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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 25 — Dec. 6, 2010
  • pp: 25922–25927
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Monte Carlo study of intrinsic linewidths in terahertz quantum cascade lasers

Christian Jirauschek  »View Author Affiliations


Optics Express, Vol. 18, Issue 25, pp. 25922-25927 (2010)
http://dx.doi.org/10.1364/OE.18.025922


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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 Optical Society of America

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

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

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]

] and potentially narrow linewidths below 1kHz [3

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]

], 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

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]

]. 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 [5

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]

8

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]

]. 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

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

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]

]. Theoretically, intrinsic linewidths in the sub-kHz regime have been predicted for THz QCLs [3

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]

]. 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.

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

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

], coupled to the EMC simulation tool. All relevant scattering mechanisms are included [13

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

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]

]. Self-consistent results for the carrier transport and the laser field are obtained by coupling the EMC simulation to an optical evolution equation [11

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

]. 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,
(1)
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

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

], with
g±(ω)=πZ0ωVn0ħn,j±Einj>0(1fjkn)|dinj|2inj(ω).
(2)
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
(3)
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

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]

, 15

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

]. 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.
(4)
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

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

] as
rijsp=πZ0n0ħ2|dij|2(1fjk)mIN=1(ωm)ij(ωm)
(5)
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

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]

, 15

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

]
δf=(4π)1K(1+α2)tN|spN1=(4π)1K(1+α2)γnspN1,
(6)
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

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]

], 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

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

]
δf=(4π)1(1+α2)γ(nsp+nth)N1.
(7)
It should be mentioned that in [3

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]

], a somewhat different result is obtained,
δf=(4π)1(1+α2)γ[nsp+(1+nsp)nth]N1.
(8)

3.1. Simulation results for 3THz QCL

In the following, simulation results are presented for a 3THz resonant phonon design [17

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]

]. 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

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]

]. 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

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

]. KP has been determined using a mode solver based on the effective-index method and the Drude model parameters given in [18

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]

], yielding KP = 1 in very good approximation. KH can be obtained from the facet reflectivities R1,2 as [15

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

]
KH=[(R1+R2)(1R1R2)R1R2ln(R1R2)]2,
(9)
giving KH = 1.0022 for the investigated design (R1 = R2 = 0.85 [17

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]

]). Thus we can set K ≈ 1.

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

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

, 19

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

]
α=N{χ(ωL)}N{χ(ωL)}=N{gc(ωL)}N{gc(ωL)}.
(10)
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

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]

].

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.

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

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]

], 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

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]

]. This value is consistent with the rate equation approach given in [3

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]

]. 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

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]

]. 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

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]

]. 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

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

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]

]. 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

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

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]

]. 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

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]

]. Assuming a coated rear facet, the extracted values for the surface plasmon (SP) waveguide [1

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]

] are R = 0.32, am = 2.8cm−1, aw = 3.7cm−1, Γ = 0.27, and for the metal-metal (MM) waveguide [2

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]

] 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

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]

], 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]

OCIS Codes
(140.3070) Lasers and laser optics : Infrared and far-infrared lasers
(140.3430) Lasers and laser optics : Laser theory
(300.3700) Spectroscopy : Linewidth
(140.5965) Lasers and laser optics : Semiconductor lasers, quantum cascade

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: October 12, 2010
Revised Manuscript: November 22, 2010
Manuscript Accepted: November 23, 2010
Published: November 26, 2010

Citation
Christian Jirauschek, "Monte Carlo study of intrinsic linewidths in terahertz quantum cascade lasers," Opt. Express 18, 25922-25927 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-25922


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References

  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, 083904 (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. Hubers, 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. Kasalynas, 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, 031115 (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, 011103 (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, 123102 (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, 013104 (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, 053106 (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, 071106 (2008). [CrossRef]

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