1. Introduction

Semiconductor quantum-dot lasers have been studied extensively during the last decades [1–3]. Compared to the well established quantum well lasers [4], properties such as the lasing threshold, temperature sensitivity, and stability against perturbations can be improved by using semiconductor quantum dots as the active medium [1,5–8].

Nevertheless, previous works have shown, that exposing quantum dot lasers to external perturbations like optical feedback or injection, a rich variety of dynamics including oscillation bursts [9], externally activated pulse trains [10], regular pulse packages [11,12], nonlinear oscillations [13], excitability [14,15] and chaos [16–18] can be observed.

Furthermore and so far unique among all lasers, simultaneous emission at two distinct wavelengths from the quantum dot ground (GS) and first excited state (ES) transition can be observed under proper operating conditions [19,20]. This so called two-state lasing arises due to the non-instantaneous charge-carrier dynamics within the quantum dots [19,21]. A subset of those two-state lasers also exhibits ground state quenching [22], i.e. a complete rollover of the GS intensity with increasing pump currents, such that lasing emission from either the GS, ES or both simultaneously can be found. A range of all-optical control schemes that allow the switching between those different lasing states has been demonstrated using either optical feedback [23,24] or optical injection [8,25–27].

Due to these features, two-state lasers have been considered for applications in self-mixing velocimetry [28], all-optical switching [25,27], enhanced modulation response [29] and artificial optical neurons [30].

This work investigates the impact of optical feedback on a two-state quantum-dot laser and describes the emergence of two-color anti-phase oscillation bursts. We restrict our investigations to the ground-state lasing regime and to a frequency selective feedback scheme [28,31], that only couples back the light emitted from the GS. Different from other works [18,23,32], we focus on the interplay between the GS and ES dynamics in the ultra-short feedback regime [33]. Our model is based on microscopically derived rate equations, which include carrier population dependent refractive index shifts and scattering rates [8,34]. We analyze the system both by direct integration and path continuation techniques.

In this manuscript, we show that short optical feedback not only can be utilized to switch between stable emission states, but can also induce two-color anti-phase oscillation bursts, where the lasing emission periodically alternates between oscillation bursts from the quantum dot ground and excited state. Previous works on two-state quantum dot lasers subject to optical injection have reported anti-phase pulsations [27,35] and two-color oscillation bursts [9], where the former has been attributed to Q-switching like behavior and the latter to thermal switching. To learn about the origin of the feedback induced oscillation bursts, we unravel the bifurcation structure and find an alternative bridge [36,37] between ECMs provided by the two-state emission, which undergoes a torus bifurcation that generates the bursts. We furthermore demonstrate, that the feedback strength quantitatively and qualitatively controls the properties of the oscillation bursts, which includes chaotic burst envelopes, and discuss the relation to low-frequency fluctuations [38,39] and regular pulse packages in single-color quantum dot laser [5,12].

2. Setup and model

In our setup, we consider a $1$ mm long edge-emitting laser based on 15 layers of self-assembled InAs/InGaAs QDs with GS and ES emission wavelengths at $1300$nm and $1210$nm respectively. A frequency selective mirror only couples light emitted by the GS back into the laser resonator. The feedback is characterized by the propagation time $\tau$ in the feedback loop, the relative feedback strength $K$ and the feedback phase $C$. Figure fig:scheme(a) shows a sketch of the setup.

Our modeling approach consists of coupled rate equations, which include microscopically calculated charge-carrier scattering rates and charge-carrier dependent refractive index shifts that are extracted from a full Bloch-equation model [2,8,34].

Figure 1(b) shows a sketch of the band structure across one quantum dot and illustrates the dynamic variables of the model. The electronic degrees of freedom are color coded in green and include: the occupation probabilities for electrons and holes in the quantum dot GS $(\rho ^\textrm {GS}_\textrm {e},\rho ^\textrm {GS}_\textrm {h})$ and in the ES $( \rho ^\textrm {ES}_\textrm {e},\rho ^\textrm {ES}_\textrm {h})$ and the area charge-carrier density in the carrier reservoir layer for electrons and holes $\left ( w_\textrm {e},w_\textrm {h} \right )$. Their interaction processes due to coulomb scattering are indicated with dark green arrows and are labeled with their respective capture $S^\textrm {cap}_\textrm {b,m}$ and relaxation $S^\textrm {rel}_\textrm {b,m}$ scattering rates. The carrier reservoir is electrically pumped by an electrical current $J$. To furthermore account for the varying sizes of the QDs and thus varying localization energies, i.e. the inhomogeneous broadening, the QDs are separated into a set of resonant and optically active QDs and a set of off-resonant and optically inactive QDs. The ratio between the two sets is extracted from a full Maxwell-Bloch model, which uses QD subgroups to describe the inhomogeneous broadening [17]. We use $\textrm {m} \in \{\rm GS,ES\}$ to denote the QD state, $\textrm {b} \in \{\rm e,h\}$ to denote the charge-carrier type and $\textrm {x} \in \{ \rm a,i\}$ to denote either the active or inactive set of QDs. The quantum dot GS and ES occupations respectively couple via stimulated recombination, depicted by red and blue arrows, to the electric field amplitudes $E^\textrm {GS}$ and $E^\textrm {ES}$ at the GS and ES transition energies. The dynamical equations of the charge-carriers read:

 

Fig. 1. (a) Sketch of the setup. A frequency selective mirror only couples back light at the frequency of the GS emission. (b) Sketch of the band structure across one quantum dot (QD). Charge carrier variables are denoted in green and their coupling via scattering processes is indicated by dark green arrows. Recombination of GS and ES charge carriers is represented by red and blue arrows respectively.

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The dynamics of the electric field amplitude is split into slowly varying envelopes $E^\textrm {GS}$ and $E^\textrm {ES}$ at the GS and ES transition energies.

The second term of (6) represents the frequency selective optical feedback and is scaled by the GS field losses $\kappa ^\textrm {GS}$ such that $K=1$ corresponds to all of the light emitted by the GS being coupled back into the laser cavity. As feedback is only applied to the GS field, only the magnitude of the ES field has to be considered in (7) and therefore the carrier dependent frequency shift of the ES field is neglected in this model. The frequency shift $\delta \omega ^\textrm {GS}$ dynamically depends on the non-resonant charge carriers and is given by

Finally, the output-power at the GS and ES frequency can be calculated according to

In summary, we end up with a system of 12 coupled nonlinear dynamical equations, which we analyze using path continuation techniques implemented in the MATLAB software package DDE-Biftool [44] and direct integration by a custom code, which implements a 4th order Runge-Kutta algorithm with Hermite interpolation of the delay array.

3. Results

3.1 Solitary and feedback dynamics

We first characterize the quantum-dot laser described by the parameters given in Table 1 without the external optical feedback loop. Figure 2(a) shows the input-output diagram where the pump current $J$ has been normalized to the lasing threshold value $J_\textrm {GS}^\textrm {th}$. The emission powers at the GS and ES frequencies are denoted by orange and blue lines respectively and the green dashed line corresponds to the total out-coupled power. Above the lasing threshold three distinct regimes are found: For pump currents below $J = 1.48J_\textrm {GS}^\textrm {th}$, only emission from the GS is observed. Above that current, the GS power experiences a roll-over as the ES starts lasing. Simultaneous emission from both states is found up to a pump current of $J = 1.55J_\textrm {GS}^\textrm {th}$, where the GS emission becomes fully quenched and only emission from the ES remains. Previous works on QD lasers have demonstrated that two-state lasing is achieved due to the non instantaneous carrier-relaxation time between the QD GS and ES [19,21,22] and that the quenching of the GS emission arises due to an asymmetry of the electron and hole dynamics [22,45]. For the investigation of the feedback induced dynamics, we choose the pump current $J = 1.40J_\textrm {GS}^\textrm {th}$ as an operating point, which is within the GS-lasing regime, but in the vicinity of the ES-lasing threshold, such that perturbations can trigger emission from the ES. Both the GS and the ES are completely overdamped at this pump current. The small-signal modulation corner frequency of the GS is at around 3.5 GHz though, which should provide a rough approximation of the relaxation oscillation frequency. The output power is $P_0 = 20.17\,\textrm {mW}$, which will be used as reference for further results.

 

Fig. 2. (a) Input-output characteristics of the solitary two-state laser with the pump current normalized to the threshold current. The GS emission (orange) exhibits a rollover after the onset of ES emission (purple) and full quenching at higher pump currents. Operating point at $J = 1.4J_\textrm {GS}^\textrm {th}$ with $P_0 = 20.17\,\textrm {mW}$. (b) Feedback induced dynamics in the feedback strength, feedback phase plane at $\tau = 150\textrm {ps}$. The steady output-power at the GS frequency is color coded in orange. Blue and green colors indicate period one and higher order oscillations. Hatches denote two-state lasing. Enclosed by a two-state lasing region, the emission completely switches from the GS to the ES. The purple line represents the parameter range shown in the bifurcation scan Fig. 3. (c) Time series of the GS (orange) and ES (purple) emission power exemplifying the complex oscillations in the green region in (b) at $C=0.849,K=0.127$.

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Table 1. Parameters used for numerical calculations unless noted otherwise.

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For our investigation of the feedback induced dynamics, we keep the feedback time fixed within the short delay regime at $\tau = 150\,\textrm {ps}$ [4,33]. Qualitatively similar dynamics exist in the feedback time range between $\tau \approx 100\,\textrm {ps}$ and $\tau \approx 300\,\textrm {ps}$, but are manifested less prominently. In order to first obtain an overview, we perform numerical sweeps of the feedback strength $K$ for different feedback phases $C$. We restrict the feedback parameters to values where dynamics other than cw-lasing from the GS are observed. The results are shown in Fig. 2(b), where the GS emission power of the cw-solutions is color coded in orange, blue regions denote simple period-one limit cycles and green regions indicate more complex oscillation patterns, i.e. oscillation bursts, which are further investigated in the following sections. One example is illustrated in Fig. 2(c). Black square hatches in Fig. 2(b) indicate regions, where the feedback has triggered two-state lasing.

At lower feedback strengths ($K \lessapprox 0.11$) and centered around the feedback phase $C \approx 1.01$, a droplet-shaped region is found, where the GS lasing cw-solution becomes unstable and the laser first switches to two-state lasing (black square hatches) and then transitions to an inner region of ES lasing and fully suppressed GS lasing (enclosed by the black square hatches). A qualitatively similar behavior has been reported for a single-mode quantum-well laser described by the Lang-Kobayashi model [46], where sufficiently short optical feedback causes the laser to turn off in a transcritical bifurcation within a droplet-shaped region. The two-state laser discussed here however, changes from GS to two-state lasing and from two-state to ES lasing in transcritical bifurcations, which is typical for multi-mode lasers [8,47]. While QD based lasers are known for a cw-emission that exhibits a large tolerance towards optical feedback [5,32], this particular laser responds to low feedback strengths around $C \approx 1$ by switching its lasing state to either two-state or excited state emission. Note that for a feedback phase $C=0$, the ground-state emission is not destabilized at all

For larger feedback strengths ($K\;\gtrapprox\;0.10$) just to the lower right of the droplet, a tail of oscillatory solutions is found. Depending on the feedback phase, period-one limit-cycles are born in Hopf-bifurcations of either the GS-cw solution ($C \lessapprox 0.88$) or the two-state cw solution ($C\;\gtrapprox\;0.881$) and thus, either GS or two-state oscillations are observed. The complex oscillations are the oscillation bursts, which are generated by Torus (secondary Hopf) bifurcations of the period-one limit-cycle, which we discuss in detail below. Following the tail of oscillatory laser emission towards lower feedback phases, the range between the limiting bifurcations becomes smaller and shifts towards larger feedback strengths up until the point ($K=0.181,C=0.182$), where the limiting bifurcations meet and annihilate, such that only GS cw emission remains.

3.2 Bifurcation structure

In this section, we further characterize the complex oscillatory dynamics and unravel the bifurcation structure leading to them, by focusing on a scan for one fixed feedback phase $C=0.849$ within the feedback strength range from $K=0.105$ to $K=0.135$, which is indicated by the purple line in Fig. 2(b). The results from direct integration and path continuation are combined and presented in Fig. 3, where (a) shows the GS emission power, (b) the corresponding ES emission power and (c) the optical frequency shift of the GS electrical field relative to the solitary lasing frequency. In all subfigures, solid lines indicate cw-solutions, dashed lines indicate limit-cycle solutions, thick lines denote stable solutions and thin lines unstable solutions. Bifurcations, which connect solutions and/or lead to a transition from stable to unstable are labeled and indicated by black markers, where squares, diamonds, circles and stars denote Andronov-Hopf (H), transcritical (TC), torus (T) and saddle-node (SN) bifurcations. The local maxima shown in Figs. 3(a) and 3(b) have been extracted from path-continuation for the cw and limit-cycle solutions and from direct integration for the region of complex oscillations. Orange, green and blue lines in Fig. 3(c) represent the frequency shifts $\delta \nu$ of the laser emission relative to the solitary frequency. Here, the field equation is solved by rotating waves, which are known as external cavity modes (ECMs) [46] $E\left (t\right ) = R\exp \left (i \frac {\delta \nu }{2 \pi } t \right )$, where $R$ is a constant amplitude and $\delta \nu$ is the feedback induced frequency shift relative to a chosen reference frame of the solitary laser. The black line shows the mean frequency shift $\langle \delta \nu \rangle$ of the oscillatory solutions, which is obtained from direct numerical integration. Furthermore, the dotted brown line shows the ratio between loops of the trajectory around two different attractors, which we discuss in more detail in Sec. 3.3. The gray vertical dashed lines labeled with [a]-[e] indicate the feedback strengths, which are chosen for a detailed characterization of the time dependent dynamics in Fig. 5. To complement the picture of the dynamics, the corresponding optical spectra and the low frequency range of the power spectra have been computed along the same parameter range as in Fig. 3 and are shown in Fig. 4.

 

Fig. 3. Bifurcation diagram along the feedback strength $K$ at $C=0.849$ (indicated by the purple line in Fig. 2). (a) and (b): local maxima of the GS and ES emission, respectively. (c): GS relative frequency shift. Solid and dashed lines indicate steady-state and periodic-orbit solutions obtained by path continuation. Thick and thin lines represent stable and unstable solutions. Orange and blue lines represent the first and second GS ECM. Green and purple lines represent the GS and ES emission of the two-state lasing solution. Local maxima of the two-state oscillation bursts solutions are obtained by direct integration and are shown in pale green and purple colors. We show Andronov-Hopf (H, squares), transcritical (TC, diamonds), torus (T, circles) and saddle-node (SN, stars) bifurcations, which connect solutions and/or cause a transition from stable to unstable. The vertical dashed lines labeled [a]-[e] indicate the parameters shown in Fig. 5.

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Fig. 4. Optical spectrum (a) and low frequency power spectrum (b) for increasing feedback strengths where the color encodes the spectral power. $\delta \nu$ denotes the relative frequency with respect to the solitary lasing frequency. Vertical dashed lines labeled [a]-[e] indicate the feedback strengths shown in Fig. 5. Vertical dotted blue lines indicate the labeled bifurcations.

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Fig. 5. Time traces and phase-space projections within the oscillation bursts regime for $K \in \{0.1210, 0.1220, 0.1250, 0.1295, 0.1305\}$. Left column (a1)-(e1): time traces of the GS (orange and green) and ES power (purple); prange and green lines correspond to the same trajectory, which is either close to the first GS ECM or to the ghost of the two-state ECM. Middle column (a2)-(e2): trajectory in a phase-space projection onto the GS inversion and instantaneous frequency shift $\langle \delta \nu \rangle _{\tau }$. Open orange circles represent the unstable first GS ECM, green circles the two-state ECM, green diamonds the transcritical bifurcation of the two-state ECM, blue stars the saddle-node bifurcation of the second GS ECM and the blue circle the stable second GS ECM as introduced in Fig. 3. Right column (a3)-(e3): trajectory projected onto the GS power, ES power, GS inversion space. Orange lines represent the Poincaré map created by the intersection of the trajectories with surfaces of constant ES powers (gray surfaces).

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Starting at the left side of Fig. 3 at $K=0.105$, the fundamental GS lasing ECM (thick solid orange line) is the only solution of the system. Increasing the feedback strength, this solution becomes unstable in a supercritical Hopf bifurcation H$_1$ at $K \approx 0.108$ and a stable period orbit (thick orange dashed line) is generated. The signature of this is seen as a second frequency appearing in the optical spectrum at $\approx 2\textrm {GHz}$ in Fig. 4(a). The repetition frequency of the intensity oscillations corresponds to the beat frequency between the two lines in the spectrum and is found to be $\approx 3.45\textrm {GHz}$, which is not the external cavity frequency, but corresponds to the undamped relaxation oscillation frequency of the laser [48]. Further increasing the feedback strength, the GS periodic orbit collides with a two-state periodic orbit (thick green dashed line in Fig. 3(a)) in a transcritical bifurcation of limit cycles (TC$_\textrm {LC}$) at $K \approx 0.113$ and exchanges stability. At this point, the emission from the ES turns on and both the GS and ES field perform near harmonic oscillations.

The ES peak power (thick purple dashed line in Fig. 3(b)) increases with the feedback strength up until the two-state periodic orbit loses its stability in a supercritical torus bifurcation T$_1$ at $K \approx 0.121$, which generates a stable quasi-periodic trajectory that is embedded on a two-dimensional torus. The resulting emission from both the GS and ES is described by two-color oscillation bursts as shown in Fig. 2(c), where the fast oscillations of the previously stable periodic orbit are modulated by a much slower envelope. Five further examples of oscillation bursts are shown in Fig. 5 and discussed in Sec. 3.3. Since continuing quasi-periodic solutions is beyond path-continuation techniques, we resort to direct integration of the system for finite times ($10\mu \textrm {s}$) at each feedback strength and plot all local maxima, which then generate the pale green and purple point clouds in Figs. 3(a) and 3(b). These point clouds appear as continuous regions, since small features such as an occasional locking of the two torus frequencies are hidden by the non-zero dot size. Locking effects however become visible and are exemplified in the zoom-in shown Sec. 3.4 in Fig. 6(a). The additional frequency introduced by the torus bifurcation T$_1$ appears as side-bands of the two main lines in the optical spectrum Fig. 4(a) and as their beat frequency in the power spectrum Fig. 4(b). The beat frequency corresponds to the inverse period of a slow modulation envelope. While the GS power oscillates at $\approx 3.43\textrm {GHz}$ at the torus bifurcation, the beat modulation is generated with a frequency of $\approx 110\textrm {MHz}$ and with an increasing feedback strength slows down to a minimum of $\approx 33\textrm {MHz}$ at $K \approx 0.1307$. Additionally, the regime of the quasi-periodic dynamics exhibits a small range of feedback strengths around $K=0.13$, where the optical and power spectrum (c. Figs. 4(a) and 4(b)) become broad and featureless, suggesting chaotic motion, which we discuss in detail in Sec. 3.4.

 

Fig. 6. Zoom of the bifurcation diagram in Fig. 3(a) and Fig. 4(b) showing local maxima of the GS power (a) and power spectra of the GS emission (b). A period-doubling cascade leads to chaotic motion of the oscillation bursts slow envelope. Vertical gray dashed lines indicate the feedback parameters used in Figs. 5(d)–5(e) and Fig. 7 respectively.

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The quasi-periodic dynamics vanishes at $K \approx 0.1308$ as the attractor collides with the unstable two-state periodic orbit (thin green dashed line in Fig. 3) in a supercritical torus bifurcation T$_2$. This two-state periodic orbit stabilizes in the torus bifurcation T$_2$, but shortly after disappears in a supercritical Hopf bifurcation H$_2$ at $K \approx 0.131$ of the two-state ECM (green solid line), which has been generated unstable in the transcritical bifurcation TC$_1$ at $K \approx 0.1285$. The two-state ECM disappears (becomes unphysical) in a transcritical bifurcation TC$_2$ with the unstable branch of the second ECM (blue thin solid line), which is born in a saddle-node bifurcation SN at $K \approx 0.1303$. Increasing the feedback strength beyond the transcritical bifurcation TC$_2$, the laser jumps to the stable branch of the second GS ECM, which then remains the only stable solution. This jump to a new ECM solution at TC$_2$ not only can be seen in the GS power in Fig. 3(a), but also in the relative frequency $\delta \nu$ in Fig. 3(c) and in the optical spectrum Fig. 4(a). Notably, between the saddle-node SN of the second GS ECM and the connecting transcritical bifurcation TC$_2$ (see Fig. 3), there exists a bistability and the laser can either show GS or two-state emission.

The unstable branch of the second ECM (blue thin line in Fig. 3) is also connected to the unstable GS periodic orbit (orange thin dashed line) via the Hopf bifurcation H$_3$. Such connections have already been found in quantum well lasers [36,37,49] and single-color QD lasers [12,50] and have been called bifurcation bridges. While this connection also exists in the two-state laser, it is unstable. Instead, the increased degrees of freedom provided by the electronic structure of the QDs and the emission from the ES yield an additional stable bridge featuring the two-color oscillation bursts. It connects the different GS ECMs by transcritical and subsequent Hopf and torus bifurcations, as discussed above.

3.3 Bursting oscillations

The bifurcation analysis revealed the existence of bursting oscillations born in torus bifurcations of a two-color periodic orbit. To characterize these oscillation bursts that occur between the torus bifurcations T$_1$ and T$_2$ in more detail, we illustrate the time evolution of the emitted light for five distinct feedback strengths in the rows of Fig. 5. These values of $K$ are indicated with vertical gray dashes lines and labeled [a]-[e] in Fig. 3 and Fig. 4.

The first column in Fig. 5 shows time series of the GS (orange and green) and ES (purple) normalized emission power, where the color of the GS emission encodes whether the trajectory encircles the unstable first GS ECM (orange) or the ghost of the second GS ECM and two-state ECM (green). The second column shows a phase-space projection onto the mean frequency shift within one delay interval $\langle \delta \nu \rangle _{\tau }$ and GS inversion $\rho _\textrm {inv} = \rho _\textrm {e}^\textrm {GS} + \rho _\textrm {h}^\textrm {GS} -1$, where the black line represents the trajectory of the bursting oscillations over a time span of 300ns. The open orange circle denotes the position of the unstable first GS ECM (thin orange solid line in Fig. 3), the blue circle the stable second GS ECM (thick blue solid line in Fig. 3), the green diamond gives the position of the lower transcritical bifurcation TC$_\textrm {1}$ of the two-state ECM and the blue star the position of the saddle-node bifurcation SN of the second GS ECM. The last column shows a projection of a finite time ($2\,\mu \textrm {s}$) trajectory (blue line) in the phase space spanned by the GS power, ES power and GS inversion. Additionally, we show a Poincaré map (orange dots), which is generated by the intersection of the trajectory with a surface of constant ES power (shown as gray transparent plane).

Starting at the feedback strength $K=0.121$ right after the torus bifurcation T$_1$ (first row), the modulation of the GS and ES oscillations is small compared to the fast oscillations and has a frequency of $\approx 111\textrm {MHz}$. In the GS inversion/frequency shift projection Fig. 5(a2), the trajectory exclusively encircles the unstable GS ECM (open orange circle). The torus surface, which is constructed by the trajectory of the quasi-periodic solution, is portrayed in blue in Fig. 5(a3). Characteristic for motion on a torus, the Poincaré map (orange lines) consists of two closed curves (top left and bottom right), which are still relatively small due to the proximity to the torus bifurcation T$_1$.

Increasing the feedback strength to $K=0.122$ (second row), the slow modulation of the GS and ES oscillations occurs at a frequency of $\approx 108\textrm {MHz}$ with a slightly larger amplitude. The GS inversion/frequency shift projection Fig. 5(a2) however reveals a qualitative change of the dynamics: The trajectory now alternates between loops around the unstable first GS ECM (orange circle) and the region in phase space, where the two-state ECM (green diamond) and second GS ECM (blue star) are generated at higher feedback strengths. The loop ratio, defined by the number of loops around each region, is $\approx 0.5$. The brown dotted line in Fig. 3(c) shows the full evolution of the loop ratio for an increasing feedback strength within the oscillation bursts regime. The alternating behavior of the GS emission is also indicated by the changing orange and green colors of the GS time series in Fig. 5(a1)–5(e1). The torus generated by the trajectory Fig. 5(b3) has increased in size due to the larger modulation depth. The Poincaré map (orange lines) now consists of two large circles, which are close, but do not connect.

Shifting further away from the torus bifurcation T$_1$ to $K=0.125$ (third row), the modulation envelope repetition rate slows down to $\approx 84\textrm {MHz}$ with stronger modulation depth and an increased ES peak power (c. Fig. 5(c1)), while the loop ratio increases to $0.78$ as the trajectory performs more green than orange loops. The larger ES peak powers result in a further increased size of the torus in Fig. 5(c3). The fourth row presents the feedback strength $K=0.1295$, which is close to the parameter region of complex dynamics and broad optical and power spectra (see Fig. 4). Here the modulation has undergone a period-doubling bifurcation and has further slowed down to $\approx 19\textrm {MHz}$. Additionally, it has become more complex as the envelope has gained multiple local maxima and the ES emission occurs in very well separated bursts. The trajectory mostly loops around the newly generated unstable two-state ECM resulting in loop ratio of $0.95$. The period-doubling bifurcation is also seen in the Poincaré map Fig. 5(d3), where the inner circle has gained an additional loop. Lastly, the fifth row in Fig. 5 shows the feedback strength $K=0.1305$, which is between the spectrally flat region and the limiting torus bifurcation T$_2$ to the right. There, the quasi-periodic motion is bistable with the second GS ECM. The slow modulation frequency is $\approx 33\textrm {MHz}$ and the Poincaré map in Fig. 5(d3) reveals that the motion lies again on the surface of a simple torus. The trajectory now exclusively loops around the unstable two-state ECM, hence the solely green color in Fig. 5(e1).

We identify the looping behavior of the trajectory as the mechanism, that mediates the shift of the optical frequencies from the first GS ECM to the two-state ECM, which then connects to the second GS ECM. At torus bifurcation T$_1$, the limit cycle only encircles the unstable first GS ECM (loop ratio $=0.0$) with a mean optical frequency close to it. After being born, the quasi-periodic oscillation bursts first almost exclusively loop around the unstable first GS ECM. With an increasing feedback strength, the bursts continuously perform more loops around the region where the two-state ECM is generated until the trajectory exclusively orbits around the unstable two-state ECM (loop ratio $=1.0$) at $K\;\gtrapprox\;0.12992$ with approximately the two-state ECM frequency. The relation between the loop ratio and the mean optical frequency shift can also be nicely seen by comparing the black and brown lines in Fig. 3(c).

While fast pulsations with regular or chaotic modulation envelopes have been studied in numerous semiconductor lasers subject to optical feedback, they where either found in different feedback regimes or created by other mechanisms. Low frequency fluctuations (LFFs) [38,51] are observed in the moderate to long feedback-time regime, where the system performs a quasi-periodic or chaotic itinerancy between the multiple attractor ruins of the ECM solutions. The oscillation bursts described in this manuscript are observed in the ultra-short feedback time regime where only one or two ECMs exist. Multi-mode LFFs [39] are also observed at longer feedback times, where anti-phase dynamics of multiple Fabry–Pérot modes can lead to quasi-periodic or chaotic motion. The two-state laser investigated here is instead considered to be single-mode at the GS and ES frequencies, while the GS and ES carriers are coupled by non-instantaneous carrier scattering. Short optical feedback applied to a single-color QD lasers can produce regular pulse packages [5,12], which appear similar to the oscillations bursts observed here, but are instead created by a homoclinic bifurcation.

3.4 Period doubling cascade to low frequency chaos

In this final section, we take a closer look at the oscillation bursts in the spectrally flat region around $K=0.13$. Hence, Fig. 6 presents a close up of the bifurcation diagram within the feedback strength range $K \in [0.129, 0.131]$, where (a) shows local maxima of the GS power and (b) the corresponding power spectra (see Fig. 4(a) and Fig. 3(a) for the zoom-outs). Vertical lines indicate the parameters discussed in Figs. 5(d)–5(e) and Fig. 7. The bifurcation diagram Fig. 6(a) exhibits multiple regions, where local maxima are less dense, indicating a locking of the torus dynamics to a periodic orbit and three discontinuities of the envelope at $K \approx 0.1298$, $0.1301$ and $0.1304$, which suggests jumps from one solution to another. The complex cascade of bifurcations, which mainly affects the envelope that shapes the oscillation bursts, is better revealed in the low frequency part of the power spectrum shown in Fig. 6(b).

 

Fig. 7. Low frequency chaos produced by the dynamics of the bursting oscillations envelope at $K=0.12985$. Time series of the GS and ES power at two different time scales (a), power spectrum of the GS at two different frequency scales (b), phase-space projection onto the GS inversion and instantaneous frequency shift $\langle \delta \nu \rangle _{\tau }$ (c) and Poincaré map of the GS power $P^\textrm {GS}$ and inversion $\rho _\textrm {inv}$ produced by the intersection with the surface of constant ES power (d).

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At $K \approx 0.1291$ the first period-doubling bifurcation PD$_1$ changes the repetition rate of the modulation envelope from $\approx 44\textrm {MHz}$ to $\approx 22\textrm {MHz}$ and creates the solution presented in Fig. 5(c). At $K \approx 0.12963$ the next period-doubling bifurcation PD$_2$ occurs and is quickly followed by more period-doubling bifurcations typical for the period-doubling route to chaos [52–55]. In between $K \approx 0.1298$ and $K \approx 0.1301$, the power spectra become broad and almost featureless, suggesting chaotic dynamics of the slow envelope. Peculiarly, within the chaotic region the trajectory locks to periodic orbits at certain feedback strengths, evident by the emergence of discrete peak structures in the power spectrum.

At $K \approx 0.1301$ the chaotic attractor collapses in a boundary crisis BC [56] onto a quasi-periodic torus attractor, indicated by the jump of local maxima in the bifurcation diagram Fig. 6(a) and the reappearance of discrete peak structure in the power spectra Fig. 6(b) with the slowest component at $\approx 37\textrm {MHz}$. Further increasing the feedback strength, the dynamics undergo another sequence of period-doubling bifurcations up until $K \approx 0.13024$, which is then directly followed by a sequence of period-halving bifurcations back to the original slowest frequency of $\approx 37\textrm {MHz}$. Another solution jump appears at $K \approx 0.130388$, where the dynamics do not qualitatively change, but the slow modulation jumps to a slightly faster frequency and the range of local maxima becomes narrower. A bistability between two different quasi-periodic orbits in the small region between $K \approx 0.130387$ and $K \approx 0.130389$ suggests, that this jump in solutions is caused by two folding saddle-node bifurcations SN$_{2}$ and SN$_{3}$.

A closer look at the low-frequency chaos at $K=0.12985$ is provided in Fig. 7, where (a) depicts a time series of the GS (orange and green) and ES (purple) power, (b) the power spectrum of the GS emission, (c) the phase-space projection of the GS inversion $\rho _\textrm {inv}$ as a function of the GS mean instantaneous frequency shift $\langle \delta \nu \rangle _{\tau }$ and (d) the Poincaré map of the GS power and inversion created by the intersection of the trajectory with the surface of constant ES power. Both the time-series and the spectrum are presented at two different time and frequency scales, where gray and white backgrounds indicate the different regimes. The time-series shows strong modulations of the oscillation bursts themselves, where each burst envelope contains three or four local maxima, which change in power from burst to burst. While the repetition rate of the individual bursts remains close to $\approx 31\textrm {MHz}$, which remains the most powerful frequency component in the low frequency range, significant contributions at all frequencies up to $200\textrm {MHz}$ exist. Nevertheless, the strongest oscillating contribution of the whole spectrum is found at $\approx 3.57\textrm {GHz}$, which corresponds to the fast oscillations of the GS and ES power within the bursts. The phase-space portrait shows that the trajectory mostly encircles the unstable two-state ECM in an irregular fashion and rarely performs an excursion around the unstable first GS ECM (loop ratio $0.99$). The excursions around the unstable first GS ECM, however, always occur at the rising edge of an oscillation burst (orange color in Fig. 7(a)) and appear to follow very similar paths. This behavior is manifested in the Poincaré section, where the excursion around the unstable first GS ECM creates the continuous half-loop at the outer left side (see Fig. 7(d)). It merges into a region of dense points produced by the chaotic trajectory of the bursting dynamics after the initial rise. The origin of the dense region in the Poincaré section is seen in Fig. 5(c3), where the first period-doubling PD$_\textrm {1}$ bifurcations leads to a twist and fold of the inner loop of the Poincaré map. The following bifurcations then repeat this process until the continuous line has transformed to the dense region of higher and fractional dimensionality.

Slightly increasing the feedback strength above $K\;\gtrapprox\;0.12992$, the regular excursion around the unstable first GS ECM disappears and only the chaotic motion around the unstable two-state ECM remains. This however, is neither related to the first locking of the trajectory at $K \approx 0.12993$ nor does it qualitatively affect the power spectrum.

4. Conclusions

Using a microscopically motivated rate equation model, we have presented the generation of two-color anti-phase oscillations bursts emitted from a two-state quantum dot laser subject to short optical feedback. Utilizing path-continuation techniques, we have shown that the oscillations bursts emerge from a torus bifurcation of a two-state periodic orbit. This two-state periodic orbit connects two stable ECMs in the feedback parameter space. This mechanism thus extends the bifurcation structure already known from single mode quantum well and quantum dot lasers and therefore provides an additional bifurcation bridge between neighboring ECMs. The change of the average optical frequency along this two-color bifurcation bridge is characterized by a shift of the loop ratio, i.e., the number of loops the trajectory performs around the ghost of the two-state ECM steadily increases with the feedback strength. Moreover, we have identified the feedback strength to be an effective control parameter to modify the bursting oscillations qualitative and quantitative properties. Within the oscillation burst regime, the repetition rate and bursting envelope change considerably. We have furthermore identified regions of two-color low-frequency optical chaos embedded within the bursting oscillations.

Appendix

The nonlinear charge-carrier in-scattering rates are implemented via (10) and (11), where the parameters $A_\textrm {m,b},B_\textrm {m,b},C_\textrm {b}$ and $D_\textrm {b}$ are fitted to the fully microscopically calculated rates [34,42].

(10)$$ S_\textrm{b,in}^\textrm{m,cap}(w_\textrm{b}) = \frac{ A_\textrm{m,b} w_\textrm{b}^{2} } {B_\textrm{m,b} + w_\textrm{b}} $$

(11)$$ S_\textrm{b,in}^\textrm{rel}(w_\textrm{b}) = \frac{ C_\textrm{b} w_\textrm{b}}{ D_\textrm{b} + w_\textrm{b}} $$

The fit parameters are presented in Table 2. The detailed balance condition [2] is used to calculate the corresponding out-scattering rates

(12)$$ S_\textrm{b,out}^\textrm{m,cap} = S_\textrm{b,in}^\textrm{m,cap} \exp \left(-\frac{E_\textrm{F,b}^\textrm{eq} - \varepsilon_\textrm{m,b}}{k_\textrm{B} T}\right) , $$

(13)$$ S_\textrm{b,out}^\textrm{rel} = S_\textrm{b,in}^\textrm{rel} \exp \left(-\frac{\varepsilon_\textrm{ES,b} - \varepsilon_\textrm{GS,b}}{k_\textrm{B} T}\right), $$

where $E_\textrm {F,b}^\textrm {eq}$ is the quasi-Fermi level of the respective quantum well band, $\varepsilon _{m,b}$ the confinement energy relative to the QW band edge of the corresponding QD state, $k_\textrm {B}$ the Boltzmann constant and $T$ the charge-carrier temperature. The quasi-Fermi levels dynamically depend upon the reservoir density $w_\textrm {b}$ and are calculated according to

(14)$$E_\textrm{F,b}^\textrm{eq} = E^\textrm{QW}_\textrm{b,0} + k_\textrm{B} T \ln \left[ \exp \left( \frac{w_\textrm{b}}{\mathcal{D}_\textrm{b}^\textrm{2D} k_\textrm{B} T} \right) - 1 \right],$$

where $E^\textrm {QW}_\textrm {b,0}$ is the energy at the QW band edge and $\mathcal {D}_\textrm {b}^\textrm {2D}$ the respective density of states.

Table 2. Fit parameters for charge-carrier scattering processes, extracted from microscopic calculations for a GS confinement energy of 64(35) meV and a GS-ES separation of 50(20) meV for electrons (holes), and T=300 K.

View Table | View all tables in this article

Funding

Deutsche Forschungsgemeinschaft (SFB787, SFB910).

Author contribution statement

KL initiated the research and BL and KL developed the model. SM, LK and JH implemented the model and carried out the simulations. The results were discussed and interpreted by all authors. SM and LK prepared the figures and SM, BL and KL and wrote the manuscript with contributions from all authors.

Disclosures

The authors declare no conflicts of interest.

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