## Multimode regimes in quantum cascade lasers with optical feedback |

Optics Express, Vol. 22, Issue 9, pp. 10105-10118 (2014)

http://dx.doi.org/10.1364/OE.22.010105

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### Abstract

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

## 1. Introduction

*THz*), high output power (> 100

*mW*), continuous wave (CW) emission, narrow linewidth (< 10

*KHz*), high speed modulation (up to several tens of

*GHz*) [1

1. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science **264**, 553–556 (1994). [CrossRef] [PubMed]

2. R. F. Curl, F. Capasso, C. Gmachl, A. A. Kosterev, B. Mc Manus, R. Lewicki, M. Pusharsky, G. Wysocki, and F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. **487**, 1–18 (2010). [CrossRef]

3. J. Faist, *Quantum Cascade Lasers* (Academic, 2013). [CrossRef]

3. J. Faist, *Quantum Cascade Lasers* (Academic, 2013). [CrossRef]

*ps*) 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 [4

4. A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, 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**, 053804 (2008). [CrossRef]

5. C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kärtner, 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**, 12929–12943 (2009). [CrossRef] [PubMed]

6. S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis,” Nat. Photonics **5**, 306–313 (2011). [CrossRef]

7. A. K. Wójcik, P. Malara, R. Blanchard, T. S. Mansuripur, F. Capasso, and A. Belyanin, “Generation of picosecond pulses and frequency combs in actively mode locked external ring cavity quantum cascade lasers,” Appl. Phys. Lett. **103**, 231102 (2013). [CrossRef]

8. N. Yu, L. Diehl, E. Cubukcu, D. Bour, S. Corzine, G. Höfler, A. K. Wójcik, K. B. Crozier, A. Belyanin, and F. Capasso, “Coherent coupling of multiple transverse modes in quantum cascade lasers,” Phys. Rev. Lett. **102**, 013901 (2009). [CrossRef] [PubMed]

9. P. Dean, Y. L. Lim, A. Valavanis, R. Kliese, M. Nikolić, S. P. Khanna, M. Lachab, D. Indjin, Z. Ikonić, P. Harrison, A. D. Rakić, E. H. Linfield, and A. G. Davies, “Terahertz imaging through self-mixing in a quantum cascade laser,” Opt. Lett. **36**, 2587–2589 (2011). [CrossRef] [PubMed]

10. Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. Linfield, A. G. Davies, S. J. Wilson, and A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. **99**, 081108 (2011). [CrossRef]

11. M. C. Phillips and S. Taubman, “Intracavity sensing via compliance voltage in an external cavity quantum cascade laser,” Opt. Lett. **37**, 2664–2666 (2012). [CrossRef] [PubMed]

12. F. P. Mezzapesa, V. Spagnolo, A. Antonio, and G. Scamarcio, “Detection of ultrafast laser ablation using quantum cascade laser-based sensing,” Appl. Phys. Lett. **101**, 171101 (2012). [CrossRef]

13. R. Paiella, R. Martini, F. Capasso, C. Gmachl, and H. Y. Hwang, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. **79**, 2526–2528 (2001). [CrossRef]

14. D. M. Kane and K. A. Shore, *Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Diode Lasers* (John Wiley, 2005). [CrossRef]

15. J. Helms and K. Petermann, “A simple analytic expression for the stable operation range of laser diodes with optical feedback,” IEEE J. Quantum Electron. **26**, 833–836 (1990). [CrossRef]

16. D. Weidmann, K. Smith, and B. Ellison, “Experimental investigation of high-frequency noise and optical feedback effects using a 9.7 μm continuous-wave distributed-feedback quantum-cascade laser,” Appl. Opt. **46**, 947–953 (2007). [CrossRef] [PubMed]

*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 [17

17. F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, and G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express **21**, 13748–13757 (2013). [CrossRef] [PubMed]

17. F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, and G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express **21**, 13748–13757 (2013). [CrossRef] [PubMed]

## 2. The model

*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 [18

18. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. **16**, 347–355 (1980). [CrossRef]

4. A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, 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**, 053804 (2008). [CrossRef]

19. T. Gensty, W. Elsäßer, and C. Mann, “Intensity noise properties of quantum cascade lasers,” Opt. Express **13**, 2032–2039 (2005). [CrossRef] [PubMed]

20. M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum cascade lasers: hidden reason for the narrow linewidth and line broadening by thermal photons,” IEEE J. Quantum Electron. **44**, 12–29 (2008). [CrossRef]

14. D. M. Kane and K. A. Shore, *Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Diode Lasers* (John Wiley, 2005). [CrossRef]

21. J. Staden, T. Gensty, W. Elsäßer, G. Giuliani, and C. Mann, “Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett. **31**, 2574–2576 (2006). [CrossRef]

22. R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. **92**, 071106 (2008). [CrossRef]

23. A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express **21**, 22194–22205 (2013). [CrossRef]

*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 [18

18. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. **16**, 347–355 (1980). [CrossRef]

14. D. M. Kane and K. A. Shore, *Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Diode Lasers* (John Wiley, 2005). [CrossRef]

*Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Diode Lasers* (John Wiley, 2005). [CrossRef]

*N*= (

*Ñ*−

*Ñ*

_{0})

*G*: where

_{n}τ_{p}*G*is the modal gain coefficient,

_{n}*τ*is the carrier density decay time from the upper laser level, the time

_{e}*t*is scaled to

*τ*, the photon lifetime. The carrier density at transparency is

_{p}*Ñ*

_{0}and the pump parameter

*I*is defined as

_{p}*I*=

_{p}*G*

_{n}τ_{p}Ñ_{0}(

*G*

_{gen}τ_{e}/Ñ_{0}− 1) where

*G*is electrical pumping term. Other parameters are

_{gen}*α*, the LEF; the photon to carrier lifetime ratio

*γ*; the free running laser frequency

*ω*

_{0}(equal to the laser cavity resonance) which will be the reference frequency; the laser cavity round trip time

*τ*, and the EC length

_{c}*L*which defines the delay time

*τ*= 2

*L/c*and of course the EFSR. The feedback parameter

*k*depends on the effective fraction of the back-reflected field re-entering the laser

*ε*, the laser exit facet reflectivity

*R*and the external mirror reflectivity

*R*trough the relation:

_{ext}*E*=

*E*exp

_{s}*i*(

*ω*−

_{F}*ω*

_{0})

*t*and

*N*=

*N*we get: As it is well known the s.c. laser with OF is an infinite dimensional system and its steady state characteristic might be strongly modified with respect to the free running laser case. We observe in particular that: 1) from Eq. (5) it directly follows that

_{s}*k*is limited by

*k*=

_{max}*τ*/2

_{c}*τ*; 2) Eq. (4) for the field frequency

_{p}*ω*is transcendental, so its multiple solutions cannot be determined in closed form.

_{F}17. F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, and G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express **21**, 13748–13757 (2013). [CrossRef] [PubMed]

*γ*, a unique feature of s.c. unipolar lasers.

*I*= 1.5),

_{p}*k*= 0.5, and a cavity length of

*L*= 146mm (i.e.

*τ*= 30 that corresponds to a cavity round trip time of ≈ 1

*ns*), we report in Fig. 2(a) the complete set of CW solutions in the intensity–frequency plane (

*ω*−

*ω*

_{0}, |

*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

21. J. Staden, T. Gensty, W. Elsäßer, G. Giuliani, and C. Mann, “Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett. **31**, 2574–2576 (2006). [CrossRef]

22. R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. **92**, 071106 (2008). [CrossRef]

*ω*

_{F}*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

*π/τ*≃ 0.21 (that corresponds to ≃ 6.45

*GHz*in physical units).

*ω*

_{0}, the CW solutions separated by ≃ 2

*π/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 [25

25. A. M. Levine, G. H. M. van Tartwijk, D. Lenstra, and T. Erneux, “Diode lasers with optical feedback: Stability of the maximum gain mode,” Phys. Rev. A **52**, 3436–3439 (1995). [CrossRef]

*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*: where

*G*and

_{th}*η*represent the values of

_{th}*G*and

*η*at threshold of the free running laser and Δ

*N*is a small variation around the value for the carrier density at threshold, the dressed laser cavity resonances

*ω*(

_{n}*N*) =

*nπc/Lη*(

*N*), change with

*N*(and thus with the field amplitude

*E*) according the formula (for a details refer to Eqs. (2.10)–(2.15) in [14

*Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Diode Lasers* (John Wiley, 2005). [CrossRef]

*α*–factor. Using Eqs. (3)–(4) the CW solutions

*ω*can be expressed as: thus in presence of feedback the CW solutions differ from the

_{F}*dressed*laser cavity resonances

*ω*by an amount that depends on the feedback strength. Since the EFSR is much smaller than the FSR of the laser, only the dressed cavity resonance

_{n}*ω*

_{1}should be considered in the previous expression. For fixed

*k*, an increase in

*α*causes an increase in the separation of

*ω*

_{1}, and consequently of

*ω*, from

_{F}*ω*

_{0}. In particular, the maximum distance between a CW frequency and

*ω*

_{0}is given by: It is thus obvious that the number of CW solutions increases with

*α*.

*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

*ω*

_{0}with halfwidth

*Max*(|

*ω*−

_{F}*ω*

_{0}|)

*= (*

_{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

*ω*

_{0}, while for

*k*→

*k*(”good external cavity limit”) the modes approach the EC resonances i.e. the ECMs. In the following we denote as CW

_{max}_{0}the CW solution closest to

*ω*

_{0}.

*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

*γ*≫ 1. Although we focus here on the LSA of the CW

_{0}solution, the same approach can be easily extended to any other CW solution.

_{0}solution (

*E*,

_{s}*ω*,

_{F}*N*) in the form: we get, after linearization, the characteristic equation for the complex eigenvalue

_{s}*λ*where:

*E*real without loss of generality. We define the critical feedback

_{s}*k*as the minimum level causing

_{c}*Max*(

*Re*(

*λ*)) > 0 and thus a CW instability.

26. T. Erneux, V. Kovanis, and A. Gavrielides, “Nonlinear dynamics of an injected quantum cascade laser,” Phys. Rev. E **88**, 032907 (2013). [CrossRef]

*λ*looking for the zeros of Eq. (9) by implementing a very precise, though CPU-intensive, minimization algorithm based on ”simplex” methods [27]. We will use its results as a reference for the following approximate analysis.

*γ*≫ 1 Eq. (9) can be reduced to the second order secular equation: where: We note that the same result, can be obtained by first adiabatically eliminating the fast variable

*N*in the LK equations and then performing the LSA of the CW solutions of this simplified model. As we demonstrated in [17

**21**, 13748–13757 (2013). [CrossRef] [PubMed]

*λ*=

*i*Ω, Ω ∈

*ℛ*in Eq. (13), that becomes: where to simplify the notation we introduce the quantities:

*θ*=

*ω*In the additional hypothesis that the instability is triggered by the competition between the stable CW and the adjacent mode and that their distance is close to 2

_{F}τ*π/τ*as discussed in the previous section, we may write: where |

*ε*| → 0 and we take

*n*= ±1.

*O*(

*ε*

^{3}) we obtain the following system of two real second order equations for

*ε*: that can be solved analytically. An instability occurs when a common solution to Eqs. (18) and (19) exists, and Fig. 3 shows the point pairs (

*k*,

_{c}*α*) that satisfy this condition and thus represent the sought CW instability boundary (dashed black line).

*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.

*α*≈ 0.35 is the critical value of

_{c}*α*below which no instability is found; it represents the limit of the ultra stable regime for THz QCL identified in [17

**21**, 13748–13757 (2013). [CrossRef] [PubMed]

*α*was described for conventional inter–band lasers in [15

15. J. Helms and K. Petermann, “A simple analytic expression for the stable operation range of laser diodes with optical feedback,” IEEE J. Quantum Electron. **26**, 833–836 (1990). [CrossRef]

*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

*k*≃ 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.

_{c}### 3.1. Variation of α_{c} with τ

*τ*implies more and denser CW solutions (see Eq. 4), we study the variation of the CW instability boundaries, and in particular of

*α*, with this parameter. In Fig. 4 we plot the numerically calculated values of

_{c}*α*versus

_{c}*τ*for

*I*= 1.5. As expected from the considerations reported in previous section, we observe that a smaller

_{p}*τ*(i.e. less solutions and more spaced ECMs) causes the onset of a CW instability for larger values of

*α*(up to

*α*= 1.7 for

_{c}*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

*α*shows a general asymptotic behaviour towards

_{c}*α*≃ 0.3. The THz QCLs ultrastability is thus confirmed over a very large range of

_{c}*τ*, or equivalently of EC lengths.

*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

*ω*

_{0}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

*th*order Adams-Bashforth-Moulton predictor-corrector algorithm with fixed pump

*I*and EC length

_{p}*L*(or, equivalently, the delay

*τ*).

*k*is increased across the threshold

*k*for different values of

_{c}*α*.

- - For values of
*α*close to zero (*α*≤*α*), typical of THz QCLs, the LSA does not predict an instability boundary for_{c}*k*<*k*, the laser never destabilizes the continuous wave solution closest to the free running laser frequency_{max}*ω*_{0}denoted here as CW_{0}. This corresponds to the ultrastable regime reported in [17**21**, 13748–13757 (2013). [CrossRef] [PubMed] - - For values of
*α*larger than*α*, but still smaller than a second critical value that we denoted as_{c}*α*, the CW_{sw}_{0}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*>*α*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_{sw}*k*overcomes*k*. A further increase of_{c}*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).

*I*= 1.5, and

_{p}*τ*= 30 (for which

*α*= 0.35).

_{c}### 4.1. CW-solution switching and multimode regimes of stationary emission

*α*=

*α*= 1.3 the maximum and minimum values of the intensity

_{sw}*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.

*α*≤

_{c}*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 CW

_{0}) 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

*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 CW

_{0}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.

*k*>

*k*the system enters a regime of regular oscillations (see region

_{c}*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 CW

_{0}and the next stationary solution denoted as CW

_{1}. 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 Δ

*t*∼ 1.2

*τ*= 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 [14

*Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Diode Lasers* (John Wiley, 2005). [CrossRef]

*k*belonging to the regions

*II*,

*III*and

*IV*. The intensity

*I*(

*t*+ Δ

*t*) is plotted against

*I*(

*t*) with Δ

*t*= 1.2

*τ*. 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.

*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

*I*and

_{p}*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

5. C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kärtner, 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**, 12929–12943 (2009). [CrossRef] [PubMed]

6. S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis,” Nat. Photonics **5**, 306–313 (2011). [CrossRef]

*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 [21

21. J. Staden, T. Gensty, W. Elsäßer, G. Giuliani, and C. Mann, “Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett. **31**, 2574–2576 (2006). [CrossRef]

*τ*= 60 and

*k*= 0.13, close to the instability threshold of the CW

_{0}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.

28. L. Columbo, M. Brambilla, M. Dabbicco, and G. Scamarcio, “Self-mixing in multi-transverse mode semiconductor lasers: model and potential application to multi-parametric sensing,” Opt. Express **20**, 6286–6305 (2012). [CrossRef] [PubMed]

28. L. Columbo, M. Brambilla, M. Dabbicco, and G. Scamarcio, “Self-mixing in multi-transverse mode semiconductor lasers: model and potential application to multi-parametric sensing,” Opt. Express **20**, 6286–6305 (2012). [CrossRef] [PubMed]

*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.

## 5. Conclusions

_{0}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.

## Acknowledgments

## References and links

1. | J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science |

2. | R. F. Curl, F. Capasso, C. Gmachl, A. A. Kosterev, B. Mc Manus, R. Lewicki, M. Pusharsky, G. Wysocki, and F. K. Tittel, “Quantum cascade lasers in chemical physics,” Chem. Phys. Lett. |

3. | J. Faist, |

4. | A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, 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 |

5. | C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kärtner, 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 |

6. | S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis,” Nat. Photonics |

7. | A. K. Wójcik, P. Malara, R. Blanchard, T. S. Mansuripur, F. Capasso, and A. Belyanin, “Generation of picosecond pulses and frequency combs in actively mode locked external ring cavity quantum cascade lasers,” Appl. Phys. Lett. |

8. | N. Yu, L. Diehl, E. Cubukcu, D. Bour, S. Corzine, G. Höfler, A. K. Wójcik, K. B. Crozier, A. Belyanin, and F. Capasso, “Coherent coupling of multiple transverse modes in quantum cascade lasers,” Phys. Rev. Lett. |

9. | P. Dean, Y. L. Lim, A. Valavanis, R. Kliese, M. Nikolić, S. P. Khanna, M. Lachab, D. Indjin, Z. Ikonić, P. Harrison, A. D. Rakić, E. H. Linfield, and A. G. Davies, “Terahertz imaging through self-mixing in a quantum cascade laser,” Opt. Lett. |

10. | Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. Linfield, A. G. Davies, S. J. Wilson, and A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. |

11. | M. C. Phillips and S. Taubman, “Intracavity sensing via compliance voltage in an external cavity quantum cascade laser,” Opt. Lett. |

12. | F. P. Mezzapesa, V. Spagnolo, A. Antonio, and G. Scamarcio, “Detection of ultrafast laser ablation using quantum cascade laser-based sensing,” Appl. Phys. Lett. |

13. | R. Paiella, R. Martini, F. Capasso, C. Gmachl, and H. Y. Hwang, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. |

14. | D. M. Kane and K. A. Shore, |

15. | J. Helms and K. Petermann, “A simple analytic expression for the stable operation range of laser diodes with optical feedback,” IEEE J. Quantum Electron. |

16. | D. Weidmann, K. Smith, and B. Ellison, “Experimental investigation of high-frequency noise and optical feedback effects using a 9.7 μm continuous-wave distributed-feedback quantum-cascade laser,” Appl. Opt. |

17. | F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, and G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express |

18. | R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. |

19. | T. Gensty, W. Elsäßer, and C. Mann, “Intensity noise properties of quantum cascade lasers,” Opt. Express |

20. | M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum cascade lasers: hidden reason for the narrow linewidth and line broadening by thermal photons,” IEEE J. Quantum Electron. |

21. | J. Staden, T. Gensty, W. Elsäßer, G. Giuliani, and C. Mann, “Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett. |

22. | R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. |

23. | A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express |

24. | F. Mezzapesa, Internal CNR-IFN report (2013). |

25. | A. M. Levine, G. H. M. van Tartwijk, D. Lenstra, and T. Erneux, “Diode lasers with optical feedback: Stability of the maximum gain mode,” Phys. Rev. A |

26. | T. Erneux, V. Kovanis, and A. Gavrielides, “Nonlinear dynamics of an injected quantum cascade laser,” Phys. Rev. E |

27. | W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, |

28. | L. Columbo, M. Brambilla, M. Dabbicco, and G. Scamarcio, “Self-mixing in multi-transverse mode semiconductor lasers: model and potential application to multi-parametric sensing,” Opt. Express |

**OCIS Codes**

(140.4050) Lasers and laser optics : Mode-locked lasers

(190.3100) Nonlinear optics : Instabilities and chaos

(140.5965) Lasers and laser optics : Semiconductor lasers, quantum cascade

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: January 22, 2014

Manuscript Accepted: March 17, 2014

Published: April 21, 2014

**Virtual Issues**

Physics and Applications of Laser Dynamics (2014) *Optics Express*

**Citation**

L. L. Columbo and M. Brambilla, "Multimode regimes in quantum cascade lasers with optical feedback," Opt. Express **22**, 10105-10118 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-10105

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### References

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