1 Introduction

Periodic pulse production regimes such as self-Q-switching, self-pulsations or passive mode-locking (PML) in interband semiconductor laser diodes (LDs) are widely used in various practical applications. However, these passive regimes are impossible in the mid-infrared (Mid-IR) Quantum Cascade Lasers (QCLs) because of their picosecond (or sub-picosecond) gain recovery time. This time is too short in comparison with the cavity round trip time so the gain saturation does not provide a “memory” mechanism to sustain such periodic regimes (Gordon et al. 2008; Wang et al. 2007). At the same time, the multimode Risken–Nummedal–Graham–Haken (RNGH) instability (Graham and Haken 1968; Risken and Nummedal 1968) in QCLs with Fabry–Pérot (FP) cavities (Gordon et al. 2008; Wang et al. 2007) may pave a way to practical approaches for ultrafast pulse production in the Mid-IR range. More specifically, the spatial hole burning (SHB) effect in FP QCLs strongly reduces the excitation threshold for RNGH self-pulsations (SPs).

The question of the origin of RNGH SPs itself is a highly debated subject (Gordon et al. 2008; Mansuripur et al. 2016; Piccardo et al. 2018; Vukovic et al. 2016a, 2017). Strong spectral broadening in single-section FP cavity QCLs (~ 20–60 cm−1) was observed both in the ridge waveguide and in the buried heterostructure design operating under CW or quasi-CW driving conditions just above the lasing threshold (Bugajski et al. 2013; Gordon et al. 2008; Mansuripur et al. 2016; Wang et al. 2007). Low threshold RNGH multimode instability was attributed to a combined effect of spatial hole burning and saturable absorption in the QCL cavity (Gordon et al. 2008) but the nature of saturable absorption which should be equally strong in a ridge waveguide and a buried heterostructure QCLs has never been fully clarified.

Another mechanism of instability in single-section QCLs has been proposed in Mansuripur et al. (2016) and Piccardo et al. (2018), where it was linked to a parametric four-wave mixing gain instability, in full analogy with the phase-locked comb production in high-finesse optically pumped micro-cavities (Herr et al. 2014) The idea stems from a parametric gain picture yielding instability of the main cavity mode, along the line of comb generation in optically pumped dielectric microdisk resonators with third-order χ(3) non-linearity. Mid-IR intersubband transitions in semiconductor QWs do produce strong third-order nonlinearity (Capasso et al. 1994; Gravé et al. 1992). However, optically passive microdisk resonators have a special cavity design to achieve a very high-quality factor (Q ~ 108). In order to reach parametric instability threshold, they employ whispering gallery modes with very small transverse sizes and operate with circulating pump power in the resonator on the order of several hundreds of watts. Electrically pumped Mid-IR QCLs with FP cavities cannot reproduce such environment to reach a parametric instability caused by material third-order nonlinearity χ(3). This conjecture is in agreement with the fact that all measurements of the second-order interferometric autocorrelation (IAC) traces reported for FP cavity QCLs reveal the peak to background ratio close to 8:3, which corresponds to a noisy multimode lasing as opposed to the ratio of 8:1 for a perfectly phase-locked comb (Diels and Rudolph 2006).

In our theoretical studies (Vukovic et al. 2016a, b, 2017) on the RNGH instability in QCLs we attribute the leading role to the gratings of population inversion and coherences (medium polarization) induced by the standing wave pattern of the cavity modes due to the spatial hole burning (SHB) effect. The formation of the carrier and coherence gratings is conditioned by the carrier lifetime T1 (longitudinal relaxation time), the carrier dephasing time T2 (transverse relaxation time) and the carrier diffusion coefficient D.

2 Model

The starting point of our analysis is the system of Maxwell–Bloch equations (Vukovic et al. 2016b, 2017):

$$\dot{\rho }_{ab} = i\omega \rho_{ab} + i\frac{\mu E}{\hbar }\Delta - \frac{{\rho_{ab} }}{{T_{2} }} + D\frac{{\partial^{2} \rho_{ab} }}{{\partial z^{2} }}$$
(1)
$$\dot{\Delta } = - 2i\frac{\mu E}{\hbar }(\rho_{ab}^{*} - \rho_{ab} ) + \frac{{\Delta_{pump} - \Delta }}{{T_{1} }} + D\frac{{\partial^{2} \Delta }}{{\partial z^{2} }}$$
(2)
$$\partial_{z}^{2} E - \frac{{n_{g}^{2} }}{{c^{2} }}\partial_{t}^{2} E = \varGamma \frac{N\mu }{{\varepsilon_{0} c^{2} }}\partial_{t}^{2} (\rho_{ab}^{*} + \rho_{ab}^{{}} ) + \frac{{n_{g} l_{0}^{{\prime }} }}{c}\partial_{t}^{{}} E$$
(3)

where \(\rho_{ab}\) is the off-diagonal element of the density matrix, \(\Delta = \rho_{bb} - \rho_{aa}\) is the population inversion, ω, and µ denote the resonant frequency and the dipole matrix element respectively. Δpump is the steady-state population inversion, E, N, and Γ stand for the optical field, the number of two-level systems per unit volume and the overlap factor between the optical mode and the active region, ng is the group refractive index (Vukovic et al. 2016a, b, 2017). Numerical simulations with the semiclassical traveling wave (TW) rate equation model (Vukovic et al. 2016a, 2017) are accomplished by introducing in Eqs. (1)–(3) two slowly varying amplitudes for the counter-propagating waves in the FP cavity \(A_{ \pm }\) and distinguishing the medium’s polarizations associated with the forward and backward traveling waves \(P_{ \pm }\) (Boiko and Vasil’ev 2012):

$$\frac{{\partial A_{ \pm } }}{\partial t} \pm \frac{c}{{n_{g} }}\frac{{\partial A_{ \pm } }}{\partial z} = \frac{1}{2}\varGamma \sqrt {\frac{{g_{0} }}{{T_{2} }}} P_{ \pm } - \frac{1}{2}\frac{c}{{n_{g} }}\alpha_{i} A_{ \pm }$$
(4)
$$\frac{{\partial P_{ \pm } }}{\partial t} = - \frac{{P_{ \pm } }}{{T_{2} }} + \sqrt {\frac{{g_{0} }}{{T_{2} }}} (n - n_{t} )A_{ \pm } + D\frac{{\partial^{2} P_{ \pm } }}{{\partial z^{2} }} + \varLambda_{ \pm } ,$$
(5)
$$\frac{\partial n}{\partial t} = - \frac{n}{{\tau_{n} }} - \sqrt {\frac{{g_{0} }}{{T_{2} }}} (A_{ + } P_{ + } + A_{ - } P_{ - } ) + \frac{J(z,t)}{ed} + D\frac{{\partial^{2} n}}{{\partial z^{2} }}.$$
(6)

Here, J(z,t) is pump current density and d is the active region thickness. The model is adapted from Boiko and Vasil’ev (2012) by removing the saturable absorber section. It incorporates Langevin force terms \(\varLambda_{ \pm }\) in Eq. (5) that seed spontaneous polarization noise into the system. The QCL gain chip parameters used in numerical simulations are listed in Table 1.

Table 1 Dynamic model parameters for InGaAs QCL considered in this paper using data from Vukovic et al. (2016a, b, 2017)
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The field amplitudes are normalized in accordance with the secondary quantization convention \(\hat{a}_{ \pm } \left| {N_{ \pm } } \right\rangle = A_{ \pm } \left| {N_{ \pm } - 1} \right\rangle\):

$${\text{E}}_{ \pm } = \sqrt {\frac{4\pi \hbar \omega }{{n_{g}^{2} }}} {\text{e}}_{ \pm } A_{ \pm } \sin (\omega t \mp kz),\quad A_{ \pm } = \sqrt {N_{ \pm } }$$
(7)

The medium’s polarization (the ensemble average for the off-diagonal elements of the density matrix) is normalized as follows:

$${\text{P}}_{ \pm } = \sqrt {\frac{{n_{g}^{2} g_{0} }}{{4\pi \omega T_{2} }}} {\text{e}}_{ \pm } P_{ \pm } \sin (\omega t \mp kz).$$
(8)

Boundary conditions for the wave amplitudes are applied at left (z = 0) and right (z = L) cavity facets in the usual way:

$$\begin{aligned} A_{ + } (0,t) & = \sqrt {R_{1} } A_{ - } (0,t), \\ A_{ - } (L,t) & = \sqrt {R_{2} } A_{ + } (L,t). \\ \end{aligned}$$
(9)

The evolution of the carrier density n, medium polarization P± (coherences) and output power at the facets is obtained by numerically integrating the system of Eqs. (4)–(9) (Boiko and Vasil’ev 2012).

3 Results and discussion

3.1 Short cavity FP QCLs

To begin with, we consider an FP QCL chip with a very short cavity (L = 0.1 mm) and T1 = 1.3 ps. Excitation of RNGH self-pulsation regime requires a perturbation which is achieved by introducing polarization noise in the model (Langevin force terms in Eq. (5)). The cavity round trip time (Tcav= 2.2 ps) is comparable to the gain recovery time and QCL produces regular RNGH self-pulsations at a current of pth2= 2.4 times above the lasing threshold (Vukovic et al. 2016a, b, 2017). Numerical simulations reveal the onset of self-pulsations within the first 8 ps interval (~ 4 cavity round trips). After 30 ps (~ 14 cavity round trips), we observe a transit to a steady regime of regular self-pulsations (Vukovic et al. 2017). The waveform example can be found in the inset of Fig. 1a. The field has a sine-wave envelope with a period being close to the cavity round trip time of 2.2 ps. The optical field and polarization of the gain medium change the sign at each passage in the cavity (at each half-period of the sine-wave). The intensity pulses are of 0.9 THz repetition rate and of 0.6 ps FWHM duration. The examples of optical and RF power spectra, 2nd order IAC traces as well as P-N attractors can be found in Vukovic et al. (2016a, b, 2017).

Fig. 1
figure 1

Results of numerical simulations with TW model for monolithic QCL of the cavity length 100 μm: a bifurcation diagrams of self-starting SPs represented by extreme values in the output power waveform Pout (red and blue symbols correspond to the maxima and minima, respectively). The insets show the steady-state Pout waveforms for different p values. b Bifurcation diagram of stable SPs (but not necessarily self-starting) obtained by adiabatically decreasing p from the initial value of p = 5.5. Note a bi-stability region p = 1.55–2.4 in (a, b). c Contour plot of Pout distribution when decreasing p (long time scale, x-axis), during time interval equal to 2 cavity round trips (short time scale, y-axis). d Behavior of the normalized Recurrence Period Density Entropy (RPDE) (Little et al. 2007) with decreasing p. The following time-embedding parameters are used: embedding dimension m = 4, embedding delay τ = 10, embedding ball radius r = 0.003. More details about these parameters can be found in Kantz and Schreiber (2004) and Little et al. (2007). (Color figure online)

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In contrast to our previous studies focused at the origin of the RNGH SPs (Vukovic et al. 2016a, b, 2017), in this paper we analyze bifurcations of the self-pulsations waveform as a function of the pump current, which we normalize to the lasing threshold p = I/Ith. We construct the bifurcation diagrams in a few different ways. In all cases, the pump current is changed in small steps of δp = 0.05. For each p value, the numerical simulations are performed over 50 cavity round trips while only the last 25 periods after the end of all transients are used in the waveform analysis.

The bifurcation diagrams of extreme values (maxima and minima) in the output power waveform Pout are plotted in Fig. 1a, b. The red (blue) circles indicate the global and local maxima (minima) in the output power waveform. For instance, the largest values designate the peak power variations while the zero values in the output power waveform are due to the sign changes in the optical field envelope.

In Fig. 1a, for each value of p, we reset the initial conditions to zero field and let the self-pulsations to occur and grow. Thus we obtain the bifurcation diagram of the self-starting SPs. For example, for p above the second threshold pth2 =  2.4, the QCL exhibits a self-starting oscillatory behavior, while below pth2, it exhibits only CW emission. The insets in Fig. 1a display the corresponding output power waveforms for different pump rates and provide a hint for interpretation of the bifurcation diagram.

In the bifurcation diagram of Fig. 1b, we study the stability of already ongoing SPs which have been excited earlier. In a particular case, we start the numerical simulations from the largest value of p = 5.5 to excite the SPs. After that, we adiabatically reduce the pump current in small δp steps. In contrast to the sequence used in Fig. 1a, for each value of p, we continue the simulations without resetting the initial conditions to zero field. Surprisingly, for the adiabatically decreasing pump rate, the SPs continue even below the self-excitation threshold pth2. There is thus a hysteresis loop, associated with a bi-stable QCL operation either in the RNGH SPs or CW lasing regimes in the range from p = 1.5–2.4 [compare Fig. 1a, b]. This hysteresis loop effect is in agreement with the results of Lyapunov stability analysis and RNGH threshold discussed in Vukovic et al. (2016a, 2017). For p > pmin, the Lyapunov exponent is positive, that is the SPs regime is possible. For instance, pmin≈ 1.3 was obtained for QCL with parameters shown in Table 1 (Vukovic et al. 2017). However, the self-starting RNGH SPs in a short-cavity QCL can build up provided the initial optical pulse perturbation is of sufficient pulse area. This condition is met at p > pth2 (pth2≈ 2.4 in considered case). Note that the longer the QCL cavity, the smaller the hysteresis width. In mm-long QCL devices, the hysteresis loop practically disappears because of pth2≈ pmin (Vukovic et al. 2016a, 2017).

With increasing pump rate p, we observe a period-doubling (at p = 4) of intensity SPs and harmonic waveform distortion.

To study whether we can trigger a sequence of the period-doubling and a bifurcation to chaos, we resolve the QCL dynamics on the fast (within one cavity round trip) and the slow (on subsequent cavity round trips) time scales, while adiabatically changing p. In Fig. 1c we show a contour plot representing the evolution of the output power waveform with decreasing p. For 10 > p>7.8, the period of intensity SPs is equal to the half of the cavity round trip time. Then for 7.8 > p>4, the period-doubling occurs (the period of SP is equal to the round trip time). For p < 4, the period of intensity SPs reduces back to the half of the cavity round trip time. We thus do not observe any sign of the period-doubling sequence.

In Fig. 1d we calculate the normalized recurrence period density entropy (RPDE) (Little et al. 2007), by applying a so-called time-delay embedding (Kantz and Schreiber 2004). Recall that a perfectly periodic waveform (and a CW signal) has the RPDE entropy value of “0”, while the RPDE value of “1” corresponds to chaos. In this manner, the RPDE indicates whether the SPs are periodic or quasi-periodic (chaotic).

For a short-cavity QCL in Fig. 1d, the RPDE entropy oscillates nearby “0” attesting for a perfectly periodic waveform even at a high pump rate p despite strong waveform anharmonicity. The spike on the RPDE curve at the lower p side of the SP/CW hysteresis loop can be attributed to the loss of stability of the SP regime at p ≤ pmin and, as a consequence, to a higher sensitivity of the waveform to polarization noise \(\varLambda_{ \pm }\) injected in the model system (Vukovic et al. 2017).

In the considered example of a short-cavity QCL, the optical gain “memory” effect (T1 = 1.3 ps) persists between the subsequent passages in the cavity (Tcav/2 = 1.1 ps), resulting in a regular pulse train.

3.2 Long cavity FP QCLs

In this subsection, we present numerically calculated optical field waveform, optical and RF power spectra, IAC trace, bifurcation diagram and RPDE plots for 1.5 mm length FP QCL chip and the same T1 = 1.3 ps as for the short cavity QCL in subsection 3.1. The cavity round-trip time is Tcav = 33 ps while all other parameters can be found in Table 1. Figure 2a shows the optical field waveform when the pump rate normalized to the lasing threshold is p = 2. It develops into square waves, in agreement with our findings in Vukovic et al. (2017). The waveform just slightly changes at each round-trip, yielding the quasi-periodic chaotic behavior. Optical spectra in Fig. 2b shows suppression of the main lasing mode (at 0 GHz) in agreement with multimode behavior observed experimentally (Gordon et al. 2008; Wang et al. 2007) and calculated numerically in Vukovic et al. (2017). RF power spectrum depicted in Fig. 2c confirms multimode comb emission. In Fig. 2d we calculate the interferometric autocorrelation trace. The peak to background ratio of the interferometric AC trace for the numerically simulated output waveform is worse than 8:3, as it can be expected for such noisy multimode lasing behavior (Diels and Rudolph 2006). However it is this 8:3 peak to background ratio that was found in experimentally measured interferometric AC traces (Antonov et al. 2017a, b; Boiko et al. 2017; Gordon et al. 2008). Quasi-periodic chaotic pulse train (Fig. 2a) with multiple pulses on the cavity round-trip is responsible for the secondary lobes seen in the IAC trace in between the cavity round-trips, (Fig. 2d). Similar IAC traces with secondary lobes have been experimentally observed in Gordon et al. 2008. In some realizations, they appear nearly at half of the cavity round-trip as in Fig. 2d [other examples can be found in Vukovic et al. (2017)].

Fig. 2
figure 2

Results of numerical simulations with TW model for monolithic QCL of the cavity length 1.5 mm (Tcav = 33 ps): a field amplitude waveform. b Optical power spectrum. Cavity modes are separated by 30 GHz and only modes with positive detuning from the main lasing mode (at zero frequency) are shown. c RF power spectrum. d IAC trace. More spectra and attractor plots can be found in Vukovic et al. (2017)

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In Fig. 3a we plot the bifurcation diagram of the waveform intensity extremes and in Fig. 3b we show the RPDE in function of adiabatically increasing pump rate p. SPs start at very low pump excess above threshold (pth2 ~ 1.1) while the bistable CW/SP region between pmin and pth2 is very narrow in QCLs with mm-long cavities. In Fig. 3a, for the pump rate higher than p = 2.5 the number of local extremes in the intensity waveform increases, as well as their amplitude, the intensity waveform becomes more chaotic.

Fig. 3
figure 3

Results of numerical simulations with TW model for monolithic QCL of the cavity length 1.5 mm: a bifurcation diagrams of self-starting SPs represented by extreme values in the output power waveform Pout (red and blue symbols correspond to the maxima and minima, respectively). b The behavior of the normalized Recurrence Period Density Entropy (RPDE) (Little et al. 2007) with increasing p. The following time-embedding parameters are used: embedding dimension m = 4, embedding delay τ = 10, embedding ball radius r = 0.02. More details about these parameters can be found in Kantz and Schreiber (2004) and Little et al. (2007). (Color figure online)

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In Fig. 3b, large values of RPDE entropy (compare to Fig. 1d) attest that the waveform becomes quasi-periodic (p > 1.2) or even chaotic (p > 2.5) when it reaches 0.7, as we suspected from Fig. 3a and the IAC trace from Fig. 2d for p = 2.

3.3 Switching the Risken–Nummedal–Graham–Haken instability on and off

In Antonov et al. (2017a, b) and Boiko et al. (2017) authors have shown experimentally that the onset of broadband multimode emission associated with RNGH instability in the FP cavity QCLs can be tailored during a transient switching-on process under the pulsed pumping. Homogeneous interaction of all cascades (periods) with the lasing mode and a small diffusion length allows for a strong SHB effect and facilitates formation of the population and coherence gratings. It was proposed that in these conditions, a sudden increase of the carrier lifetime T1 e.g. from 0.2 to 0.5 ps as a result of the lasing transition change causes a drop in the second threshold. A QCL starting initially to operate in the usual emission regime with a few lasing cavity modes switch to broadband multimode RNGH self-pulsations (Antonov et al. 2017a, b).

The second mechanism proposed in Antonov et al. (2017a, b) is for the switching-off of the RNGH instability at high currents. It has been attributed to the change of the contrast of the SHB-induced grating as a result of inhomogeneous interaction of the cascades with the lasing field when driving a QCL on the unstable part of I-V curve leads to a formation of electric field domains (EFDs) with non-uniform charge accumulation and depletion across the QCL periods (Lu et al. 2006; Wienold et al. 2011). The contrast reduction of the population grating as a result of inhomogeneous broadening reduces the efficiency of the mode coupling through the induced carrier grating and leads to an increase of the second threshold to prohibitively high currents (Antonov et al. 2017a).

These experimentally observed changes in the dynamic regimes were explained using a simple analytical expression for the second threshold in QCLs (Vukovic et al. 2016a). Here we verify both mechanisms of RNGH instability switching in numerical simulations. As a reference testbed, we use a FP QCL with a very short cavity (L = 0.1 mm), λ = 8 μm, T1 = 0.5 ps, 1/4Dk2 = 2 ps, and all other parameters being as listed in Table 1. We label it as “QCL#1” (The upper state lifetime T1 = 0.5 ps is more realistic in Mid-IR QCLs). In Fig. 4 we study the evolution of the output intensity waveform in QCL#1 and resolve its dynamics on the fast (within two cavity round trips) and the slow (on subsequent cavity round trips) time scales. In difference to Fig. 1c obtained with adiabatically decreasing pump current, in Fig. 4 the bifurcation map is built for the pump rate increasing adiabatically in small steps of δp = 0.05. Self-starting RNGH self-pulsations occur at a current of Ith2= 0.45 A corresponding to the normalized pump rate of pth2= 2.1 times above the lasing threshold.

Fig. 4
figure 4

Numerical simulations with the TW model for monolithic QCL of the cavity length 100 μm, QCL#1. Color map of Pout distribution when adiabatically increasing p (long time scale, x-axis), during a time interval equal to 2 cavity round trips (short time scale, y-axis). Carrier relaxation time T1 = 0.5 ps, carrier decoherence time T2 = 0.14 ps, diffusion coefficient D = 180 cm2/s, the emission wavelength λ = 8 μm. Note that in comparison with Fig. 1c here we increase p while resetting field to zero, and use lower T1

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The period of self-starting intensity self-pulsations is equal to the cavity round trip time (2.2 ps). Output power waveform develops into two pulses per round trip, where width of the pulses is approximately equal to T1.

To verify the suppression of RNGH SPs for a transition with a shorter gain recovery time, we perform a second test. In a model device labeled as QCL#2 we decrease the carrier relaxation time T1 down to 0.2 ps, and take all other parameters as in the reference QCL#1. The waveform bifurcation diagram for QCL#2 is plotted as a color map in Fig. 5. The pump excess above threshold required to excite RNGH SP increases from pth2= 2.1 (in QCL#1) to pth2= 2.5 indicating that RNGH instability is indeed more difficult to achieve if the lasing transition has a very short gain recovery time. This conclusion is in qualitative agreement with experimental results from Antonov et al. (2017a) showing no RNGH SPs for transient lasing on a transition from an upper subband with reduced carrier lifetime.

Fig. 5
figure 5

Numerical simulations with the TW model for monolithic QCL of the cavity length 100 μm, QCL#2. Color map of Pout distribution when adiabatically increasing p (long time scale, x-axis), during a time interval equal to 2 cavity round trips (short time scale, y-axis). Carrier relaxation time T1 = 0.2 ps, while all other parameters are the same as in Fig. 4, for QCL#1

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The reduction of the SHB is modeled by introducing strong carrier diffusion such that 4Dk2T1 ≥ 1. In the third numerical test (QCL#3) we increase the diffusion coefficient D by a factor of 5 (to the value D = 900 cm2/s) in order to make the population grating relaxation time comparable to T1, i.e. 1/4Dk2 = 0.4 ps, while other parameters are the same as in QCL#1. As a result, the contrast of the SHB-induced grating is significantly reduced. The waveform bifurcation diagram for QCL#3 is shown in Fig. 6. The second threshold current increases in 1.4 times as compared to QCL#1 in Fig. 4. This test confirms that RNGH SPs can be suppressed at a high pump rate if SHB strength is reduced. This provides a possible interpretation of the experimental results from Antonov et al. (2017a) and Bugajski et al. (2013) where RNGH SPs cease at high currents. Once QCL bias exceeds the optimal one for the tunneling resonance, QCL enters into the part of the I-V curve characterized by unstable electric field domain formation (EFD) (Lu et al. 2006). It results in inhomogeneous broadening of the gain across different cascades and hence in the weakening of the SHB effect.

Fig. 6
figure 6

Numerical simulations with the TW model for monolithic QCL of the cavity length 100 μm, QCL#3. Color map of Pout distribution when adiabatically increasing p (long time scale, x-axis), during a time interval equal to 2 cavity round trips (short time scale, y-axis). Diffusion coefficient D = 900 cm2/s, while all other parameters are the same as in Fig. 4

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4 Summary

In this paper, we provide a review of our recent theoretical studies on the origin of Risken–Nummedal–Graham–Haken self-pulsations and large spectral broadening in mid-infrared FP QCLs. We report new results on bifurcation analysis of the waveforms and the recurrence period density entropy used to distinguish periodic self-pulsations from the chaotic self-pulsations. We provide numerical simulations supporting our interpretation of switching the RNGH self-pulsations on and off observed in the experiment. Our findings open up a new point of view on understanding RNGH instability and the possibility of controlling it for specific applications.