Frequency comb formation in quantum cascade lasers is studied theoretically using a Maxwell-Bloch formalism based on a modal decomposition, where dispersion is considered. In the mid-infrared, comb formation persists in the presence of weak cavity dispersion (500 fs2 mm−1) but disappears when much larger values are used (30’000 fs2 mm−1). Active modulation at the round-trip frequency is found to induce mode-locking in THz devices, where the upper state lifetime is in the tens of picoseconds. Our results show that mode-locking based on four-wave mixing in broadband gain, low dispersion cavities is the most promising way of achieving broadband quantum cascade laser frequency combs.
© 2015 Optical Society of America
In a mode-locked laser, a fixed phase relationship is established between the longitudinal modes of the laser cavity . This phase relation is generally established by either a subtile compensation between a negative dispersion and the Kerr effect, giving rise to the propagation of a temporal soliton, or by a saturable absorber opening a time-window of low loss for a pulsed output. In both cases, the net result is the production of short pulses that will be produced at the roundtrip frequency of the cavity and, in well designed laser, an average power commensurate to the one of a continuous wave laser.
In fact, the terminology of mode-locked laser is now tacitly used to design lasers which do not only have a fixed relative phase between the modes, but one where this phase difference between adjacent modes is constant, such that the solution is a single pulse as described above. In such pulsed lasers, the upper state lifetime of the laser transition should be much longer than the round-trip time, so that the energy could be accumulated in the upper state in-between successive pulses.
Quantum cascade lasers (QCL) , being based on intersubband transitions in quantum wells, exhibit very short upper state lifetime τ2 , in the sub picosecond range (τ2 ≈ 0.6 ps) in high performance devices operating at room temperature. This time is much shorter than the typical cavity round trip time τrt, 64 ps for a 3 mm long device, such that the product ωτ2 << 1, where ω = 2π/τrt is the angular frequency corresponding to the longitudinal mode spacing. As a result, passive mode-locking in the sense described above is not possible. Active mode-locking has been achieved in devices operating at cryogenic temperature in the THz [4, 5] or with photon-assisted tunneling transitions  such that the condition ωτ2 ≈ 1 was satisfied. However, compared to passive mode-locking in which the pulse length can routinely reach the inverse of the gain bandwidth Ωg, fundamental active mode-locking leads to much longer pulses since the pulse length τ is now a geometrical average of the gain bandwidth and the round-trip frequency ω as given by the Kuizenga-Siegman formula [1, 7]:6], the formula above predicts a pulse length of τ = 1.1 ps, still significantly shorter than the observed minimum pulse length of 3 ps as Eq. (1) assumes ωτ2 >> 1. All in all, the reduced resulting mode-locking bandwidth therefore limits the usefulness of such devices for broadband spectroscopy.
In contrast, it was recently shown that broadband quantum cascade laser, engineered in low dispersion waveguides, can also operate as frequency combs such that the phase relation between the modes is also fixed , and preliminary results using these devices demonstrated the possibility of achieving dual-comb spectroscopy . In the case of QCLs, however, locking between the modes is achieved by four-wave mixing (FWM) , and the phase relation between the modes is such that the optical intensity is approximately constant in the cavity. In this picture, the beating between two modes create sidebands  that injection-lock the adjacent modes. Such model of multimode laser and the resulting locking to modes equally spaced was already explained theoretically by Lamb in the limit of three modes . Recently, such a Maxwell-Bloch model in frequency space was extended to the case of multimode operation, in order to describe the nature of the comb behavior of quantum cascade laser and contrast it to the case of semiconductor lasers with long recovery times . In this study, some simplifications were assumed in the numerical implementation such as neglecting dispersion altogether, assuming that the gain was due to a single active region stack, and not allowing the possibility of modulating a section of the device as was done in [6, 5].
In this paper, we extend the same model to broadband gain to take into account the dispersion caused by both the gain medium as well as by the cavity. We also examine the influence of an amplitude modulation applied to a section of the device. Finally, we also look at the case of multiband stacks.
Simple time-dependent rate equations [13, 14] are sufficient to model the propagation of pulses in a quantum cascade laser cavity and agreed with ultrafast pump and probe measurements . In particular, these rate equations predict, as shown in Fig. 1(a), a strong damping of pulses upon propagation in the active region of a QCL device illustrating the difficulty of sustaining mode-locked pulses. However, these models are not able to predict the dynamics of multimode operation since there is not coherent coupling between the populations and the electric field.
The Maxwell-Bloch model followed here uses a modal decomposition of the cavity field  as shown schematically in Fig. 1(b), and not a propagation in the time domain, as was done in many early theoretical studies of multimode instabilities in quantum cascade lasers [15, 16]. While the latter technique is very well adapted to follow the propagation of a pulse inside the laser cavity, it is ill-conditioned to study the output of a laser that exhibits mostly a frequency modulated output.
2.1. Density matrix
The model is based on the evolution of a density matrix where the active region is approximated by a two-level system, following the evolution equation:Eqs. (8), (10) and (11) in Eq. (6). The rotating wave approximation consists, in this context, to neglect the terms that would lead to e±i2ω21t time dependence. We have therefore for the first equation:
We then solve these equations using a perturbative approach on Δρr and σ21,k. The zeroth order for yields from Eq. (14):Eq. (15) then yields: Eq. (16) in Eq. (12) yields (for values of r ≠ (−1, 0, −1)):
We now apply the substitution k′ = k − r, k′ = k + 1 − r and k′ = k − 1 − r for the first terms of the first, second and third lines respectively, which yields:Eq. (15): Eq. (23) into Eq. (24), we have: 12].
To ease the implementation, a number of simplifications are introduced. In general, we have that γ12 ≫ ω, as the gain is broad enough to sustain many modes. Therefore, we can assume that:
The same result may be rewritten using normalized coefficients (and with the approximation mentioned above):
2.2. Cavity modes
In this section, we give the expression of the intracavity field E(t), which is a solution of Maxwell’s equation and has to obey:Eq. (8):
We prescribe that all the modes must be eigenmodes of the cavity, assumed to have no losses through the mirrors. This is a simplification in semiconductor lasers since usually at least one of the mirror has a relatively low reflectivity and therefore the field amplitude close to the facet is not a perfect standing wave. We define then the wavevector kn for the n-th eigenmode by:
To go further, we take Eq. (45), multiply both sides by sin(knz)2, divide by the cavity length lc, and integrate over the cavity length:Eq. (33) and (32).
Focusing first on the integral on the spatial coordinate, and recalling the relationship between Ωl and Al, we observe that the term will lead to a integral with the form:
It is well known that active mode-locking is achieved in semiconductor lasers by modulating only a section of the active region. For this reason, we assume now that the modulation term Δρp,M has a spatial dependence so that a fraction 0 < lm < 1 of the total length lc is modulated:
As a result, the factor κ′n,l,k,n+k−l(lm) depends explicitly on lm. An important consequence is that κ′n,n±1 is non-zero and has the value:
2.3. Equation for the mode amplitudes
We obtain finally:12] to the exception of the addition of the dispersion term and of the modulation.
2.4. Specific cases and simplifications
The expression is simplified when the whole device is uniformly modulated, i.e. lm = 1. In that case, κ′n,n±1 = 0 and we have:Eq. (65) is now
Althought it was not discussed in , the formalism developed allows to take into account dispersion. The latter will arise from both the gain Gn as well as from the four-wave mixing and modulation terms M±, Ckl, Bkl that are all complex quantities. Because of the dispersion, the frequencies are pulled from their equidistant values, and as a result the phase of the mode amplitudes An will be rotating with time. Actually, after solving for the time-dependent amplitude An, the frequency offset δωn of a specific mode n from its reference value is obtained directly using:
The resonance condition is that, for the mode number n, we have:Eq. (78) is then:
2.6. Broad gain devices
MIR comb operation was actually demonstrated in broad-gain devices. To simulate such a structure, one should think of a number of two-level systems coupled to the same cavity modes. Each two-level system will create its own polarization that has to be summed afterwards. We concentrate here on the simplest structure comprising two active regions with equal gain, each offset from ω0 by a quantity ±noffω. As such, the total gain of such a stack would be:(1) tuned to (n − noff)ω will yield:
A possibility to reduce the computation load is to make the following approximation when computing the FWM term:equation 65 is unchanged, only the parameters Gn and Bkl must be recomputed.
3.1. Dispersion in MIR QCL frequency combs
As already mentioned, the influence of dispersion on the behaviour of the modes was not considered in the first attempt to simulate QCL frequency combs . Nevertheless, dispersion plays a crucial role on mode-locking of lasers . Dispersion in QCL may arise from the dispersion of the different materials used in active region and in the cladding layers, from the dispersion created by the cavity (also called modal dispersion). Ultimately, an effective dispersion is induced by the laser gain as well as by the FWM gain, as the terms Gn, δ, M±, Ckl and Bkl can assume complex values and their imaginary parts can therefore induce dispersion. In this section we use our model to investigate the impact of all these sources of dispersion on the modal behaviour of a MIR QCL. For this purpose, we simulate a typical MIR QCL by setting τ2 = 0.5 ps, τcoh = 0.1 ps. The device studied is 6 mm long (τrt = 128 ps) and the photon lifetime in the cavity τc = 26 ps, corresponding to total losses of 4.1 cm −1. The device is assumed to be free-running and no active modulation or saturable absorber is considered. However, all the dispersion terms are taken into account in the simulation. Therefore, Eq. (65) reduces now to:
The results of a typical MIR QCL frequency comb with GVD = −500 fs2mm−1 are shown in Fig. 2. The optical amplitude spectrum as well as the phase spectrum are represented in Fig. 2(a). The amplitudes of the modes are distributed in a somehow random way, characteristic of MIR QCL frequency combs. In addition, the phase difference between adjacent modes is not the same for all pairs of modes, as it is the case for lasers emitting transform limited pulses. Instead, the amplitudes and the phases are distributed in way such that the total power is nearly constant, as shown in Fig. 2(b), where the output power is represented as a function of time. This behaviour is characteristic of a frequency-modulated laser and is also illustrated in Fig. 2(c), where the oscillations of the instantaneous frequency with time are represented. Finally, Fig. 2(d) shows the amplitude spectrum of the detected power (corresponding to the spectrum of the photocurrent when measured by a spectrum analyser). We observe that all the harmonics of the detected power are around 3 orders of magnitudes lower than the direct current (DC) value, corresponding to a constant detected power. The phase-locking mechanism of such a free-running, continuous-wave QCL is four-wave mixing which, combined with the short gain recovery time of a QCL, leads to a phase signature comparable to a frequency modulated laser.
We characterise the impact of the dispersion by computing the value corresponding to the average of the frequency of the mode n, expressed by:Fig. 3(a).
Figure 3(b) shows the frequency offset of the MIR QCL previously described. We observe that is identically zero over the entire laser spectrum when GVD = −500 fs2.mm−1, showing that this value of GVD is relatively low and does not disturb the comb formation. In addition, we study the same devices with a stronger value of dispersion (GVD = ±30000 fs2mm−1). These two results are also shown in Fig. 3(b). In agreement with our predictions, we observe a curvature of , demonstrating that these values of dispersion are enough to disturb the frequency comb operation of the QCL. Moreover, we see that changing the sign of the GVD modifies the sign of the curvature of .
Finally, as the first QCL frequency comb was demonstrated using a broad gain device , we use our model to simulate a MIR QCL containing two active regions with different central frequencies. We assume the same values for τ2, τcoh and τrt as the device previously studied. The total losses are assumed to be slightly higher than the previous case, corresponding to a cavity lifetime of τc = 26 ps, as broad gain devices usually experience higher total losses than single active region devices. The two active regions have equal gain and are centered at 1375 cm−1 and 1436 cm−1, corresponding to an offset of 61 cm−1. Figure 4(a) shows the gain profile as well as the optical amplitude spectrum of such a broad gain device. The flat broad gain profile allows multimode emission, as more than 150 modes are present in the optical amplitude spectrum. Figure 4(b) shows the amplitude spectrum of the detected power, which indicates that no amplitude modulation is observed in this broad gain device. The results of our model agree qualitatively with the measurements reported in , which are represented in Fig. 4(a) and (b).
3.2. THz QCL active mode-locking
In the mid-infrared, active mode-locking of ”conventional” QCL devices, i.e. devices with sub picosecond upper state lifetimes, remains extremely unfavorable because of the short upper state lifetime. In recent experiments, the beatnote of a mid-infrared QCL device was locked by electrical injection up to a frequency of 14GHz  but later measurements of interferometric autocorrelation did not show evidence for pulses, much like earlier studies . Actually, RF injection at the round-trip frequency in a mid-IR QCL comb was found to destabilize the comb formation above a certain injection power level .
In contrast, active mode-locking of a THz QCL has been experimentally demonstrated by modulating the QCL bias current at the round-trip frequency on a small section of the device . A coherent sampling technique - which can be seen as an optical sampling oscilloscope acquisition scheme - was used to measure transform-limited pulses with a duration of ∼10 ps. In this section we show that our model can also predict active mode-locking of THZ QCLs. As laser action in THz QCLs occurs between two states with energy spacing below the optical phonon energy, the typical lifetimes in THz QCLs are significantly higher than of MIR QCLs. This has a significant impact on the mode-locking mechanism. For a typical THz QCL, we assume τ2 = 12 ps, τcoh = 0.7 ps. The device studied is 3 mm long (τrt = 63 ps) and the photon lifetime is τc = 21 ps, corresponding to a total losses of 5.1 cm −1. The device is actively modulated at the round trip frequency ω. The device is DC-biased (1.2G0) over 90% of its length and a modulation (3.5G0) at the round trip frequency ω is applied on the other section of the device. Moreover, all the dispersion terms are taken into account in the simulation. In this case, all the terms of Eq. (65) are taken into account in this study.
Figure 5(a) shows the optical amplitude spectrum as well as the phase spectrum of an active mode-locked THz QCL. We observe a somehow gaussian distribution for the amplitude of the different modes and the phases show a linear variation across the spectrum, in agreement with the behaviour of an active mode-locked laser. The instantaneous power is plotted in Fig. 5(b). Narrow pulses with a FWHM of 7.3 ps are predicted in this case, in agreement with the value measured in . The amplitude spectrum of the detected power, shown in Fig. 5(c), also is characteristic of a pulsed output with high power harmonics. Figure 5(d) shows the time domain evolution of the amplitude of each mode. We observe a transient regime (< 5ns) where the laser turns on and therefore the amplitudes are not stable. After this transient regime, we observe a steady-state regime where all the mode amplitudes are extremely stable with time.
One very specific feature of quantum cascade lasers, due to the very fast scattering time, is the fact that the transport is driven by the photon field up to very high frequencies. As a result, any amplitude modulation of the optical field will be reflected on the driving current of the device. This fact was observed in many experiments in both the mid- and far-infrared frequencies where the same beatnote was measured either using an external detector or a probe on the current of the device. For this reason, we do have a positive feedback loop, where the (self) generation of an amplitude modulation does create a current modulation of the device.
Because our model is both able to predict the amount of amplitude modulation (by beating of the mode amplitude) and the effect of a current, the effect of such a feedback can be computed. In our model, the driving current is proportional to g0, and an uniteless photon flux can be written as:18]. For this purpose, we assumed the same THz QCL investigated before and we replace the active modulation at the round trip frequency by the self-generated current modulation expressed by Eq. (93). The results show an optical amplitude spectrum consisting of 2 modes, similar to the amplitude spectrum of the same device when no modulation (external or self-generated) is applied. Also, no sign of pulse generation is observed.
We have theoretically studied the frequency comb operation of quantum cascade lasers using a coupled-mode formalism based on Maxwell-Bloch equations. Cavity dispersion and external modulation were taken into account in the model. The model agreed with the experimental results both in the mid- and far-infrared. In particular, it confirmed that active mode-locking could be induced by modulating a short section of the device, when the lifetime of the upper state is long enough. The model, however, confirmed the experimental facts that such active mode-locking occurs in a rather short range of operating conditions and lead to a relatively limited bandwidth. Even more difficult seems the achievement of mode-locking using self-induced transparency as gain and loss have to be carefully calibrated in the same cavity [19, 20]. For this reason, it appears that mode-locking based on four-wave mixing in broadband gain, low dispersion cavities is the most promising way of achieving broadband quantum cascade laser optical frequency combs. A very interesting extension of that model would be to include the Langevin noise terms in order to predict the frequency and amplitude noise, for which data are becoming available .
References and links
1. H. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1173–1185 (2000). [CrossRef]
4. S. Barbieri, P. Gellie, G. Santarelli, L. Ding, W. Maineult, C. Sirtori, R. Colombelli, H. Beere, and D. Ritchie, “Phase-locking of a 2.7-THz quantum cascade laser to a mode-locked erbium-doped fibre laser,” Nat. Photonics 4(9), 636–640 (2010). [CrossRef]
5. S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis (vol 5, pg 306, 2011),” Nat. Photonics 5(6), 378 (2011).
6. C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kaertner, M. A. Belkin, A. Belyanin, X. Li, D. Ham, H. Schneider, P. Grant, C. Y. Song, S. Haffouz, Z. R. Wasilewski, H. C. Liu, and F. Capasso, “Mode-locked pulses from mid-infrared quantum cascade lasers,” Opt. Express 17(15), 12929–12943 (2009). [CrossRef] [PubMed]
7. D. J. Kuizenga and A. Siegman, “FM and AM mode locking of homogeneous Laser-Part I: Theory,” IEEE J. Quantum Electron. 6(11), 694 (1970). [CrossRef]
8. A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature (London) 492(7428), 229–233 (2012). [CrossRef]
9. G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 1–9 (2014). [CrossRef]
10. P. Friedli, H. Sigg, B. Hinkov, A. Hugi, S. Riedi, M. Beck, and J. Faist, “Four-wave mixing in a quantum cascade laser amplifier,” Appl. Phys. Lett. 102(22), 222104 (2013). [CrossRef]
11. W. E. Lamb Jr, “Theory of an optical maser,” Phys. Rev. 134(6A), A1429–A1450 (1964). [CrossRef]
12. J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104(8), 081118 (2014). [CrossRef]
13. J. Faist, Quantum Cascade Lasers, 1st ed. (Oxford University, 2013). [CrossRef]
14. H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. Norris, “Time-resolved investigations of electronic transport dynamics in quantum cascade lasers based on diagonal lasing transition,” IEEE J. Quantum Electron. 45(4), 307–321 (2009). [CrossRef]
15. A. Gordon, C. Y. Wang, L. Diehl, F. X. Kaertner, A. Belyanin, D. Bour, S. Corzine, G. Hoefler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77(5), 053804 (2008). [CrossRef]
16. V. M. Gkortsas, C. Wang, L. Kuznetsova, L. Diehl, A. Gordon, C. Jirauschek, M. A. Belkin, A. Belyanin, F. Capasso, and F. X. Kartner, “Dynamics of actively mode-locked quantum cascade lasers,” Opt. Express 18(13), 13616–13630 (2010). [CrossRef] [PubMed]
17. M. R. St-Jean, M. I. Amanti, A. Bernard, A. Calvar, A. Bismuto, E. Gini, M. Beck, J. Faist, H. C. Liu, and C. Sirtori, “Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation,” Laser Photon. Rev. 8(3), 443–449 (2014). [CrossRef]
18. R. Terazzi and J. Faist, “A density matrix model of transport and radiation in quantum cascade lasers,” New Journal of Physics 12, 033045 (2010). [CrossRef]
19. V V Kozlov, “Self-induced transparency soliton laser via coherent mode locking,” Phys. Rev. A 56(2), 1607– 1612 (1997). [CrossRef]