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

Nonlinear Dynamics of Two-State Quantum Dot Lasers under Optical Feedback hv photonics Article Nonlinear Dynamics of Two-State Quantum Dot Lasers under Optical Feedback 1 1 1 , 2 1 , Xiang-Hui Wang , Zheng-Mao Wu , Zai-Fu Jiang and Guang-Qiong Xia * Chongqing City Key Laboratory of Micro & Nano Structure Optoelectronics, School of Physical Science and Technology, Southwest University, Chongqing 400715, China; wangxianghui@email.swu.edu.cn (X.-H.W.); zmwu@swu.edu.cn (Z.-M.W.); jzf23003@email.swu.edu.cn (Z.-F.J.) School of Mathematics and Physics, Jingchu University of Technology, Jingmen 448000, China * Correspondence: gqxia@swu.edu.cn Abstract: 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. Keywords: nonlinear dynamics; quantum dot lasers; optical feedback; chaotic; linewidth enhance- Citation: Wang, X.-H.; Wu, Z.-M.; ment factor (LEF) Jiang, Z.-F.; Xia, G.-Q. Nonlinear Dynamics of Two-State Quantum Dot Lasers under Optical Feedback. Photonics 2021, 8, 300. https:// 1. Introduction doi.org/10.3390/photonics8080300 After introducing external perturbations, semiconductor lasers (SLs) can exhibit rich nonlinear dynamics [1,2], which can be applied in many fields such as random number Received: 17 May 2021 generation, secure communication, photonic microwave signal generation, all-optical logic Accepted: 23 July 2021 gates, and reservoir computing [3–7]. Published: 27 July 2021 Quantum dot (QD) lasers are self-assembled nanostructured SLs. Compared with tra- ditional quantum well (QW) SLs, QD lasers have many advantages such as low threshold Publisher’s Note: MDPI stays neutral current density [8], high temperature stability [9], low chirp [10], and large modulation with regard to jurisdictional claims in bandwidth [11]. Such unique characteristics make QD lasers become excellent candidate published maps and institutional affil- light sources in optical communication, optical interconnection, silicon photonic integrated iations. circuits, and photonic microwave generation, etc. [12–16]. Due to strong three-dimension quantum confinement of the carriers, QD lasers have discrete energy levels and state densi- ties, 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 Copyright: © 2021 by the authors. to the first threshold, QD lasers can emit on the ground-state (GS). Continuously increasing Licensee MDPI, Basel, Switzerland. the bias current, the number of carriers at the excited-state (ES) increases rapidly. Once the This article is an open access article bias current exceeds a certain value (the second threshold), QD lasers can simultaneously distributed under the terms and emit on GS and ES. Correspondingly, such QD lasers are named as two-state QD lasers conditions of the Creative Commons (TSQDLs) [17,18]. Via some technologies, QD lasers can emit solely on GS or ES, and the Attribution (CC BY) license (https:// corresponding QD lasers are named as GS-QD lasers and ES-QD lasers, respectively [19,20]. creativecommons.org/licenses/by/ 4.0/). Photonics 2021, 8, 300. https://doi.org/10.3390/photonics8080300 https://www.mdpi.com/journal/photonics Photonics 2021, 8, 300 2 of 11 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 op- tical 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 [23–26]. 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. [29–32]. In recent years, the investigations on the nonlinear dy- namics 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 phenom- ena 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. 2. 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 WL captured into ES with a captured time t . Some carriers relaxed directly into GS with a ES spon WL relaxation time t . The rest of the carriers recombined spontaneously with a time t . GS WL ES For the carriers in ES, some of them relaxed into GS with a relaxation time t and the GS spon other carriers recombined spontaneously with an emission time t . On the other hand, ES owing to the thermal excitation effect, some carriers were excited into WL with an escape ES GS time t . Similarly, the carriers in GS were excited into ES with an escape time t , and WL ES spon some carriers also recombined spontaneously with an emission time t . Based on the GS 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: dN hI N N N N WL ES WL WL WL = + (1 r ) (1 r ) (1) ES GS spon ES WL WL dt q t t t WL ES GS WL Photonics 2021, 8, 300 3 of 11 dN N N N N N ES WL GS ES ES ES = (1 r ) + (1 r ) (1 r ) G v g S (2) ES ES GS P g ES ES spon WL GS ES ES dt t t t t t ES ES WL GS ES dN N N N N GS WL ES GS GS = (1 r ) + (1 r ) (1 r ) G v g S (3) GS GS ES P g GS GS spon WL ES GS dt t t t t GS GS ES GS dS 1 N k GS GS = G v g S + b + 2 S (t)S (t t) cos(Df ) (4) g sp P GS GS GS GS GS spon dt t t p t in GS dS 1 N k ES ES = G v g S + b + 2 S (t)S (t t) cos(Df ) (5) P g ES ES sp ES ES ES spon dt t t p in ES df a 1 k S (t t) GS GS = G v g sin(Df ) (6) P g GS GS dt 2 t t S (t) p in GS df a 1 k S (t t) ES ES = G v g sin(Df ) (7) P ES ES dt 2 t t S (t) in ES where WL, ES, GS are the wetting layer, excited-state, and ground-state, respectively, and the superscript spon represents the spontaneous emission. N, S, f are the carrier number, photon number, and phase, respectively. I is the injection current, h is the current injection efficiency, and q is the electron charge. G is the optical confinement factor, u (= c/n , where p g r c is the light speed in vacuum and n the refractive index) is the group velocity. t is the r p photon lifetime, t is the round-trip time in the laser cavity, and t (= 2 l /c, where l the ex ex in external cavity length) is the round-trip time of external cavity. k is the feedback strength, and a 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]: N N GS ES r = ; r = (8) GS ES 2N 4N B B a N GS g = (2r 1) (9) GS GS GS 1 + x B GS a N ES B g = (2r 1) (10) ES ES ES 1 + x ES where N is the number of quantum dots. a and a are the differential gain, x and x B GS ES GS ES are the gain compression factor, vs. is the volume of the laser field inside the cavity, and V is the volume of the active region. The feedback phase variation can be described as: Df = f (t) f (t t) + w t (11) GS GS GS GS Df = f (t) f (t t) + w t (12) ES ES ES ES where w and w are the angular frequencies for GS and ES lasing, respectively. GS ES 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 [42]: Photonics 2021, 8, x FOR PEER REVIEW 4 of 11 Table 1. Simulation parameters of the QD lasers. Symbol Parameter Value WL τ Capture time from WL to ES 12.6 ps ES ES τ Capture time from ES to GS 8 ps GS WL τ Relaxation time from WL to GS 15 ps GS GS τ Escape time from GS to ES 10.4 ps ES ES τ Escape time from ES to WL 5.4 ns WL spon τ Spontaneous emission time from WL 0.5 ns WL spon τ Spontaneous emission time from ES 0.5 ns ES spon τ Spontaneous emission time from GS 1.2 ps GS τP Photon lifetime 4.1 ps NB Total number of QD 1.0  10 Гp Optical confinement factor 0.06 nr Refractive index 3.5 τin Round-trip time 10 ps −15 2 aGS Differential gain from GS 5.0  10 cm −15 2 aES Differential gain from ES 10.0  10 cm −16 3 ξGS Gain compression factor from GS 1.0  10 cm −16 3 ξES Gain compression factor from ES 8.0  10 cm −6 βsp Spontaneous emission factor 5.0  10 ωGS Angular frequency from GS 1.446  10 rad/s ωES Angular frequency from ES 1.529  10 rad/s −11 3 VB Active region volume 5.0  10 cm −15 3 VS Resonant cavity volume 0.833  10 cm η Injection efficiency 0.25 −19 Photonics 2021, 8, 300 q Elementary charge 1.6 × 10 C 4 of 11 τ Feedback delay time 100 ps α Linewidth enhancement factor 3.5 Figure 1. Schematic diagram of the carrier dynamics for QD lasers based on the three-level model. Figure 1. Schematic diagram of the carrier dynamics for QD lasers based on the three-level model. WL: wetting layer; GS: ground state; ES: excited state. WL: wetting layer; GS: ground state; ES: excited state. Table 1. Simulation parameters of the QD lasers. 3. Results and Discussion Symbol Parameter Value Figure 2 shows the normalized output power of the GS and ES lasing as a function of Capture time from WL to ES 12.6 ps WL the injection current for a TSQDL under free-running (solid lines) or optical feedback with ES ES Capture time from ES to GS 8 ps a feedback strength of k = 0.11 (dotted lines). For the TSQDL operating at free-running, GS Relaxation time from WL to GS 15 ps WL GS GS Escape time from GS to ES 10.4 ps ES Escape time from ES to WL 5.4 ns ES WL spon Spontaneous emission time from WL 0.5 ns WL spon Spontaneous emission time from ES 0.5 ns ES spon Spontaneous emission time from GS 1.2 ps GS Photon lifetime 4.1 ps Total number of QD 1.0  10 Optical confinement factor 0.06 Refractive index 3.5 Round-trip time 10 ps in 15 2 Differential gain from GS 5.0  10 cm GS 15 2 Differential gain from ES 10.0  10 cm ES 16 3 Gain compression factor from GS 1.0  10 cm GS 16 3 Gain compression factor from ES 8.0  10 cm ES Spontaneous emission factor 5.0  10 sp Angular frequency from GS 1.446  10 rad/s GS Angular frequency from ES 1.529  10 rad/s ES 11 3 Active region volume 5.0  10 cm 15 3 Resonant cavity volume 0.833  10 cm Injection efficiency 0.25 Elementary charge 1.6  10 C Feedback delay time 100 ps Linewidth enhancement factor 3.5 3. Results and Discussion 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 Photonics 2021, 8, 300 5 of 11 Photonics 2021, 8, x FOR PEER REVIEW 5 of 11 a feedback strength of k = 0.11 (dotted lines). For the TSQDL operating at free-running, the GS ES threshold currents of the GS and ES lasing were 36 mA (I ) and GS 88 mA (I ), respectively ES . th th the threshold currents of the GS and ES lasing were 36 mA (I ) and 88 mA (I ), respec- th th With the increase of the current from 36 mA to 88 mA, the power of GS lasing gradually tively. With the increase of the current from 36 mA to 88 mA, the power of GS lasing increased while the ES lasing was always in a suppressed state. However, once the injection gradually increased while the ES lasing was always in a suppressed state. However, once current was exceeded 88 mA, the ES lasing could be observed. Further increasing the the injection current was exceeded 88 mA, the ES lasing could be observed. Further in- current, the power of the ES lasing rapidly increased while the power of the GS lasing creasing the current, the power of the ES lasing rapidly increased while the power of the increased slowly. Above results are in agreement with those reported in Ref. [43]. After GS lasing increased slowly. Above results are in agreement with those reported in Ref. introducing an optical feedback of k = 0.11, the threshold current for GS slightly decreased, [43]. After introducing an optical feedback of k = 0.11, the threshold current for GS slightly which is similar with that observed in a single-mode distributed feedback semiconductor decreased, which is similar with that observed in a single-mode distributed feedback sem- laser under optical feedback. However, optical feedback raises the threshold of ES. The iconductor laser under optical feedback. However, optical feedback raises the threshold reason is that the predominant component in the feedback light is originating from GS of ES. The reason is that the predominant component in the feedback light is originating lasing, and therefore the optical feedback enhances the competitiveness of the GS lasing. from GS lasing, and therefore the optical feedback enhances the competitiveness of the GS Correspondingly, a higher current is needed for ES to start oscillation. In the following, we lasing. Correspondingly, a higher current is needed for ES to start oscillation. In the fol- fixed the current of the TSQDL at 120 mA, at which the power of GS lasing was more than lowing, we fixed the current of the TSQDL at 120 mA, at which the power of GS lasing that of ES lasing. was more than that of ES lasing. Figure 2. Normalized output power as a function of the injection current for a TSQDL under Figure 2. Normalized output power as a function of the injection current for a TSQDL under free- free-running (solid lines) or optical feedback with a feedback strength of k = 0.11 (dotted lines). running (solid lines) or optical feedback with a feedback strength of k = 0.11 (dotted lines). Figure 3 displays the time series, power spectra, and phase portraits of typical dy- Figure 3 displays the time series, power spectra, and phase portraits of typical dy- namic state output from GS lasing and ES lasing of a TSQDL biased at 120 mA under namic state output from GS lasing and ES lasing of a TSQDL biased at 120 mA under optical feedback with t = 100 ps and different k. For k = 0.03, the output intensity of GS 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 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, (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 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 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 por- of about 6.3 GHz obtained from the power spectrum (Figure 3(b2)), and the phase portrait trait is a dense dot (Figure 3(b3)). Based on these characteristics, the dynamic state of is a dense dot (Figure 3(b3)). Based on these characteristics, the dynamic state of GS lasing GS lasing can be judged as a period-one (P1) state. For k = 0.092, the time series of GS can be judged as a period-one (P1) state. For k = 0.092, the time series of GS lasing (Figure lasing (Figure 3(c1)) behaves periodic oscillation with two peak intensities, both the sub- 3(c1)) behaves periodic oscillation with two peak intensities, both the sub-harmonic fre- harmonic frequency (about 3.1 GHz) and the fundamental frequency (about 6.3 GHz) quency (about 3.1 GHz) and the fundamental frequency (about 6.3 GHz) present clearly present clearly in the power spectrum (Figure 3(c2)), and the corresponding phase portrait in the power spectrum (Figure 3(c2)), and the corresponding phase portrait (Figure 3(c3)) (Figure 3(c3)) is two closed circles, which are typical characteristics of period-two (P2) state. is two closed circles, which are typical characteristics of period-two (P2) state. For k = 0.097, For k = 0.097, the time series of GS lasing (Figure 3(d1)) exhibited multiple different peaks, the time series of GS lasing (Figure 3(d1)) exhibited multiple different peaks, a quarter- a quarter-harmonic frequency component appeared in the power spectrum (Figure 3(d2)), harmonic frequency component appeared in the power spectrum (Figure 3(d2)), and the and the phase portrait (Figure 3(d3)) showed multiple loops. These features mean that phase portrait (Figure 3(d3)) showed multiple loops. These features mean that the dynam- the dynamical state of GS lasing is a multi-period (MP) state. For k = 0.154, the time series ical state of GS lasing is a multi-period (MP) state. For k = 0.154, the time series of GS lasing of GS lasing (Figure 3(e1)) showed a disordered oscillation, and the power spectra were (Figure 3(e1)) showed a disordered oscillation, and the power spectra were broadened broadened (Figure 3(e2)). In addition, the corresponding phase portrait (Figure 3(e3)) (Figure 3(e2)). In addition, the corresponding phase portrait (Figure 3(e3)) showed a showed a strange attractor. Therefore, the dynamic state of GS lasing can be determined to strange attractor. Therefore, the dynamic state of GS lasing can be determined to be the be the chaotic (C) state. Through comparing the characteristics of ES lasing with those of chaotic (C) state. Through comparing the characteristics of ES lasing with those of GS las- ing, it can be seen that the dynamical states of ES lasing are always the same as those of GS lasing. Photonics 2021, 8, 300 6 of 11 Photonics 2021, 8, x FOR PEER REVIEW 6 of 11 GS lasing, it can be seen that the dynamical states of ES lasing are always the same as those of GS lasing. Figure 3. Time series, power spectra, and phase portraits output from GS lasing (red) and ES lasing (blue) in a TSQDL biased Figure 3. Time series, power spectra, and phase portraits output from GS lasing (red) and ES lasing (blue) in a TSQDL at 120 mA under optical feedback with t = 100 ps and k = 0.03 (a), 0.07 (b), 0.092 (c), 0.097 (d), and 0.154 (e), respectively. biased at 120 mA under optical feedback with τ = 100 ps and k = 0.03 (a), 0.07 (b), 0.092 (c), 0.097 (d), and 0.154 (e), respec- tively. 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 Above results show that, through setting feedback parameters at different values, inspect the evolution route of dynamical state with the feedback strength, Figure 4 presents some typical dynamical states can be observed for both ES and GS lasing. In order to in- the bifurcation diagrams of the power extreme and largest Lyapunov exponent (LLE) of the spect the evolution route of dynamical state with the feedback strength, Figure 4 presents GS lasing and ES lasing as a function of feedback strength. LLE is an important indicator the bifurcation diagrams of the power extreme and largest Lyapunov exponent (LLE) of to measure the stability of a laser nonlinear dynamical system [49]. A positive LLE value the GS lasing and ES lasing as a function of feedback strength. LLE is an important indi- means that the laser operates at a chaotic state while a negative LLE value corresponds to a cator to measure the stability of a laser nonlinear dynamical system [49]. A positive LLE steady state. For a laser operating at periodic states, the LLE value tends to approach zero. value means that the laser operates at a chaotic state while a negative LLE value corre- From this diagram, it can be seen that, with the increase of k from 0 to 0.043, the output sponds to a steady state. For a laser operating at periodic states, the LLE value tends to of GS lasing and ES lasing remains in a stable state due to the relatively low feedback approach zero. From this diagram, it can be seen that, with the increase of k from 0 to strength. Further increasing the feedback strength, the external cavity modes compete with 0.043, the output of GS lasing and ES lasing remains in a stable state due to the relatively the intrinsic oscillation frequency of the laser, and the dynamic states of GS lasing and ES low feedback strength. Further increasing the feedback strength, the external cavity lasing transform into periodic states including P1, P2, and MP. When the feedback strength modes compete with the intrinsic oscillation frequency of the laser, and the dynamic states exceeds 0.11, the TSQDL enters into the C state due to coherent collapse. As a result, the of GS lasing and ES lasing transform into periodic states including P1, P2, and MP. When dynamics evolution routes of S-P1-P2-MP-C of the GS lasing and ES lasing are presented. the feedback strength exceeds 0.11, the TSQDL enters into the C state due to coherent Continuously increasing the feedback strength, the laser enters into the chaos state through collapse. As a result, the dynamics evolution routes of S-P1-P2-MP-C of the GS lasing and period-doubling bifurcation, and such an evolution process repeats continuously. 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. Photonics 2021, 8, x FOR PEER REVIEW 7 of 11 Photonics 2021, 8, 300 7 of 11 Photonics 2021, 8, x FOR PEER REVIEW 7 of 11 (a) (b) (a) (b) Figure 4. Bifurcation diagrams of power extreme and largest Lyapunov exponent (LLE) as a function of feedback strength Figure 4. Bifurcation diagrams of power extreme and largest Lyapunov exponent (LLE) as a function of feedback strength of Figure the GS las 4. Bifur ing ( cation a) and diagrams ES lasing of (power b) in a extr TSQDL eme bia and selar d at 120 mA under gest Lyapunov exponent optical feed (LLE) bac as k wit a function h τ = 100 ofps. feedback strength of the GS lasing (a) and ES lasing (b) in a TSQDL biased at 120 mA under optical feedback with τ = 100 ps. of the GS lasing (a) and ES lasing (b) in a TSQDL biased at 120 mA under optical feedback with t = 100 ps. Next, we discuss the influences of the round-trip time (τ) of the external cavity under Next, we discuss the influences of the round-trip time (τ) of the external cavity under Next, we discuss the influences of the round-trip time (t) of the external cavity under a given feedback strength of k = 0.1. Here, we only consider the case that τ is varied around a given feedback strength of k = 0.1. Here, we only consider the case that τ is varied around a given feedback strength of k = 0.1. Here, we only consider the case that t is varied τ0 = 100 ps within a very small range, in which the offset (Δτ) of τ from τ0 = 100 ps satisfies τ ar 0 = ound 100 ps t w =i100 thin ps a ve within ry smal a very l rang small e, in which range, th ine which offset (the Δτ)of of fset τ fr(om Dt) τof 0 = t100 from ps t sati = sfies 100 0 0 –π/ωGS ≤ Δτ ≤ π/ωGS. Under this case, the phase offset φ(=ΔτωGS) of GS lasing is varied –π/ωGS ≤ Δτ ≤ π/ωGS. Under this case, the phase offset φ(=ΔτωGS) of GS lasing is varied ps satisfies –/w  Dt  /w . Under this case, the phase offset j(=Dtw ) of GS within (−π, π), and GSthe correspondin GS g phase offset of ES lasing is varied withinGS (−1.06π, within (−π, π), and the corresponding phase offset of ES lasing is varied within (−1.06π, lasing is varied within (, ), and the corresponding phase offset of ES lasing is varied 1.06π). Figure 5 presents the bifurcation diagrams of the power extreme and LLE of the 1.06 within π). F ( igur 1.06 e  5 , pre 1.06 sents ). Figur the bi e 5 fu pr rcesents ation di the agrams bifur cation of the diagrams power extof reme the an power d LLE extr of eme the GS lasing and ES lasing as a function of phase offset under k = 0.1. With the increase of GS andlasing LLE of an the d ES GS lasing lasing as and a f ES unction lasing o as f ph a funct ase o ion ffset of under phase k of= fset 0.1under . With kth =e 0.1. incW rea ith se the of phase offset φ from −π to π, the dynamics evolution routes are more diverse. There exist increase of phase offset j from  to , the dynamics evolution routes are more diverse. 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- multipl There exist e chao multiple tic evolchaotic ution routes evolution for GS routes lasing for and GS E lasing S lasing and inc ES lud lasing ing Pincluding 1-S-C, P2-P1-S-C, P1-P2- C, and C -MP-P2-C. P2-P1-P2-C, and C -MP-P2-C. C, and C -MP-P2-C. (a) (b) (a) (b) Figure 5. Bifurcation diagrams of the power extreme; LLE as a function of phase offset of the GS lasing (a) and ES lasing Figure 5. Bifurcation diagrams of the power extreme; LLE as a function of phase offset of the GS lasing (a) and ES lasing (b) Figure 5. Bifurcation diagrams of the power extreme; LLE as a function of phase offset of the GS lasing (a) and ES lasing (b) in a TSQDL under I = 120 mA and k = 0.1. in a TSQDL under I = 120 mA and k = 0.1. (b) in a TSQDL under I = 120 mA and k = 0.1. The above results demonstrate that the feedback strength and the round-trip time τ The above results demonstrate that the feedback strength and the round-trip time 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 t (equivalent to phase offset) of the external cavity are two crucial parameters affecting (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 dynam- the nonlinear dynamics of TSQDLs. Therefore, it is essential to investigate the overall nonlinear dynamics of TSQDLs. Therefore, it is essential to investigate the overall dynam- ical evolution in the parameter space of feedback strength and phase offset. Figure 6 pre- dynamical evolution in the parameter space of feedback strength and phase offset. Figure 6 ical evolution in the parameter space of feedback strength and phase offset. Figure 6 pre- sents the mapping of the dynamical states for GS lasing (a) and ES lasing (b) in the param- presents the mapping of the dynamical states for GS lasing (a) and ES lasing (b) in the sents the mapping of the dynamical states for GS lasing (a) and ES lasing (b) in the param- eter space of feedback strength and phase offset. There are rich dynamic states including parameter space of feedback strength and phase offset. There are rich dynamic states eter 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 including S, P1, P2, MP, and C in the parameter space. With the increase of feedback 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 strength, the phase offset required for achieving a chaotic state is gradually widened. 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 dif- Although the dynamic state distributions of GS lasing and ES lasing are similar, there exist dynamic state distributions of GS lasing and ES lasing are similar, there exist subtle dif- ferences at the boundary between two modes. Through observing this diagram carefully, subtle differences at the boundary between two modes. Through observing this diagram ferences at the boundary between two modes. Through observing this diagram carefully, Photonics 2021, 8, x FOR PEER REVIEW 8 of 11 Photonics 2021, 8, 300 8 of 11 Photonics 2021, 8, x FOR PEER REVIEW 8 of 11 it can be found that there are multiple evolution routes for driving the laser into the cha- carefully, it can be found that there are multiple evolution routes for driving the laser into it can be found that there are multiple evolution routes for driving the laser into the cha- otic state such as S-P1-P2-MP-C, P1-P2-MP-C, and P1-MP-C. the otic chaotic state suc state h as such S-P1as -P2 S-P1-P2-MP-C, -MP-C, P1-P2-MP P1-P2-MP-C, -C, and P1and -MPP1-MP-C. -C. Figure 6. Mapping of the dynamical states for GS lasing (a) and ES lasing (b) of a TSQDL in the Figure 6. Mapping of the dynamical states for GS lasing (a) and ES lasing (b) of a TSQDL in the Figure 6. Mapping of the dynamical states for GS lasing (a) and ES lasing (b) of a TSQDL in the parameter space of feedback strength and phase offset. S: stable, P1: period-one, P2: period-two, parameter space of feedback strength and phase offset. S: stable, P1: period-one, P2: period-two, MP: parameter space of feedback strength and phase offset. S: stable, P1: period-one, P2: period-two, MP: multi-period, and C: chaos. multi-period, and C: chaos. MP: multi-period, and C: chaos. Relevant research shows that the linewidth enhancement factor (LEF) α plays an im- Relevant research shows that the linewidth enhancement factor (LEF) a plays an Relevant research shows that the linewidth enhancement factor (LEF) α plays an im- portant role for the nonlinear dynamics of SLs under external perturbations [50,51]. The important role for the nonlinear dynamics of SLs under external perturbations [50,51]. The portant 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 above results were obtained under a fixed a taken as 3.5. Finally, we discuss the influences 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 of LEF on the dynamical state distribution of a TSQDL under optical feedback. Figure 7 of LEF on the dynamical state distribution of a TSQDL under optical feedback. Figure 7 depicts depicts mappings mappings o of f dyn dynamic amic states statesoof f GS GSlasin lasing g an and d ES ES lasing lasing under under diff dif erfer enent t α. For a. For α = depicts mappings of dynamic states of GS lasing and ES lasing under different α. For α = a 0.5 = ( 0.5 Fig(Figur ure 7(a1,a2 e 7(a1,a2), ), the d the ynamic dynamical al states states of GS of anGS d ES and are ES rela ar tivel e rel y atively simple,simple, which incl which ude 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 include S, P1, and C. In the whole parameter space, most of the region is in a stable state, S, P1, and C. In the whole parameter space, most of the region is in a stable state, and only and a sm only all reg a io small n is r in egion the ch isaotic in the state. chaotic For state α = 2.5, . For as ashown = 2.5, as in shown Figure 7 in (b1 Figur ,b2), e 7 th (b1,b2), ere are a small region is in the chaotic state. For α = 2.5, as shown in Figure 7 (b1,b2), there are ther much e ar richer e much dynam richer ic dynamic states involv states ing involving P2 and MP P2 . F and or a MP lar.ger For αa o lar f 4.5 ger asa show of 4.5n as inshown Figure much richer dynamic states involving P2 and MP. For a larger α of 4.5 as shown in Figure in Figure 7(c1,c2), the chaotic state occupies a large area. Therefore, a large a is helpful for 7 (c1,c2), the chaotic state occupies a large area. Therefore, a large α is helpful for achieving 7 (c1,c2), the chaotic state occupies a large area. Therefore, a large α is helpful for achieving achieving chaotic state output. chaotic state output. chaotic state output. Figure 7. Mappings of the dynamical states of GS lasing (the first row) and ES lasing (the second Figure 7. Mappings of the dynamical states of GS lasing (the first row) and ES lasing (the second Figure 7. Mappings of the dynamical states of GS lasing (the first row) and ES lasing (the second row) row) in the parameter space of feedback strength and phase offset under different α, where (a) α = row) in the parameter space of feedback strength and phase offset under different α, where (a) α = in the parameter space of feedback strength and phase offset under different a, where (a) a = 0.5, (b) 0.5, (b) α = 2.5, (c) α = 4.5. S: stable, P1: period-one, P2: period-two, MP: multi-period, and C: chaos. 0.5, (b) α = 2.5, (c) α = 4.5. S: stable, P1: period-one, P2: period-two, MP: multi-period, and C: chaos. a = 2.5, (c) a = 4.5. S: stable, P1: period-one, P2: period-two, MP: multi-period, and C: chaos. Photonics 2021, 8, 300 9 of 11 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. 4. 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. Author Contributions: X.-H.W. and Z.-F.J. were responsible for the numerical simulation, analyzing the results, and the writing of the paper. Z.-M.W. and G.-Q.X. were responsible for the discussion of the results and reviewing/editing/revising/proof-reading of the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China (Grant Nos. 61775184 and 61875167). Data Availability Statement: Not applicable. 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Nonlinear Dynamics of Two-State Quantum Dot Lasers under Optical Feedback

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hv photonics Article Nonlinear Dynamics of Two-State Quantum Dot Lasers under Optical Feedback 1 1 1 , 2 1 , Xiang-Hui Wang , Zheng-Mao Wu , Zai-Fu Jiang and Guang-Qiong Xia * Chongqing City Key Laboratory of Micro & Nano Structure Optoelectronics, School of Physical Science and Technology, Southwest University, Chongqing 400715, China; wangxianghui@email.swu.edu.cn (X.-H.W.); zmwu@swu.edu.cn (Z.-M.W.); jzf23003@email.swu.edu.cn (Z.-F.J.) School of Mathematics and Physics, Jingchu University of Technology, Jingmen 448000, China * Correspondence: gqxia@swu.edu.cn Abstract: 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. Keywords: nonlinear dynamics; quantum dot lasers; optical feedback; chaotic; linewidth enhance- Citation: Wang, X.-H.; Wu, Z.-M.; ment factor (LEF) Jiang, Z.-F.; Xia, G.-Q. Nonlinear Dynamics of Two-State Quantum Dot Lasers under Optical Feedback. Photonics 2021, 8, 300. https:// 1. Introduction doi.org/10.3390/photonics8080300 After introducing external perturbations, semiconductor lasers (SLs) can exhibit rich nonlinear dynamics [1,2], which can be applied in many fields such as random number Received: 17 May 2021 generation, secure communication, photonic microwave signal generation, all-optical logic Accepted: 23 July 2021 gates, and reservoir computing [3–7]. Published: 27 July 2021 Quantum dot (QD) lasers are self-assembled nanostructured SLs. Compared with tra- ditional quantum well (QW) SLs, QD lasers have many advantages such as low threshold Publisher’s Note: MDPI stays neutral current density [8], high temperature stability [9], low chirp [10], and large modulation with regard to jurisdictional claims in bandwidth [11]. Such unique characteristics make QD lasers become excellent candidate published maps and institutional affil- light sources in optical communication, optical interconnection, silicon photonic integrated iations. circuits, and photonic microwave generation, etc. [12–16]. Due to strong three-dimension quantum confinement of the carriers, QD lasers have discrete energy levels and state densi- ties, 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 Copyright: © 2021 by the authors. to the first threshold, QD lasers can emit on the ground-state (GS). Continuously increasing Licensee MDPI, Basel, Switzerland. the bias current, the number of carriers at the excited-state (ES) increases rapidly. Once the This article is an open access article bias current exceeds a certain value (the second threshold), QD lasers can simultaneously distributed under the terms and emit on GS and ES. Correspondingly, such QD lasers are named as two-state QD lasers conditions of the Creative Commons (TSQDLs) [17,18]. Via some technologies, QD lasers can emit solely on GS or ES, and the Attribution (CC BY) license (https:// corresponding QD lasers are named as GS-QD lasers and ES-QD lasers, respectively [19,20]. creativecommons.org/licenses/by/ 4.0/). Photonics 2021, 8, 300. https://doi.org/10.3390/photonics8080300 https://www.mdpi.com/journal/photonics Photonics 2021, 8, 300 2 of 11 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 op- tical 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 [23–26]. 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. [29–32]. In recent years, the investigations on the nonlinear dy- namics 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 phenom- ena 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. 2. 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 WL captured into ES with a captured time t . Some carriers relaxed directly into GS with a ES spon WL relaxation time t . The rest of the carriers recombined spontaneously with a time t . GS WL ES For the carriers in ES, some of them relaxed into GS with a relaxation time t and the GS spon other carriers recombined spontaneously with an emission time t . On the other hand, ES owing to the thermal excitation effect, some carriers were excited into WL with an escape ES GS time t . Similarly, the carriers in GS were excited into ES with an escape time t , and WL ES spon some carriers also recombined spontaneously with an emission time t . Based on the GS 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: dN hI N N N N WL ES WL WL WL = + (1 r ) (1 r ) (1) ES GS spon ES WL WL dt q t t t WL ES GS WL Photonics 2021, 8, 300 3 of 11 dN N N N N N ES WL GS ES ES ES = (1 r ) + (1 r ) (1 r ) G v g S (2) ES ES GS P g ES ES spon WL GS ES ES dt t t t t t ES ES WL GS ES dN N N N N GS WL ES GS GS = (1 r ) + (1 r ) (1 r ) G v g S (3) GS GS ES P g GS GS spon WL ES GS dt t t t t GS GS ES GS dS 1 N k GS GS = G v g S + b + 2 S (t)S (t t) cos(Df ) (4) g sp P GS GS GS GS GS spon dt t t p t in GS dS 1 N k ES ES = G v g S + b + 2 S (t)S (t t) cos(Df ) (5) P g ES ES sp ES ES ES spon dt t t p in ES df a 1 k S (t t) GS GS = G v g sin(Df ) (6) P g GS GS dt 2 t t S (t) p in GS df a 1 k S (t t) ES ES = G v g sin(Df ) (7) P ES ES dt 2 t t S (t) in ES where WL, ES, GS are the wetting layer, excited-state, and ground-state, respectively, and the superscript spon represents the spontaneous emission. N, S, f are the carrier number, photon number, and phase, respectively. I is the injection current, h is the current injection efficiency, and q is the electron charge. G is the optical confinement factor, u (= c/n , where p g r c is the light speed in vacuum and n the refractive index) is the group velocity. t is the r p photon lifetime, t is the round-trip time in the laser cavity, and t (= 2 l /c, where l the ex ex in external cavity length) is the round-trip time of external cavity. k is the feedback strength, and a 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]: N N GS ES r = ; r = (8) GS ES 2N 4N B B a N GS g = (2r 1) (9) GS GS GS 1 + x B GS a N ES B g = (2r 1) (10) ES ES ES 1 + x ES where N is the number of quantum dots. a and a are the differential gain, x and x B GS ES GS ES are the gain compression factor, vs. is the volume of the laser field inside the cavity, and V is the volume of the active region. The feedback phase variation can be described as: Df = f (t) f (t t) + w t (11) GS GS GS GS Df = f (t) f (t t) + w t (12) ES ES ES ES where w and w are the angular frequencies for GS and ES lasing, respectively. GS ES 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 [42]: Photonics 2021, 8, x FOR PEER REVIEW 4 of 11 Table 1. Simulation parameters of the QD lasers. Symbol Parameter Value WL τ Capture time from WL to ES 12.6 ps ES ES τ Capture time from ES to GS 8 ps GS WL τ Relaxation time from WL to GS 15 ps GS GS τ Escape time from GS to ES 10.4 ps ES ES τ Escape time from ES to WL 5.4 ns WL spon τ Spontaneous emission time from WL 0.5 ns WL spon τ Spontaneous emission time from ES 0.5 ns ES spon τ Spontaneous emission time from GS 1.2 ps GS τP Photon lifetime 4.1 ps NB Total number of QD 1.0  10 Гp Optical confinement factor 0.06 nr Refractive index 3.5 τin Round-trip time 10 ps −15 2 aGS Differential gain from GS 5.0  10 cm −15 2 aES Differential gain from ES 10.0  10 cm −16 3 ξGS Gain compression factor from GS 1.0  10 cm −16 3 ξES Gain compression factor from ES 8.0  10 cm −6 βsp Spontaneous emission factor 5.0  10 ωGS Angular frequency from GS 1.446  10 rad/s ωES Angular frequency from ES 1.529  10 rad/s −11 3 VB Active region volume 5.0  10 cm −15 3 VS Resonant cavity volume 0.833  10 cm η Injection efficiency 0.25 −19 Photonics 2021, 8, 300 q Elementary charge 1.6 × 10 C 4 of 11 τ Feedback delay time 100 ps α Linewidth enhancement factor 3.5 Figure 1. Schematic diagram of the carrier dynamics for QD lasers based on the three-level model. Figure 1. Schematic diagram of the carrier dynamics for QD lasers based on the three-level model. WL: wetting layer; GS: ground state; ES: excited state. WL: wetting layer; GS: ground state; ES: excited state. Table 1. Simulation parameters of the QD lasers. 3. Results and Discussion Symbol Parameter Value Figure 2 shows the normalized output power of the GS and ES lasing as a function of Capture time from WL to ES 12.6 ps WL the injection current for a TSQDL under free-running (solid lines) or optical feedback with ES ES Capture time from ES to GS 8 ps a feedback strength of k = 0.11 (dotted lines). For the TSQDL operating at free-running, GS Relaxation time from WL to GS 15 ps WL GS GS Escape time from GS to ES 10.4 ps ES Escape time from ES to WL 5.4 ns ES WL spon Spontaneous emission time from WL 0.5 ns WL spon Spontaneous emission time from ES 0.5 ns ES spon Spontaneous emission time from GS 1.2 ps GS Photon lifetime 4.1 ps Total number of QD 1.0  10 Optical confinement factor 0.06 Refractive index 3.5 Round-trip time 10 ps in 15 2 Differential gain from GS 5.0  10 cm GS 15 2 Differential gain from ES 10.0  10 cm ES 16 3 Gain compression factor from GS 1.0  10 cm GS 16 3 Gain compression factor from ES 8.0  10 cm ES Spontaneous emission factor 5.0  10 sp Angular frequency from GS 1.446  10 rad/s GS Angular frequency from ES 1.529  10 rad/s ES 11 3 Active region volume 5.0  10 cm 15 3 Resonant cavity volume 0.833  10 cm Injection efficiency 0.25 Elementary charge 1.6  10 C Feedback delay time 100 ps Linewidth enhancement factor 3.5 3. Results and Discussion 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 Photonics 2021, 8, 300 5 of 11 Photonics 2021, 8, x FOR PEER REVIEW 5 of 11 a feedback strength of k = 0.11 (dotted lines). For the TSQDL operating at free-running, the GS ES threshold currents of the GS and ES lasing were 36 mA (I ) and GS 88 mA (I ), respectively ES . th th the threshold currents of the GS and ES lasing were 36 mA (I ) and 88 mA (I ), respec- th th With the increase of the current from 36 mA to 88 mA, the power of GS lasing gradually tively. With the increase of the current from 36 mA to 88 mA, the power of GS lasing increased while the ES lasing was always in a suppressed state. However, once the injection gradually increased while the ES lasing was always in a suppressed state. However, once current was exceeded 88 mA, the ES lasing could be observed. Further increasing the the injection current was exceeded 88 mA, the ES lasing could be observed. Further in- current, the power of the ES lasing rapidly increased while the power of the GS lasing creasing the current, the power of the ES lasing rapidly increased while the power of the increased slowly. Above results are in agreement with those reported in Ref. [43]. After GS lasing increased slowly. Above results are in agreement with those reported in Ref. introducing an optical feedback of k = 0.11, the threshold current for GS slightly decreased, [43]. After introducing an optical feedback of k = 0.11, the threshold current for GS slightly which is similar with that observed in a single-mode distributed feedback semiconductor decreased, which is similar with that observed in a single-mode distributed feedback sem- laser under optical feedback. However, optical feedback raises the threshold of ES. The iconductor laser under optical feedback. However, optical feedback raises the threshold reason is that the predominant component in the feedback light is originating from GS of ES. The reason is that the predominant component in the feedback light is originating lasing, and therefore the optical feedback enhances the competitiveness of the GS lasing. from GS lasing, and therefore the optical feedback enhances the competitiveness of the GS Correspondingly, a higher current is needed for ES to start oscillation. In the following, we lasing. Correspondingly, a higher current is needed for ES to start oscillation. In the fol- fixed the current of the TSQDL at 120 mA, at which the power of GS lasing was more than lowing, we fixed the current of the TSQDL at 120 mA, at which the power of GS lasing that of ES lasing. was more than that of ES lasing. Figure 2. Normalized output power as a function of the injection current for a TSQDL under Figure 2. Normalized output power as a function of the injection current for a TSQDL under free- free-running (solid lines) or optical feedback with a feedback strength of k = 0.11 (dotted lines). running (solid lines) or optical feedback with a feedback strength of k = 0.11 (dotted lines). Figure 3 displays the time series, power spectra, and phase portraits of typical dy- Figure 3 displays the time series, power spectra, and phase portraits of typical dy- namic state output from GS lasing and ES lasing of a TSQDL biased at 120 mA under namic state output from GS lasing and ES lasing of a TSQDL biased at 120 mA under optical feedback with t = 100 ps and different k. For k = 0.03, the output intensity of GS 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 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, (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 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 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 por- of about 6.3 GHz obtained from the power spectrum (Figure 3(b2)), and the phase portrait trait is a dense dot (Figure 3(b3)). Based on these characteristics, the dynamic state of is a dense dot (Figure 3(b3)). Based on these characteristics, the dynamic state of GS lasing GS lasing can be judged as a period-one (P1) state. For k = 0.092, the time series of GS can be judged as a period-one (P1) state. For k = 0.092, the time series of GS lasing (Figure lasing (Figure 3(c1)) behaves periodic oscillation with two peak intensities, both the sub- 3(c1)) behaves periodic oscillation with two peak intensities, both the sub-harmonic fre- harmonic frequency (about 3.1 GHz) and the fundamental frequency (about 6.3 GHz) quency (about 3.1 GHz) and the fundamental frequency (about 6.3 GHz) present clearly present clearly in the power spectrum (Figure 3(c2)), and the corresponding phase portrait in the power spectrum (Figure 3(c2)), and the corresponding phase portrait (Figure 3(c3)) (Figure 3(c3)) is two closed circles, which are typical characteristics of period-two (P2) state. is two closed circles, which are typical characteristics of period-two (P2) state. For k = 0.097, For k = 0.097, the time series of GS lasing (Figure 3(d1)) exhibited multiple different peaks, the time series of GS lasing (Figure 3(d1)) exhibited multiple different peaks, a quarter- a quarter-harmonic frequency component appeared in the power spectrum (Figure 3(d2)), harmonic frequency component appeared in the power spectrum (Figure 3(d2)), and the and the phase portrait (Figure 3(d3)) showed multiple loops. These features mean that phase portrait (Figure 3(d3)) showed multiple loops. These features mean that the dynam- the dynamical state of GS lasing is a multi-period (MP) state. For k = 0.154, the time series ical state of GS lasing is a multi-period (MP) state. For k = 0.154, the time series of GS lasing of GS lasing (Figure 3(e1)) showed a disordered oscillation, and the power spectra were (Figure 3(e1)) showed a disordered oscillation, and the power spectra were broadened broadened (Figure 3(e2)). In addition, the corresponding phase portrait (Figure 3(e3)) (Figure 3(e2)). In addition, the corresponding phase portrait (Figure 3(e3)) showed a showed a strange attractor. Therefore, the dynamic state of GS lasing can be determined to strange attractor. Therefore, the dynamic state of GS lasing can be determined to be the be the chaotic (C) state. Through comparing the characteristics of ES lasing with those of chaotic (C) state. Through comparing the characteristics of ES lasing with those of GS las- ing, it can be seen that the dynamical states of ES lasing are always the same as those of GS lasing. Photonics 2021, 8, 300 6 of 11 Photonics 2021, 8, x FOR PEER REVIEW 6 of 11 GS lasing, it can be seen that the dynamical states of ES lasing are always the same as those of GS lasing. Figure 3. Time series, power spectra, and phase portraits output from GS lasing (red) and ES lasing (blue) in a TSQDL biased Figure 3. Time series, power spectra, and phase portraits output from GS lasing (red) and ES lasing (blue) in a TSQDL at 120 mA under optical feedback with t = 100 ps and k = 0.03 (a), 0.07 (b), 0.092 (c), 0.097 (d), and 0.154 (e), respectively. biased at 120 mA under optical feedback with τ = 100 ps and k = 0.03 (a), 0.07 (b), 0.092 (c), 0.097 (d), and 0.154 (e), respec- tively. 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 Above results show that, through setting feedback parameters at different values, inspect the evolution route of dynamical state with the feedback strength, Figure 4 presents some typical dynamical states can be observed for both ES and GS lasing. In order to in- the bifurcation diagrams of the power extreme and largest Lyapunov exponent (LLE) of the spect the evolution route of dynamical state with the feedback strength, Figure 4 presents GS lasing and ES lasing as a function of feedback strength. LLE is an important indicator the bifurcation diagrams of the power extreme and largest Lyapunov exponent (LLE) of to measure the stability of a laser nonlinear dynamical system [49]. A positive LLE value the GS lasing and ES lasing as a function of feedback strength. LLE is an important indi- means that the laser operates at a chaotic state while a negative LLE value corresponds to a cator to measure the stability of a laser nonlinear dynamical system [49]. A positive LLE steady state. For a laser operating at periodic states, the LLE value tends to approach zero. value means that the laser operates at a chaotic state while a negative LLE value corre- From this diagram, it can be seen that, with the increase of k from 0 to 0.043, the output sponds to a steady state. For a laser operating at periodic states, the LLE value tends to of GS lasing and ES lasing remains in a stable state due to the relatively low feedback approach zero. From this diagram, it can be seen that, with the increase of k from 0 to strength. Further increasing the feedback strength, the external cavity modes compete with 0.043, the output of GS lasing and ES lasing remains in a stable state due to the relatively the intrinsic oscillation frequency of the laser, and the dynamic states of GS lasing and ES low feedback strength. Further increasing the feedback strength, the external cavity lasing transform into periodic states including P1, P2, and MP. When the feedback strength modes compete with the intrinsic oscillation frequency of the laser, and the dynamic states exceeds 0.11, the TSQDL enters into the C state due to coherent collapse. As a result, the of GS lasing and ES lasing transform into periodic states including P1, P2, and MP. When dynamics evolution routes of S-P1-P2-MP-C of the GS lasing and ES lasing are presented. the feedback strength exceeds 0.11, the TSQDL enters into the C state due to coherent Continuously increasing the feedback strength, the laser enters into the chaos state through collapse. As a result, the dynamics evolution routes of S-P1-P2-MP-C of the GS lasing and period-doubling bifurcation, and such an evolution process repeats continuously. 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. Photonics 2021, 8, x FOR PEER REVIEW 7 of 11 Photonics 2021, 8, 300 7 of 11 Photonics 2021, 8, x FOR PEER REVIEW 7 of 11 (a) (b) (a) (b) Figure 4. Bifurcation diagrams of power extreme and largest Lyapunov exponent (LLE) as a function of feedback strength Figure 4. Bifurcation diagrams of power extreme and largest Lyapunov exponent (LLE) as a function of feedback strength of Figure the GS las 4. Bifur ing ( cation a) and diagrams ES lasing of (power b) in a extr TSQDL eme bia and selar d at 120 mA under gest Lyapunov exponent optical feed (LLE) bac as k wit a function h τ = 100 ofps. feedback strength of the GS lasing (a) and ES lasing (b) in a TSQDL biased at 120 mA under optical feedback with τ = 100 ps. of the GS lasing (a) and ES lasing (b) in a TSQDL biased at 120 mA under optical feedback with t = 100 ps. Next, we discuss the influences of the round-trip time (τ) of the external cavity under Next, we discuss the influences of the round-trip time (τ) of the external cavity under Next, we discuss the influences of the round-trip time (t) of the external cavity under a given feedback strength of k = 0.1. Here, we only consider the case that τ is varied around a given feedback strength of k = 0.1. Here, we only consider the case that τ is varied around a given feedback strength of k = 0.1. Here, we only consider the case that t is varied τ0 = 100 ps within a very small range, in which the offset (Δτ) of τ from τ0 = 100 ps satisfies τ ar 0 = ound 100 ps t w =i100 thin ps a ve within ry smal a very l rang small e, in which range, th ine which offset (the Δτ)of of fset τ fr(om Dt) τof 0 = t100 from ps t sati = sfies 100 0 0 –π/ωGS ≤ Δτ ≤ π/ωGS. Under this case, the phase offset φ(=ΔτωGS) of GS lasing is varied –π/ωGS ≤ Δτ ≤ π/ωGS. Under this case, the phase offset φ(=ΔτωGS) of GS lasing is varied ps satisfies –/w  Dt  /w . Under this case, the phase offset j(=Dtw ) of GS within (−π, π), and GSthe correspondin GS g phase offset of ES lasing is varied withinGS (−1.06π, within (−π, π), and the corresponding phase offset of ES lasing is varied within (−1.06π, lasing is varied within (, ), and the corresponding phase offset of ES lasing is varied 1.06π). Figure 5 presents the bifurcation diagrams of the power extreme and LLE of the 1.06 within π). F ( igur 1.06 e  5 , pre 1.06 sents ). Figur the bi e 5 fu pr rcesents ation di the agrams bifur cation of the diagrams power extof reme the an power d LLE extr of eme the GS lasing and ES lasing as a function of phase offset under k = 0.1. With the increase of GS andlasing LLE of an the d ES GS lasing lasing as and a f ES unction lasing o as f ph a funct ase o ion ffset of under phase k of= fset 0.1under . With kth =e 0.1. incW rea ith se the of phase offset φ from −π to π, the dynamics evolution routes are more diverse. There exist increase of phase offset j from  to , the dynamics evolution routes are more diverse. 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- multipl There exist e chao multiple tic evolchaotic ution routes evolution for GS routes lasing for and GS E lasing S lasing and inc ES lud lasing ing Pincluding 1-S-C, P2-P1-S-C, P1-P2- C, and C -MP-P2-C. P2-P1-P2-C, and C -MP-P2-C. C, and C -MP-P2-C. (a) (b) (a) (b) Figure 5. Bifurcation diagrams of the power extreme; LLE as a function of phase offset of the GS lasing (a) and ES lasing Figure 5. Bifurcation diagrams of the power extreme; LLE as a function of phase offset of the GS lasing (a) and ES lasing (b) Figure 5. Bifurcation diagrams of the power extreme; LLE as a function of phase offset of the GS lasing (a) and ES lasing (b) in a TSQDL under I = 120 mA and k = 0.1. in a TSQDL under I = 120 mA and k = 0.1. (b) in a TSQDL under I = 120 mA and k = 0.1. The above results demonstrate that the feedback strength and the round-trip time τ The above results demonstrate that the feedback strength and the round-trip time 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 t (equivalent to phase offset) of the external cavity are two crucial parameters affecting (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 dynam- the nonlinear dynamics of TSQDLs. Therefore, it is essential to investigate the overall nonlinear dynamics of TSQDLs. Therefore, it is essential to investigate the overall dynam- ical evolution in the parameter space of feedback strength and phase offset. Figure 6 pre- dynamical evolution in the parameter space of feedback strength and phase offset. Figure 6 ical evolution in the parameter space of feedback strength and phase offset. Figure 6 pre- sents the mapping of the dynamical states for GS lasing (a) and ES lasing (b) in the param- presents the mapping of the dynamical states for GS lasing (a) and ES lasing (b) in the sents the mapping of the dynamical states for GS lasing (a) and ES lasing (b) in the param- eter space of feedback strength and phase offset. There are rich dynamic states including parameter space of feedback strength and phase offset. There are rich dynamic states eter 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 including S, P1, P2, MP, and C in the parameter space. With the increase of feedback 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 strength, the phase offset required for achieving a chaotic state is gradually widened. 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 dif- Although the dynamic state distributions of GS lasing and ES lasing are similar, there exist dynamic state distributions of GS lasing and ES lasing are similar, there exist subtle dif- ferences at the boundary between two modes. Through observing this diagram carefully, subtle differences at the boundary between two modes. Through observing this diagram ferences at the boundary between two modes. Through observing this diagram carefully, Photonics 2021, 8, x FOR PEER REVIEW 8 of 11 Photonics 2021, 8, 300 8 of 11 Photonics 2021, 8, x FOR PEER REVIEW 8 of 11 it can be found that there are multiple evolution routes for driving the laser into the cha- carefully, it can be found that there are multiple evolution routes for driving the laser into it can be found that there are multiple evolution routes for driving the laser into the cha- otic state such as S-P1-P2-MP-C, P1-P2-MP-C, and P1-MP-C. the otic chaotic state suc state h as such S-P1as -P2 S-P1-P2-MP-C, -MP-C, P1-P2-MP P1-P2-MP-C, -C, and P1and -MPP1-MP-C. -C. Figure 6. Mapping of the dynamical states for GS lasing (a) and ES lasing (b) of a TSQDL in the Figure 6. Mapping of the dynamical states for GS lasing (a) and ES lasing (b) of a TSQDL in the Figure 6. Mapping of the dynamical states for GS lasing (a) and ES lasing (b) of a TSQDL in the parameter space of feedback strength and phase offset. S: stable, P1: period-one, P2: period-two, parameter space of feedback strength and phase offset. S: stable, P1: period-one, P2: period-two, MP: parameter space of feedback strength and phase offset. S: stable, P1: period-one, P2: period-two, MP: multi-period, and C: chaos. multi-period, and C: chaos. MP: multi-period, and C: chaos. Relevant research shows that the linewidth enhancement factor (LEF) α plays an im- Relevant research shows that the linewidth enhancement factor (LEF) a plays an Relevant research shows that the linewidth enhancement factor (LEF) α plays an im- portant role for the nonlinear dynamics of SLs under external perturbations [50,51]. The important role for the nonlinear dynamics of SLs under external perturbations [50,51]. The portant 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 above results were obtained under a fixed a taken as 3.5. Finally, we discuss the influences 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 of LEF on the dynamical state distribution of a TSQDL under optical feedback. Figure 7 of LEF on the dynamical state distribution of a TSQDL under optical feedback. Figure 7 depicts depicts mappings mappings o of f dyn dynamic amic states statesoof f GS GSlasin lasing g an and d ES ES lasing lasing under under diff dif erfer enent t α. For a. For α = depicts mappings of dynamic states of GS lasing and ES lasing under different α. For α = a 0.5 = ( 0.5 Fig(Figur ure 7(a1,a2 e 7(a1,a2), ), the d the ynamic dynamical al states states of GS of anGS d ES and are ES rela ar tivel e rel y atively simple,simple, which incl which ude 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 include S, P1, and C. In the whole parameter space, most of the region is in a stable state, S, P1, and C. In the whole parameter space, most of the region is in a stable state, and only and a sm only all reg a io small n is r in egion the ch isaotic in the state. chaotic For state α = 2.5, . For as ashown = 2.5, as in shown Figure 7 in (b1 Figur ,b2), e 7 th (b1,b2), ere are a small region is in the chaotic state. For α = 2.5, as shown in Figure 7 (b1,b2), there are ther much e ar richer e much dynam richer ic dynamic states involv states ing involving P2 and MP P2 . F and or a MP lar.ger For αa o lar f 4.5 ger asa show of 4.5n as inshown Figure much richer dynamic states involving P2 and MP. For a larger α of 4.5 as shown in Figure in Figure 7(c1,c2), the chaotic state occupies a large area. Therefore, a large a is helpful for 7 (c1,c2), the chaotic state occupies a large area. Therefore, a large α is helpful for achieving 7 (c1,c2), the chaotic state occupies a large area. Therefore, a large α is helpful for achieving achieving chaotic state output. chaotic state output. chaotic state output. Figure 7. Mappings of the dynamical states of GS lasing (the first row) and ES lasing (the second Figure 7. Mappings of the dynamical states of GS lasing (the first row) and ES lasing (the second Figure 7. Mappings of the dynamical states of GS lasing (the first row) and ES lasing (the second row) row) in the parameter space of feedback strength and phase offset under different α, where (a) α = row) in the parameter space of feedback strength and phase offset under different α, where (a) α = in the parameter space of feedback strength and phase offset under different a, where (a) a = 0.5, (b) 0.5, (b) α = 2.5, (c) α = 4.5. S: stable, P1: period-one, P2: period-two, MP: multi-period, and C: chaos. 0.5, (b) α = 2.5, (c) α = 4.5. S: stable, P1: period-one, P2: period-two, MP: multi-period, and C: chaos. a = 2.5, (c) a = 4.5. S: stable, P1: period-one, P2: period-two, MP: multi-period, and C: chaos. Photonics 2021, 8, 300 9 of 11 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. 4. 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. Author Contributions: X.-H.W. and Z.-F.J. were responsible for the numerical simulation, analyzing the results, and the writing of the paper. Z.-M.W. and G.-Q.X. were responsible for the discussion of the results and reviewing/editing/revising/proof-reading of the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China (Grant Nos. 61775184 and 61875167). Data Availability Statement: Not applicable. 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Journal

PhotonicsMultidisciplinary Digital Publishing Institute

Published: Jul 27, 2021

Keywords: nonlinear dynamics; quantum dot lasers; optical feedback; chaotic; linewidth enhancement factor (LEF)

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