Abstract

Based on a coupled simulation of carrier transport and optical cavity field, the intrinsic linewidth in resonant phonon terahertz quantum cascade lasers is self-consistently analyzed. For high power structures, values on the order of Hz are obtained. Thermal photons are found to play a considerable role at elevated temperatures. A linewidth enhancement factor of 0.5 is calculated for the investigated designs.

© 2010 OSA

1. Introduction

The terahertz (THz) quantum cascade laser (QCL) is a versatile source of coherent THz radiation, offering the advantages of a solid state device, like robustness and compactness. Featuring high achievable output powers of several 100mW [1, 2] and potentially narrow linewidths below 1kHz [3], the THz QCL is a promising tool for various applications, such as telecommunications or high resolution THz spectroscopy. In this context, the intrinsic linewidth due to quantum noise plays a fundamental role, ultimately limiting the achievable spectral resolution and coherence length. Recently, the intrinsic linewidth has been measured for a 4.33 μm QCL, yielding ∼ 500Hz [4]. To date, such comprehensive data are still lacking for THz structures. In fact, the measured linewidths are dominated by extrinsic noise and exceed the Schawlow-Townes limit by far [58]. These noise sources can be suppressed by locking the THz QCL to an external reference, and frequency stabilization to the Hz level has recently been demonstrated [9,10]. Theoretically, intrinsic linewidths in the sub-kHz regime have been predicted for THz QCLs [3]. However, these results are based on a rate equation approach containing intrinsic laser parameters like the upper laser level lifetime and the linewidth enhancement factor, which had to be estimated from available experimental data.

Recently we have developed a method to perform self-consistent coupled simulations of the carrier transport and the optical dynamics in QCLs [11]. This approach is based on a well established Monte Carlo carrier transport simulation supplemented by evolution equations for the optical cavity field, and only relies on well known material parameters and the device specifications. Here we extend this approach to also account for spontaneous emission and black-body radiation, allowing us to self-consistently investigate optical fluctuations due to these effects. We study the intrinsic linewidth and related properties like the linewidth enhancement factor for various resonant phonon THz QCL designs. For high power structures, especially low intrinsic linewidths in the Hz range are obtained.

2. Method

The carrier transport in the QCL structure is modeled using a semiclassical ensemble Monte Carlo (EMC) approach. The subband wave functions and energies are computed with a Schrödinger-Poisson solver [12], coupled to the EMC simulation tool. All relevant scattering mechanisms are included [13, 14]. Self-consistent results for the carrier transport and the laser field are obtained by coupling the EMC simulation to an optical evolution equation [11]. For studying optical fluctuations and the intrinsic linewidth, we have to consider thermal photons and spontaneous photon emission events in addition to the stimulated transitions. The evolution of the photon number N in the laser mode is then given by

(n0/c)tN=(Γga)N+Γgnsp+anth,
where a = aw + am describes the total cavity loss (comprising waveguide loss aw and outcoupling at the end mirrors am), and Γ is the overlap factor. Furthermore, n0, g and nsp are the refractive index, net power gain coefficient and spontaneous emission factor of the gain medium at the (angular) lasing frequency ω = 2πf = ωL, and c is the vacuum speed of light. The term (Γga)N describes amplification in the gain medium and cavity damping, Γgnsp contains spontaneous emission processes in the gain medium, and anth describes the injection of thermal photons from the cavity into the lasing mode. Here, nth = [exp (ħω/(kBT)) – 1]−1 is the photon occupation number in thermal equilibrium at temperature T, with kB and ħ denoting the Boltzmann and reduced Planck constant, respectively.

The optical evolution equation Eq. (1) is coupled to the carrier transport via the quantities g and nsp, which are self-consistently computed in the EMC simulation. There, the inter- and intrasubband scattering events are stochastically evaluated for a large ensemble of discrete particles. Each carrier n is at a given time described by its quantum state |in, kn〉, i.e., subband in and in-plane wave vector kn. The net gain g = g+g is then evaluated by summing over all available final states |j, kn〉 and performing an ensemble average over all carriers n [11], with

g±(ω)=πZ0ωVn0ħn,j±Einj>0(1fjkn)|dinj|2inj(ω).
Here, g+ and g are the material gain and loss in the gain medium due to stimulated emission and absorption, respectively. Z0 denotes the impedance of free space, and V is the simulated volume. Moreover, Einj = EinEj = ħωinj corresponds to the energy difference between the initial and final state, dinj denotes the optical transition matrix element,
inj(ω)={1π1ω|ωinj|+iγinj}=1πγinjγinj2+(ω|ωinj|)2
is the Lorentzian lineshape function with linewidth γinj, and fjkn is the occupation probability of state |j, kn〉. The spontaneous emission rate into an optical mode (c/n0) Γgnsp equals the stimulated emission rate per one photon in this mode (c/n0) Γg+ [3, 15]. Thus we obtain nsp = g+ (ωL)/g (ωL), allowing us to extract nsp directly from the EMC simulation via Eq. (2).

The back-action of the optical cavity field on the carrier transport is considered by adding scattering rates for photon emission and absorption to EMC. The optical intensity in a gain medium with volume Vg is obtained from the photon number N in Eq. (1) by

IN(ω)=(ΓN/Vg)ħωc/n0.
As mentioned above, the spontaneous emission rate equals the stimulated emission rate for a single photon, corresponding to N = 1 in Eq. (4). Thus, the spontaneous transition rate for an electron in a state |i, k〉 to a state |j, k〉 follows from the stimulated electron-photon scattering rate given in [11] as
rijsp=πZ0n0ħ2|dij|2(1fjk)mIN=1(ωm)ij(ωm)
for Eij > 0, and rijsp=0 otherwise. To account for the total spontaneous emission, we sum in Eq. (5) over all cavity modes, characterized by their frequencies ωm. However, for the investigated QCLs, rijsp is far too low to contribute significantly to the carrier transport.

3. Intrinsic linewidth analysis

The intrinsic linewidth is given by the modified Schawlow-Townes formula [3, 15]

δf=(4π)1K(1+α2)tN|spN1=(4π)1K(1+α2)γnspN1,
valid for single-mode lasing. The second expression is obtained from the first one by inserting the spontaneous emission rate into the lasing mode ∂tN|sp = Γgnspc/n0 (see Eq. (1)) and substituting the gain-loss balance relation γ ≈ Γgc/n0, where γ = ac/n0 is the photon decay rate. The excess spontaneous emission factor K is a correction factor for non-orthogonal cavity modes, and the linewidth enhancement factor α quantifies the strength of amplitude-phase coupling in the gain medium. As pointed out in [3], blackbody radiation, i.e., thermal photons, play a role in the THz regime, where the thermal photon number nth reaches unity around or even below room temperature. The inclusion of thermal photons leads to a further modified linewidth formula, which is for K = 1 given by [16]
δf=(4π)1(1+α2)γ(nsp+nth)N1.
It should be mentioned that in [3], a somewhat different result is obtained,
δf=(4π)1(1+α2)γ[nsp+(1+nsp)nth]N1.

3.1. Simulation results for 3THz QCL

In the following, simulation results are presented for a 3THz resonant phonon design [17]. The gain medium cross section and cavity length are A = 10μm × 23μm and L = 1.22mm, respectively; furthermore, Γ = 0.93, aw = 18.7cm−1, and am = 1.3cm−1 [17]. The bias is set to 10.9kV/cm, where the simulation yields maximum output power.

The excess spontaneous emission factor can be written as K = KPKH, where KP and KH account for the non-orthogonality of transverse and longitudinal modes due to gain guiding and mirror outcoupling, respectively [15]. KP has been determined using a mode solver based on the effective-index method and the Drude model parameters given in [18], yielding KP = 1 in very good approximation. KH can be obtained from the facet reflectivities R1,2 as [15]

KH=[(R1+R2)(1R1R2)R1R2ln(R1R2)]2,
giving KH = 1.0022 for the investigated design (R1 = R2 = 0.85 [17]). Thus we can set K ≈ 1.

Next, the linewidth enhancement factor α is computed at ω = ωL using [16, 19]

α=N{χ(ωL)}N{χ(ωL)}=N{gc(ωL)}N{gc(ωL)}.
Here, χ(ω) = −in0cgc (ω)/ω is the contribution of the intersubband transitions to the gain medium susceptibility χ+n021. The complex gain gc is evaluated based on Eq. (2), replacing the lineshape function ℒ by the full complex expression in Eq. (3). Simulations are performed at slightly varied cavity losses, yielding gc for different photon numbers N. Various lattice temperatures TL between 10 and 170K are considered. In Fig. 1(a), χ is shown for different values of N at TL = 100K. Eq. (10) is evaluated at the position of the peak gain, i.e., the frequency ωL where ℑ{χ} reaches its minimum. For all simulated temperatures, we obtain α ≈ 0.5, which agrees with the only experimental value available for THz QCLs [20].

 

Fig. 1 (a) Susceptibility χ vs frequency f = ω/(2π) for internal powers of 145, 160 and 175mW, corresponding to N = 1.01 × 109, 1.11 × 109, and 1.21 × 109, respectively. (b) Intrinsic linewidth as a function of temperature without and with thermal photons included.

Download Full Size | PPT Slide | PDF

In the following, the temperature dependence of the intrinsic linewidth δf is investigated for the 3THz design, using K = 1 and α = 0.5. In Fig. 1(b), δf is shown, as obtained without considering thermal photons (Eq. (6)) and with thermal photons included (Eqs. (7), (8)). At elevated temperatures, the laser operation is impaired, and the photon number N decreases strongly from 1.52 × 109 (10K) to 3.08 × 108 (170K), while nsp only changes slightly from 3.94 to 3.55, and γ and α are constant in good approximation. According to Eq. (6), this results in an increase of δf from 40.5Hz to 180.6Hz (solid curve). Thermal photons cause additional broadening of 18% at 170K according to Eq. (7) (dashed curve), and even 80% according to Eq. (8) (dotted curve). At this point, it would be desirable to clarify which of the two models Eqs. (7), (8) is valid for THz QCLs; however, this goes beyond the scope of the present paper.

As discussed in [3], no comprehensive experimental results are available for the intrinsic linewidth of THz QCLs. However, recently measurements have been performed for a 4.33 μm design, yielding δf ≈ 500Hz [4]. This value is consistent with the rate equation approach given in [3]. Thus we compare our results to that model, yielding δf=(4π)1γβτtτr1(I0/Ith1)1 for THz QCLs at moderate electron temperatures up to Te ≈ 100K [3]. Here, K = 1 and α = 0 is assumed, and thermal photons are neglected. Estimating an above threshold current I0/Ith = 1.2, a photon decay rate γ = 2 × 1011 s−1, a coupling efficiency β = 10−2 and upper laser level lifetimes τr = 10μs and τt = 10ps due to spontaneous emission and non-lasing transitions, respectively, δf = 800Hz is obtained for a 3THz QCL [3]. We compare this value to our EMC simulation at TL = 10K (corresponding to Te ≈ 100K), yielding a much smaller value δf = 32.4Hz for our investigated 3THz design. However, using parameter values directly extracted from this simulation (I0/Ith = 2.5, τt = 1.83ps, βτr1=1950s1, γ = ac/n0 = 1.58 ×1011 s−1), above formula yields δf = 30Hz in excellent agreement with our numerical result.

3.2. Simulation results for high power QCLs

As evident from Eq. (6), the intrinsic linewidth δf can be reduced by increasing the photon number N, i.e., the generated optical power. This can be achieved by designing the waveguide and active region correspondingly [1, 2]. For example, N can be directly enhanced by increasing the waveguide width w, without affecting γ and nsp much. Besides, QCLs work best at low temperatures, also eliminating the influence of thermal photons (nth ≈ 0). In the following, δf is investigated for two high power QCLs at TL = 10K, providing a lower estimate for the obtainable intrinsic linewidth of current THz QCL designs. Both structures are based on the same active region design, but use different types of waveguides with comparable widths w ≈ 100μm, lengths L ≈ 2mm, and active region thicknesses d ≈ 10μm [1, 2]. The cavity parameters have been determined at 4.5THz using a mode solver based on the effective-index method, along with the Drude model parameters and facet reflectivities R given in [18]. Assuming a coated rear facet, the extracted values for the surface plasmon (SP) waveguide [1] are R = 0.32, am = 2.8cm−1, aw = 3.7cm−1, Γ = 0.27, and for the metal-metal (MM) waveguide [2] R = 0.7, am = 0.9cm−1, aw = 11.5cm−1, Γ = 1.0. Furthermore, we obtain KP = 1.0 for both structures, and Eq. (9) yields KH = 1.11 and KH = 1.01 for the SP- and MM-based QCL, respectively. We simulate δf and the linewidth enhancement factor α as described in Section 3.1. We obtain again α ≈ 0.5 for both devices, furthermore δf = 3.0Hz for the MM structure and even δf = 0.64Hz for the SP structure, as compared to δf = 40.5Hz for the 3THz MM QCL (see Fig. 1(b)). The ∼ 7 times higher gain medium volume of the high power QCLs results in high photon numbers N = 3.13 × 1010 and 2.07 × 1010 for the SP- and MM-based design, as compared to N = 1.52 × 109 for the 3THz QCL. Furthermore, the low photon decay rate γ = 5.1×1010 s−1 of the SP-based device contributes directly to its extremely narrow linewidth (see Eqs. (6)(8)), and also indirectly through the increased photon number. We note that there are uncertainties associated with determining aw [17], affecting the exact value of δf. However, our results clearly indicate that intrinsic linewidths on the order of Hz or even below can be obtained for high power THz QCLs.

4. Conclusion

Based on coupled carrier transport and optical simulations, a method is developed to self-consistently investigate optical fluctuations due to spontaneous emission and thermal photons in QCLs. The intrinsic linewidth in resonant phonon THz designs is analyzed, yielding values on the order of Hz or even below for high power structures. At elevated temperatures, thermal photons are found to contribute significantly to linewidth broadening. For the investigated designs, a linewidth enhancement factor of around α ≈ 0.5 is extracted.

Acknowledgments

We acknowledge support from P. Lugli at the TUM. This work was funded by the Emmy Noether program of the German Research Foundation under Grant No. DFG, JI115/1-1.

References and links

1. B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “High-power terahertz quantum-cascade lasers,” Electron. Lett. 42, 89–90 (2006). [CrossRef]  

2. A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “High-power and high-temperature THz quantum-cascade lasers based on lens-coupled metal-metal waveguides,” Opt. Lett. 32, 2840–2842 (2007). [CrossRef]  

3. 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]  

4. S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. de Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: Beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104, 083 904 (2010). [CrossRef]  

5. A. Barkan, F. K. Tittel, D. M. Mittleman, R. Dengler, P. H. Siegel, G. Scalari, L. Ajili, J. Faist, H. E. Beere, E. H. Linfield, A. G. Davies, and D. A. Ritchie, “Linewidth and tuning characteristics of terahertz quantum cascade lasers,” Opt. Lett. 29, 575–577 (2004). [CrossRef]   [PubMed]  

6. H. Hübers, S. G. Pavlov, A. D. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and E. H. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005). [CrossRef]   [PubMed]  

7. A. Baryshev, J. N. Hovenier, A. J. L. Adam, I. Kašalynas, J. R. Gao, T. O. Klaassen, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Phase locking and spectral linewidth of a two-mode terahertz quantum cascade laser,” Appl. Phys. Lett. 89, 031 115 (2006). [CrossRef]  

8. A. A. Danylov, T. M. Goyette, J. Waldman, M. J. Coulombe, A. J. Gatesman, R. H. Giles, W. D. Goodhue, X. Qian, and W. E. Nixon, “Frequency stabilization of a single mode terahertz quantum cascade laser to the kilohertz level,” Opt. Express 17, 7525–7532 (2009). [CrossRef]   [PubMed]  

9. D. Rabanus, U. U. Graf, M. Philipp, O. Ricken, J. Stutzki, B. Vowinkel, M. C. Wiedner, C. Walther, M. Fischer, and J. Faist, “Phase locking of a 15 Terahertz quantum cascade laser and use as a local oscillator in a heterodyne HEB receiver,” Opt. Express 17, 1159–1168 (2009). [CrossRef]   [PubMed]  

10. 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, 636–640 (2010). [CrossRef]  

11. C. Jirauschek, “Monte Carlo study of carrier-light coupling in terahertz quantum cascade lasers,” Appl. Phys. Lett. 96, 011 103 (2010). [CrossRef]  

12. C. Jirauschek, “Accuracy of transfer matrix approaches for solving the effective mass Schrödinger equation,” IEEE J. Quantum Electron. 45, 1059–1067 (2009). [CrossRef]  

13. C. Jirauschek and P. Lugli, “Monte-Carlo-based spectral gain analysis for terahertz quantum cascade lasers,” J. Appl. Phys. 105, 123 102 (2009). [CrossRef]  

14. C. Jirauschek, A. Matyas, and P. Lugli, “Modeling bound-to-continuum terahertz quantum cascade lasers: The role of Coulomb interactions,” J. Appl. Phys. 107, 013 104 (2010). [CrossRef]  

15. G. Grau and W. Freude, Optische Nachrichtentechnik - Eine Einführung (Springer, 1991).

16. H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys. 204, 262–275 (1967). [CrossRef]  

17. B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Operation of terahertz quantum-cascade lasers at 164 K in pulsed mode and at 117 K in continuous-wave mode,” Opt. Express 13, 3331–3339 (2005). [CrossRef]   [PubMed]  

18. S. Kohen, B. S. Williams, and Q. Hu, “Electromagnetic modeling of terahertz quantum cascade laser waveguides and resonators,” J. Appl. Phys. 97, 053 106 (2005). [CrossRef]  

19. C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982). [CrossRef]  

20. R. P. Green, J. 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, 071 106 (2008). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “High-power terahertz quantum-cascade lasers,” Electron. Lett. 42, 89–90 (2006).
    [CrossRef]
  2. A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “High-power and high-temperature THz quantum-cascade lasers based on lens-coupled metal-metal waveguides,” Opt. Lett. 32, 2840–2842 (2007).
    [CrossRef]
  3. 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]
  4. S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. de Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: Beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104, 083 904 (2010).
    [CrossRef]
  5. A. Barkan, F. K. Tittel, D. M. Mittleman, R. Dengler, P. H. Siegel, G. Scalari, L. Ajili, J. Faist, H. E. Beere, E. H. Linfield, A. G. Davies, and D. A. Ritchie, “Linewidth and tuning characteristics of terahertz quantum cascade lasers,” Opt. Lett. 29, 575–577 (2004).
    [CrossRef] [PubMed]
  6. H. Hübers, S. G. Pavlov, A. D. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and E. H. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).
    [CrossRef] [PubMed]
  7. A. Baryshev, J. N. Hovenier, A. J. L. Adam, I. Kašalynas, J. R. Gao, T. O. Klaassen, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Phase locking and spectral linewidth of a two-mode terahertz quantum cascade laser,” Appl. Phys. Lett. 89, 031 115 (2006).
    [CrossRef]
  8. A. A. Danylov, T. M. Goyette, J. Waldman, M. J. Coulombe, A. J. Gatesman, R. H. Giles, W. D. Goodhue, X. Qian, and W. E. Nixon, “Frequency stabilization of a single mode terahertz quantum cascade laser to the kilohertz level,” Opt. Express 17, 7525–7532 (2009).
    [CrossRef] [PubMed]
  9. D. Rabanus, U. U. Graf, M. Philipp, O. Ricken, J. Stutzki, B. Vowinkel, M. C. Wiedner, C. Walther, M. Fischer, and J. Faist, “Phase locking of a 15 Terahertz quantum cascade laser and use as a local oscillator in a heterodyne HEB receiver,” Opt. Express 17, 1159–1168 (2009).
    [CrossRef] [PubMed]
  10. 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, 636–640 (2010).
    [CrossRef]
  11. C. Jirauschek, “Monte Carlo study of carrier-light coupling in terahertz quantum cascade lasers,” Appl. Phys. Lett. 96, 011 103 (2010).
    [CrossRef]
  12. C. Jirauschek, “Accuracy of transfer matrix approaches for solving the effective mass Schrödinger equation,” IEEE J. Quantum Electron. 45, 1059–1067 (2009).
    [CrossRef]
  13. C. Jirauschek and P. Lugli, “Monte-Carlo-based spectral gain analysis for terahertz quantum cascade lasers,” J. Appl. Phys. 105, 123 102 (2009).
    [CrossRef]
  14. C. Jirauschek, A. Matyas, and P. Lugli, “Modeling bound-to-continuum terahertz quantum cascade lasers: The role of Coulomb interactions,” J. Appl. Phys. 107, 013 104 (2010).
    [CrossRef]
  15. G. Grau and W. Freude, Optische Nachrichtentechnik - Eine Einführung (Springer, 1991).
  16. H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys. 204, 262–275 (1967).
    [CrossRef]
  17. B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Operation of terahertz quantum-cascade lasers at 164 K in pulsed mode and at 117 K in continuous-wave mode,” Opt. Express 13, 3331–3339 (2005).
    [CrossRef] [PubMed]
  18. S. Kohen, B. S. Williams, and Q. Hu, “Electromagnetic modeling of terahertz quantum cascade laser waveguides and resonators,” J. Appl. Phys. 97, 053 106 (2005).
    [CrossRef]
  19. C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
    [CrossRef]
  20. R. P. Green, J. 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, 071 106 (2008).
    [CrossRef]

2010 (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, 636–640 (2010).
[CrossRef]

C. Jirauschek, “Monte Carlo study of carrier-light coupling in terahertz quantum cascade lasers,” Appl. Phys. Lett. 96, 011 103 (2010).
[CrossRef]

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. de Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: Beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104, 083 904 (2010).
[CrossRef]

C. Jirauschek, A. Matyas, and P. Lugli, “Modeling bound-to-continuum terahertz quantum cascade lasers: The role of Coulomb interactions,” J. Appl. Phys. 107, 013 104 (2010).
[CrossRef]

2009 (4)

2008 (2)

R. P. Green, J. 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, 071 106 (2008).
[CrossRef]

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]

2007 (1)

2006 (2)

B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “High-power terahertz quantum-cascade lasers,” Electron. Lett. 42, 89–90 (2006).
[CrossRef]

A. Baryshev, J. N. Hovenier, A. J. L. Adam, I. Kašalynas, J. R. Gao, T. O. Klaassen, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Phase locking and spectral linewidth of a two-mode terahertz quantum cascade laser,” Appl. Phys. Lett. 89, 031 115 (2006).
[CrossRef]

2005 (3)

2004 (1)

1982 (1)

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

1967 (1)

H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys. 204, 262–275 (1967).
[CrossRef]

Adam, A. J. L.

A. Baryshev, J. N. Hovenier, A. J. L. Adam, I. Kašalynas, J. R. Gao, T. O. Klaassen, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Phase locking and spectral linewidth of a two-mode terahertz quantum cascade laser,” Appl. Phys. Lett. 89, 031 115 (2006).
[CrossRef]

Ajili, L.

Akikusa, N.

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]

Barbieri, S.

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, 636–640 (2010).
[CrossRef]

Barkan, A.

Bartalini, S.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. de Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: Beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104, 083 904 (2010).
[CrossRef]

Baryshev, A.

A. Baryshev, J. N. Hovenier, A. J. L. Adam, I. Kašalynas, J. R. Gao, T. O. Klaassen, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Phase locking and spectral linewidth of a two-mode terahertz quantum cascade laser,” Appl. Phys. Lett. 89, 031 115 (2006).
[CrossRef]

Beere, H.

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, 636–640 (2010).
[CrossRef]

Beere, H. E.

Beltram, F.

R. P. Green, J. 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, 071 106 (2008).
[CrossRef]

Borri, S.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. de Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: Beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104, 083 904 (2010).
[CrossRef]

Cancio, P.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. de Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: Beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104, 083 904 (2010).
[CrossRef]

Castrillo, A.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. de Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: Beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104, 083 904 (2010).
[CrossRef]

Colombelli, R.

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, 636–640 (2010).
[CrossRef]

Coulombe, M. J.

Danylov, A. A.

Davies, A. G.

de Natale, P.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. de Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: Beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104, 083 904 (2010).
[CrossRef]

Dengler, R.

Ding, L.

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, 636–640 (2010).
[CrossRef]

Edamura, T.

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]

Faist, J.

Fischer, M.

Freude, W.

G. Grau and W. Freude, Optische Nachrichtentechnik - Eine Einführung (Springer, 1991).

Fujita, K.

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]

Galli, I.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. de Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: Beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104, 083 904 (2010).
[CrossRef]

Gao, J. R.

A. Baryshev, J. N. Hovenier, A. J. L. Adam, I. Kašalynas, J. R. Gao, T. O. Klaassen, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Phase locking and spectral linewidth of a two-mode terahertz quantum cascade laser,” Appl. Phys. Lett. 89, 031 115 (2006).
[CrossRef]

Gatesman, A. J.

Gellie, P.

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, 636–640 (2010).
[CrossRef]

Gianfrani, L.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. de Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: Beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104, 083 904 (2010).
[CrossRef]

Giles, R. H.

Giuliani, G.

R. P. Green, J. 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, 071 106 (2008).
[CrossRef]

Giusfredi, G.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. de Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: Beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104, 083 904 (2010).
[CrossRef]

Goodhue, W. D.

Goyette, T. M.

Graf, U. U.

Grau, G.

G. Grau and W. Freude, Optische Nachrichtentechnik - Eine Einführung (Springer, 1991).

Green, R. P.

R. P. Green, J. 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, 071 106 (2008).
[CrossRef]

Haken, H.

H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys. 204, 262–275 (1967).
[CrossRef]

Haug, H.

H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys. 204, 262–275 (1967).
[CrossRef]

Henry, C.

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

Hovenier, J. N.

A. Baryshev, J. N. Hovenier, A. J. L. Adam, I. Kašalynas, J. R. Gao, T. O. Klaassen, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Phase locking and spectral linewidth of a two-mode terahertz quantum cascade laser,” Appl. Phys. Lett. 89, 031 115 (2006).
[CrossRef]

Hu, Q.

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “High-power and high-temperature THz quantum-cascade lasers based on lens-coupled metal-metal waveguides,” Opt. Lett. 32, 2840–2842 (2007).
[CrossRef]

A. Baryshev, J. N. Hovenier, A. J. L. Adam, I. Kašalynas, J. R. Gao, T. O. Klaassen, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Phase locking and spectral linewidth of a two-mode terahertz quantum cascade laser,” Appl. Phys. Lett. 89, 031 115 (2006).
[CrossRef]

B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “High-power terahertz quantum-cascade lasers,” Electron. Lett. 42, 89–90 (2006).
[CrossRef]

S. Kohen, B. S. Williams, and Q. Hu, “Electromagnetic modeling of terahertz quantum cascade laser waveguides and resonators,” J. Appl. Phys. 97, 053 106 (2005).
[CrossRef]

B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Operation of terahertz quantum-cascade lasers at 164 K in pulsed mode and at 117 K in continuous-wave mode,” Opt. Express 13, 3331–3339 (2005).
[CrossRef] [PubMed]

Hübers, H.

Jirauschek, C.

C. Jirauschek, A. Matyas, and P. Lugli, “Modeling bound-to-continuum terahertz quantum cascade lasers: The role of Coulomb interactions,” J. Appl. Phys. 107, 013 104 (2010).
[CrossRef]

C. Jirauschek, “Monte Carlo study of carrier-light coupling in terahertz quantum cascade lasers,” Appl. Phys. Lett. 96, 011 103 (2010).
[CrossRef]

C. Jirauschek and P. Lugli, “Monte-Carlo-based spectral gain analysis for terahertz quantum cascade lasers,” J. Appl. Phys. 105, 123 102 (2009).
[CrossRef]

C. Jirauschek, “Accuracy of transfer matrix approaches for solving the effective mass Schrödinger equation,” IEEE J. Quantum Electron. 45, 1059–1067 (2009).
[CrossRef]

Kan, H.

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]

Kašalynas, I.

A. Baryshev, J. N. Hovenier, A. J. L. Adam, I. Kašalynas, J. R. Gao, T. O. Klaassen, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Phase locking and spectral linewidth of a two-mode terahertz quantum cascade laser,” Appl. Phys. Lett. 89, 031 115 (2006).
[CrossRef]

Klaassen, T. O.

A. Baryshev, J. N. Hovenier, A. J. L. Adam, I. Kašalynas, J. R. Gao, T. O. Klaassen, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Phase locking and spectral linewidth of a two-mode terahertz quantum cascade laser,” Appl. Phys. Lett. 89, 031 115 (2006).
[CrossRef]

Kohen, S.

S. Kohen, B. S. Williams, and Q. Hu, “Electromagnetic modeling of terahertz quantum cascade laser waveguides and resonators,” J. Appl. Phys. 97, 053 106 (2005).
[CrossRef]

Köhler, R.

Kumar, S.

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “High-power and high-temperature THz quantum-cascade lasers based on lens-coupled metal-metal waveguides,” Opt. Lett. 32, 2840–2842 (2007).
[CrossRef]

A. Baryshev, J. N. Hovenier, A. J. L. Adam, I. Kašalynas, J. R. Gao, T. O. Klaassen, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Phase locking and spectral linewidth of a two-mode terahertz quantum cascade laser,” Appl. Phys. Lett. 89, 031 115 (2006).
[CrossRef]

B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “High-power terahertz quantum-cascade lasers,” Electron. Lett. 42, 89–90 (2006).
[CrossRef]

B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Operation of terahertz quantum-cascade lasers at 164 K in pulsed mode and at 117 K in continuous-wave mode,” Opt. Express 13, 3331–3339 (2005).
[CrossRef] [PubMed]

Lee, A. W. M.

Linfield, E. H.

Lugli, P.

C. Jirauschek, A. Matyas, and P. Lugli, “Modeling bound-to-continuum terahertz quantum cascade lasers: The role of Coulomb interactions,” J. Appl. Phys. 107, 013 104 (2010).
[CrossRef]

C. Jirauschek and P. Lugli, “Monte-Carlo-based spectral gain analysis for terahertz quantum cascade lasers,” J. Appl. Phys. 105, 123 102 (2009).
[CrossRef]

Mahler, L.

R. P. Green, J. 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, 071 106 (2008).
[CrossRef]

H. Hübers, S. G. Pavlov, A. D. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and E. H. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).
[CrossRef] [PubMed]

Maineult, W.

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, 636–640 (2010).
[CrossRef]

Matyas, A.

C. Jirauschek, A. Matyas, and P. Lugli, “Modeling bound-to-continuum terahertz quantum cascade lasers: The role of Coulomb interactions,” J. Appl. Phys. 107, 013 104 (2010).
[CrossRef]

Mazzotti, D.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. de Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: Beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104, 083 904 (2010).
[CrossRef]

Mittleman, D. M.

Nixon, W. E.

Pavlov, S. G.

Philipp, M.

Qian, X.

Qin, Q.

Rabanus, D.

Reno, J. L.

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “High-power and high-temperature THz quantum-cascade lasers based on lens-coupled metal-metal waveguides,” Opt. Lett. 32, 2840–2842 (2007).
[CrossRef]

A. Baryshev, J. N. Hovenier, A. J. L. Adam, I. Kašalynas, J. R. Gao, T. O. Klaassen, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Phase locking and spectral linewidth of a two-mode terahertz quantum cascade laser,” Appl. Phys. Lett. 89, 031 115 (2006).
[CrossRef]

B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “High-power terahertz quantum-cascade lasers,” Electron. Lett. 42, 89–90 (2006).
[CrossRef]

B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Operation of terahertz quantum-cascade lasers at 164 K in pulsed mode and at 117 K in continuous-wave mode,” Opt. Express 13, 3331–3339 (2005).
[CrossRef] [PubMed]

Ricken, O.

Ritchie, D.

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, 636–640 (2010).
[CrossRef]

Ritchie, D. A.

Santarelli, G.

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, 636–640 (2010).
[CrossRef]

Scalari, G.

Semenov, A. D.

Siegel, P. H.

Sirtori, C.

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, 636–640 (2010).
[CrossRef]

Stutzki, J.

Tittel, F. K.

Tredicucci, A.

R. P. Green, J. 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, 071 106 (2008).
[CrossRef]

H. Hübers, S. G. Pavlov, A. D. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and E. H. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).
[CrossRef] [PubMed]

Vowinkel, B.

Waldman, J.

Walther, C.

Wiedner, M. C.

Williams, B. S.

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “High-power and high-temperature THz quantum-cascade lasers based on lens-coupled metal-metal waveguides,” Opt. Lett. 32, 2840–2842 (2007).
[CrossRef]

A. Baryshev, J. N. Hovenier, A. J. L. Adam, I. Kašalynas, J. R. Gao, T. O. Klaassen, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Phase locking and spectral linewidth of a two-mode terahertz quantum cascade laser,” Appl. Phys. Lett. 89, 031 115 (2006).
[CrossRef]

B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “High-power terahertz quantum-cascade lasers,” Electron. Lett. 42, 89–90 (2006).
[CrossRef]

S. Kohen, B. S. Williams, and Q. Hu, “Electromagnetic modeling of terahertz quantum cascade laser waveguides and resonators,” J. Appl. Phys. 97, 053 106 (2005).
[CrossRef]

B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Operation of terahertz quantum-cascade lasers at 164 K in pulsed mode and at 117 K in continuous-wave mode,” Opt. Express 13, 3331–3339 (2005).
[CrossRef] [PubMed]

Xu, J.

R. P. Green, J. 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, 071 106 (2008).
[CrossRef]

Yamanishi, M.

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]

Appl. Phys. Lett. (3)

A. Baryshev, J. N. Hovenier, A. J. L. Adam, I. Kašalynas, J. R. Gao, T. O. Klaassen, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Phase locking and spectral linewidth of a two-mode terahertz quantum cascade laser,” Appl. Phys. Lett. 89, 031 115 (2006).
[CrossRef]

C. Jirauschek, “Monte Carlo study of carrier-light coupling in terahertz quantum cascade lasers,” Appl. Phys. Lett. 96, 011 103 (2010).
[CrossRef]

R. P. Green, J. 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, 071 106 (2008).
[CrossRef]

Electron. Lett. (1)

B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “High-power terahertz quantum-cascade lasers,” Electron. Lett. 42, 89–90 (2006).
[CrossRef]

IEEE J. Quantum Electron. (3)

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

C. Jirauschek, “Accuracy of transfer matrix approaches for solving the effective mass Schrödinger equation,” IEEE J. Quantum Electron. 45, 1059–1067 (2009).
[CrossRef]

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]

J. Appl. Phys. (3)

C. Jirauschek and P. Lugli, “Monte-Carlo-based spectral gain analysis for terahertz quantum cascade lasers,” J. Appl. Phys. 105, 123 102 (2009).
[CrossRef]

C. Jirauschek, A. Matyas, and P. Lugli, “Modeling bound-to-continuum terahertz quantum cascade lasers: The role of Coulomb interactions,” J. Appl. Phys. 107, 013 104 (2010).
[CrossRef]

S. Kohen, B. S. Williams, and Q. Hu, “Electromagnetic modeling of terahertz quantum cascade laser waveguides and resonators,” J. Appl. Phys. 97, 053 106 (2005).
[CrossRef]

Nat. Photonics (1)

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, 636–640 (2010).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. de Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: Beyond the Schawlow-Townes limit,” Phys. Rev. Lett. 104, 083 904 (2010).
[CrossRef]

Z. Phys. (1)

H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys. 204, 262–275 (1967).
[CrossRef]

Other (1)

G. Grau and W. Freude, Optische Nachrichtentechnik - Eine Einführung (Springer, 1991).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

(a) Susceptibility χ vs frequency f = ω/(2π) for internal powers of 145, 160 and 175mW, corresponding to N = 1.01 × 109, 1.11 × 109, and 1.21 × 109, respectively. (b) Intrinsic linewidth as a function of temperature without and with thermal photons included.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

( n 0 / c ) t N = ( Γ g a ) N + Γ g n sp + a n th ,
g ± ( ω ) = π Z 0 ω V n 0 ħ n , j ± E i n j > 0 ( 1 f j k n ) | d i n j | 2 i n j ( ω ) .
i n j ( ω ) = { 1 π 1 ω | ω i n j | + i γ i n j } = 1 π γ i n j γ i n j 2 + ( ω | ω i n j | ) 2
I N ( ω ) = ( Γ N / V g ) ħ ω c / n 0 .
r i j sp = π Z 0 n 0 ħ 2 | d i j | 2 ( 1 f j k ) m I N = 1 ( ω m ) i j ( ω m )
δ f = ( 4 π ) 1 K ( 1 + α 2 ) t N | sp N 1 = ( 4 π ) 1 K ( 1 + α 2 ) γ n sp N 1 ,
δ f = ( 4 π ) 1 ( 1 + α 2 ) γ ( n sp + n th ) N 1 .
δ f = ( 4 π ) 1 ( 1 + α 2 ) γ [ n sp + ( 1 + n sp ) n th ] N 1 .
K H = [ ( R 1 + R 2 ) ( 1 R 1 R 2 ) R 1 R 2 ln ( R 1 R 2 ) ] 2 ,
α = N { χ ( ω L ) } N { χ ( ω L ) } = N { g c ( ω L ) } N { g c ( ω L ) } .

Metrics