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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 10 — May. 7, 2012
  • pp: 11207–11217
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Lateral distributed-feedback gratings for single-mode, high-power terahertz quantum-cascade lasers

M. Wienold, A. Tahraoui, L. Schrottke, R. Sharma, X. Lü, K. Biermann, R. Hey, and H. T. Grahn  »View Author Affiliations


Optics Express, Vol. 20, Issue 10, pp. 11207-11217 (2012)
http://dx.doi.org/10.1364/OE.20.011207


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Abstract

We report on terahertz quantum-cascade lasers (THz QCLs) based on first-order lateral distributed-feedback (lDFB) gratings, which exhibit continuous-wave operation, high output powers (>8 mW), and single-mode emission at 3.3–3.4 THz. A general method is presented to determine the coupling coefficients of lateral gratings in terms of the coupled-mode theory, which demonstrates that large coupling strengths are obtained in the presence of corrugated metal layers. The experimental spectra are in agreement with simulations of the lDFB cavities, which take into account the reflective end facets.

© 2012 OSA

1. Introduction

2. Modeling of lateral gratings

In a DFB grating, the dielectric function is modulated in the ridge direction (labeled z), where in the following Δε(z) denotes the modulation of the dielectric function with respect to a reference waveguide. In Fig. 1(a), a schematic diagram of an lDFB QCL with its two end facets is shown. The grating period Λ corresponds to the vacuum Bragg wavelength of λB = 2Λn/m, where n denotes the real part of the effective refractive index of the waveguide (neff = n + ik) and m the grating order. Figure 1(b) depicts an lDFB mesa after dry etching.

Fig. 1 (a) Schematic diagram of a THz QCL with a lDFB grating. L denotes the length of the laser, Λ the grating period, and ϕF (0) and ϕF (L) the facet phases with respect to the grating comb. (b) Scanning electron microscopy (SEM) image of an lDFB mesa after dry etching.

In Ref. [22

22. A. Laakso, M. Dumitrescu, J. Viheriälä, J. Karinen, M. Suominen, and M. Pessa, “Optical modeling of laterally-corrugated ridge-waveguide gratings,” Opt. Quant. Electron. 40, 907–920 (2008). [CrossRef]

], the coupling coefficients for interband DFB lasers based on lateral gratings are determined by assuming a reference waveguide with an average refractive index in the lateral corrugation region. However, this approach does not account for changes of the intensity profile of modes, which are located in different grating sections. In our case, different mode profiles in the wide- and narrow-ridge sections are caused by the presence of a corrugated metal layer. A more appropriate way to calculate the coupling coefficients in this case arises from the correspondence of uniform DFB gratings and one-dimensional photonic crystals.

In the simple case of an index DFB grating, which consists of two alternating homogeneous slabs, i.e. a grating with a square modulation of the dielectric constant and no waveguide dispersion (ng = n), the expression for K becomes a very simple analytic one:
K=Δε2Λn2=2ΔnλB,
(3)
where Δε = 2nΔn denotes the difference of the dielectric constant between the sections. Equation (3) can be easily derived from the general Eqs. (1) and (2) cited in [22

22. A. Laakso, M. Dumitrescu, J. Viheriälä, J. Karinen, M. Suominen, and M. Pessa, “Optical modeling of laterally-corrugated ridge-waveguide gratings,” Opt. Quant. Electron. 40, 907–920 (2008). [CrossRef]

]; other derivations can be found in many textbooks (cf. citations in [15

15. B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Distributed-feedback terahertz quantum-cascade lasers with laterally corrugated metal waveguides,” Opt. Lett. 30, 2909–2911 (2005). [CrossRef] [PubMed]

, 22

22. A. Laakso, M. Dumitrescu, J. Viheriälä, J. Karinen, M. Suominen, and M. Pessa, “Optical modeling of laterally-corrugated ridge-waveguide gratings,” Opt. Quant. Electron. 40, 907–920 (2008). [CrossRef]

]). Using Λ = 12.5 μm, Δε = 0.2, and n = 3.584, Eq. (3) results in K = 6.23 cm−1. This is exactly the same value as obtained via the numerical determination of ω and ω+ for the corresponding one-dimensional photonic crystal, which confirms the validity of the photonic-crystal approach. Since a stack of homogeneous slabs is considered here, the numerical result for K does not depend on the lateral and vertical dimensions of the simulated unit cell.

Fig. 2 (a) Geometry and initial finite-element grid of the unit cell (periodic in the z-direction) used in the simulation of the uniform lDFB grating. Top and bottom metallizations are yellow colored, the active region red, the highly doped bottom contact layer blue, and the substrate gray. Perfectly matched layers are applied at x = 100 μm (transparent boundary conditions). (b) Projections of the intensity distribution of the ω and ω+ TM00 modes to the xz and xy plane.

Table 1. Real and imaginary part of the eigenfrequencies ω and ω+ and corresponding coupling coefficient K + iKg (ng = 3.8) for the TM00, TM01, and TM02 modes for the grating geometry shown in Fig. 2(a).

table-icon
View This Table

For known coupling coefficients, the CMT allows for a determination of the complex eigenvalues of the DFB cavity, i.e. the frequencies and threshold gain for the different longitudinal modes. Note that in case of reflective end facets the cavity eigenvalues depend also on the facet phases ϕF [cf. Fig. 1(a)]. In the case of cleaved facets, these are not known a priori due to the limited accuracy of the cleaving process. This causes an uncertainty of the frequency values, which is approximately equal to the mode spacing of the corresponding Fabry-Pérot cavity.

3. Experimental results and discussion

The THz QCL active region is based on the GaAs/Al0.25Ga0.75As heterostructure reported as sample B in Ref. [25

25. M. Wienold, L. Schrottke, M. Giehler, R. Hey, W. Anders, and H. T. Grahn, “Low-threshold terahertz quantum-cascade lasers based on GaAs/Al0.25Ga0.75As heterostructures,” Appl. Phys. Lett. 97, 071113 (2010). [CrossRef]

]. The processing is similar to single-plasmon Fabry-Pérot lasers. In the first step, the metal grating is defined in a lift-off process (10/100 nm Ti/Au). This is followed by a dry etching step to form the corrugated mesa and a second metallization/lift-off step, which defines the bottom contact and reinforces the bonding area of the top contact (300 nm AuGe/Ni). For dry etching, we used a silicon oxide hard mask and a reactive ion etch system based on an inductively coupled plasma (SAMCO RIE-140iP). The grating duty cycle is 0.5, and the widths of the wide- and narrow-ridge sections of the etched mesa are approximately 120 and 107 μm. These are also the parameters, which we used for the simulations (neglecting a small technological gap of about 1 μm between the top metallization and the etch mask).

In the following, the experimental results for three laser stripes from a single die with grating periods of 12.4, 12.5, and 12.6 μm and as-cleaved facets are discussed. The corresponding Bragg frequencies fall within the emission range of 3.15–3.45 THz found for Fabry-Pérot lasers of the same wafer. Since the three laser stripes are located on the same die, they have all the same ridge length (1.454 mm). Figure 3(a) shows the light-current-voltage (L-I-V) characteristics of the three lasers for driving currents up to 0.75 A (455 A cm−2). In this range, the emission of the lasers is dominated by a single spectral mode. Due to the large dissipated electrical power, only temperatures above 20 K can be maintained over the whole range of driving currents. The three lasers exhibit similar L-I-V characteristics with threshold current densities Jth of 290, 290, and 270 A cm−2 at 30 K for Λ = 12.4, 12.5, and 12.6 μm, respectively. The maximum operating temperature is about 55 K for Λ = 12.4 and 12.5 μm and about 60 K for Λ = 12.6 μm. The slope efficiency at threshold (30 K) is approximately 30 mW/A for all three lasers. The output powers at a driving current of 0.75 A are 8.7, 8.0, and 8.7 mW at 30 K and 3.6, 3.3, and 4.8 mW at 50 K for Λ = 12.4, 12.5, and 12.6 μm, respectively. For comparison, a Fabry-Pérot laser ridge of the same wafer with similar dimensions (0.10 × 1.52 mm2) operated in cw mode up to 55 K. For this laser, values for the threshold current density, slope efficiency, and maximum cw output power at 30 K are 320 A cm−2, 20 mW/A, and 6.5 mW, respectively. The smaller threshold current densities of the lDFB QCLs in comparison with Fabry-Pérot QCLs are qualitatively explained by a smaller threshold gain as a consequence of the feedback provided by the grating in addition to the feedback of the reflective end facets. However, the effect is stronger than expected. While the levels of cw output power are comparable for lDFB and Fabry-Pérot laser ridges, the spectral power density for the lDFB QCLs is much higher due to single-mode operation.

Fig. 3 (a) Continuous-wave L-I-V characteristics of three lDFB QCLs (ridge length 1.454 mm) with grating periods of 12.4, 12.5, and 12.6 μm at different temperatures. The nominal temperature of 10 K is maintained only up to current values of 0.46 A. (b) Typical emission spectra of the three lasers in their single-mode regime (linear intensity scale). Insets: Corresponding power spectra on a logarithmic scale for a wide frequency range.

From beam-profile measurements for one of the lDFB lasers operated in a Stirling cooler, we found that the far field is very similar to single-plasmon lasers without a grating (cf. Ref. [6

6. H. Richter, M. Greiner-Bär, S. G. Pavlov, A. D. Semenov, M. Wienold, L. Schrottke, M. Giehler, R. Hey, H. T. Grahn, and H.-W. Hübers, “A compact, continuous-wave terahertz source based on a quantum-cascade laser and a miniature cryocooler,” Opt. Express 18, 10177–10187 (2010). [CrossRef] [PubMed]

]). For a driving current of 600 mA and operating temperature of 50 K, the beam profile of the Λ = 12.4 μm laser consists mainly of a single lobe, covering approximately 80% of the power, for which we estimate a full width at half maximum divergence of about 24° in both, the lateral and the vertical, directions. The beam divergence appears to be comparable to the one of third-order DFB QCLs, for which values of 15 to 30° have been reported [16

16. M. I. Amanti, G. Scalari, F. Castellano, M. Beck, and J. Faist, “Low divergence Terahertz photonic-wire laser,” Opt. Express 18, 6390–6395 (2010). [CrossRef] [PubMed]

].

Fig. 4 (a) Continuous-wave L-I-V characteristics for the lDFB QCL with Λ = 12.5 μm including the multi-mode emission regime (disabled temperature control loop). (b) Corresponding emission spectra at four different driving currents.

The emission spectra can be interpreted within the framework of the coupled-mode equations for a DFB cavity (see Appendix) with the boundary condition of two reflective end facets, for which the facet phases as defined in Fig. 1(a) enter the eigenvalue problem. The coupled-mode equations have been solved numerically for this case. In Fig. 5, the calculated threshold gain for the TM00 mode is plotted versus the frequency eigenvalues for the three lDFB QCLs with Λ = 12.4, 12.5, and 12.6 μm. We have performed simulations either for a complex coupling coefficient or for a real coupling coefficient (Kg = 0). The experimentally observed single-mode emission and the calculated longitudinal modes with the lowest threshold gain occur at almost the same frequencies, where the quantitative agreement appears to be better for the simulations with a real coupling coefficient. In this case, an agreement better than 9 GHz is found for all three lasers, while in the case of a complex coupling coefficient the calculated mode with the second lowest threshold gain coincides with the lasing frequency for Λ = 12.6 μm. While the Bragg frequency (depicted as dashed lines in Fig. 5) decreases with increasing grating period, the finite facet reflectance and the different facet phases cause lasing in the longitudinal mode below the Bragg frequency for Λ = 12.4 and 12.6 μm and above the Bragg frequency for Λ = 12.5 μm. The differences in the calculated threshold gain explain the different experimentally observed threshold current densities. However, the calculation predicts the lowest threshold gain for Λ = 12.5 μm, while experimentally the lowest value of Jth is found for Λ = 12.6 μm [cf. Fig. 3(a)]. Differences between the experimental and simulated results are likely due to the uncertainty of the involved parameters such as the coupling coefficients. For instance, the occurrence of a small displacement between the etch mask and the metallization mask might result in a phase difference between the index and gain coupling, which has been neglected in the simulations.

Fig. 5 Calculated threshold gain gth vs. frequency of the eigenmodes of the lDFB lasers (Λ = 12.4, 12.5, and 12.6 μm) in units of gth of the reference Fabry-Pérot cavity. The vertical dashed lines indicate the Bragg frequency using n = 3.58. The facet phases ϕF as defined in Fig. 1(a) have been determined by SEM imaging and are given in radians. Circles refer to a complex coupling coefficient [K + iKg = (3.0 + 0.8i) cm−1] and squares to a real coupling coefficient (K = 3.0 cm−1). Triangles indicate the experimentally observed frequencies.

While the single-mode operation regime is quite well explained by the coupled-mode equations for the fundamental lateral TM00 mode, the situation becomes more complex for the multi-mode emission regime. In Fig. 4(b), the emission of a second mode at 3.40 THz for the laser with Λ = 12.5 μm can be understood by the Stark shift of the gain with increasing bias, since the emission frequency agrees with the calculated frequency of the mode with the second lowest threshold gain. However, the multi-mode emission pattern for the highest driving current seems to exhibit a stop band between 3.40 and 3.50 THz. This cannot be explained by the spectral position and threshold gain of the longitudinal modes with TM00 symmetry. A likely explanation is the presence of higher-order lateral modes such as TM01 and TM02, for which the coupling coefficient is much larger and the Bragg frequency is increased due to the smaller effective index of these modes. For lasing of higher-order lateral modes in favor of the fundamental TM00 mode, the threshold gain has to be smaller as compared to the TM00 mode, i.e. the larger coupling coefficient has to compensate the higher waveguide losses.

4. Summary and conclusions

We have demonstrated THz QCLs based on first-order lateral DFB gratings and single-plasmon waveguides, which operate in continuous-wave mode with high output powers and single-mode emission around 3.3 THz. For the single-mode regime, cw output powers exceeding 8 mW have been obtained, while the maximum output power including the multi-mode regime can exceed even 12 mW. A method has been developed to calculate the coupling strength of DFB gratings in the presence of corrugated metal layers, which demonstrates that the origin of the large coupling strength in the investigated lasers is the strong plasmonic metal-light interaction at the lateral grating edges. The emission frequency of the lasers is determined by the Bragg frequency of the grating and the positions of the cleaved facets with respect to the grating comb. By taking into account the reflective end facets, a quantitative agreement between the experimental spectra and simulations based on the coupled-mode equations of DFB lasers is obtained. The present approach is limited by the accuracy of the cleaving process, which causes an uncertainty of the emission frequency approximately equal to the mode spacing of the corresponding Fabry-Pérot cavity. However, the number of mounted laser dies, which have to be tested to obtain a particular target frequency, can be kept small if laser ridges with different grating periods are located on the same die.

5. Appendix

In order to derive Eq. (1), we start with the coupled-mode equations following the notation of Ref. [21

21. S. L. Chuang, Physics of Photonic Devices (Wiley, New York, 2009), 2nd ed.

]:
ddz(A(z)B(z))=i(ΔβKabKbaΔβ)(A(z)B(z)).
(4)
A(z) and B(z) denote the amplitudes of the forward and backward propagating wave, respectively. Kab and Kba refer to the coupling coefficients of the forward and backward propagating mode, respectively, and Δβ = β0βB corresponds to the difference of the propagation constant β0 = ωneff/c and the Bragg wavevector βB = π/Λ. The coupled-mode equations in Eq. (4) are formally solved by
(A(z)B(z))=(A±B±)e±iqz
(5)
with
q=Δβ2KabKba.
(6)
The latter equation is the dispersion relation for the coupled system of forward and backward propagating waves of a uniform grating of infinite length. The propagation constant of the coupled system are given by β± = π/Λ ± q, while β0 and Δβ refer to the reference waveguide. Note that q, β0, and Δβ are complex quantities in the current notation due to the presence of gain and losses, while the frequency ω is a real quantity.

In the case of index and gain coupling, the coupling coefficients can be written as Kab = K + iKg exp(g) and Kba = K + iKg exp(−g) with K denoting the index coupling coefficient, Kg the gain coupling coefficient, and ϕg the phase difference between index and gain coupling. We obtain the relation
KabKba=K2Kg2+2iKKgcos(ϕg).
(7)
Here, only symmetric gratings are considered, for which the origin in the ridge direction z can be chosen such that Δε(z) = Δε(−z). The only possible values for ϕg are 0 and π, KabKba = (K ± iKg)2, where we chose the plus sign in the following by allowing for negative amplitudes Kg. We rewrite Eq. (6) using ω = β0c/neff = β0c/(n + ik):
ω±(q)=cn+ik(πΛ±q2+KabKba).
(8)
In the following, the wavevector π± q is treated as a real quantity, which causes β0 and Δβ to be real, while the frequency ω becomes complex. The case q = 0 correspond to the eigenfrequencies of a one-dimensional photonic crystal at the edge of the Brillouin zone. Assuming a symmetric grating with KabKba = (K + iKg)2, the complex frequencies of the two eigenmodes at q = 0 are given by Eq. (1).

Acknowledgments

The authors would like to thank M. Höricke for sample growth, W. Anders for sample processing, H. Ogiya (SAMCO Inc.) for dry etching, H. Wenzel (FBH Berlin) for the numerical tool to solve the coupled-mode equations, and H. Richter (DLR Berlin) for providing the beam profile measurements. We also acknowledge partial financial support by the European Commission through the ProFIT program of the Investitionsbank Berlin.

References and links

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H.-W. 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]

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3.

P. Khosropanah, W. Zhang, J. N. Hovenier, J. R. Gao, T. M. Klapwijk, M. I. Amanti, G. Scalari, and J. Faist, “3.4 THz heterodyne receiver using a hot electron bolometer and a distributed feedback quantum cascade laser,” J. Appl. Phys. 104, 113106 (2008). [CrossRef]

4.

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 1.5 Terahertz quantum cascade laser and use as a local oscillator in a heterodyne HEB receiver,” Opt. Express 17, 1159–1168 (2009). [CrossRef] [PubMed]

5.

Y. Ren, J. N. Hovenier, R. Higgins, J. R. Gao, T. M. Klapwijk, S. C. Shi, B. Klein, T.-Y. Kao, Q. Hu, and J. L. Reno, “High-resolution heterodyne spectroscopy using a tunable quantum cascade laser around 3.5 THz,” Appl. Phys. Lett. 98, 231109 (2011). [CrossRef]

6.

H. Richter, M. Greiner-Bär, S. G. Pavlov, A. D. Semenov, M. Wienold, L. Schrottke, M. Giehler, R. Hey, H. T. Grahn, and H.-W. Hübers, “A compact, continuous-wave terahertz source based on a quantum-cascade laser and a miniature cryocooler,” Opt. Express 18, 10177–10187 (2010). [CrossRef] [PubMed]

7.

J. Faist, C. Gmachl, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon, and A. Y. Cho, “Distributed feedback quantum cascade lasers,” Appl. Phys. Lett. 70, 2670–2672 (1997). [CrossRef]

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L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, and D. A. Ritchie, “High-power surface emission from terahertz distributed feedback lasers with a dual-slit unit cell,” Appl. Phys. Lett. 96, 191109 (2010). [CrossRef]

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16.

M. I. Amanti, G. Scalari, F. Castellano, M. Beck, and J. Faist, “Low divergence Terahertz photonic-wire laser,” Opt. Express 18, 6390–6395 (2010). [CrossRef] [PubMed]

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21.

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22.

A. Laakso, M. Dumitrescu, J. Viheriälä, J. Karinen, M. Suominen, and M. Pessa, “Optical modeling of laterally-corrugated ridge-waveguide gratings,” Opt. Quant. Electron. 40, 907–920 (2008). [CrossRef]

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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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25.

M. Wienold, L. Schrottke, M. Giehler, R. Hey, W. Anders, and H. T. Grahn, “Low-threshold terahertz quantum-cascade lasers based on GaAs/Al0.25Ga0.75As heterostructures,” Appl. Phys. Lett. 97, 071113 (2010). [CrossRef]

26.

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]

OCIS Codes
(140.3070) Lasers and laser optics : Infrared and far-infrared lasers
(140.3490) Lasers and laser optics : Lasers, distributed-feedback
(140.5965) Lasers and laser optics : Semiconductor lasers, quantum cascade

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: December 15, 2011
Revised Manuscript: March 4, 2012
Manuscript Accepted: April 10, 2012
Published: May 1, 2012

Citation
M. Wienold, A. Tahraoui, L. Schrottke, R. Sharma, X. Lü, K. Biermann, R. Hey, and H. T. Grahn, "Lateral distributed-feedback gratings for single-mode, high-power terahertz quantum-cascade lasers," Opt. Express 20, 11207-11217 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-10-11207


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References

  1. H.-W. 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. Express13, 5890–5896 (2005). [CrossRef] [PubMed]
  2. H. Richter, A. D. Semenov, S. G. Pavlov, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, K. S. Il’in, M. Siegel, and H.-W. Hübers, “Terahertz heterodyne receiver with quantum cascade laser and hot electron bolometer mixer in a pulse tube cooler,” Appl. Phys. Lett.93, 141108 (2008). [CrossRef]
  3. P. Khosropanah, W. Zhang, J. N. Hovenier, J. R. Gao, T. M. Klapwijk, M. I. Amanti, G. Scalari, and J. Faist, “3.4 THz heterodyne receiver using a hot electron bolometer and a distributed feedback quantum cascade laser,” J. Appl. Phys.104, 113106 (2008). [CrossRef]
  4. 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 1.5 Terahertz quantum cascade laser and use as a local oscillator in a heterodyne HEB receiver,” Opt. Express17, 1159–1168 (2009). [CrossRef] [PubMed]
  5. Y. Ren, J. N. Hovenier, R. Higgins, J. R. Gao, T. M. Klapwijk, S. C. Shi, B. Klein, T.-Y. Kao, Q. Hu, and J. L. Reno, “High-resolution heterodyne spectroscopy using a tunable quantum cascade laser around 3.5 THz,” Appl. Phys. Lett.98, 231109 (2011). [CrossRef]
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