Abstract

We present the design of mid-infrared and THz quantum cascade laser cavities formed from planar photonic crystals with a complete in-plane photonic bandgap. The design is based on a honeycomb lattice, and achieves a full in-plane photonic gap for transverse-magnetic polarized light while preserving a connected pattern for efficient electrical injection. Candidate defects modes for lasing are identified. This lattice is then used as a model system to demonstrate a novel effect: under certain conditions -that are typically satisfied in the THz range - a complete photonic gap can be obtained by the sole patterning of the top metal contact. This possibility greatly reduces the required fabrication complexity and avoids potential damage of the semiconductor active region.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]

2006 (8)

L. Diehl, D. Bour, S. Corzine, J. Zhu, G. Hfler, M. Loncar, M. Troccoli, and F. Capasso, "High-power quantum cascade lasers grown by low-pressure metal organic vapor-phase epitaxy operating in continuous wave above 400 K," Appl. Phys. Lett. 88, 201115 (2006).
[CrossRef]

S. R. Darvish, S. Slivken, A. Evans, J. S. Yu, and M. Razeghi, "Room-temperature, high-power, and continuouswave operation of distributed-feedback quantum-cascade lasers at lambda ≈ 9.6 μm," Appl. Phys. Lett. 88, 201114 (2006).
[CrossRef]

M. Schubert and F. Rana, "Analysis of Terahertz surface-emitting Quantum Cascade Lasers," IEEE J. Quantum Electron 42, 257-265 (2006).
[CrossRef]

M. Bahriz, V. Moreau, J. Palomo, R. Colombelli, D. Austin, J. Cockburn, L. Wilson, A. Krysa, and J. Roberts, "Room-temperature operation of λ = 7.5 μm surface-plasmon quantum cascade lasers," Appl. Phys. Lett. 88, 181103 (2006).
[CrossRef]

O. Demichel et al., "Surface plasmon photonic structures in terahertz quantum cascade lasers," Opt. Express 14, 5337-5345 (2006).
[CrossRef]

S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, "Terahretz Surface Plasmon-Polariton propagation and focusing on periodically corrugated metal wires," Phys. Rev. Lett. 97, 176805 (2006).
[CrossRef] [PubMed]

S. Maier, "Plasmonic field enhancement and SERS in the effective mode volume picture," Opt. Express 14, 1957-1964 (2006).
[CrossRef] [PubMed]

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, 11672-11680 (2006).
[CrossRef] [PubMed]

2005 (4)

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

L. Mahler, A. Redicucci, R. K¨ohler, F. Beltram, H. E. Beere, E. H. Linfield, and D. A. Ritchie, "High-performance operation of single-mode terahertz quantum cascade lasers with metallic gratings," Appl. Phys. Lett. 87, 181101 (2005).
[CrossRef]

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]

S. Takayama, H. Kitagawa, Y. Tanaka, T. Asano, and S. Noda, "Experimental demonstration of complete photonic band gap in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 87, 061107 (2005).
[CrossRef]

2004 (3)

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. Gmachl, D. Tennant, A. Sergent, D. Sivco, A. Cho, and F. Capasso, "Fabrication technologies for quantum cascade photonic-crystal microlasers," IOP Nanotechnology 15, 675 (2004).

R. Colombelli et al., "Quantum Cascade Photonic-Crystal Surface-Emitting Laser," Science 302, 1374 (2004).
[CrossRef]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004).
[CrossRef] [PubMed]

2003 (3)

I. Vurgaftman and J. Meyer, "Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers," IEEE J. Quantum Electron 39, 689-700 (2003).
[CrossRef]

O. Painter and K. Srinivasan, "Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis," Phys. Rev. B 68, 035110 (2003).
[CrossRef]

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

2002 (4)

K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Hwang, A. Sergent, D. Sivco, and A. Cho, "Quantum cascade lasers with double metal-semiconductor waveguide resonators," Appl. Phys. Lett. 80, 3060-3062 (2002).
[CrossRef]

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, "Multidirectionally distributed feedback photonic crystal lasers," Phys. Rev. B 65, 195306 (2002).
[CrossRef]

R. Ruppin, "Electromagnetic energy density in a dispersive and absorptive material," Phys. Lett. A 299, 309-312 (2002).
[CrossRef]

K. Srinivasan and O. Painter, "Momentum space design of high-Q photonic crystal optical cavities," Opt. Express 10, 670-684 (2002).
[PubMed]

2001 (2)

2000 (1)

W. Schrenk, N. Finger, S. Gianordoli, L. Hvozdara, G. Strasser, and E. Gornik, "Surface-emitting distributed feedback quantum-cascade lasers," Appl. Phys. Lett. 77, 2086-2088 (2000).
[CrossRef]

1999 (2)

D. Hofstetter, J. Faist, M. Beck, and U. Oesterle, "Surface-emitting 10.1 μm quantum-cascade distributed feedback lasers," Appl. Phys. Lett. 75, 3769-3771 (1999).
[CrossRef]

O. Painter, J. Vučković, and A. Scherer, "Defect modes of a two-dimensional Photonic Crystal in an optically thin dielectric slab," J. Opt. Soc. Am. B 16, 275-285 (1999).
[CrossRef]

1998 (2)

1993 (1)

1991 (1)

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant," Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

1983 (1)

1981 (1)

G. Mur, "Asorbing Boundary Conditions for the finite-difference approximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

Alexander, R.

Andrews, S. R.

S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, "Terahretz Surface Plasmon-Polariton propagation and focusing on periodically corrugated metal wires," Phys. Rev. Lett. 97, 176805 (2006).
[CrossRef] [PubMed]

Asano, T.

S. Takayama, H. Kitagawa, Y. Tanaka, T. Asano, and S. Noda, "Experimental demonstration of complete photonic band gap in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 87, 061107 (2005).
[CrossRef]

Austin, D.

M. Bahriz, V. Moreau, J. Palomo, R. Colombelli, D. Austin, J. Cockburn, L. Wilson, A. Krysa, and J. Roberts, "Room-temperature operation of λ = 7.5 μm surface-plasmon quantum cascade lasers," Appl. Phys. Lett. 88, 181103 (2006).
[CrossRef]

Bahriz, M.

M. Bahriz, V. Moreau, J. Palomo, R. Colombelli, D. Austin, J. Cockburn, L. Wilson, A. Krysa, and J. Roberts, "Room-temperature operation of λ = 7.5 μm surface-plasmon quantum cascade lasers," Appl. Phys. Lett. 88, 181103 (2006).
[CrossRef]

Beck, M.

D. Hofstetter, J. Faist, M. Beck, and U. Oesterle, "Surface-emitting 10.1 μm quantum-cascade distributed feedback lasers," Appl. Phys. Lett. 75, 3769-3771 (1999).
[CrossRef]

Belkin, M. A.

Bell, R.

Bell, R. R.

Bell, S. E.

Bour, D.

L. Diehl, D. Bour, S. Corzine, J. Zhu, G. Hfler, M. Loncar, M. Troccoli, and F. Capasso, "High-power quantum cascade lasers grown by low-pressure metal organic vapor-phase epitaxy operating in continuous wave above 400 K," Appl. Phys. Lett. 88, 201115 (2006).
[CrossRef]

Brommer, K. D.

Callebaut, H.

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

Capasso, F.

L. Diehl, D. Bour, S. Corzine, J. Zhu, G. Hfler, M. Loncar, M. Troccoli, and F. Capasso, "High-power quantum cascade lasers grown by low-pressure metal organic vapor-phase epitaxy operating in continuous wave above 400 K," Appl. Phys. Lett. 88, 201115 (2006).
[CrossRef]

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, 11672-11680 (2006).
[CrossRef] [PubMed]

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. Gmachl, D. Tennant, A. Sergent, D. Sivco, A. Cho, and F. Capasso, "Fabrication technologies for quantum cascade photonic-crystal microlasers," IOP Nanotechnology 15, 675 (2004).

K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Hwang, A. Sergent, D. Sivco, and A. Cho, "Quantum cascade lasers with double metal-semiconductor waveguide resonators," Appl. Phys. Lett. 80, 3060-3062 (2002).
[CrossRef]

C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho, Rep. Prog. Phys 64, 1533 (2001).
[CrossRef]

C. Sirtori, C. Gmachl, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, "Long-wavelength (λ≈8−11.5 μm) semiconductor lasers with waveguides based on surface plasmons," Opt. Lett. 23, 1366 (1998).
[CrossRef]

Cho, A.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. Gmachl, D. Tennant, A. Sergent, D. Sivco, A. Cho, and F. Capasso, "Fabrication technologies for quantum cascade photonic-crystal microlasers," IOP Nanotechnology 15, 675 (2004).

K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Hwang, A. Sergent, D. Sivco, and A. Cho, "Quantum cascade lasers with double metal-semiconductor waveguide resonators," Appl. Phys. Lett. 80, 3060-3062 (2002).
[CrossRef]

Cho, A. Y.

Chutinan, A.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, "Multidirectionally distributed feedback photonic crystal lasers," Phys. Rev. B 65, 195306 (2002).
[CrossRef]

Cockburn, J.

M. Bahriz, V. Moreau, J. Palomo, R. Colombelli, D. Austin, J. Cockburn, L. Wilson, A. Krysa, and J. Roberts, "Room-temperature operation of λ = 7.5 μm surface-plasmon quantum cascade lasers," Appl. Phys. Lett. 88, 181103 (2006).
[CrossRef]

Colombelli, R.

M. Bahriz, V. Moreau, J. Palomo, R. Colombelli, D. Austin, J. Cockburn, L. Wilson, A. Krysa, and J. Roberts, "Room-temperature operation of λ = 7.5 μm surface-plasmon quantum cascade lasers," Appl. Phys. Lett. 88, 181103 (2006).
[CrossRef]

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. Gmachl, D. Tennant, A. Sergent, D. Sivco, A. Cho, and F. Capasso, "Fabrication technologies for quantum cascade photonic-crystal microlasers," IOP Nanotechnology 15, 675 (2004).

R. Colombelli et al., "Quantum Cascade Photonic-Crystal Surface-Emitting Laser," Science 302, 1374 (2004).
[CrossRef]

K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Hwang, A. Sergent, D. Sivco, and A. Cho, "Quantum cascade lasers with double metal-semiconductor waveguide resonators," Appl. Phys. Lett. 80, 3060-3062 (2002).
[CrossRef]

Corzine, S.

L. Diehl, D. Bour, S. Corzine, J. Zhu, G. Hfler, M. Loncar, M. Troccoli, and F. Capasso, "High-power quantum cascade lasers grown by low-pressure metal organic vapor-phase epitaxy operating in continuous wave above 400 K," Appl. Phys. Lett. 88, 201115 (2006).
[CrossRef]

D’Urso, B.

Darvish, S. R.

S. R. Darvish, S. Slivken, A. Evans, J. S. Yu, and M. Razeghi, "Room-temperature, high-power, and continuouswave operation of distributed-feedback quantum-cascade lasers at lambda ≈ 9.6 μm," Appl. Phys. Lett. 88, 201114 (2006).
[CrossRef]

Davies, A. G.

Demichel, O.

O. Demichel et al., "Surface plasmon photonic structures in terahertz quantum cascade lasers," Opt. Express 14, 5337-5345 (2006).
[CrossRef]

Diehl, L.

L. Diehl, D. Bour, S. Corzine, J. Zhu, G. Hfler, M. Loncar, M. Troccoli, and F. Capasso, "High-power quantum cascade lasers grown by low-pressure metal organic vapor-phase epitaxy operating in continuous wave above 400 K," Appl. Phys. Lett. 88, 201115 (2006).
[CrossRef]

Evans, A.

S. R. Darvish, S. Slivken, A. Evans, J. S. Yu, and M. Razeghi, "Room-temperature, high-power, and continuouswave operation of distributed-feedback quantum-cascade lasers at lambda ≈ 9.6 μm," Appl. Phys. Lett. 88, 201114 (2006).
[CrossRef]

Faist, J.

D. Hofstetter, J. Faist, M. Beck, and U. Oesterle, "Surface-emitting 10.1 μm quantum-cascade distributed feedback lasers," Appl. Phys. Lett. 75, 3769-3771 (1999).
[CrossRef]

C. Sirtori, C. Gmachl, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, "Long-wavelength (λ≈8−11.5 μm) semiconductor lasers with waveguides based on surface plasmons," Opt. Lett. 23, 1366 (1998).
[CrossRef]

Fan, J. A.

Finger, N.

W. Schrenk, N. Finger, S. Gianordoli, L. Hvozdara, G. Strasser, and E. Gornik, "Surface-emitting distributed feedback quantum-cascade lasers," Appl. Phys. Lett. 77, 2086-2088 (2000).
[CrossRef]

Garcia-Vidal, F. J.

S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, "Terahretz Surface Plasmon-Polariton propagation and focusing on periodically corrugated metal wires," Phys. Rev. Lett. 97, 176805 (2006).
[CrossRef] [PubMed]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004).
[CrossRef] [PubMed]

Gianordoli, S.

W. Schrenk, N. Finger, S. Gianordoli, L. Hvozdara, G. Strasser, and E. Gornik, "Surface-emitting distributed feedback quantum-cascade lasers," Appl. Phys. Lett. 77, 2086-2088 (2000).
[CrossRef]

Gmachl, C.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. Gmachl, D. Tennant, A. Sergent, D. Sivco, A. Cho, and F. Capasso, "Fabrication technologies for quantum cascade photonic-crystal microlasers," IOP Nanotechnology 15, 675 (2004).

K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Hwang, A. Sergent, D. Sivco, and A. Cho, "Quantum cascade lasers with double metal-semiconductor waveguide resonators," Appl. Phys. Lett. 80, 3060-3062 (2002).
[CrossRef]

C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho, Rep. Prog. Phys 64, 1533 (2001).
[CrossRef]

C. Sirtori, C. Gmachl, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, "Long-wavelength (λ≈8−11.5 μm) semiconductor lasers with waveguides based on surface plasmons," Opt. Lett. 23, 1366 (1998).
[CrossRef]

Gornik, E.

W. Schrenk, N. Finger, S. Gianordoli, L. Hvozdara, G. Strasser, and E. Gornik, "Surface-emitting distributed feedback quantum-cascade lasers," Appl. Phys. Lett. 77, 2086-2088 (2000).
[CrossRef]

Hfler, G.

L. Diehl, D. Bour, S. Corzine, J. Zhu, G. Hfler, M. Loncar, M. Troccoli, and F. Capasso, "High-power quantum cascade lasers grown by low-pressure metal organic vapor-phase epitaxy operating in continuous wave above 400 K," Appl. Phys. Lett. 88, 201115 (2006).
[CrossRef]

Hofstetter, D.

D. Hofstetter, J. Faist, M. Beck, and U. Oesterle, "Surface-emitting 10.1 μm quantum-cascade distributed feedback lasers," Appl. Phys. Lett. 75, 3769-3771 (1999).
[CrossRef]

Hu, Q.

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

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]

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

Hutchinson, A. L.

Hvozdara, L.

W. Schrenk, N. Finger, S. Gianordoli, L. Hvozdara, G. Strasser, and E. Gornik, "Surface-emitting distributed feedback quantum-cascade lasers," Appl. Phys. Lett. 77, 2086-2088 (2000).
[CrossRef]

Hwang, H.

K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Hwang, A. Sergent, D. Sivco, and A. Cho, "Quantum cascade lasers with double metal-semiconductor waveguide resonators," Appl. Phys. Lett. 80, 3060-3062 (2002).
[CrossRef]

Imada, M.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, "Multidirectionally distributed feedback photonic crystal lasers," Phys. Rev. B 65, 195306 (2002).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Khanna, S.

Kitagawa, H.

S. Takayama, H. Kitagawa, Y. Tanaka, T. Asano, and S. Noda, "Experimental demonstration of complete photonic band gap in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 87, 061107 (2005).
[CrossRef]

Kohen, S.

S. Kohen, B. S. Williams, and Q. Hu, "Electromagnetic modeling of terahertz quantum cascade laser waveguides and resonators," J. Appl. Phys. 97, 053106 (2005).
[CrossRef]

Krysa, A.

M. Bahriz, V. Moreau, J. Palomo, R. Colombelli, D. Austin, J. Cockburn, L. Wilson, A. Krysa, and J. Roberts, "Room-temperature operation of λ = 7.5 μm surface-plasmon quantum cascade lasers," Appl. Phys. Lett. 88, 181103 (2006).
[CrossRef]

Kumar, S.

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

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

Lachab, M.

Linfield, E. H.

Loncar, M.

L. Diehl, D. Bour, S. Corzine, J. Zhu, G. Hfler, M. Loncar, M. Troccoli, and F. Capasso, "High-power quantum cascade lasers grown by low-pressure metal organic vapor-phase epitaxy operating in continuous wave above 400 K," Appl. Phys. Lett. 88, 201115 (2006).
[CrossRef]

Long, L.

Mahler, L.

L. Mahler, A. Redicucci, R. K¨ohler, F. Beltram, H. E. Beere, E. H. Linfield, and D. A. Ritchie, "High-performance operation of single-mode terahertz quantum cascade lasers with metallic gratings," Appl. Phys. Lett. 87, 181101 (2005).
[CrossRef]

Maier, S.

Maier, S. A.

S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, "Terahretz Surface Plasmon-Polariton propagation and focusing on periodically corrugated metal wires," Phys. Rev. Lett. 97, 176805 (2006).
[CrossRef] [PubMed]

Martin-Moreno, L.

S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, "Terahretz Surface Plasmon-Polariton propagation and focusing on periodically corrugated metal wires," Phys. Rev. Lett. 97, 176805 (2006).
[CrossRef] [PubMed]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004).
[CrossRef] [PubMed]

Meade, R. D.

Meyer, J.

I. Vurgaftman and J. Meyer, "Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers," IEEE J. Quantum Electron 39, 689-700 (2003).
[CrossRef]

Mochizuki, M.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, "Multidirectionally distributed feedback photonic crystal lasers," Phys. Rev. B 65, 195306 (2002).
[CrossRef]

Moreau, V.

M. Bahriz, V. Moreau, J. Palomo, R. Colombelli, D. Austin, J. Cockburn, L. Wilson, A. Krysa, and J. Roberts, "Room-temperature operation of λ = 7.5 μm surface-plasmon quantum cascade lasers," Appl. Phys. Lett. 88, 181103 (2006).
[CrossRef]

Mur, G.

G. Mur, "Asorbing Boundary Conditions for the finite-difference approximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

Mysyrowicz, A.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant," Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

Noda, S.

S. Takayama, H. Kitagawa, Y. Tanaka, T. Asano, and S. Noda, "Experimental demonstration of complete photonic band gap in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 87, 061107 (2005).
[CrossRef]

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, "Multidirectionally distributed feedback photonic crystal lasers," Phys. Rev. B 65, 195306 (2002).
[CrossRef]

O’Brien, J.

Oesterle, U.

D. Hofstetter, J. Faist, M. Beck, and U. Oesterle, "Surface-emitting 10.1 μm quantum-cascade distributed feedback lasers," Appl. Phys. Lett. 75, 3769-3771 (1999).
[CrossRef]

Ordal, M.

Painter, O.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. Gmachl, D. Tennant, A. Sergent, D. Sivco, A. Cho, and F. Capasso, "Fabrication technologies for quantum cascade photonic-crystal microlasers," IOP Nanotechnology 15, 675 (2004).

O. Painter and K. Srinivasan, "Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis," Phys. Rev. B 68, 035110 (2003).
[CrossRef]

K. Srinivasan and O. Painter, "Momentum space design of high-Q photonic crystal optical cavities," Opt. Express 10, 670-684 (2002).
[PubMed]

O. Painter, J. Vučković, and A. Scherer, "Defect modes of a two-dimensional Photonic Crystal in an optically thin dielectric slab," J. Opt. Soc. Am. B 16, 275-285 (1999).
[CrossRef]

B. D’Urso, O. Painter, J. O’Brien, T. Tombrello, A. Scherer, and A. Yariv, "Modal reflectivity in finite-depth two-dimensional photonic-crystal microcavitites," J. Opt. Soc. Am. B 15, 1155-1159 (1998).
[CrossRef]

Palomo, J.

M. Bahriz, V. Moreau, J. Palomo, R. Colombelli, D. Austin, J. Cockburn, L. Wilson, A. Krysa, and J. Roberts, "Room-temperature operation of λ = 7.5 μm surface-plasmon quantum cascade lasers," Appl. Phys. Lett. 88, 181103 (2006).
[CrossRef]

Pendry, J. B.

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004).
[CrossRef] [PubMed]

Prade, B.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant," Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

Rana, F.

M. Schubert and F. Rana, "Analysis of Terahertz surface-emitting Quantum Cascade Lasers," IEEE J. Quantum Electron 42, 257-265 (2006).
[CrossRef]

Rappe, A. M.

Razeghi, M.

S. R. Darvish, S. Slivken, A. Evans, J. S. Yu, and M. Razeghi, "Room-temperature, high-power, and continuouswave operation of distributed-feedback quantum-cascade lasers at lambda ≈ 9.6 μm," Appl. Phys. Lett. 88, 201114 (2006).
[CrossRef]

Redicucci, A.

L. Mahler, A. Redicucci, R. K¨ohler, F. Beltram, H. E. Beere, E. H. Linfield, and D. A. Ritchie, "High-performance operation of single-mode terahertz quantum cascade lasers with metallic gratings," Appl. Phys. Lett. 87, 181101 (2005).
[CrossRef]

Reno, J.

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

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

Roberts, J.

M. Bahriz, V. Moreau, J. Palomo, R. Colombelli, D. Austin, J. Cockburn, L. Wilson, A. Krysa, and J. Roberts, "Room-temperature operation of λ = 7.5 μm surface-plasmon quantum cascade lasers," Appl. Phys. Lett. 88, 181103 (2006).
[CrossRef]

Ruppin, R.

R. Ruppin, "Electromagnetic energy density in a dispersive and absorptive material," Phys. Lett. A 299, 309-312 (2002).
[CrossRef]

Scherer, A.

Schrenk, W.

W. Schrenk, N. Finger, S. Gianordoli, L. Hvozdara, G. Strasser, and E. Gornik, "Surface-emitting distributed feedback quantum-cascade lasers," Appl. Phys. Lett. 77, 2086-2088 (2000).
[CrossRef]

Schubert, M.

M. Schubert and F. Rana, "Analysis of Terahertz surface-emitting Quantum Cascade Lasers," IEEE J. Quantum Electron 42, 257-265 (2006).
[CrossRef]

Sergent, A.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. Gmachl, D. Tennant, A. Sergent, D. Sivco, A. Cho, and F. Capasso, "Fabrication technologies for quantum cascade photonic-crystal microlasers," IOP Nanotechnology 15, 675 (2004).

K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Hwang, A. Sergent, D. Sivco, and A. Cho, "Quantum cascade lasers with double metal-semiconductor waveguide resonators," Appl. Phys. Lett. 80, 3060-3062 (2002).
[CrossRef]

Sirtori, C.

Sivco, D.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. Gmachl, D. Tennant, A. Sergent, D. Sivco, A. Cho, and F. Capasso, "Fabrication technologies for quantum cascade photonic-crystal microlasers," IOP Nanotechnology 15, 675 (2004).

K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Hwang, A. Sergent, D. Sivco, and A. Cho, "Quantum cascade lasers with double metal-semiconductor waveguide resonators," Appl. Phys. Lett. 80, 3060-3062 (2002).
[CrossRef]

Sivco, D. L.

Slivken, S.

S. R. Darvish, S. Slivken, A. Evans, J. S. Yu, and M. Razeghi, "Room-temperature, high-power, and continuouswave operation of distributed-feedback quantum-cascade lasers at lambda ≈ 9.6 μm," Appl. Phys. Lett. 88, 201114 (2006).
[CrossRef]

Srinivasan, K.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. Gmachl, D. Tennant, A. Sergent, D. Sivco, A. Cho, and F. Capasso, "Fabrication technologies for quantum cascade photonic-crystal microlasers," IOP Nanotechnology 15, 675 (2004).

O. Painter and K. Srinivasan, "Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis," Phys. Rev. B 68, 035110 (2003).
[CrossRef]

K. Srinivasan and O. Painter, "Momentum space design of high-Q photonic crystal optical cavities," Opt. Express 10, 670-684 (2002).
[PubMed]

Strasser, G.

W. Schrenk, N. Finger, S. Gianordoli, L. Hvozdara, G. Strasser, and E. Gornik, "Surface-emitting distributed feedback quantum-cascade lasers," Appl. Phys. Lett. 77, 2086-2088 (2000).
[CrossRef]

Takayama, S.

S. Takayama, H. Kitagawa, Y. Tanaka, T. Asano, and S. Noda, "Experimental demonstration of complete photonic band gap in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 87, 061107 (2005).
[CrossRef]

Tanaka, Y.

S. Takayama, H. Kitagawa, Y. Tanaka, T. Asano, and S. Noda, "Experimental demonstration of complete photonic band gap in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 87, 061107 (2005).
[CrossRef]

Tennant, D.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. Gmachl, D. Tennant, A. Sergent, D. Sivco, A. Cho, and F. Capasso, "Fabrication technologies for quantum cascade photonic-crystal microlasers," IOP Nanotechnology 15, 675 (2004).

Tombrello, T.

Troccoli, M.

L. Diehl, D. Bour, S. Corzine, J. Zhu, G. Hfler, M. Loncar, M. Troccoli, and F. Capasso, "High-power quantum cascade lasers grown by low-pressure metal organic vapor-phase epitaxy operating in continuous wave above 400 K," Appl. Phys. Lett. 88, 201115 (2006).
[CrossRef]

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. Gmachl, D. Tennant, A. Sergent, D. Sivco, A. Cho, and F. Capasso, "Fabrication technologies for quantum cascade photonic-crystal microlasers," IOP Nanotechnology 15, 675 (2004).

Unterrainer, K.

K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Hwang, A. Sergent, D. Sivco, and A. Cho, "Quantum cascade lasers with double metal-semiconductor waveguide resonators," Appl. Phys. Lett. 80, 3060-3062 (2002).
[CrossRef]

Vinet, J. Y.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant," Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

Vuckovic, J.

Vurgaftman, I.

I. Vurgaftman and J. Meyer, "Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers," IEEE J. Quantum Electron 39, 689-700 (2003).
[CrossRef]

Ward, C. A.

Williams, B.

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

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

Williams, B. S.

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]

Wilson, L.

M. Bahriz, V. Moreau, J. Palomo, R. Colombelli, D. Austin, J. Cockburn, L. Wilson, A. Krysa, and J. Roberts, "Room-temperature operation of λ = 7.5 μm surface-plasmon quantum cascade lasers," Appl. Phys. Lett. 88, 181103 (2006).
[CrossRef]

Yariv, A.

Yu, J. S.

S. R. Darvish, S. Slivken, A. Evans, J. S. Yu, and M. Razeghi, "Room-temperature, high-power, and continuouswave operation of distributed-feedback quantum-cascade lasers at lambda ≈ 9.6 μm," Appl. Phys. Lett. 88, 201114 (2006).
[CrossRef]

Zhu, J.

L. Diehl, D. Bour, S. Corzine, J. Zhu, G. Hfler, M. Loncar, M. Troccoli, and F. Capasso, "High-power quantum cascade lasers grown by low-pressure metal organic vapor-phase epitaxy operating in continuous wave above 400 K," Appl. Phys. Lett. 88, 201115 (2006).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (9)

L. Mahler, A. Redicucci, R. K¨ohler, F. Beltram, H. E. Beere, E. H. Linfield, and D. A. Ritchie, "High-performance operation of single-mode terahertz quantum cascade lasers with metallic gratings," Appl. Phys. Lett. 87, 181101 (2005).
[CrossRef]

D. Hofstetter, J. Faist, M. Beck, and U. Oesterle, "Surface-emitting 10.1 μm quantum-cascade distributed feedback lasers," Appl. Phys. Lett. 75, 3769-3771 (1999).
[CrossRef]

W. Schrenk, N. Finger, S. Gianordoli, L. Hvozdara, G. Strasser, and E. Gornik, "Surface-emitting distributed feedback quantum-cascade lasers," Appl. Phys. Lett. 77, 2086-2088 (2000).
[CrossRef]

L. Diehl, D. Bour, S. Corzine, J. Zhu, G. Hfler, M. Loncar, M. Troccoli, and F. Capasso, "High-power quantum cascade lasers grown by low-pressure metal organic vapor-phase epitaxy operating in continuous wave above 400 K," Appl. Phys. Lett. 88, 201115 (2006).
[CrossRef]

S. R. Darvish, S. Slivken, A. Evans, J. S. Yu, and M. Razeghi, "Room-temperature, high-power, and continuouswave operation of distributed-feedback quantum-cascade lasers at lambda ≈ 9.6 μm," Appl. Phys. Lett. 88, 201114 (2006).
[CrossRef]

S. Takayama, H. Kitagawa, Y. Tanaka, T. Asano, and S. Noda, "Experimental demonstration of complete photonic band gap in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 87, 061107 (2005).
[CrossRef]

M. Bahriz, V. Moreau, J. Palomo, R. Colombelli, D. Austin, J. Cockburn, L. Wilson, A. Krysa, and J. Roberts, "Room-temperature operation of λ = 7.5 μm surface-plasmon quantum cascade lasers," Appl. Phys. Lett. 88, 181103 (2006).
[CrossRef]

K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Hwang, A. Sergent, D. Sivco, and A. Cho, "Quantum cascade lasers with double metal-semiconductor waveguide resonators," Appl. Phys. Lett. 80, 3060-3062 (2002).
[CrossRef]

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

IEEE J. Quantum Electron (2)

I. Vurgaftman and J. Meyer, "Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers," IEEE J. Quantum Electron 39, 689-700 (2003).
[CrossRef]

M. Schubert and F. Rana, "Analysis of Terahertz surface-emitting Quantum Cascade Lasers," IEEE J. Quantum Electron 42, 257-265 (2006).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

G. Mur, "Asorbing Boundary Conditions for the finite-difference approximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

IOP Nanotechnology (1)

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. Gmachl, D. Tennant, A. Sergent, D. Sivco, A. Cho, and F. Capasso, "Fabrication technologies for quantum cascade photonic-crystal microlasers," IOP Nanotechnology 15, 675 (2004).

J. Appl. Phys. (1)

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]

J. Opt. Soc. Am. B (3)

Opt. Express (6)

Opt. Lett. (1)

Phys. Lett. A (1)

R. Ruppin, "Electromagnetic energy density in a dispersive and absorptive material," Phys. Lett. A 299, 309-312 (2002).
[CrossRef]

Phys. Rev. B (3)

O. Painter and K. Srinivasan, "Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis," Phys. Rev. B 68, 035110 (2003).
[CrossRef]

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant," Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, "Multidirectionally distributed feedback photonic crystal lasers," Phys. Rev. B 65, 195306 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, "Terahretz Surface Plasmon-Polariton propagation and focusing on periodically corrugated metal wires," Phys. Rev. Lett. 97, 176805 (2006).
[CrossRef] [PubMed]

Rep. Prog. Phys (1)

C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho, Rep. Prog. Phys 64, 1533 (2001).
[CrossRef]

Science (2)

R. Colombelli et al., "Quantum Cascade Photonic-Crystal Surface-Emitting Laser," Science 302, 1374 (2004).
[CrossRef]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004).
[CrossRef] [PubMed]

Other (6)

The symmetry of the mode is that of a ŷ-polarized dipole in the plane of the MIM waveguide. It is degenerate with a second dipole-like mode with x-polarization.

P. Yeh, Optical Waves in Layered Media (John Wiley and Sons, 2005).

K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. Tennant, A. Sergent, D. Sivco, A. Cho, M. Troccoli, and C. F, "Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser," Appl. Phys. Lett. 84, 4164-4166 (2004).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals (Princeton University Press, Princeton, 1995).

A similar phenomenon occurs in guided membrane PC structures, where it is known that the extent of the photonic gap depends on the membrane thickness. However, in the dielectric membrane structures, beyond a critical membrane thickness further reduction in thickness does not increase the bandgap due to a loss of mode localization in the dielectric membrane. The double-metal waveguide structure does not suffer from such a loss of confinement.

H. Raether, Surface Plasmons, Springer-Verlag Tracts in Modern Physics (Springer-Verlag, New York, 1988) Vol. 3.

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Figures (14)

Fig. 1.
Fig. 1.

Structure of the honeycomb lattice . (a) Two-dimensional honeycomb lattice with two “dielectric atoms” (A and B) per unit cell. a 1 = (a,0) and a 2 = (a/2,√3a/2) are the principal lattice vectors (a = |a 1| = |a 2|),r is the hole radius. (b) Two-dimensional reciprocal space for the honeycomb lattice. The reduced Brillouin zone is shaded in blue. Reciprocal lattice vectors are superpositions of G 1 = (0,4π/π3a) and G 2 = (2π/a,2π/√3a).

Fig. 2.
Fig. 2.

Photonic band calculations for a 2D honeycomb lattice of air holes in a dielectric lattice (ε = 11.22). (a) Photonic bandstructure for TM polarization with r/a = 0.234. (b) Complete band gap (shaded blue region) as a function of r/a. A full TM gap exists only for r/a > 0.18

Fig. 3.
Fig. 3.

(a) Illustration of the 2D honeycomb lattice defect cavity supercell (background dielectric material shown as grey, air holes shown as white). The principle lattice vectors are a 1, a 2, where |a 1| = |a 2| = a. The defect region consists of removal of the central hexagon of air holes (central hexagon shown as a dashed line). (b) Folded TM photonic bandstructure for a 20-period supercell of the defect cavity shown in (a) with r/a = 0.24. The blue-lines correspond to the folded bandstructure of the honeycomb lattice. The red lines are the defect frequency levels. Field plots of the Ez component of the localized (c) hexapole and (d-e) dipole-like defect modes.

Fig. 4.
Fig. 4.

Normalized intensity profile of the fundamental optical mode computed for a quantum cascade laser dielectric waveguide (a) and a surface-plasmon waveguide (b) designed for a wavelength of 8 μm. The shaded green areas indicate the stack of active regions and injectors. The vertical dashed line indicates the position of the device surface. In the dielectric case the thickness of the epitaxially grown material is ≈6 μm. The origin of the abscissa is at the air-semiconductor interface. In the surface-plasmon case the thickness is instead ≈ 3 μm. The optical confinement factor is indicated by Γ.

Fig. 5.
Fig. 5.

(a) Surface-plasmon optical mode profile superimposed on a sketch of the vertical section of a mid-IR photonic crystal QC laser with air holes extending 4.7 μm deep into the semiconductor. (b) 3D-FDTD calculated electric field intensity cross-section of a defect mode (top-view shown in Fig. 6(a)) of the deeply patterned honeycomb photonic lattice. The vertical mode profile shows a similar behavior as the 1D surface-plasmon simulation of (a), whereas the confinement in the in-plane direction is provided by distributed Bragg reflection of the photonic lattice.

Fig. 6.
Fig. 6.

In-plane mode profiles (Ez ) for the (a) x-dipole, (b)y-dipole, and (c) hexapole defect modes of the surface-plasmon vertical waveguide structure with an “etched” hexagonal defect cavity in the honeycomb lattice.

Fig. 7.
Fig. 7.

(a) Schematic showing the cross-section of a MIM waveguide (metal layers are assumed semi-infinite). (b) Ez normal electric field component, (c) Ex longitudinal field component, and (c) Hy magnetic field plot of the even parity guided surface mode at λ = 100 μm for an active region core thickness of La = 1 μm. (e) Dispersion diagram of the double-metal (Au) waveguide structure (La = 1 μm). The core is modeled with a constant refractive index of na = 3.59, while the metal layers are modeled using a Drude-Lorentz model for Au (background dielectric constant ε b = 9.54, plasmon frequency ω p = 1.35 × 1016 rad/s, and relaxation time 8 × 10-15 s).

Fig. 8.
Fig. 8.

Schematic of the honeycomb lattice and MIM structure used in FDTD modeling. (a) Top view of the lattice with a central defect consisting of the removing of the central hexagon of air holes. (b) Cross-section of the simulated structure. Only the top metal layer (perfect conductor) is patterned. The displayed layer thicknesses are for an operating wavelength of λ0 = 100 μm and a normalized frequency of a0 = 0.17 within the bandgap of the honeycomb lattice.

Fig. 9.
Fig. 9.

FDTD calculated bandstructure of a patterned MIM waveguide with a normalized dielectric core thickness of L̄ a =La /a = 0.176 and a normalized air hole radius of r/a = 0.25 in the top metal contact. Metal layers are simulated as perfect conducting boundaries. At an operating wavelength of λ0 = 100 μm within the photonic bandgap (a0 = 0.17), the corresponding physical sizes of the patterned double-metal waveguide are {a, r, La } = {17, 4.25, 3}μm.

Fig. 10.
Fig. 10.

(a) Valence and conduction band dispersion between the X and J points for varying active region core thickness (L̄ a = {0.088(◦),0.176(×),0.294(◇),0.412(▫)}; nominal physical thicknesses La = {1.5(◦),3(×),5(◇),7(▫)} μm). The symbols correspond to data points from FDTD simulations and the lines are guides to the eye. The normalized hole radius for all simulations was fixed at r/a = 0.25. Note that the complete band gap shrinks as device thickness increases, and is closed for L̄ a > 0.294 (La > 5 μm).

Fig. 11.
Fig. 11.

High-frequency (a-c) and low-frequency (d-f) gap mode field plots (Ez ) at the X-point of the honeycomb lattice for the patterned double-metal waveguide structure of Fig. 10 with L̄ a = 0.294 (La = 5 μm). The in-plane field plots of (a,d) correspond to a section through the core of the MIM structure just below the patterned top metal contact. The dashed white lines in (a,d) and (b,e) indicate the position of the sections used in the field plots of (b,e) and (c,f), respectively.

Fig. 12.
Fig. 12.

In-plane mode profile of the ŷ-polarized dipole-like defect mode for a point defect consisting of the removal of the central 6 air-hole patterns in the top metal contact (see Fig. 8). (a) Real space profile of Ez and (b) magnitude of the spatial Fourier transform of Ez for an in-plane cut through the middle of the MIM waveguide.

Fig. 13.
Fig. 13.

Increase of Q with the number of PC periods for an MIM structure with L a = 3 μm (L̄ a = 0.176) thick active region. The diamonds show the results of FDTD simulations and the solid line is a least squares fit to these results. The PC cavities were chosen to be approximately square, so that the number of lattice periods (nx × ny ) used in the FDTD simulations were 3×2, 5×3, 7×4, 9×5 and 11×6 (referring to Fig. 8(a), a “period” in the x̂ direction is taken as a, whereas in the ŷ direction it is √3a). The abscissa of the plot refers to the number of periods (nx ) in the x̂ direction.

Fig. 14.
Fig. 14.

Effective Q-factor due to metal waveguide loss (Qm , blue line) and effective energy velocity index (nE = C E , red line) versus active region thickness (La ) for a 1D MIM waveguide with Au metal guiding layers at λ0 = 100 μm.

Tables (3)

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Table 1. Layer structure for the mid-IR SP QC laser[19] modeled in the 3D FDTD simulations. Nominal operating wavelength is λ ≈ 8 μm.

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Table 2. Comparison between 2D (PWE) and 3D (FDTD) simulated defect modes.

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Table 3. Effective volume (V eff), metal-absorption-limited loss coefficient (α m ), Q-factor (Qm ), and guided wave energy velocity νE as a function of the active region thickness at a nominal vacuum wavelength of λ = 100 μm.

Equations (5)

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α 4 π λ n m n d 3 k m 3 ,
1 Q rad = 1 Q + 1 Q ,
V eff = ε E 2 dV max [ ε E 2 ] ,
Q m = ω 0 2 ν E Im ( β ) ,
ν E S d z W d z ,

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