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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 5 — Mar. 11, 2013
  • pp: 6180–6185
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Monte Carlo study of terahertz difference frequency generation in quantum cascade lasers

Christian Jirauschek, Alpar Matyas, Paolo Lugli, and Markus-Christian Amann  »View Author Affiliations


Optics Express, Vol. 21, Issue 5, pp. 6180-6185 (2013)
http://dx.doi.org/10.1364/OE.21.006180


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Abstract

We present an extended ensemble Monte Carlo approach, allowing for the self-consistent modeling of terahertz difference frequency generation in quantum cascade lasers. Our simulations are validated against available experimental data for a current room temperature design. Tera-hertz output powers in the mW range are predicted for ideal light extraction.

© 2013 OSA

1. Introduction

Quantum cascade laser (QCL) sources based on difference frequency generation (DFG) constitute a very promising approach for the generation of coherent terahertz (THz) radiation [1

1. M. Belkin, F. Capasso, A. Belyanin, D. Sivco, A. Cho, D. Oakley, C. Vineis, and G. Turner, “Terahertz quantum-cascade-laser source based on intracavity difference-frequency generation,” Nat. Photonics 1, 288–292 (2007) [CrossRef] .

]. The DFG process is here implemented by dual-wavelength mid-infrared (MIR) QCLs pumping a giant optical nonlinearity, which is commonly directly integrated into the QCL gain medium [2

2. M. Belkin, F. Capasso, F. Xie, A. Belyanin, M. Fischer, A. Wittmann, and J. Faist, “Room temperature terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett. 92, 201101 (2008) [CrossRef] .

5

5. K. Vijayraghavan, R. W. Adams, A. Vizbaras, M. Jang, C. Grasse, G. Boehm, M. C. Amann, and M. A. Belkin, “Terahertz sources based on Čerenkov difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 100, 251104 (2012) [CrossRef] .

]. With such intracavity DFG sources, room temperature operation has been demonstrated [2

2. M. Belkin, F. Capasso, F. Xie, A. Belyanin, M. Fischer, A. Wittmann, and J. Faist, “Room temperature terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett. 92, 201101 (2008) [CrossRef] .

, 4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

, 5

5. K. Vijayraghavan, R. W. Adams, A. Vizbaras, M. Jang, C. Grasse, G. Boehm, M. C. Amann, and M. A. Belkin, “Terahertz sources based on Čerenkov difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 100, 251104 (2012) [CrossRef] .

], whereas in conventional THz QCLs operating temperatures are still restricted to below 200 K [6

6. S. Fathololoumi, E. Dupont, C. Chan, Z. Wasilewski, S. Laframboise, D. Ban, A. Mátyás, C. Jirauschek, Q. Hu, and H. Liu, “Terahertz quantum cascade lasers operating up to ∼200 K with optimized oscillator strength and improved injection tunneling,” Opt. Express 20, 3866–3876 (2012) [CrossRef] [PubMed] .

]. On the other hand, while output powers of several 100mW have been obtained with THz QCLs [7

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

], DFG structures still suffer from low THz powers in the μW range [2

2. M. Belkin, F. Capasso, F. Xie, A. Belyanin, M. Fischer, A. Wittmann, and J. Faist, “Room temperature terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett. 92, 201101 (2008) [CrossRef] .

, 4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

, 5

5. K. Vijayraghavan, R. W. Adams, A. Vizbaras, M. Jang, C. Grasse, G. Boehm, M. C. Amann, and M. A. Belkin, “Terahertz sources based on Čerenkov difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 100, 251104 (2012) [CrossRef] .

]. For a further improvement of these sources, a detailed understanding of the MIR generation, nonlinear frequency conversion and THz outcoupling is essential. In this context, self-consistent state of the art carrier transport simulations which do not rely on free fitting parameters or require experimental input are especially valuable.

Here we employ the ensemble Monte Carlo (EMC) method, which has proven very helpful for the design, analysis and optimization of MIR [8

8. R. C. Iotti and F. Rossi, “Carrier thermalization versus phonon-assisted relaxation in quantum-cascade lasers: A Monte Carlo approach,” Appl. Phys. Lett. 78, 2902–2904 (2001) [CrossRef] .

10

10. A. Mátyás, P. Lugli, and C. Jirauschek, “Photon-induced carrier transport in high efficiency midinfrared quantum cascade lasers,” J. Appl. Phys. 110, 013108 (2011) [CrossRef] .

] and THz [11

11. R. Köhler, R. C. Iotti, A. Tredicucci, and F. Rossi, “Design and simulation of terahertz quantum cascade lasers,” Appl. Phys. Lett. 79, 3920–3922 (2001) [CrossRef] .

16

16. A. Mátyás, M. Belkin, P. Lugli, and C. Jirauschek, “Temperature performance analysis of terahertz quantum cascade lasers: Vertical versus diagonal designs,” Appl. Phys. Lett. 96, 201110 (2010).

] QCLs. For the investigation of DFG-based THz sources, an inclusion of the optical dynamics is crucial, which is typically neglected in EMC and other advanced carrier transport simulation methods. Recently we have developed an extended EMC approach, accounting for the carrier transport and the optical cavity field on an equal footing [10

10. A. Mátyás, P. Lugli, and C. Jirauschek, “Photon-induced carrier transport in high efficiency midinfrared quantum cascade lasers,” J. Appl. Phys. 110, 013108 (2011) [CrossRef] .

, 17

17. C. Jirauschek, “Monte Carlo study of carrier-light coupling in terahertz quantum cascade lasers,” Appl. Phys. Lett. 96, 011103 (2010) [CrossRef] .

, 18

18. C. Jirauschek, “Monte Carlo study of intrinsic linewidths in terahertz quantum cascade lasers,” Opt. Express 18, 25922–25927 (2010) [CrossRef] [PubMed] .

]. Here we adapt our tool to the simulation of DFG-based THz sources, self-consistently including the nonlinear frequency conversion process. We employ our approach to analyze a recent DFG structure which achieved record THz output powers at room temperature [4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

]. The simulation yields good agreement with available experimental data. Specifically, our approach allows us to identify the main factors limiting the emitted THz power, and to explore the theoretical limit for optimum outcoupling. From our calculations, room temperature output powers in the mW range are found to be accomplishable.

2. Modeling of difference frequency generation

As mentioned above, in DFG structures two MIR QCLs at frequencies ω1 and ω2 are used to pump a medium with giant optical nonlinearity, generating THz radiation at a frequency ω = ω1ω2. The effective second order susceptibility in the nonlinear medium is given by [19

19. Y. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, 1984).

]
χ(2)=12ε0LP,m,ndmdmndnnE2D0f(KmnKmn)dε,Kmn=(1ωniγnω+1ωnm+iγnm+ω)(1ωmiγm+ω2+1ωmiγmω1).
(1)
Here, nE2D=m*/(π2) is the two-dimensional density of states. The constants m*, ε0 and ħ denote the effective mass, vacuum permittivity and reduced Planck constant, respectively. Furthermore, ωmn = (EmEn) / ħ and dmn are the resonance frequency and the dipole matrix element of the optical transition between the subbands i = m, n with corresponding eigenenergies Ei. The optical linewidth γmn (ε) and the occupation probability fi (ε) depend on the kinetic electron energy ε.

To evaluate Eq. (1), we can proceed in analogy to EMC-based optical gain calculations [20

20. C. Jirauschek and P. Lugli, “Monte-Carlo-based spectral gain analysis for terahertz quantum cascade lasers,” J. Appl. Phys. 105, 123102 (2009) [CrossRef] .

]: The occupation probability is directly extracted from the simulation, and the linewidth is evaluated based on lifetime broadening, yielding γmn (ε) = [γm (ε) +γn (ε)]/2 with the intersubband outscattering rates γi (ε) [20

20. C. Jirauschek and P. Lugli, “Monte-Carlo-based spectral gain analysis for terahertz quantum cascade lasers,” J. Appl. Phys. 105, 123102 (2009) [CrossRef] .

]. For periodic structures such as QCLs, it is sufficient to consider the states of a single central period, along with all available states m, n (including those in neighboring periods); LP is then the length of a QCL period. Nonparabolicity effects are included in our Schrödinger-Poisson solver [21

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

], and are also considered in the EMC simulation by assigning individual effective masses m* to each subband [10

10. A. Mátyás, P. Lugli, and C. Jirauschek, “Photon-induced carrier transport in high efficiency midinfrared quantum cascade lasers,” J. Appl. Phys. 110, 013108 (2011) [CrossRef] .

]. We implement the different m* into Eq. (1) by taking nE,2D=m*/(π2), and using modified definitions ωmn (ε) = (Em + εmEnεn) / ħ, γmn (ε) = [γm (εm) + γn (εn)] / 2 with εi=εm*/mi*.

For the optical propagation in the resonator along the z direction, we write the electric field as Ei (x, y, z) = Ai (z) Fi (x, y)exp(ikiz − iωit) + c.c. [22

22. G. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

], with (∬|Fi|2 dxdy)1/2 = 1. Ai and Fi denote the complex amplitude and transverse mode distribution of the field component with frequency ωi and wavenumber ki, and c.c. refers to the complex conjugate. The optical power at frequency ωi is then given by Pi = 20ni |Ai|2, with the effective refractive index of the corresponding mode ni and the vacuum speed of light c. The propagation of the THz field component generated by the DFG process is in the slowly varying amplitude approximation described by [23

23. N. Bloembergen, Nonlinear Optics (World Scientific, 1996).

]
zA3=iω2cn3f321χ(2)A1A2*exp(iΔkz)a2A3,
(2)
where Δk = k3k1 + k2 is the phase mismatch, a denotes the THz resonator loss coefficient at frequency ω, and χ(2) is the peak value of the nonlinear susceptibility χ(2) (x, y) in the QCL waveguide. Furthermore, the asterisk denotes the complex conjugate. The overlap integral f321 is defined as f321=(χ(2))1χ(2)(x,y)F3*F2*F1dxdy[1

1. M. Belkin, F. Capasso, A. Belyanin, D. Sivco, A. Cho, D. Oakley, C. Vineis, and G. Turner, “Terahertz quantum-cascade-laser source based on intracavity difference-frequency generation,” Nat. Photonics 1, 288–292 (2007) [CrossRef] .

, 22

22. G. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

].

For solving Eq. (2), P1 and P2 are commonly taken to be constant over the resonator length L, which holds for moderate MIR outcoupling at the facets. Furthermore, the depletion of the MIR pump beams due to the DFG process is negligible. The calculation is greatly simplified for La−1. With a ∼ 100cm−1 and L in the mm range [1

1. M. Belkin, F. Capasso, A. Belyanin, D. Sivco, A. Cho, D. Oakley, C. Vineis, and G. Turner, “Terahertz quantum-cascade-laser source based on intracavity difference-frequency generation,” Nat. Photonics 1, 288–292 (2007) [CrossRef] .

,2

2. M. Belkin, F. Capasso, F. Xie, A. Belyanin, M. Fischer, A. Wittmann, and J. Faist, “Room temperature terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett. 92, 201101 (2008) [CrossRef] .

,4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

], this condition is typically fulfilled for DFG structures. Introducing the effective interaction area Seff = |f321|−2 and the coherence length Lcoh=(Δk2+a2/4)1/2, the THz power at the facet Pf = P3 (L) is then obtained as [1

1. M. Belkin, F. Capasso, A. Belyanin, D. Sivco, A. Cho, D. Oakley, C. Vineis, and G. Turner, “Terahertz quantum-cascade-laser source based on intracavity difference-frequency generation,” Nat. Photonics 1, 288–292 (2007) [CrossRef] .

]
Pf=ω28ε0c3n1n2n3|χ(2)|2P1P2SeffLcoh2.
(3)
Insight into the total generated and lost THz power can be obtained by multiplying Eq. (2) with A3* from left and adding the complex conjugate. The term zP3|a = −aP3 in the resulting equation corresponds to the THz power lost in the waveguide. Integration over the propagation length yields the total absorbed power Pa. For La−1, we can approximate P3 = Pf and thus obtain Pa = −2LaPf, where 2L is the cavity roundtrip length and Pf is given by Eq. (3). The total generated power is then
Pg=Pa=2LaPf.
(4)

The large losses due to the strong THz absorption in the waveguide can be reduced by out-coupling the generated THz radiation along the resonator surface rather than at the facet. This can for example be achieved by a grating etched into the laser ridges [3

3. C. Pflügl, M. Belkin, Q. Wang, M. Geiser, A. Belyanin, M. Fischer, A. Wittmann, J. Faist, and F. Capasso, “Surface-emitting terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett. 93, 161110 (2008) [CrossRef] .

], or in a Cherenkov emission scheme [5

5. K. Vijayraghavan, R. W. Adams, A. Vizbaras, M. Jang, C. Grasse, G. Boehm, M. C. Amann, and M. A. Belkin, “Terahertz sources based on Čerenkov difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 100, 251104 (2012) [CrossRef] .

]. For surface-emitting structures characterized by a THz outcoupling coefficient κ, the theoretical description can easily be adapted by replacing a with a + κ in Eq. (2). In analogy to Eq. (4), we obtain for L ≫ (a + κ)−1 the generated power
P˜g=2L(a+κ)Pf.
(5)
Here Pf is given by Eq. (3) with a modified coherence length Lcoh=[Δk2+(a+κ)2/4]1/2. In Eq. (5), the first contribution 2LaPf corresponds to the power absorbed in the waveguide, and the second contribution is the power outcoupled at the surface
Ps=2LκPf.
(6)

3. Analysis of a terahertz source based on difference frequency generation

In the following, we present a self-consistent analysis of a room temperature DFG source, producing a THz output power of 8.5μW in single mode operation [4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

]. The active region consists of a bound-to-continuum (BTC) section which also contains the giant optical nonlinearity for the DFG process (see Fig. 1(a)), and a double-resonant-phonon (DRP) section. A distributed feedback grating is integrated into the resonator to enforce single mode lasing of the BTC and DRP design at frequencies of 32 and 28THz, respectively, resulting in DFG at 4THz [4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

].

Fig. 1 (a) Conduction band diagram of one period of the BTC structure at 300K for a bias of 38kV/cm. The three energy levels marked in red indicate the level subset contributing most to χ(2) in Eq. (1). (b) Tapered ridge waveguide structure, viewed from top.

3.1. Device simulation

The carrier transport in the QCL structure is modeled based on a semiclassical EMC approach, specifically adapted to the simulation of MIR QCLs [10

10. A. Mátyás, P. Lugli, and C. Jirauschek, “Photon-induced carrier transport in high efficiency midinfrared quantum cascade lasers,” J. Appl. Phys. 110, 013108 (2011) [CrossRef] .

]. All relevant scattering mechanisms are considered [10

10. A. Mátyás, P. Lugli, and C. Jirauschek, “Photon-induced carrier transport in high efficiency midinfrared quantum cascade lasers,” J. Appl. Phys. 110, 013108 (2011) [CrossRef] .

,20

20. C. Jirauschek and P. Lugli, “Monte-Carlo-based spectral gain analysis for terahertz quantum cascade lasers,” J. Appl. Phys. 105, 123102 (2009) [CrossRef] .

,24

24. 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] .

], and the dynamics due to the optical cavity field is taken into account [10

10. A. Mátyás, P. Lugli, and C. Jirauschek, “Photon-induced carrier transport in high efficiency midinfrared quantum cascade lasers,” J. Appl. Phys. 110, 013108 (2011) [CrossRef] .

, 17

17. C. Jirauschek, “Monte Carlo study of carrier-light coupling in terahertz quantum cascade lasers,” Appl. Phys. Lett. 96, 011103 (2010) [CrossRef] .

, 18

18. C. Jirauschek, “Monte Carlo study of intrinsic linewidths in terahertz quantum cascade lasers,” Opt. Express 18, 25922–25927 (2010) [CrossRef] [PubMed] .

]. The carrier transport simulation is coupled to a Schrödinger-Poisson solver [21

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

], providing the subband energies and wave functions. The simulations are self-consistent, only relying on the device specifications and well-established material parameters.

Based on the given layer sequences of the BTC and DRP design [2

2. M. Belkin, F. Capasso, F. Xie, A. Belyanin, M. Fischer, A. Wittmann, and J. Faist, “Room temperature terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett. 92, 201101 (2008) [CrossRef] .

,4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

], the EMC simulations provide the MIR powers and the electric current. The waveguide length is L = 3mm and the width is w0 = 16μm, tapered to wf = 60μm toward the front facet with a small taper angle of 1° [4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

], so that the mode can expand adiabatically (see Fig. 1(b)). This results in a contact area of A = 0.076mm2 and an average waveguide width of w̄ = A/L = 25μm, used to calculate the current and MIR powers from the current density and optical intensities delivered by EMC. Furthermore, we assume a confinement factor of 0.4 in both gain sections [2

2. M. Belkin, F. Capasso, F. Xie, A. Belyanin, M. Fischer, A. Wittmann, and J. Faist, “Room temperature terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett. 92, 201101 (2008) [CrossRef] .

,4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

], and a combined waveguide and mirror loss of 6.4cm−1 + 2.1cm−1 for both MIR modes. Here the common assumption is made that the optical power in each mode is equally distributed over the resonator length [17

17. C. Jirauschek, “Monte Carlo study of carrier-light coupling in terahertz quantum cascade lasers,” Appl. Phys. Lett. 96, 011103 (2010) [CrossRef] .

]. The 32THz mode generated in the BTC section influences the carrier dynamics in the DRP gain medium, and analogously for the 28THz mode. This is considered by performing alternate simulations of the BTC and DRP sections: For the DRP region, the intensity of the 32THz mode is fixed to the value extracted from the BTC region simulation at the same current density, and vice versa. The simulations are repeated until convergence is reached.

The DFG process is simulated using the converged MIR EMC simulation results, since the generated THz power is so weak that its influence on the carrier transport can be neglected. The second order susceptibility is calculated from Eq. (1). Based on waveguide simulations using the effective refractive index approximation [25

25. K. Chiang, “Performance of the effective-index method for the analysis of dielectric waveguides,” Opt. Lett. 16, 714–716 (1991) [CrossRef] [PubMed] .

], Seff, a, and the facet transmittance Tf[26

26. J. Butler and J. Zoroofchi, “Radiation fields of GaAs-(AlGa)As injection lasers,” IEEE J. Quantum Electron. 10, 809–815 (1974) [CrossRef] .

] are calculated for the fundamental THz mode. Here, the Drude model is employed for the bulk layers [27

27. 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] .

], while the complex permittivity of the gain medium is directly extracted from the EMC simulation [18

18. C. Jirauschek, “Monte Carlo study of intrinsic linewidths in terahertz quantum cascade lasers,” Opt. Express 18, 25922–25927 (2010) [CrossRef] [PubMed] .

]. With THz losses of a ≈ 150cm−1[4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

], only radiation generated at ≲ 0.1mm from the facet contributes to the THz emission. Thus, Seff, a and Tf are calculated for the waveguide width at the facet wf = 60μm. Furthermore, a phase mismatch of Δk = 7cm−1a is assumed [4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

], corresponding to near-perfect phase matching. The THz power outcoupled through the facet can then be calculated using Pout = TfPf, with Pf given by Eq. (3).

3.2. Results

To validate our modeling approach, we compare our simulation results at 300K to available experimental data for a recent room temperature DFG source [4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

]. For an optimum bias where the THz output power Pout reaches its maximum value, the EMC simulation yields a current of 11.7A, which agrees very well with the measured value of 12A [4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

]. Also the simulated MIR powers P1 = 1.55W (32THz) and P2 = 0.60W (28THz) agree reasonably well with the measured values of 1.2 and 0.8W [4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

]. The THz performance of the DFG source is characterized by Pout and the conversion efficiency η = Pout/ (P1P2) [4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

]. With Seff = 5300μm2, Tf = 0.23, |χ(2)| = 44nm/V and a = 150cm−1, we obtain Pout = 12μW and η = 13 μW/W2, as compared to 8.5μW and 10μW/W2 found in experiment [4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

]. Furthermore, the simulation confirms the experimentally estimated values for the nonlinear susceptibility (|χ(2)|≈ 40nm/V) and the waveguide loss (a ≈ 150cm−1) [4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

]. Thus, good agreement with experiment is obtained. Remaining deviations are attributed to uncertainties in the MIR waveguide and interface roughness parameter values [10

10. A. Mátyás, P. Lugli, and C. Jirauschek, “Photon-induced carrier transport in high efficiency midinfrared quantum cascade lasers,” J. Appl. Phys. 110, 013108 (2011) [CrossRef] .

], as well as the implementation of the MIR mirror outcoupling by a distributed loss coefficient as mentioned above. In this context, we note that the experimental MIR and THz powers seem to strongly depend on the exact design parameters [2

2. M. Belkin, F. Capasso, F. Xie, A. Belyanin, M. Fischer, A. Wittmann, and J. Faist, “Room temperature terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett. 92, 201101 (2008) [CrossRef] .

, 4

4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

].

Figure 2 contains temperature dependent simulation results for the investigated DFG source. In Fig. 2(a), the MIR powers P1 and P2 are displayed for optimum bias where Pout is maximum. The corresponding outcoupled THz power Pout and conversion efficiency η are shown in Fig. 2(b). In agreement with experiment [2

2. M. Belkin, F. Capasso, F. Xie, A. Belyanin, M. Fischer, A. Wittmann, and J. Faist, “Room temperature terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett. 92, 201101 (2008) [CrossRef] .

], P1, P2 and Pout strongly degrade with increasing temperature, while η is insensitive over a wide temperature range.

Fig. 2 Temperature dependent simulation results: (a) MIR powers; (b) and (c) THz powers and conversion efficiencies for (b) facet emission and (c) surface outcoupling.

As shown in Section 2, the generated THz power can be outcoupled more efficiently by using a surface outcoupling scheme, furthermore offering improved beam collimation [3

3. C. Pflügl, M. Belkin, Q. Wang, M. Geiser, A. Belyanin, M. Fischer, A. Wittmann, J. Faist, and F. Capasso, “Surface-emitting terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett. 93, 161110 (2008) [CrossRef] .

]. The corresponding surface-emitted power Ps in Eq. (6) is maximized for near-perfect phase matching Δka and a coupling coefficient κ = a. In the following, we assume ideal surface emission from our investigated structure in order to estimate the maximum extractable THz power. The corresponding surface grating is here considered to have a negligible effect on the MIR laser modes [3

3. C. Pflügl, M. Belkin, Q. Wang, M. Geiser, A. Belyanin, M. Fischer, A. Wittmann, J. Faist, and F. Capasso, “Surface-emitting terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett. 93, 161110 (2008) [CrossRef] .

]. Thus, the EMC simulation results remain unaffected. For better comparability, we furthermore assume that also the THz mode profile and loss are the same as for the facet emitting device. Since the outcoupling now occurs across the whole surface area A rather than at the facet, the average waveguide width w̄ = A/L instead of wf is used for the waveguide simulations, yielding a reduced Seff = 2200μm2 while a does not change significantly. In Fig. 2(c), the emitted THz power Ps and conversion efficiency ηs = Ps/ (P1P2) are shown. The outcoupled power at 300K is now Ps = 3.1mW, suggesting that room temperature output powers in the mW range should be feasible with an improved outcoupling scheme. From Eqs. (5) and (6) it follows that for κ = a, 50% of the generated THz power is emitted. By contrast, for the case of facet emission shown in Fig. 2(b), the outcoupling ratio is only Pout/Pg ≈ 0.1%. On a practical note, experimental structures with grating-based outcoupling have up to now only yielded moderate improvement [3

3. C. Pflügl, M. Belkin, Q. Wang, M. Geiser, A. Belyanin, M. Fischer, A. Wittmann, J. Faist, and F. Capasso, “Surface-emitting terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett. 93, 161110 (2008) [CrossRef] .

], partly because of the difficulties in implementing a strong grating with κa[5

5. K. Vijayraghavan, R. W. Adams, A. Vizbaras, M. Jang, C. Grasse, G. Boehm, M. C. Amann, and M. A. Belkin, “Terahertz sources based on Čerenkov difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 100, 251104 (2012) [CrossRef] .

]. An alternative surface emission scheme based on the Cherenkov effect has recently shown promising results [5

5. K. Vijayraghavan, R. W. Adams, A. Vizbaras, M. Jang, C. Grasse, G. Boehm, M. C. Amann, and M. A. Belkin, “Terahertz sources based on Čerenkov difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 100, 251104 (2012) [CrossRef] .

].

4. Conclusion

An extended EMC approach for the self-consistent modeling of terahertz difference frequency generation in QCLs is developed and applied to a current experimental room temperature design. Good agreement with available experimental data is obtained. For optimized light extraction via surface outcoupling, room temperature output powers in the mW range are predicted.

Acknowledgments

This work was funded by the Nanosystems Initiative Munich and the Emmy Noether program of the DFG ( JI115/1-1). We thank M. Belkin for fruitful discussions.

References and links

1.

M. Belkin, F. Capasso, A. Belyanin, D. Sivco, A. Cho, D. Oakley, C. Vineis, and G. Turner, “Terahertz quantum-cascade-laser source based on intracavity difference-frequency generation,” Nat. Photonics 1, 288–292 (2007) [CrossRef] .

2.

M. Belkin, F. Capasso, F. Xie, A. Belyanin, M. Fischer, A. Wittmann, and J. Faist, “Room temperature terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett. 92, 201101 (2008) [CrossRef] .

3.

C. Pflügl, M. Belkin, Q. Wang, M. Geiser, A. Belyanin, M. Fischer, A. Wittmann, J. Faist, and F. Capasso, “Surface-emitting terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett. 93, 161110 (2008) [CrossRef] .

4.

Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 99, 131106 (2011) [CrossRef] .

5.

K. Vijayraghavan, R. W. Adams, A. Vizbaras, M. Jang, C. Grasse, G. Boehm, M. C. Amann, and M. A. Belkin, “Terahertz sources based on Čerenkov difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett. 100, 251104 (2012) [CrossRef] .

6.

S. Fathololoumi, E. Dupont, C. Chan, Z. Wasilewski, S. Laframboise, D. Ban, A. Mátyás, C. Jirauschek, Q. Hu, and H. Liu, “Terahertz quantum cascade lasers operating up to ∼200 K with optimized oscillator strength and improved injection tunneling,” Opt. Express 20, 3866–3876 (2012) [CrossRef] [PubMed] .

7.

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

8.

R. C. Iotti and F. Rossi, “Carrier thermalization versus phonon-assisted relaxation in quantum-cascade lasers: A Monte Carlo approach,” Appl. Phys. Lett. 78, 2902–2904 (2001) [CrossRef] .

9.

X. Gao, D. Botez, and I. Knezevic, “X-valley leakage in GaAs-based midinfrared quantum cascade lasers: A Monte Carlo study,” J. Appl. Phys. 101, 063101 (2007) [CrossRef] .

10.

A. Mátyás, P. Lugli, and C. Jirauschek, “Photon-induced carrier transport in high efficiency midinfrared quantum cascade lasers,” J. Appl. Phys. 110, 013108 (2011) [CrossRef] .

11.

R. Köhler, R. C. Iotti, A. Tredicucci, and F. Rossi, “Design and simulation of terahertz quantum cascade lasers,” Appl. Phys. Lett. 79, 3920–3922 (2001) [CrossRef] .

12.

H. Callebaut, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Analysis of transport properties of terahertz quantum cascade lasers,” Appl. Phys. Lett. 83, 207–209 (2003) [CrossRef] .

13.

O. Bonno, J.-L. Thobel, and F. Dessenne, “Modeling of electron-electron scattering in Monte Carlo simulation of quantum cascade lasers,” J. Appl. Phys. 97, 043702 (2005) [CrossRef] .

14.

J. T. Lü and J. C. Cao, “Coulomb scattering in the Monte Carlo simulation of terahertz quantum-cascade lasers,” Appl. Phys. Lett. 89, 211115 (2006) [CrossRef] .

15.

C. Jirauschek, G. Scarpa, P. Lugli, M. S. Vitiello, and G. Scamarcio, “Comparative analysis of resonant phonon THz quantum cascade lasers,” J. Appl. Phys. 101, 086109 (2007) [CrossRef] .

16.

A. Mátyás, M. Belkin, P. Lugli, and C. Jirauschek, “Temperature performance analysis of terahertz quantum cascade lasers: Vertical versus diagonal designs,” Appl. Phys. Lett. 96, 201110 (2010).

17.

C. Jirauschek, “Monte Carlo study of carrier-light coupling in terahertz quantum cascade lasers,” Appl. Phys. Lett. 96, 011103 (2010) [CrossRef] .

18.

C. Jirauschek, “Monte Carlo study of intrinsic linewidths in terahertz quantum cascade lasers,” Opt. Express 18, 25922–25927 (2010) [CrossRef] [PubMed] .

19.

Y. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, 1984).

20.

C. Jirauschek and P. Lugli, “Monte-Carlo-based spectral gain analysis for terahertz quantum cascade lasers,” J. Appl. Phys. 105, 123102 (2009) [CrossRef] .

21.

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

22.

G. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

23.

N. Bloembergen, Nonlinear Optics (World Scientific, 1996).

24.

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] .

25.

K. Chiang, “Performance of the effective-index method for the analysis of dielectric waveguides,” Opt. Lett. 16, 714–716 (1991) [CrossRef] [PubMed] .

26.

J. Butler and J. Zoroofchi, “Radiation fields of GaAs-(AlGa)As injection lasers,” IEEE J. Quantum Electron. 10, 809–815 (1974) [CrossRef] .

27.

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] .

OCIS Codes
(140.3070) Lasers and laser optics : Infrared and far-infrared lasers
(140.3430) Lasers and laser optics : Laser theory
(190.2620) Nonlinear optics : Harmonic generation and mixing
(140.5965) Lasers and laser optics : Semiconductor lasers, quantum cascade

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: January 17, 2013
Revised Manuscript: February 20, 2013
Manuscript Accepted: February 23, 2013
Published: March 4, 2013

Citation
Christian Jirauschek, Alpar Matyas, Paolo Lugli, and Markus-Christian Amann, "Monte Carlo study of terahertz difference frequency generation in quantum cascade lasers," Opt. Express 21, 6180-6185 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-6180


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References

  1. M. Belkin, F. Capasso, A. Belyanin, D. Sivco, A. Cho, D. Oakley, C. Vineis, and G. Turner, “Terahertz quantum-cascade-laser source based on intracavity difference-frequency generation,” Nat. Photonics1, 288–292 (2007). [CrossRef]
  2. M. Belkin, F. Capasso, F. Xie, A. Belyanin, M. Fischer, A. Wittmann, and J. Faist, “Room temperature terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett.92, 201101 (2008). [CrossRef]
  3. C. Pflügl, M. Belkin, Q. Wang, M. Geiser, A. Belyanin, M. Fischer, A. Wittmann, J. Faist, and F. Capasso, “Surface-emitting terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett.93, 161110 (2008). [CrossRef]
  4. Q. Y. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Room temperature single-mode terahertz sources based on intracavity difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett.99, 131106 (2011). [CrossRef]
  5. K. Vijayraghavan, R. W. Adams, A. Vizbaras, M. Jang, C. Grasse, G. Boehm, M. C. Amann, and M. A. Belkin, “Terahertz sources based on Čerenkov difference-frequency generation in quantum cascade lasers,” Appl. Phys. Lett.100, 251104 (2012). [CrossRef]
  6. S. Fathololoumi, E. Dupont, C. Chan, Z. Wasilewski, S. Laframboise, D. Ban, A. Mátyás, C. Jirauschek, Q. Hu, and H. Liu, “Terahertz quantum cascade lasers operating up to ∼200 K with optimized oscillator strength and improved injection tunneling,” Opt. Express20, 3866–3876 (2012). [CrossRef] [PubMed]
  7. B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “High-power terahertz quantum-cascade lasers,” Electron. Lett.42, 89–90 (2006). [CrossRef]
  8. R. C. Iotti and F. Rossi, “Carrier thermalization versus phonon-assisted relaxation in quantum-cascade lasers: A Monte Carlo approach,” Appl. Phys. Lett.78, 2902–2904 (2001). [CrossRef]
  9. X. Gao, D. Botez, and I. Knezevic, “X-valley leakage in GaAs-based midinfrared quantum cascade lasers: A Monte Carlo study,” J. Appl. Phys.101, 063101 (2007). [CrossRef]
  10. A. Mátyás, P. Lugli, and C. Jirauschek, “Photon-induced carrier transport in high efficiency midinfrared quantum cascade lasers,” J. Appl. Phys.110, 013108 (2011). [CrossRef]
  11. R. Köhler, R. C. Iotti, A. Tredicucci, and F. Rossi, “Design and simulation of terahertz quantum cascade lasers,” Appl. Phys. Lett.79, 3920–3922 (2001). [CrossRef]
  12. H. Callebaut, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Analysis of transport properties of terahertz quantum cascade lasers,” Appl. Phys. Lett.83, 207–209 (2003). [CrossRef]
  13. O. Bonno, J.-L. Thobel, and F. Dessenne, “Modeling of electron-electron scattering in Monte Carlo simulation of quantum cascade lasers,” J. Appl. Phys.97, 043702 (2005). [CrossRef]
  14. J. T. Lü and J. C. Cao, “Coulomb scattering in the Monte Carlo simulation of terahertz quantum-cascade lasers,” Appl. Phys. Lett.89, 211115 (2006). [CrossRef]
  15. C. Jirauschek, G. Scarpa, P. Lugli, M. S. Vitiello, and G. Scamarcio, “Comparative analysis of resonant phonon THz quantum cascade lasers,” J. Appl. Phys.101, 086109 (2007). [CrossRef]
  16. A. Mátyás, M. Belkin, P. Lugli, and C. Jirauschek, “Temperature performance analysis of terahertz quantum cascade lasers: Vertical versus diagonal designs,” Appl. Phys. Lett.96, 201110 (2010).
  17. C. Jirauschek, “Monte Carlo study of carrier-light coupling in terahertz quantum cascade lasers,” Appl. Phys. Lett.96, 011103 (2010). [CrossRef]
  18. C. Jirauschek, “Monte Carlo study of intrinsic linewidths in terahertz quantum cascade lasers,” Opt. Express18, 25922–25927 (2010). [CrossRef] [PubMed]
  19. Y. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, 1984).
  20. C. Jirauschek and P. Lugli, “Monte-Carlo-based spectral gain analysis for terahertz quantum cascade lasers,” J. Appl. Phys.105, 123102 (2009). [CrossRef]
  21. C. Jirauschek, “Accuracy of transfer matrix approaches for solving the effective mass Schrödinger equation,” IEEE J. Quantum Electron.45, 1059–1067 (2009). [CrossRef]
  22. G. Agrawal, Nonlinear Fiber Optics (Academic, 2001).
  23. N. Bloembergen, Nonlinear Optics (World Scientific, 1996).
  24. 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]
  25. K. Chiang, “Performance of the effective-index method for the analysis of dielectric waveguides,” Opt. Lett.16, 714–716 (1991). [CrossRef] [PubMed]
  26. J. Butler and J. Zoroofchi, “Radiation fields of GaAs-(AlGa)As injection lasers,” IEEE J. Quantum Electron.10, 809–815 (1974). [CrossRef]
  27. 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]

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