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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 11 — May. 23, 2011
  • pp: 10707–10713
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Complex-coupled photonic crystal THz lasers with independent loss and refractive index modulation

Hua Zhang, Giacomo Scalari, Mattias Beck, Jérôme Faist, and Romuald Houdré  »View Author Affiliations


Optics Express, Vol. 19, Issue 11, pp. 10707-10713 (2011)
http://dx.doi.org/10.1364/OE.19.010707


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Abstract

Compared to near infra-red photonic crystal (PhC) band-edge lasers, achieving vertical emission with quantum cascade (QC) material operating in the THz range needs dedicated engineering because the TM polarized emission of QCLs favors in-plane emitting schemes and the currently used double plasmon waveguide, prevents vertical light extraction. We present an approach with independent refractive index and extraction losses modulation. The extraction losses are obtained with small extracting holes located at appropriate positions. The modal operation of the PhC is shown to critically depend on the external losses introduced. Very high surface emission power for optimum loss extractor design is achieved.

© 2011 OSA

1. Introduction

Surface emission is always a challenging task for semiconductor lasers [1

1. R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302(5649), 1374–1377 (2003). [CrossRef] [PubMed]

,2

2. O. P. Marshall, V. Apostolopoulos, J. R. Freeman, R. Rungsawang, H. E. Beere, and D. A. Ritchie, “Surface-emitting photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. 93(17), 171112 (2008). [CrossRef]

], especially with quantum cascade (QC) active material operating in the terahertz (THz) range. Photonic crystal (PhC) structures are good candidates to address this issue. Previously reported surface emitting THz PhC QC lasers were based on deep etched hole arrays or on patterning holes in metal while keeping the active area unetched [3

3. L. Sirigu, R. Terazzi, M. I. Amanti, M. Giovannini, J. Faist, L. A. Dunbar, and R. Houdré, “Terahertz quantum cascade lasers based on two-dimensional photonic crystal resonators,” Opt. Express 16(8), 5206–5217 (2008). [CrossRef] [PubMed]

,4

4. Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature 457(7226), 174–178 (2009). [CrossRef] [PubMed]

]. The full potential of PhC was not exploited as these devices were operating around an incomplete photonic-bandgap (PBG) or experienced only a weak refractive index contrast. In this work we present a complex-coupled THz surface emitting structure with periodic extraction losses on a pillar-type PhC pattern, which allows us to independently tailor the refractive index and the extraction loss. As a consequence, low average current densities and very high surface emitting power (in excess of 30 mW) is obtained through such extractors.

The periodic arrangement of an optical medium achieved in PhC structure leads to Bloch mode eigenstates with remarkable properties. The band-edge modes are of particular interest as they are responsible of slow light properties and a large density-of-states. The low group velocity of these states is associated with a significant gain enhancement, proportional to the product of the group index and the confinement factor of the field energy in the gain medium [5

5. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser - a new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994). [CrossRef]

]. In a Distributed Feedback (DFB) laser, the complex coupling between the refractive index and gain is based on 1D modulation [6

6. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972). [CrossRef]

,7

7. E. Kapon, A. Hardy, and A. Katzir, “The effect of complex coupling-coefficients on distributed feedback lasers,” IEEE J. Quantum Electron. 18(1), 66–71 (1982). [CrossRef]

]. Adding up dimensions when going from 1D DFB to 2D PhC structures allows a potentially easier scaling of the output power as well as producing symmetric, low divergence beam. A gain enhancement factor larger than in 1D DFB was also predicted [8

8. S. Nojima, “Optical-gain enhancement in two-dimensional active photonic crystals,” J. Appl. Phys. 90(2), 545–551 (2001). [CrossRef]

].

The first reported THz surface emitting devices were a second order DFB laser [9

9. O. Demichel, L. Mahler, T. Losco, C. Mauro, R. Green, A. Tredicucci, J. Xu, F. Beltram, H. E. Beere, D. A. Ritchie, and V. Tamosinuas, “Surface plasmon photonic structures in terahertz quantum cascade lasers,” Opt. Express 14(12), 5335–5345 (2006). [CrossRef] [PubMed]

11

11. S. Kumar, B. S. Williams, Q. Qin, A. W. M. Lee, Q. Hu, and J. L. Reno, “Surface-emitting distributed feedback terahertz quantum-cascade lasers in metal-metal waveguides,” Opt. Express 15(1), 113–128 (2007). [CrossRef] [PubMed]

] or weakly modulated 2D structures [3

3. L. Sirigu, R. Terazzi, M. I. Amanti, M. Giovannini, J. Faist, L. A. Dunbar, and R. Houdré, “Terahertz quantum cascade lasers based on two-dimensional photonic crystal resonators,” Opt. Express 16(8), 5206–5217 (2008). [CrossRef] [PubMed]

] exhibiting an incomplete bandgap and where the entire area is pumped. Instead, we will focus on an approach based on deeply etched pillars that provide a full TM gap, and excellent performances in terms of drive current and high temperature operation thanks to propagation losses reduction combined with the benefit that only the area with a large field overlap is pumped [12

12. H. Zhang, L. A. Dunbar, G. Scalari, R. Houdré, and J. Faist, “Terahertz photonic crystal quantum cascade lasers,” Opt. Express 15(25), 16818–16827 (2007). [CrossRef] [PubMed]

]. However, such devices do not allow surface emission as the slow light mode lies outside of the light cone [12

12. H. Zhang, L. A. Dunbar, G. Scalari, R. Houdré, and J. Faist, “Terahertz photonic crystal quantum cascade lasers,” Opt. Express 15(25), 16818–16827 (2007). [CrossRef] [PubMed]

]. We show here structures that combine slow light enhancement with vertical emission. Moreover the slow light regime is achieved over large k-vector ranges, which is a necessary condition to achieve small device size over a small number of unit cells.

Compared to near infra-red PhC band-edge lasers, achieving vertical emission with quantum cascade (QC) [13

13. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994). [CrossRef] [PubMed]

] material operating in the THz range [14

14. R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002). [CrossRef] [PubMed]

,15

15. B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics 1(9), 517–525 (2007). [CrossRef]

] needs dedicated engineering because, even if the operating Bloch states lies within the light cone, the TM polarized emission of QCLs favors in-plane emitting schemes and secondly because the currently used [16

16. B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, and J. L. Reno, “Terahertz quantum-cascade laser at λ≈100 μm using metal waveguide for mode confinement,” Appl. Phys. Lett. 83(11), 2124–2126 (2003). [CrossRef]

] double plasmon waveguide, which provides nearly unitary mode confinement, prevents vertical light extraction [12

12. H. Zhang, L. A. Dunbar, G. Scalari, R. Houdré, and J. Faist, “Terahertz photonic crystal quantum cascade lasers,” Opt. Express 15(25), 16818–16827 (2007). [CrossRef] [PubMed]

] unless the top metal layer is appropriately turned semi-transparent. In this work we make use of this apparent drawback to design a vertically emitting structure with independent index and extraction losses modulation.

For TM polarized light, the pillar-type PhC laser is the most suited structure as it exhibits complete in-plane PBGs. After BCB planarization [17

17. H. Zhang, G. Scalari, J. Faist, L. A. Dunbar, and R. Houdré, “Design and fabrication technology for high performance electrical pumped terahertz photonic crystal band edge lasers with complete photonic band gap,” J. Appl. Phys. 108(9), 093104 (2010). [CrossRef]

], a top continuous metallic layer can be deposited and provides several functionalities: i) It defines the top plasmon layer. ii) It is also used as an electrical contact, which does not suffer from potential electrical leakage [3

3. L. Sirigu, R. Terazzi, M. I. Amanti, M. Giovannini, J. Faist, L. A. Dunbar, and R. Houdré, “Terahertz quantum cascade lasers based on two-dimensional photonic crystal resonators,” Opt. Express 16(8), 5206–5217 (2008). [CrossRef] [PubMed]

,4

4. Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature 457(7226), 174–178 (2009). [CrossRef] [PubMed]

] as the pillars define a non-connected surface. iii) Finally it allows the introduction of an independent extraction loss modulation by opening small holes into the top metal layer at proper selected positions (relative to the pillar positions). These holes introduce periodic losses to the existing system while barely disturbing the overall dispersion.

2. Design

3. Device fabrication and characterization

Both circular and elliptical pillars were designed with several different photonic-lattice constants in order to ensure the overlap between the gain bandwidth and the band-edge states at the Γ-point. All the fabricated samples are from the same QC active layer whose wavelength is centred around 3.3 THz, with a gain bandwidth of 0.8 THz [22

22. M. I. Amanti, G. Scalari, R. Terazzi, M. Fischer, M. Beck, J. Faist, A. Rudra, P. Gallo, and E. Kapon, “Bound-to-continuum terahertz quantum cascade laser with a single-quantum-well phonon extraction/injection stage,” N. J. Phys. 11(12), 125022 (2009). [CrossRef]

]. The fabrication and characterization techniques are similar to that described in Refs. [12

12. H. Zhang, L. A. Dunbar, G. Scalari, R. Houdré, and J. Faist, “Terahertz photonic crystal quantum cascade lasers,” Opt. Express 15(25), 16818–16827 (2007). [CrossRef] [PubMed]

], [17

17. H. Zhang, G. Scalari, J. Faist, L. A. Dunbar, and R. Houdré, “Design and fabrication technology for high performance electrical pumped terahertz photonic crystal band edge lasers with complete photonic band gap,” J. Appl. Phys. 108(9), 093104 (2010). [CrossRef]

] and [20

20. M. I. Amanti, M. Fischer, G. Scalari, M. Beck, and J. Faist, “Low-divergence single-mode terahertz quantum cascade laser,” Nat. Photonics 3(10), 586–590 (2009). [CrossRef]

]. All the lasers were fabricated with two extraction schemes. The hole extractor is 5 μm wide in diameter, which is assumed to have no effect on the current injection. Note that rather selecting an optimized active material with narrow gain bandwidth for single mode operation, the active material structure was deliberately selected with a very broad gain bandwidth in order to ensure a proof of principle demonstration as well as a PhC singular point spectroscopy around the PBG2. As predicted, the spectra (Fig. 2(a)) of circular pillar PhC laser shows lasing at Γ band-edge frequencies but with undesired multimode behaviour (Figs. 2(a) and 3(b)
Fig. 3 Light–current–voltage (LIV) a circular pillar type photonic-crystal laser with lattice constant a = 40 μm, with extractor scheme of Holes-on-BCB. The maximum power collected from the top of the device exceeds 31 mW.
). On the contrary, elliptical pillar PhC lasers (Fig. 2(b)) show a much more efficient mode control. For smaller lattice constant a = 36 μm, the gain bandwidth is broad enough to excite both the split Γ band-edge states. The measured mode splitting energy between Γ2 and Γ3 states corresponds well to the calculated Δu = 0.0224. For larger lattice constant a = 38 μm, where the gain shifts to higher energy and overlaps with the Γ3 state only, a single mode lasing is observed. For both the circular and elliptical pillar lasers very high surface emission power from the holes-on-BCB extraction scheme was achieved. Figure 3 shows the LIV characterization for circular pillar laser of a = 40 μm. All power measurements have been performed with a terahertz absolute power meter (Thomas Keating Ltd.), which has a broad surface (6x4 cm2), placed directly in front of the cryostat window at a distance of 2.5 cm from the laser. This gives a prism of angular aperture 100 deg x 76 deg. The maximum power collected from the top of the device via the extractors is 31 mW with a differential efficiency value of 67 mW/A, which represents a significant improvement by a factor 4 compared to a double plasmon Fabry-Perot fabricated with the same active material (3 mW and 15 mW/A respectively obtained on a 0.15 mm2 device).

Finally, in this section, we discuss experimental results on a = 32 µm elliptical pillar structure to investigate the extraction loss modulation scheme. The LIV curves in Fig. 4(a)
Fig. 4 Light–current–voltage (LIV) and spectral characterizations of elliptical pillar type photonic crystal quantum cascade lasers. (a) and (b), Light–current–voltage (LIV) characteristics as a function of temperature for the two type of extractors. The maximum power collected from the top of the devices are 18.2 mW and 13.4 mW, for holes on pillars and holes on BCB extraction schemes, respectively. The maximum operating temperature for both devices is 120 K, which is the same as the standard FP ridge lasers (embedded with BCB) fabricated on the same batch with the same material. (c) and (d), Dispersion curves and lasing spectra of elliptical pillar-type photonic-crystal lasers at lattice constant a = 32 µm, with holes on pillars and holes on BCB configurations, as shown insert SEM images for (c) and (d), respectively. The intense line at u = 0.35 correspond to lasing at a M3* point.
and Fig. 4(b) show similar threshold currents and maximum operation temperatures. The difference of maximum emitted power between Fig. 3 and 4 can originate from the different lattice constant in both structures, leading to a gain material not operating at the same wavelength. The laser spectra shown in Fig. 4(c) and Fig. 4(d) are measured at different injection currents to investigate the impact of losses on the mode selection. When the injection current is low (≤ 590 mA in both cases), both lasers show mode control as expected from the elliptical pillar dispersion (refer to the elliptical dispersion in Fig. 2(b)). The spectral line positions correspond to the Γ2 mode and the M3* mode (with the elliptical pillar structure the 6-fold degenerated M points, split in two group of a 2-fold (4-fold) M (M*) points, see Fig. 2(b)). However for larger injection currents (> 590 mA), the two structures show different behaviors, the holes-on-pillars scheme turns to multimode rapidly when the injection current is increased. As previously discussed, the holes-on-pillars scheme introduces more losses to the lasing modes, and eventually increases the losses to a comparable level to the losses of the modes induced by any defect cavity or FP-like modes. As a consequence the PhC mode control is lost and the laser becomes extremely multimode. On the contrary, the PhC lasers with holes-on-BCB extraction scheme display mode control across the entire dynamic range. Preliminary far-fields were performed. Angular narrowing, more pronounced in one direction for the elliptical patterns, could be observed, however some complex structures were observed and further analysis and spectrally resolved data are necessary. It is also well known that achievement of good far-field characteristics relies strongly on a fine optimization of the overall pad shape and boundary conditions, which was not done in the present samples that were mainly designed for a proof of feasibility purpose.

4. Conclusion

Acknowledgments

We thank Maria I. Amanti for the assistant on 3D modelling and Oscar Marchat for his help on device characterization. This work was supported by the Swiss National Science Foundation and the National Center of Competence in Research, Quantum Photonics.

References and links

1.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302(5649), 1374–1377 (2003). [CrossRef] [PubMed]

2.

O. P. Marshall, V. Apostolopoulos, J. R. Freeman, R. Rungsawang, H. E. Beere, and D. A. Ritchie, “Surface-emitting photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. 93(17), 171112 (2008). [CrossRef]

3.

L. Sirigu, R. Terazzi, M. I. Amanti, M. Giovannini, J. Faist, L. A. Dunbar, and R. Houdré, “Terahertz quantum cascade lasers based on two-dimensional photonic crystal resonators,” Opt. Express 16(8), 5206–5217 (2008). [CrossRef] [PubMed]

4.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature 457(7226), 174–178 (2009). [CrossRef] [PubMed]

5.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser - a new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994). [CrossRef]

6.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972). [CrossRef]

7.

E. Kapon, A. Hardy, and A. Katzir, “The effect of complex coupling-coefficients on distributed feedback lasers,” IEEE J. Quantum Electron. 18(1), 66–71 (1982). [CrossRef]

8.

S. Nojima, “Optical-gain enhancement in two-dimensional active photonic crystals,” J. Appl. Phys. 90(2), 545–551 (2001). [CrossRef]

9.

O. Demichel, L. Mahler, T. Losco, C. Mauro, R. Green, A. Tredicucci, J. Xu, F. Beltram, H. E. Beere, D. A. Ritchie, and V. Tamosinuas, “Surface plasmon photonic structures in terahertz quantum cascade lasers,” Opt. Express 14(12), 5335–5345 (2006). [CrossRef] [PubMed]

10.

J. A. Fan, M. A. Belkin, F. Capasso, S. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Surface emitting terahertz quantum cascade laser with a double-metal waveguide,” Opt. Express 14(24), 11672–11680 (2006). [CrossRef] [PubMed]

11.

S. Kumar, B. S. Williams, Q. Qin, A. W. M. Lee, Q. Hu, and J. L. Reno, “Surface-emitting distributed feedback terahertz quantum-cascade lasers in metal-metal waveguides,” Opt. Express 15(1), 113–128 (2007). [CrossRef] [PubMed]

12.

H. Zhang, L. A. Dunbar, G. Scalari, R. Houdré, and J. Faist, “Terahertz photonic crystal quantum cascade lasers,” Opt. Express 15(25), 16818–16827 (2007). [CrossRef] [PubMed]

13.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994). [CrossRef] [PubMed]

14.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002). [CrossRef] [PubMed]

15.

B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics 1(9), 517–525 (2007). [CrossRef]

16.

B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, and J. L. Reno, “Terahertz quantum-cascade laser at λ≈100 μm using metal waveguide for mode confinement,” Appl. Phys. Lett. 83(11), 2124–2126 (2003). [CrossRef]

17.

H. Zhang, G. Scalari, J. Faist, L. A. Dunbar, and R. Houdré, “Design and fabrication technology for high performance electrical pumped terahertz photonic crystal band edge lasers with complete photonic band gap,” J. Appl. Phys. 108(9), 093104 (2010). [CrossRef]

18.

H. Zhang, G. Scalari, R. Houdré, and J. Faist, “In-plane and surface emitting high performance THz pillar type photonic crystal lasers with complete photonic bandgaps” in Proceedings of the IEEE Lasers and Electro-Optics 2009 and European Quantum Electronics Conference (CLEO Europe - EQEC2009), DOI 10.1109/CLEOE-EQEC.2009.5192585.

19.

COMSOL-Multiphysics, http://www.comsol.com.

20.

M. I. Amanti, M. Fischer, G. Scalari, M. Beck, and J. Faist, “Low-divergence single-mode terahertz quantum cascade laser,” Nat. Photonics 3(10), 586–590 (2009). [CrossRef]

21.

K. Inoue, M. Sasada, J. Kawamata, K. Sakoda, and J. W. Haus, “A two-dimensional photonic crystal laser,” Jpn. J. Appl. Phys. 38(Part 2, No. 2B), L157–L159 (1999). [CrossRef]

22.

M. I. Amanti, G. Scalari, R. Terazzi, M. Fischer, M. Beck, J. Faist, A. Rudra, P. Gallo, and E. Kapon, “Bound-to-continuum terahertz quantum cascade laser with a single-quantum-well phonon extraction/injection stage,” N. J. Phys. 11(12), 125022 (2009). [CrossRef]

OCIS Codes
(220.4000) Optical design and fabrication : Microstructure fabrication
(260.3090) Physical optics : Infrared, far
(050.5298) Diffraction and gratings : Photonic crystals
(140.5965) Lasers and laser optics : Semiconductor lasers, quantum cascade

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: March 9, 2011
Revised Manuscript: May 13, 2011
Manuscript Accepted: May 13, 2011
Published: May 17, 2011

Citation
Hua Zhang, Giacomo Scalari, Mattias Beck, Jérôme Faist, and Romuald Houdré, "Complex-coupled photonic crystal THz lasers with independent loss and refractive index modulation," Opt. Express 19, 10707-10713 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-11-10707


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References

  1. R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302(5649), 1374–1377 (2003). [CrossRef] [PubMed]
  2. O. P. Marshall, V. Apostolopoulos, J. R. Freeman, R. Rungsawang, H. E. Beere, and D. A. Ritchie, “Surface-emitting photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. 93(17), 171112 (2008). [CrossRef]
  3. L. Sirigu, R. Terazzi, M. I. Amanti, M. Giovannini, J. Faist, L. A. Dunbar, and R. Houdré, “Terahertz quantum cascade lasers based on two-dimensional photonic crystal resonators,” Opt. Express 16(8), 5206–5217 (2008). [CrossRef] [PubMed]
  4. Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature 457(7226), 174–178 (2009). [CrossRef] [PubMed]
  5. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser - a new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994). [CrossRef]
  6. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972). [CrossRef]
  7. E. Kapon, A. Hardy, and A. Katzir, “The effect of complex coupling-coefficients on distributed feedback lasers,” IEEE J. Quantum Electron. 18(1), 66–71 (1982). [CrossRef]
  8. S. Nojima, “Optical-gain enhancement in two-dimensional active photonic crystals,” J. Appl. Phys. 90(2), 545–551 (2001). [CrossRef]
  9. O. Demichel, L. Mahler, T. Losco, C. Mauro, R. Green, A. Tredicucci, J. Xu, F. Beltram, H. E. Beere, D. A. Ritchie, and V. Tamosinuas, “Surface plasmon photonic structures in terahertz quantum cascade lasers,” Opt. Express 14(12), 5335–5345 (2006). [CrossRef] [PubMed]
  10. J. A. Fan, M. A. Belkin, F. Capasso, S. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Surface emitting terahertz quantum cascade laser with a double-metal waveguide,” Opt. Express 14(24), 11672–11680 (2006). [CrossRef] [PubMed]
  11. S. Kumar, B. S. Williams, Q. Qin, A. W. M. Lee, Q. Hu, and J. L. Reno, “Surface-emitting distributed feedback terahertz quantum-cascade lasers in metal-metal waveguides,” Opt. Express 15(1), 113–128 (2007). [CrossRef] [PubMed]
  12. H. Zhang, L. A. Dunbar, G. Scalari, R. Houdré, and J. Faist, “Terahertz photonic crystal quantum cascade lasers,” Opt. Express 15(25), 16818–16827 (2007). [CrossRef] [PubMed]
  13. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994). [CrossRef] [PubMed]
  14. R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002). [CrossRef] [PubMed]
  15. B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics 1(9), 517–525 (2007). [CrossRef]
  16. B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, and J. L. Reno, “Terahertz quantum-cascade laser at λ≈100 μm using metal waveguide for mode confinement,” Appl. Phys. Lett. 83(11), 2124–2126 (2003). [CrossRef]
  17. H. Zhang, G. Scalari, J. Faist, L. A. Dunbar, and R. Houdré, “Design and fabrication technology for high performance electrical pumped terahertz photonic crystal band edge lasers with complete photonic band gap,” J. Appl. Phys. 108(9), 093104 (2010). [CrossRef]
  18. H. Zhang, G. Scalari, R. Houdré, and J. Faist, “In-plane and surface emitting high performance THz pillar type photonic crystal lasers with complete photonic bandgaps” in Proceedings of the IEEE Lasers and Electro-Optics 2009 and European Quantum Electronics Conference (CLEO Europe - EQEC2009), DOI 10.1109/CLEOE-EQEC.2009.5192585.
  19. COMSOL-Multiphysics, http://www.comsol.com .
  20. M. I. Amanti, M. Fischer, G. Scalari, M. Beck, and J. Faist, “Low-divergence single-mode terahertz quantum cascade laser,” Nat. Photonics 3(10), 586–590 (2009). [CrossRef]
  21. K. Inoue, M. Sasada, J. Kawamata, K. Sakoda, and J. W. Haus, “A two-dimensional photonic crystal laser,” Jpn. J. Appl. Phys. 38(Part 2, No. 2B), L157–L159 (1999). [CrossRef]
  22. M. I. Amanti, G. Scalari, R. Terazzi, M. Fischer, M. Beck, J. Faist, A. Rudra, P. Gallo, and E. Kapon, “Bound-to-continuum terahertz quantum cascade laser with a single-quantum-well phonon extraction/injection stage,” N. J. Phys. 11(12), 125022 (2009). [CrossRef]

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