optical feedback. We created a 2D bifurcation diagram of the QD laser dynamics when the injection current and feedback strength were changed. Two different regions of LFF dynamics appeared in the bifurcation diagram. Fast LFF dynamics were observed at the sub-GHz frequency region, which was considerably faster than the ordinary LFF dynamics at MHz frequencies. We compared the dynamics of the QD laser with those of single- and multi-mode QW semiconductor lasers. We found that fast LFF dynamics are unique characteristics of QD lasers with optical feedback.
The findings about the fast LFF dynamics in the sub-GHz region in QD lasers are significantly important for understanding the nonlinear dynamics of QD lasers with optical feedback, and to avoid instabilities in optical communications. Furthermore, such fast LFF dynamics could be useful for applications in fast physical random number generation and photonic decision making.
Funding
Core Research for Evolutional Science and Technology (JPMJCR17N2); Japan Society for the Promotion of Science (JP19H00868, JP20H00233, JP20K15185); Telecommunications Advancement Foundation.
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.
References
1. Y. Arakawa and H. Sakaki, “Multidimensional quantum well lasers and temperature dependence of its threshold current,” Appl. Phys. Lett. 40(11), 939–941 (1982). [CrossRef]
2. D. G. Deppe, K. Shavritranuruk, G. Ozgur, H. Chen, and S. Freisem, “Quantum dot laser diode with low threshold and low internal loss,” Electron. Lett. 45(1), 54–55 (2009). [CrossRef]
3. A. Y. Liu, C. Zhang, J. Norman, A. Snyder, D. Lubyshev, J. M. Fastenau, A. W. K. Liu, A. C. Gossard, and J. E. Bowers, “High performance continuous wave 1.3 µm quantum dot lasers on silicon,” Appl. Phys. Lett. 104(4), 041104 (2014). [CrossRef]
4. K. Nishi, K. Takemasa, M. Sugawara, and Y. Arakawa, “Development of quantum dot lasers for data-com and silicon photonics applications,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017). [CrossRef]
5. N. Yamamoto, K. Akahane, T. Kawanishi, H. Sotobayashi, Y. Yoshioka, and H. Takai, “Characterization of wavelength-tunable quantum dot external cavity laser for 1.3-µm-waveband coherent light sources,” Jpn. J. Appl. Phys. 51(2S), 02BG08 (2012). [CrossRef]
6. G. Huyet, D. O’Brien, S. P. Hegarty, J. G. McInerney, A. V. Uskov, D. Bimberg, C. Ribbat, V. M. Ustinov, A. E. Zhukov, S. S. Mikhrin, A. R. Kovsh, J. K. White, K. Hinzer, and A. J. SpringThorpe, “Quantum dot semiconductor lasers with optical feedback,” Phys. Stat. Sol. (a) 201(2), 345–352 (2004). [CrossRef]
7. C. Otto, K. Lüdge, and E. Schöll, “Modeling quantum dot lasers with optical feedback: sensitivity of bifurcation scenarios,” Phys. Stat. Sol. (b) 247(4), 829–845 (2010). [CrossRef]
8. T. Erneux, E. A. Viktorov, and P. Mandel, “Time scales and relaxation dynamics in quantum dot lasers,” Phys. Rev. A 76(2), 023819 (2007). [CrossRef]
9. B. Lingnau, W. W. Chow, E. Schöll, and K. Lüdge, “Feedback and injection locking instabilities in quantum dot lasers: a microscopically based bifurcation analysis,” New J. Phys. 15(9), 093031 (2013). [CrossRef]
10. J. Pausch, C. Otto, E. Tylaite, N. Majer, E. Schöll, and K. Lüdge, “Optically injected quantum dot lasers: impact of nonlinear carrier lifetimes on frequency-locking dynamics,” New J. Phys. 14(5), 053018 (2012). [CrossRef]
11. B. Lingnau, K. Lüdge, W. W. Chow, and E. Schöll, “Influencing modulation properties of quantum-dot semiconductor lasers by electron lifetime engineering,” Appl. Phys. Lett. 101(13), 131107 (2012). [CrossRef]
12. A. Röhm, B. Lingnau, and K. Lüdge, “Understanding ground-state quenching in quantum dot lasers,” IEEE J. Quantum Electron. 51(1), 2000211 (2015). [CrossRef]
13. B. Lingnau and K. Lüdge, “Ultrafast gain recovery and large nonlinear optical response in submonolayer quantum dots,” Phys. Rev. B 94(1), 014305 (2016). [CrossRef]
14. M. Virte, S. Breuer, M. Sciamanna, and K. Panajotov, “Switching between ground and excited states by optical feedback in a quantum dot laser diode,” Appl. Phys. Lett. 105(12), 121109 (2014). [CrossRef]
15. H. Huang, D. Arsenijević, K. Schires, T. Sadeev, D. Bimberg, and F. Grillot, “Multimode optical feedback dynamics of InAs/GaAs quantum dot lasers emitting on different lasing states,” AIP Adv. 6(12), 125114 (2016). [CrossRef]
16. H. Huang, L. Lin, C. Chen, D. Arsenijevic, D. Bimberg, F. Lin, and F. Grillot, “Multimode optical feedback dynamics in InAs/GaAs quantum dot lasers emitting exclusively on ground or excited states: transition from short- to long-delay regimes,” Opt. Express 26(2), 1743–1751 (2018). [CrossRef]
17. B. Tykalewicz, D. Goulding, S. P. Hegarty, G. Huyet, T. Erneux, B. Kelleher, and E. A. Viktorov, “Emergence of resonant mode-locking via delayed feedback in quantum dot semiconductor lasers,” Opt. Express 24(4), 4239–4246 (2016). [CrossRef]
18. B. Kelleher, M. J. Wishon, A. Locquet, D. Goulding, B. Tykalewicz, G. Huyet, and E. A. Viktorov, “Delay induced high order locking effects in semiconductor lasers,” Chaos 27(11), 114325 (2017). [CrossRef]
19. J. Ohtsubo, Semiconductor Lasers, Stability, Instability and Chaos, 4th ed., Chap. 8.6 (Springer, 2017).
20. T. Sano, “Antimode dynamics and chaotic itinerancy in the coherence collapse of semiconductor lasers with optical feedback,” Phys. Rev. A 50(3), 2719–2726 (1994). [CrossRef]
21. I. Fischer, G. H. M. van Tartwijk, A. M. Levine, W. Elsässer, E. Göbel, and D. Lenstra, “Fast pulsing and chaotic itinerancy with a drift in the coherence collapse of semiconductor lasers,” Phys. Rev. Lett. 76(2), 220–223 (1996). [CrossRef]
22. T. Heil, I. Fischer, and W. Elsäßer, “Coexistence of low-frequency fluctuations and stable emission on a single high-gain mode in semiconductor lasers with external optical feedback,” Phys. Rev. A 58(4), R2672–R2675 (1998). [CrossRef]
23. A. Uchida, Optical Communication with Chaotic Lasers, Applications of Nonlinear Dynamics and Synchronization (Wiley-VCH, 2012).
24. A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008). [CrossRef]
25. A. Karsaklian Dal Bosco, N. Sato, Y. Terashima, S. Ohara, A. Uchida, T. Harayama, and M. Inubushi, “Random number generation from intermittent optical chaos,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1801208 (2017). [CrossRef]
26. M. Naruse, Y. Terashima, A. Uchida, and S.-J. Kim, “Ultrafast photonic reinforcement learning based on laser chaos,” Sci. Rep. 7(1), 8772 (2017). [CrossRef]
27. T. Mihana, Y. Mitsui, M. Takabayashi, K. Kanno, S. Sunada, M. Naruse, and A. Uchida, “Decision making for the multi-armed bandit problem using lag synchronization of chaos in mutually coupled semiconductor lasers,” Opt. Express 27(19), 26989–27008 (2019). [CrossRef]
28. T. Mihana, K. Fujii, K. Kanno, M. Naruse, and A. Uchida, “Laser network decision making by lag synchronization of chaos in a ring configuration,” Opt. Express 28(26), 40112–40130 (2020). [CrossRef]
29. K. Yamasaki, K. Kanno, A. Matsumoto, M. Naruse, N. Yamamoto, and A. Uchida, “Experimental investigation of nonlinear dynamics and bifurcation in a quantum-dot laser with optical feedback,” Frontiers in Optics / Laser Science, OSA Technical Digest, paper JTu3A.59 (2019).
30. T. Heil, I. Fischer, and W. Elsäßer, “Dynamics of semiconductor lasers subject to delayed optical feedback: The short cavity regime,” Phys. Rev. Lett. 87(24), 243901 (2001). [CrossRef]
31. A. Karsaklian Dal Bosco, S. Ohara, N. Sato, Y. Akizawa, A. Uchida, T. Harayama, and M. Inubushi, “Dynamics versus feedback delay time in photonic integrated circuits: Mapping the short cavity regime,” IEEE Photonics J. 9(2), 6600512 (2017). [CrossRef]
32. R. Heitz, H. Born, F. Guffarth, O. Stier, A. Schliwa, A. Hoffmann, and D. Bimberg, “Existence of a phonon bottleneck for excitons in quantum dots,” Phys. Rev. B 64(24), 241305 (2001). [CrossRef]
33. K. Mukai, N. Ohtsuka, H. Shoji, and M. Sugawara, “Phonon bottleneck in self-formed InxGa1-xAs/GaAs quantum dots by electroluminescence and time-resolved photoluminescence,” Phys. Rev. B 54(8), R5243–R5246 (1996). [CrossRef]
34. J. Urayama, T. B. Norris, J. Singh, and P. Bhattacharya, “Observation of phonon bottleneck in quantum dot electronic relaxation,” Phys. Rev. Lett. 86(21), 4930–4933 (2001). [CrossRef]
