Nonlinear Dynamics of Two-State Quantum Dot Lasers under Optical Feedback

: A modiﬁed rate equation model was presented to theoretically investigate the nonlinear dynamics of solitary two-state quantum dot lasers (TSQDLs) under optical feedback. The simulated results showed that, for a TSQDL biased at a relatively high current, the ground-state (GS) and excited-state (ES) lasing of the TSQDL can be stimulated simultaneously. After introducing optical feedback, both GS lasing and ES lasing can exhibit rich nonlinear dynamic states including steady state (S), period one (P1), period two (P2), multi-period (MP), and chaotic (C) state under different feedback strength and phase offset, respectively, and the dynamic states for the two lasing types are always identical. Furthermore, the inﬂuences of the linewidth enhancement factor (LEF) on the nonlinear dynamical state distribution of TSQDLs in the parameter space of feedback strength and phase offset were also analyzed. For a TSQDL with a larger LEF, much more dynamical states can be observed, and the parameter regions for two lasing types operating at chaotic state are widened after introducing optical feedback.

Quantum dot (QD) lasers are self-assembled nanostructured SLs. Compared with traditional quantum well (QW) SLs, QD lasers have many advantages such as low threshold current density [8], high temperature stability [9], low chirp [10], and large modulation bandwidth [11]. Such unique characteristics make QD lasers become excellent candidate light sources in optical communication, optical interconnection, silicon photonic integrated circuits, and photonic microwave generation, etc. [12][13][14][15][16]. Due to strong three-dimension quantum confinement of the carriers, QD lasers have discrete energy levels and state densities, which lead to their unique emission performances. Related studies have shown that there exist two current thresholds in ordinary QD lasers. When the bias current is increased to the first threshold, QD lasers can emit on the ground-state (GS). Continuously increasing the bias current, the number of carriers at the excited-state (ES) increases rapidly. Once the bias current exceeds a certain value (the second threshold), QD lasers can simultaneously emit on GS and ES. Correspondingly, such QD lasers are named as two-state QD lasers (TSQDLs) [17,18]. Via some technologies, QD lasers can emit solely on GS or ES, and the corresponding QD lasers are named as GS-QD lasers and ES-QD lasers, respectively [19,20].
Previous studies have shown that different types of QD lasers can exhibit different performances. GS-QD lasers possess a low threshold current and low sensitivity to optical feedback owing to relatively low energy levels and strong damping of relaxation oscillation [21,22]. Compared with GS-QD lasers, ES-QD lasers possess larger modulation bandwidths and richer nonlinear dynamics under external perturbations owing to faster carrier capture rates [23][24][25][26]. Different from GS-QD lasers and ES-QD lasers, TSQDLs can lase at two wavelengths separated by several tens of nanometers [27] and exhibit lower intensity noise [28], which can be applied in many fields such as terahertz (THz) signal generation, two-color light sources, two color mode-locking, all-optical processing, and artificial optical neurons, etc. [29][30][31][32]. In recent years, the investigations on the nonlinear dynamics of TSQDLs under external perturbations have attracted special attention. Through introducing optical injection into GS, the ES emission in TSQDLs can be suppressed and the mode switching from ES to GS is triggered [33,34]. Through scanning the optical power of injection light along different varying routes, a bistable phenomenon can be observed [35,36]. After introducing optical feedback to TSQDLs, many interesting phenomena can be observed such as mode switching and mode competition between the GS and ES [37,38], energy exchanging among longitudinal modes [39], two-color oscillating [40], and anti-phase low frequency fluctuating [41]. However, to our knowledge, the nonlinear dynamical state evolution of TSQDLs under optical feedback has not been reported.
In this work, based on three-level model of QD lasers [42,43], a modified theoretical model for TSQDLs under optical feedback was presented to numerically investigate the nonlinear dynamical characteristics of TSQDLs under optical feedback. Moreover, the influences of the linewidth enhancement factor (LEF) on the nonlinear dynamical state distribution of TSQDLs in the parameter space of feedback strength and phase offset were also analyzed.

Rate Equation Model
The theoretical model in this work was based on the three-level model of QD lasers, which has been adopted to analyze the static and dynamic behaviors, noise characteristics of QD lasers operating at free-running [42,43], and the small-signal modulation response and relative intensity noise of QD lasers under optical injection-locking conditions [44]. Figure 1 shows the simplified schematic diagram of the carrier dynamics for two-state QD lasers (TSQDLs) based on the three-level model [45]. In this system, two relatively low energy levels involving ground state (GS) and the first excited state (ES) were taken into account. The electrons and holes were treated as neutral excitons (electron-hole pairs), and the stimulated emission can occur in GS and ES. It was assumed that all QDs had the same size and the active region consisted of only one QD ensemble. Therefore, the inhomogeneous broadening effect was ignored. As shown in the figure, the carriers were injected directly into the wetting layer (WL) from the electrodes. In the WL, owing to Auger recombination and phonon-assisted scattering processes [46,47], some carriers were captured into ES with a captured time τ W L ES . Some carriers relaxed directly into GS with a relaxation time τ W L GS . The rest of the carriers recombined spontaneously with a time τ spon W L . For the carriers in ES, some of them relaxed into GS with a relaxation time τ ES GS and the other carriers recombined spontaneously with an emission time τ spon ES . On the other hand, owing to the thermal excitation effect, some carriers were excited into WL with an escape time τ ES W L . Similarly, the carriers in GS were excited into ES with an escape time τ GS ES , and some carriers also recombined spontaneously with an emission time τ spon GS . Based on the three-level model, after referring to the optical feedback processing methods in Ref. [48], we propose modified rate equations for describing the nonlinear dynamics of TSQDLs under optical feedback as follows: where WL, ES, GS are the wetting layer, excited-state, and ground-state, respectively, and the superscript spon represents the spontaneous emission. N, S, φ are the carrier number, photon number, and phase, respectively. I is the injection current, η is the current injection efficiency, and q is the electron charge. Γ p is the optical confinement factor, υ g (= c/n r , where c is the light speed in vacuum and n r the refractive index) is the group velocity. τ p is the photon lifetime, τ in is the round-trip time in the laser cavity, and τ (= 2 l ex /c, where l ex the external cavity length) is the round-trip time of external cavity. k is the feedback strength, and α is the linewidth enhancement factor. Considering that GS and ES have twofold degeneration and fourfold degeneration, respectively, the carrier occupation probabilities and the gains of GS and ES can be expressed as [42]: where N B is the number of quantum dots. a GS and a ES are the differential gain, ξ GS and ξ ES are the gain compression factor, vs. is the volume of the laser field inside the cavity, and V B is the volume of the active region. The feedback phase variation can be described as: where ω GS and ω ES are the angular frequencies for GS and ES lasing, respectively. The rate equations can be numerically solved by the fourth-order Runge-Kutta method via MATLAB software. During the calculations, the used parameters and their values are given in Table 1 Figure 2 shows the normalized output power of the GS and ES lasing as a function of the injection current for a TSQDL under free-running (solid lines) or optical feedback with a feedback strength of k = 0.11 (dotted lines). For the TSQDL operating at free-running,   a feedback strength of k = 0.11 (dotted lines). For the TSQDL operating at free-running, the threshold currents of the GS and ES lasing were 36 mA (I GS th ) and 88 mA (I ES th ), respectively. With the increase of the current from 36 mA to 88 mA, the power of GS lasing gradually increased while the ES lasing was always in a suppressed state. However, once the injection current was exceeded 88 mA, the ES lasing could be observed. Further increasing the current, the power of the ES lasing rapidly increased while the power of the GS lasing increased slowly. Above results are in agreement with those reported in Ref. [43]. After introducing an optical feedback of k = 0.11, the threshold current for GS slightly decreased, which is similar with that observed in a single-mode distributed feedback semiconductor laser under optical feedback. However, optical feedback raises the threshold of ES. The reason is that the predominant component in the feedback light is originating from GS lasing, and therefore the optical feedback enhances the competitiveness of the GS lasing. Correspondingly, a higher current is needed for ES to start oscillation. In the following, we fixed the current of the TSQDL at 120 mA, at which the power of GS lasing was more than that of ES lasing. , respectively. With the increase of the current from 36 mA to 88 mA, the power of GS lasing gradually increased while the ES lasing was always in a suppressed state. However, once the injection current was exceeded 88 mA, the ES lasing could be observed. Further increasing the current, the power of the ES lasing rapidly increased while the power of the GS lasing increased slowly. Above results are in agreement with those reported in Ref. [43]. After introducing an optical feedback of k = 0.11, the threshold current for GS slightly decreased, which is similar with that observed in a single-mode distributed feedback semiconductor laser under optical feedback. However, optical feedback raises the threshold of ES. The reason is that the predominant component in the feedback light is originating from GS lasing, and therefore the optical feedback enhances the competitiveness of the GS lasing. Correspondingly, a higher current is needed for ES to start oscillation. In the following, we fixed the current of the TSQDL at 120 mA, at which the power of GS lasing was more than that of ES lasing.   (Figure 3(b1)) exhibited a stable periodic oscillation with a fundamental frequency of about 6.3 GHz obtained from the power spectrum (Figure 3(b2)), and the phase portrait is a dense dot (Figure 3(b3)). Based on these characteristics, the dynamic state of GS lasing can be judged as a period-one (P1) state. For k = 0.092, the time series of GS lasing ( Figure  3(c1)) behaves periodic oscillation with two peak intensities, both the sub-harmonic frequency (about 3.1 GHz) and the fundamental frequency (about 6.3 GHz) present clearly in the power spectrum (Figure 3(c2)), and the corresponding phase portrait (Figure 3(c3)) is two closed circles, which are typical characteristics of period-two (P2) state. For k = 0.097, the time series of GS lasing (Figure 3(d1)) exhibited multiple different peaks, a quarterharmonic frequency component appeared in the power spectrum (Figure 3(d2)), and the phase portrait (Figure 3(d3)) showed multiple loops. These features mean that the dynamical state of GS lasing is a multi-period (MP) state. For k = 0.154, the time series of GS lasing (Figure 3(e1)) showed a disordered oscillation, and the power spectra were broadened (Figure 3(e2)). In addition, the corresponding phase portrait (Figure 3(e3)) showed a strange attractor. Therefore, the dynamic state of GS lasing can be determined to be the chaotic (C) state. Through comparing the characteristics of ES lasing with those of GS lasing, it can be seen that the dynamical states of ES lasing are always the same as those of GS lasing.  Figure 3 displays the time series, power spectra, and phase portraits of typical dynamic state output from GS lasing and ES lasing of a TSQDL biased at 120 mA under optical feedback with τ = 100 ps and different k. For k = 0.03, the output intensity of GS lasing (Figure 3(a1)) was nearly a constant, the power spectrum was relatively smooth (Figure 3(a2)), and the phase portrait was a dot (Figure 3(a3)). Obviously, under this case, the dynamical state of GS lasing is a stable (S) state. For k = 0.07, the time series of GS lasing (Figure 3(b1)) exhibited a stable periodic oscillation with a fundamental frequency of about 6.3 GHz obtained from the power spectrum (Figure 3(b2)), and the phase portrait is a dense dot (Figure 3(b3)). Based on these characteristics, the dynamic state of GS lasing can be judged as a period-one (P1) state. For k = 0.092, the time series of GS lasing (Figure 3(c1)) behaves periodic oscillation with two peak intensities, both the subharmonic frequency (about 3.1 GHz) and the fundamental frequency (about 6.3 GHz) present clearly in the power spectrum (Figure 3(c2)), and the corresponding phase portrait (Figure 3(c3)) is two closed circles, which are typical characteristics of period-two (P2) state. For k = 0.097, the time series of GS lasing (Figure 3(d1)) exhibited multiple different peaks, a quarter-harmonic frequency component appeared in the power spectrum (Figure 3(d2)), and the phase portrait (Figure 3(d3)) showed multiple loops. These features mean that the dynamical state of GS lasing is a multi-period (MP) state. For k = 0.154, the time series of GS lasing (Figure 3(e1)) showed a disordered oscillation, and the power spectra were broadened (Figure 3(e2)). In addition, the corresponding phase portrait (Figure 3(e3)) showed a strange attractor. Therefore, the dynamic state of GS lasing can be determined to be the chaotic (C) state. Through comparing the characteristics of ES lasing with those of  Above results show that, through setting feedback parameters at different values, some typical dynamical states can be observed for both ES and GS lasing. In order to inspect the evolution route of dynamical state with the feedback strength, Figure 4 presents the bifurcation diagrams of the power extreme and largest Lyapunov exponent (LLE) of the GS lasing and ES lasing as a function of feedback strength. LLE is an important indicator to measure the stability of a laser nonlinear dynamical system [49]. A positive LLE value means that the laser operates at a chaotic state while a negative LLE value corresponds to a steady state. For a laser operating at periodic states, the LLE value tends to approach zero. From this diagram, it can be seen that, with the increase of k from 0 to 0.043, the output of GS lasing and ES lasing remains in a stable state due to the relatively low feedback strength. Further increasing the feedback strength, the external cavity modes compete with the intrinsic oscillation frequency of the laser, and the dynamic states of GS lasing and ES lasing transform into periodic states including P1, P2, and MP. When the feedback strength exceeds 0.11, the TSQDL enters into the C state due to coherent collapse. As a result, the dynamics evolution routes of S-P1-P2-MP-C of the GS lasing and ES lasing are presented. Continuously increasing the feedback strength, the laser enters into the chaos state through period-doubling bifurcation, and such an evolution process repeats continuously. Above results show that, through setting feedback parameters at different values, some typical dynamical states can be observed for both ES and GS lasing. In order to inspect the evolution route of dynamical state with the feedback strength, Figure 4 presents the bifurcation diagrams of the power extreme and largest Lyapunov exponent (LLE) of the GS lasing and ES lasing as a function of feedback strength. LLE is an important indicator to measure the stability of a laser nonlinear dynamical system [49]. A positive LLE value means that the laser operates at a chaotic state while a negative LLE value corresponds to a steady state. For a laser operating at periodic states, the LLE value tends to approach zero. From this diagram, it can be seen that, with the increase of k from 0 to 0.043, the output of GS lasing and ES lasing remains in a stable state due to the relatively low feedback strength. Further increasing the feedback strength, the external cavity modes compete with the intrinsic oscillation frequency of the laser, and the dynamic states of GS lasing and ES lasing transform into periodic states including P1, P2, and MP. When the feedback strength exceeds 0.11, the TSQDL enters into the C state due to coherent collapse. As a result, the dynamics evolution routes of S-P1-P2-MP-C of the GS lasing and ES lasing are presented. Continuously increasing the feedback strength, the laser enters into the chaos state through period-doubling bifurcation, and such an evolution process repeats continuously.  Next, we discuss the influences of the round-trip time (τ) of the external cavity under a given feedback strength of k = 0.1. Here, we only consider the case that τ is varied around τ0 = 100 ps within a very small range, in which the offset (Δτ) of τ from τ0 = 100 ps satisfies -π/ωGS ≤ Δτ ≤ π/ωGS. Under this case, the phase offset φ(=ΔτωGS) of GS lasing is varied within (−π, π), and the corresponding phase offset of ES lasing is varied within (−1.06π, 1.06π). Figure 5 presents the bifurcation diagrams of the power extreme and LLE of the GS lasing and ES lasing as a function of phase offset under k = 0.1. With the increase of phase offset φ from −π to π, the dynamics evolution routes are more diverse. There exist multiple chaotic evolution routes for GS lasing and ES lasing including P1-S-C, P2-P1-P2-C, and C -MP-P2-C. The above results demonstrate that the feedback strength and the round-trip time τ (equivalent to phase offset) of the external cavity are two crucial parameters affecting the nonlinear dynamics of TSQDLs. Therefore, it is essential to investigate the overall dynamical evolution in the parameter space of feedback strength and phase offset. Figure 6 presents the mapping of the dynamical states for GS lasing (a) and ES lasing (b) in the parameter space of feedback strength and phase offset. There are rich dynamic states including S, P1, P2, MP, and C in the parameter space. With the increase of feedback strength, the phase offset required for achieving a chaotic state is gradually widened. Although the dynamic state distributions of GS lasing and ES lasing are similar, there exist subtle differences at the boundary between two modes. Through observing this diagram carefully, Next, we discuss the influences of the round-trip time (τ) of the external cavity under a given feedback strength of k = 0.1. Here, we only consider the case that τ is varied around τ 0 = 100 ps within a very small range, in which the offset (∆τ) of τ from τ 0 = 100 ps satisfies -π/ω GS ≤ ∆τ ≤ π/ω GS . Under this case, the phase offset ϕ(=∆τω GS ) of GS lasing is varied within (−π, π), and the corresponding phase offset of ES lasing is varied within (−1.06π, 1.06π). Figure 5 presents the bifurcation diagrams of the power extreme and LLE of the GS lasing and ES lasing as a function of phase offset under k = 0.1. With the increase of phase offset ϕ from −π to π, the dynamics evolution routes are more diverse. There exist multiple chaotic evolution routes for GS lasing and ES lasing including P1-S-C, P2-P1-P2-C, and C -MP-P2-C.  Next, we discuss the influences of the round-trip time (τ) of the external cavity under a given feedback strength of k = 0.1. Here, we only consider the case that τ is varied around τ0 = 100 ps within a very small range, in which the offset (Δτ) of τ from τ0 = 100 ps satisfies -π/ωGS ≤ Δτ ≤ π/ωGS. Under this case, the phase offset φ(=ΔτωGS) of GS lasing is varied within (−π, π), and the corresponding phase offset of ES lasing is varied within (−1.06π, 1.06π). Figure 5 presents the bifurcation diagrams of the power extreme and LLE of the GS lasing and ES lasing as a function of phase offset under k = 0.1. With the increase of phase offset φ from −π to π, the dynamics evolution routes are more diverse. There exist multiple chaotic evolution routes for GS lasing and ES lasing including P1-S-C, P2-P1-P2-C, and C -MP-P2-C. The above results demonstrate that the feedback strength and the round-trip time τ (equivalent to phase offset) of the external cavity are two crucial parameters affecting the nonlinear dynamics of TSQDLs. Therefore, it is essential to investigate the overall dynamical evolution in the parameter space of feedback strength and phase offset. Figure 6 presents the mapping of the dynamical states for GS lasing (a) and ES lasing (b) in the parameter space of feedback strength and phase offset. There are rich dynamic states including S, P1, P2, MP, and C in the parameter space. With the increase of feedback strength, the phase offset required for achieving a chaotic state is gradually widened. Although the dynamic state distributions of GS lasing and ES lasing are similar, there exist subtle differences at the boundary between two modes. Through observing this diagram carefully, The above results demonstrate that the feedback strength and the round-trip time τ (equivalent to phase offset) of the external cavity are two crucial parameters affecting the nonlinear dynamics of TSQDLs. Therefore, it is essential to investigate the overall dynamical evolution in the parameter space of feedback strength and phase offset. Figure 6 presents the mapping of the dynamical states for GS lasing (a) and ES lasing (b) in the parameter space of feedback strength and phase offset. There are rich dynamic states including S, P1, P2, MP, and C in the parameter space. With the increase of feedback strength, the phase offset required for achieving a chaotic state is gradually widened. Although the dynamic state distributions of GS lasing and ES lasing are similar, there exist subtle differences at the boundary between two modes. Through observing this diagram it can be found that there are multiple evolution routes for driving the laser into the chaotic state such as S-P1-P2-MP-C, P1-P2-MP-C, and P1-MP-C. Relevant research shows that the linewidth enhancement factor (LEF) α plays an important role for the nonlinear dynamics of SLs under external perturbations [50,51]. The above results were obtained under a fixed α taken as 3.5. Finally, we discuss the influences of LEF on the dynamical state distribution of a TSQDL under optical feedback. Figure 7 depicts mappings of dynamic states of GS lasing and ES lasing under different α. For α = 0.5 (Figure 7(a1,a2), the dynamical states of GS and ES are relatively simple, which include S, P1, and C. In the whole parameter space, most of the region is in a stable state, and only a small region is in the chaotic state. For α = 2.5, as shown in Figure 7 (b1,b2), there are much richer dynamic states involving P2 and MP. For a larger α of 4.5 as shown in Figure  7 (c1,c2), the chaotic state occupies a large area. Therefore, a large α is helpful for achieving chaotic state output.  Relevant research shows that the linewidth enhancement factor (LEF) α plays an important role for the nonlinear dynamics of SLs under external perturbations [50,51]. The above results were obtained under a fixed α taken as 3.5. Finally, we discuss the influences of LEF on the dynamical state distribution of a TSQDL under optical feedback. Figure 7 depicts mappings of dynamic states of GS lasing and ES lasing under different α. For α = 0.5 (Figure 7(a1,a2), the dynamical states of GS and ES are relatively simple, which include S, P1, and C. In the whole parameter space, most of the region is in a stable state, and only a small region is in the chaotic state. For α = 2.5, as shown in Figure 7(b1,b2), there are much richer dynamic states involving P2 and MP. For a larger α of 4.5 as shown in Figure 7(c1,c2), the chaotic state occupies a large area. Therefore, a large α is helpful for achieving chaotic state output. it can be found that there are multiple evolution routes for driving the laser into the chaotic state such as S-P1-P2-MP-C, P1-P2-MP-C, and P1-MP-C. Relevant research shows that the linewidth enhancement factor (LEF) α plays an important role for the nonlinear dynamics of SLs under external perturbations [50,51]. The above results were obtained under a fixed α taken as 3.5. Finally, we discuss the influences of LEF on the dynamical state distribution of a TSQDL under optical feedback. Figure 7 depicts mappings of dynamic states of GS lasing and ES lasing under different α. For α = 0.5 (Figure 7(a1,a2), the dynamical states of GS and ES are relatively simple, which include S, P1, and C. In the whole parameter space, most of the region is in a stable state, and only a small region is in the chaotic state. For α = 2.5, as shown in Figure 7 (b1,b2), there are much richer dynamic states involving P2 and MP. For a larger α of 4.5 as shown in Figure  7 (c1,c2), the chaotic state occupies a large area. Therefore, a large α is helpful for achieving chaotic state output.  Additionally, it should be pointed out that above results were obtained under the condition that the spontaneous emission noises were ignored. In fact, after considering the influence of spontaneous emission noise, the boundary of dynamical states may be changed slightly.

Conclusions
In summary, via a rate equation model used to characterize TSQDLs with optical feedback, the nonlinear dynamics of TSQDLs subject to optical feedback were investigated theoretically. For a TSQDL biased at 120 mA, both GS and ES lasing could be stimulated simultaneously, and the output power of GS emission was slightly larger than that of ES emission. After introducing optical feedback, multiple nonlinear dynamical states including S, P1, P2, MP, and C were observed for GS lasing and ES lasing under suitable feedback strengths and phase offset. Through mapping the evolution of dynamics state in the parameter space of feedback strength and phase offset, different evolution routes were revealed. In addition, the influences of the linewidth enhanced factor (LEF) on the dynamic state distribution of TSQDLs in the space parameter of feedback strength and phase shift were also presented. For a larger LEF, the parameter regions for GS lasing and ES lasing operating at chaotic state were wider. Although the dynamical behaviors of TSQDLs under optical feedback were similar to those observed in quantum well lasers under optical feedback, TSQDLs under optical feedback have the ability to provide two-channel chaotic signals with different lasing wavelengths, which are more promising for high-speed random number generation, wavelength-division multiplexing secure communication, and parallel-reservoir computing.