We study the instability thresholds of the stationary emission of a quantum cascade laser with optical feedback described by the Lang Kobayashi model. We introduce an exact linear stability analysis and an approximated one for an unipolar lasers, who does not exhibit relaxation oscillations, and investigate the regimes of the emitter beyond the continuous wave instability threshold, depending on the number and density of the external cavity modes. We then show that a unipolar laser with feedback can exhibit coherent multimode oscillations that indicate spontaneous phase-locking.
© 2014 Optical Society of America
Since long Quantum Cascade Lasers (QCLs) have triggered a widespread interest for several reasons: wavelength agility (1 – 100THz), high output power (> 100mW), continuous wave (CW) emission, narrow linewidth (< 10KHz), high speed modulation (up to several tens of GHz) [1, 2, 3]. Such features led to a row of highly prized applications in imaging, communications technology, sensing, astrophysics and space-science .
The search for mode-locked regimes in multimode QCLs, aiming at applications such as frequency comb generation, time-resolved measurement, and nonlinear wavelength conversion is in its initial stages. Contrary to the case of conventional bipolar semiconductor (s.c.) lasers, the fast gain recovery time of QCLs (∼ 1ps) due to intrasubands transitions (non-radiative phonon scattering) seems to favor the onset of multimode regimes such as those associated with the spatial hole burning and with a coherent instability similar to the Risken-Nummedal-Graham-Haken instability, although preventing spontaneous mode-locking . So far, few examples of active mode-locking among longitudinal modes obtained by modulating the bias current have been recently demonstrated and used to generate ultrashort picosecond pulses [5, 6, 7]. Also, an example of stable phase coherence of multiple transverse modes is reported in .
From a theoretical point of view a complete understanding of the coherent multimode dynamics in QCLs is still to be achieved and this would eventually help to identify favorable phase relations for pulsed emission.
On the other side, QCLs entered with success the community of self-mixing interferometric sensing in which the laser source is used to simultaneously generate and detect electromagnetic radiation [9, 10, 11, 12]. As an emitter, a QCL is characterized by a class A dynamics, where the carriers and the polarization variables are enslaved by the electromagnetic field. This is due to high values of the ratio between photon and carrier lifetime in unipolar lasers where the latter is dominated by non-radiative phonon scattering. This unique feature leads to the absence of relaxation oscillations in the evolution of a free running QCLs towards its steady-state . In the case of conventional s.c. lasers subject to optical feedback (OF) it is well known that the amplification of relaxation oscillations under increasing feedback power is the fundamental mechanism of CW destabilization, leading to chaotic dynamics . Also, the linewidth enhancement factor (LEF) or Henry factor known to play a role in conventional s.c. lasers as evidenced in [15, 16], favoring stability for low values.
Recently, we predicted that a THz single mode QCL with OF, due precisely to the absence of relaxation oscillations and negligible LEF (α < 1) can exhibit an absolute stability against OF and experiments proved that it can sustain a feedback strength ∼ 70 times larger than that typically leading to the onset of chaos in a diode laser .
In this work we focus on the instabilities and dynamical behavior of the retroinjected QCLs, and investigate in particular the occurrence of regular oscillatory dynamics, that can be qualified as form of phase synchronization among modes. This can be an important heralding feature towards mode-locking and pulse generation. In absence of relaxation oscillations, the instability we deal with is characterized by the competition of modes, not defined by the free running laser, but by the external cavity (EC) formed by the feedback mirror and the laser output facet. As we will show, the LEF and EC length strongly influence the multimode dynamics and can either remove the instability altogether or favor a chaotic dynamics. We are of course interested in regimes comprised between the two, where regular dynamics can ensue from mode competition. To address this issue, we will adopt an essential and simple modelization of a QCL with OF, that allows us to focus on crucial dynamical parameters such as the LEF and characteristic time scales of the field and carriers dynamics. This is important to gain a physical insight towards the complex dynamics originating by the destabilization of the single mode CW emission, typical of lasers with OF .
The paper begins with the description and motivation of the adopted model (Sec. 2), it then proceeds with the analysis of the CW solutions and their spectral properties, with respect to the dependence on critical parameters (LEF and EC length). A Linear Stability Analysis (LSA) in Sec. 3 identifies the conditions for the loss of stability of the CW solutions and provides a simpler approximation valid for unipolar lasers. Section 4 is devoted to the analysis of the laser behavior above the instability threshold and shows the existence of regular oscillations where a number of CW solutions contributes to the emission. The conclusions are drawn in Sec. 5.
2. The model
We consider a QCL to which OF is provided by an external mirror placed at a distance L from the laser output facet as sketched in Fig. 1. The intensity of the back reflected radiation can be varied by an attenuator (not shown in the picture). We describe the system dynamics in the framework of the well known Lang-Kobayashi (LK) approach . With respect to more complex descriptions of the QCL dynamical behavior based on Maxwell-Bloch equations for two  o three level systems [19, 20], it keeps a relatively simple formalism while still having remarkable success in describing the actual behavior of retroinjected lasers . Actually it has been recently used to derive relevant parameters of Mid-IR and THz quantum cascade lasers such as the LEF or to measure dynamical features of the target [21, 22, 23].
The model consists in two rate equations for the spatio-temporal evolution of the electric field and the carrier density in the laser cavity; it considers a phenomenological refractive index and gain, independent from the frequency and linearly dependent from the carriers density and a single delay term in the equation for the field. The flat gain is a sustainable approximation at this investigatory stage, since the number of external cavity modes (ECMs) we will consider and the free spectral range of the EC (EFSR) will in general allow this approximation for a QCL, whose gain width easily exceeds few THz. Another fundamental limiting assumption in the LK model is the single roundtrip approximation, where multiple reflections in the EC are neglected assuming a moderate feedback . Of course, when the strength of the OF grows this approximation fails and more refined approaches must be used . We nevertheless checked that even when corrections accounting for multiple reflections are introduced, the relevant dynamics remains unchanged.
The standard LK model in Eqs. (7.4) of  can be written in an adimensional form by introducing the field , the carrier density N = (Ñ − Ñ0)Gnτp:Eq. (5) it directly follows that k is limited by kmax = τc/2τp; 2) Eq. (4) for the field frequency ωF is transcendental, so its multiple solutions cannot be determined in closed form.
If not otherwise specified we use a set of parameters for the THz QCL model reported in Table 1. They refer to typical GaAs/AlGaAs heterostructures and have been derived from experimental data [24, 17]. Note the large value of the photon to carrier lifetime ratio γ, a unique feature of s.c. unipolar lasers.
While the complex structure of the CW stationary solutions of this model has been studied for decades, we now dwell to some extent thereon, in order to clarify the relation among the number of solutions, their ordering with respect to the ECMs, the feedback strength and to highlighten the role of the LEF in this model.
Choosing a pump 50% above free running laser thershold (Ip = 1.5), k = 0.5, and a cavity length of L = 146mm (i.e. τ = 30 that corresponds to a cavity round trip time of ≈ 1ns), we report in Fig. 2(a) the complete set of CW solutions in the intensity–frequency plane (ω − ω0, |E|2) for three different values of the LEF α = 0.35, 1.3, 3. Roughly speaking, the first two values can be considered appropriate for a THz QCL, a Mid-IR QCL [21, 22], while 3 is more typical of diode lasers. Moreover, in Fig. 2(b) we plot the frequencies of the CW solutions ωF vs k for the chosen values of α. We remind that the upper limit of k in Fig. 2(b) is imposed by the self-consistency of the LK model and it is given by .
Figure 2(a) shows that the CW solutions are close but not exactly coincident with the ECMs (vertical dashed lines) that are separated by the EFSR = 2π/τ ≃ 0.21 (that corresponds to ≃ 6.45GHz in physical units).
Starting from the CW closest to ω0, the CW solutions separated by ≃ 2π/τ and associated with higher intensities are called ”modes” of the system and correspond to in–phase interference between the electric field in the laser cavity and the reinjected delayed field. The other solutions are called ”antimodes” and correspond to destructive interference; such antimodes are always unstable in standard operation as demonstrated by T. Erneux and coworkers for a diode laser . Modes and antimodes appears in pairs via a saddle-node bifurcation as we increase the feedback strength.
It is important to remark that the number of CW solutions increases with increasing α. This can be understood by considering that in the hypothesis of a linear dependence of the gain G and refractive index η from the carrier density N:14]): Eqs. (3)–(4) the CW solutions ωF can be expressed as:
We now turn our attention to the dependence of the CW solution frequencies on the feedback parameter k. A qualitative picture can be garnered from Fig. 2(b). We now fix α and, in agreement with Eq. (6), we observe that a line parallel to the x–axis drawn for growing values of k, will exhibit an increasing number of intersections with the e.g. red curve (α=3); this means that the number of CW solutions increases with k and their frequencies always belong to an interval centered in ω0 with halfwidth Max(|ωF − ω0|)kmax = (α/2 + 0.5). Moreover we observe that, for k ≪ 1 (”bad external cavity limit”) the CW solutions reduce to a single one, close the free running laser frequency ω0, while for k → kmax (”good external cavity limit”) the modes approach the EC resonances i.e. the ECMs. In the following we denote as CW0 the CW solution closest to ω0.
The behavior of the steady state CW solutions indicates that the number of modes that can be involved in laser emission is, on the one hand, clearly linked to the EFSR, but on the other hand it can be strongly affected by the feedback strength k and by the LEF. The CW solutions stability clearly depends on such crucial parameters and a study of its boundary is presented in the next section.
3. Linear stability analysis
While the CW solutions do not depend on the field and carrier decay rates, the linear stability analysis (LSA) we will perform in this section is pivotally centered on the QCL property γ ≫ 1. Although we focus here on the LSA of the CW0 solution, the same approach can be easily extended to any other CW solution.
By considering perturbations to the CW0 solution (Es, ωF, Ns) in the form:
We observe that because of the phase invariance of Eqs. (1) and (2) we can choose Es real without loss of generality. We define the critical feedback kc as the minimum level causing Max(Re(λ)) > 0 and thus a CW instability.
At difference from T. Erneux and coworkers that in  are able to analytically qualify the roots of the secular equation and the associated CW instability, in the case of a single mode QCL with OF, the delay-associated nonlocality renders Eq. (9) transcendental and this prevents any analytical solution. We found the complex eigenvalues λ looking for the zeros of Eq. (9) by implementing a very precise, though CPU-intensive, minimization algorithm based on ”simplex” methods . We will use its results as a reference for the following approximate analysis.
In the limit γ ≫ 1 Eq. (9) can be reduced to the second order secular equation:17], in this case the CW instability is due to the competition among the CW solutions close to the ECMs, because the fast medium suppresses the well known mechanism of destabilization via amplification of the relaxation oscillations. In the following we now further characterize the coherent character of this instability.
The instability boundaries are found by setting λ = iΩ, Ω ∈ ℛ in Eq. (13), that becomes:
Inserting the previous expansion in the complex Eq. (16) and neglecting terms of order O(ε3) we obtain the following system of two real second order equations for ε:Eqs. (18) and (19) exists, and Fig. 3 shows the point pairs (kc, α) that satisfy this condition and thus represent the sought CW instability boundary (dashed black line).
We compare this prediction to that obtained from the complete characteristic equation (9) (full black line) and to the instability onset obtained numerically by integrating the LK equations (open square symbols). We observe a difference of few percents or less for α values in (1.5 – 2.5) (suitable for Mid-IR QCLs), among the complete and approximated LSA, while the numerics confirm the validity of the former one everywhere.
Finally, αc ≈ 0.35 is the critical value of α below which no instability is found; it represents the limit of the ultra stable regime for THz QCL identified in .
While the decrease of the threshold with α was described for conventional inter–band lasers in , we stress here that the underlying physics leading to the instability is quite different.
Results from the previous section (Fig. 2), allow to better interpret Fig. 3: in particular, we ascribe the asymptotic threshold decrease to the fact that increasing α, increments the number of CW solutions existing and this favors a multimode competition. Although not reported in this paper, we extended the analysis for α > 3 and we found that the curves in Fig. 3 approach the value kc ≃ 0.1, suggesting that, even in a flat gain scheme, once the number of CW solutions is large enough and their separation small enough, the destabilization takes place with a role determined (initially) by just a limited number of modes near the stable CW mode.
3.1. Variation of αc with τ
Since an increasing EC round trip time τ implies more and denser CW solutions (see Eq. 4), we study the variation of the CW instability boundaries, and in particular of αc, with this parameter. In Fig. 4 we plot the numerically calculated values of αc versus τ for Ip = 1.5. As expected from the considerations reported in previous section, we observe that a smaller τ (i.e. less solutions and more spaced ECMs) causes the onset of a CW instability for larger values of α (up to αc = 1.7 for τ = 7.5). Moreover when τ becomes larger than ≃ 80, the ECMs are so close that the number of competing CW modes leading to the CW instabilty does not really depend on α, so that αc shows a general asymptotic behaviour towards αc ≃ 0.3. The THz QCLs ultrastability is thus confirmed over a very large range of τ, or equivalently of EC lengths.
In Fig. 4 though, some values of τ (evidenced by circles) can be noted, where the threshold drops (i.e. the system is more prone to destabilize the stationary emission). By analyzing the values τ = 22.5, 52.5 and 90, we noted that the frequency of the free running laser frequency ω0 there falls close (less than 1/3 of the EFSR) to midway between two adjacent ECMs. This is a situation where the competition between the modes is strongest and thus leads to easier destabilization. In fact, by studying the distance of the closest (to the emitted one) CW mode from the neighboring ECMs, one sees that it diminishes with increasing k, keeping always smaller than for neighboring values of τ. This also tells us that the instability of this system is a complex mechanism where the number of solutions and the EFSR isn’t the only criterion involved, but the relative positions of the frequencies of the free-running laser, CW solutions and ECMs play a role.
4. Dynamical simulations
In this section we study the dynamical behavior of the laser when the instability, studied in the previous section, occurs and the steady states the laser achieves thereupon. Equations (1) and (2) were integrated with a 6th order Adams-Bashforth-Moulton predictor-corrector algorithm with fixed pump Ip and EC length L (or, equivalently, the delay τ).
We studied the dynamical regimes when the feedback k is increased across the threshold kc for different values of α.
As a general rule, we identify different dynamical regimes.
- - For values of α close to zero (α≤αc), typical of THz QCLs, the LSA does not predict an instability boundary for k < kmax, the laser never destabilizes the continuous wave solution closest to the free running laser frequency ω0 denoted here as CW0. This corresponds to the ultrastable regime reported in 
- - For values of α larger than αc, but still smaller than a second critical value that we denoted as αsw, the CW0 destabilization leads the laser to switch to an adjacent CW mode, and the emission is still constant (see subsection 4.1 for details)
- - For α > αsw we observe a Hopf bifurcation leading to the onset of a regime of regular oscillations given by the locking among few (usually two) CW modes as soon as the feedback strength k overcomes kc. A further increase of k leads to the destabilization of this regular regime to a stronger multimode competition that is generally associated with a chaotic dynamics. For even higher feedback the system shows one of the following dynamical behaviors: 1. a single chaos crisis leading to the restoration of single mode operation on a different CW mode with constant intensity; 2. windows of chaotic behavior alternated with single CW operation or regular oscillations; 3. a chaotic system dynamics (see subsection 4.2).
4.1. CW-solution switching and multimode regimes of stationary emission
In Fig. 5 we plot for α = αsw = 1.3 the maximum and minimum values of the intensity I = |E|2 during a ramp in k from 0 to 0.5. To avoid transient latency, the system was allowed to relax to steady state before changing k by each step.
On the overall, in all cases where αc ≤ α ≤ 1.3, all instabilities met by increasing k cause the switching of a stationary emission from a CW mode (for k ≪ 1 the laser starts from CW0) to the adjacent one. This corresponds to the system transitions from region I to region II and from region II to region III in Fig. 5. The switches are accompanied by a transient oscillation that can be seen in Fig. 5 immediately after the thresholds.
4.2. Multimode regimes: irregular dynamics and locked states of regular high contrast oscillations
When we depart from the regime of CW-switching, e.g. by increasing α, we observe a dynamics where a limited number of modes appear in the optical spectrum. The regular oscillations appearing in the intensity proves that the modes have a constant phase relation. As an example Fig. 6(a) shows the system bifurcation diagram obtained as described in the previous paragraph for α = 3. While in region I, the solution CW0 is stable, in the grey areas (regions II, III, IV, V) the CW emission is unstable showing regular oscillations or a chaotic dynamics. In Fig. 6(b) we show the power spectra for fixed values of k in the CW unstable regimes as marked by the corresponding roman numbers.
In particular we observe that when k > kc the system enters a regime of regular oscillations (see region II in Fig. 6(a) and in Fig. 6(b). An inspection of the spectrum (see Fig. 6(c) for k = 0.12), reveals a main peak at the frequency difference between the CW0 and the next stationary solution denoted as CW1. The period of these oscillations varies with k and α according to the dependency of the CW modes on k as illustrated in Fig. 2(b) and thus can be somewhat different from the cavity round trip time τ. For the parameters in Fig. 6 the period of the oscillations is Δt ∼ 1.2τ = 35. By further increasing k the system enters a chaotic regime characterized by continuous and multipeaked spectra with still dominant contributions linked to the CW modes (see regions III and V in Figs. 6(a) and 6(b)). An example of this irregular dynamics is shown in the spectrum and intensity plot in Fig. 6(d) for k = 0.163 where the ECMs are indicated by vertical dashed lines. Interspersed within the chaos region, one meets windows of regular oscillations (see region IV in Figs. 6(a) and 6(b) where again the multimode coherent competition shows a regular dynamics with a period slightly variable with k. This is known to occur also in bipolar lasers with phase–conjugated feedback (see chap.2 of ), where windows of periodic orbits are separated from each other by ”bubbles” of irregular dynamics, and the onset of chaos is associated with a torus breakup.
A detailed behavior of the oscillations in the regular regimes is shown in Fig. 6(f) where we plot the correlation diagram for three different values of k belonging to the regions II, III and IV. The intensity I(t + Δt) is plotted against I(t) with Δt = 1.2τ. The points corresponding to regular intensity oscillations in region II and IV depict two distinct limit cycles corresponding to slightly different periods, while those corresponding to the irregular oscillations of region III are more scattered but their density is higher around the two limit cycles, since the dominant contribution in system dynamics is linked to the CW modes.
The stable periodic oscillations in Fig. 6(e) for k = 0.2 represent the phenomenon of coherent dynamics involving the largest number of modes (≃ 5 in the first decade of the power spectrum) that we were able to simulate for the chosen values of Ip and τ and we believe that it can be considered an interesting phenomenon of coherent synchronization of multimode emission. Here, without external modulation of laser gain (as in active mode-locking [5, 6]), the system shows a self-organization of its emission with a (still limited) comb of ECMs. This is a condition that may hint towards a spontaneous mode-locking in a QCL with feedback.
Since we are interested in QCLs, we scanned smaller values of α. Of course the reduction in the number of solutions this implies, must be compensated by an increase of the EC length (i.e. of τ) in order to have a sizable number of modes to compete and possibly lock in phase. We found that the regime of coherent phase synchronization is rather widespread and it even occurs close to the first instability window. We report here on α = 2, a value that is still compatible with experimental evidences (a value as high as 2.5 is reported in ) and shows phase coherence to appear for still moderate EC lengths.
As shown in Fig. 7 for τ = 60 and k = 0.13, close to the instability threshold of the CW0 solution and before any chaotic crisis, a regime of nontrivial regular oscillations appears where the ≃ 5 CW modes are present in the first decade. For k > 0.2 a complex multimode competition causes the first abrupt transition to a highly irregular regime.
As anticipated, we verified that the regimes of coherent phase synchronization we met in different parametric conditions, are sustained even when multiple reflections in the EC are taken into account. While the model extension is outside the scope of this paper, we relied on previous knowledge gathered in  and adopted an expression of the feedback field generalized to an arbitrary number of reflections (see Eqs. (15) and (17) in ). The general indications of the extended model is that, of course, instability threshold values for k may vary, but the dynamical scenery remains qualitatively the same: at threshold one meets the onset of laser regular oscillations, linked to the competition of two adjacent ECMs, which upon increasing k show first the occurrence of more modes, locked in phase and still producing a regular intensity pulsation, and then the system abruptly plunges into a chaotic behavior, which again is interrupted by windows of regular dynamics.
In conclusion, we analyzed the stationary solutions of a QCL with optical feedback and showed their dependencies on critical parameters such as the LEF and the EC length, we provided an exact LSA and a simpler, approximated one, valid for unipolar lasers which allows to study the destabilization of the CW0 mode and also validates and extends the prediction of a regime of absolute stability of THz QCLs against OF. By studying the behavior of the laser above the instability threshold we could evidence the multimode dynamics typical of unipolar lasers where the mechanism of amplification of relaxation oscillations is absent and the emission is determined by the competition of several modes; in particular at threshold or in the windows of regular dynamics between chaotic islands, we could prove the existences of regimes of coherent multimode oscillations emerging from very simple physical processes in a 2-level model with feedback, which could possibly indicate a path towards spontaneous mode-locking.
This research has been funded by the Italian Ministry of Research (MIUR) through the Futuro in Ricerca FIRB-grant PHOCOS ( RBFR08E7VA). The authors also acknowledge support from MIURPON02-0576 INNOVHEAD and MASSIME.
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