Abstract

Gain/loss dispersion characteristics of photonic modes in a two-dimensional photonic crystal (PC) are demonstrated for the first time, to our knowledge. The dispersion analysis is based on an improved plane-wave expansion method that includes the gain/loss factor along the propagation direction of each plane wave and treats surface-emission properties. PC lasers operating at terahertz frequency are considered for the numerical calculation. Our analysis indicates that although PC lasers with different lattice periods possess almost the same photonic band structure, their gain/loss dispersion characteristics are significantly different. The systematic study on gain/loss dispersion characteristics reveals the optimum PC structure to obtain high-performance surface-emitting lasers.

© 2010 Optical Society of America

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2008 (3)

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445–447 (2008).
[CrossRef]

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

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, 171112 (2008).
[CrossRef]

2007 (3)

G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101, 081726 (2007).
[CrossRef]

S. Brand, R. A. Abram, and M. A. Kaliteevski, “Complex photonic band structure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. B 75, 035102 (2007).
[CrossRef]

K. Sakai, E. Miyai, and S. Noda, “Two-dimensional coupling wave theory for square-lattice photonic crystal lasers with TM-polarization,” Opt. Express 15, 3981–3990 (2007).
[CrossRef] [PubMed]

2005 (1)

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun. 23, 1335–1340 (2005).
[CrossRef]

2003 (4)

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, 1374–1377 (2003).
[CrossRef] [PubMed]

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

B. S. Williams, H. Callebaut, S. Kumar, Q. Hu, and J. L. Reno, “3.4-THz quantum cascade laser based on longitudinal-optical-phonon scattering for depopulation,” Appl. Phys. Lett. 82, 1015–1017 (2003).
[CrossRef]

S.-C. Lee and A. Wacker, “Theoretical analysis of spectral gain in a terahertz quantum-cascade laser: Prospects for gain at 1 THz,” Appl. Phys. Lett. 83, 2506–2508 (2003).
[CrossRef]

2002 (1)

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

2001 (1)

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

2000 (1)

I. El-Kady, M. M. Sigalas, R. Biswas, K. H. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[CrossRef]

1999 (1)

V. Yannopapas, A. Modinos, and N. Strfanos, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B 60, 5359–5365 (1999).
[CrossRef]

1998 (1)

S. Nojima, “Enhancement of optical gain in two-dimensional photonic crystals with active lattice points,” Jpn. J. Appl. Phys., Part 2 37, L565–L567 (1998).
[CrossRef]

1994 (2)

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[CrossRef]

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994).
[CrossRef]

1993 (1)

A. R. McGurn and A. A. Maradudin, “Photonic band structures of two- and three-dimensional periodic metal or semiconductor arrays,” Phys. Rev. B 48, 17576–17579 (1993).
[CrossRef]

1991 (1)

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

1990 (2)

1985 (2)

S. Adachi, “GaAs, AlAs, and AlxGa1−xAs: Material parameters for use in research and device applications,” J. Appl. Phys. 58, R1–R29 (1985).
[CrossRef]

R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21, 144–150 (1985).
[CrossRef]

1983 (1)

1977 (1)

R. T. Holm, J. W. Gibson, and E. D. Palik, “Infrared reflectance studies of bulk and epitaxial-film n-type GaAs,” J. Appl. Phys. 48, 212–223 (1977).
[CrossRef]

1969 (1)

C. H. Gooch, Gallium Arsenide Lasers (Wiley-Interscience, 1969), pp. 64–65.

Abram, R. A.

S. Brand, R. A. Abram, and M. A. Kaliteevski, “Complex photonic band structure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. B 75, 035102 (2007).
[CrossRef]

Adachi, S.

S. Adachi, “GaAs, AlAs, and AlxGa1−xAs: Material parameters for use in research and device applications,” J. Appl. Phys. 58, R1–R29 (1985).
[CrossRef]

Alexander, R. W.

Amanti, M. I.

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

G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101, 081726 (2007).
[CrossRef]

Apostolopoulos, V.

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, 171112 (2008).
[CrossRef]

Beere, H.

G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101, 081726 (2007).
[CrossRef]

Beere, H. E.

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, 171112 (2008).
[CrossRef]

Bell, R. J.

Bell, R. R.

Bell, S. E.

Biswas, R.

I. El-Kady, M. M. Sigalas, R. Biswas, K. H. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[CrossRef]

Brand, S.

S. Brand, R. A. Abram, and M. A. Kaliteevski, “Complex photonic band structure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. B 75, 035102 (2007).
[CrossRef]

Callebaut, H.

B. S. Williams, H. Callebaut, S. Kumar, Q. Hu, and J. L. Reno, “3.4-THz quantum cascade laser based on longitudinal-optical-phonon scattering for depopulation,” Appl. Phys. Lett. 82, 1015–1017 (2003).
[CrossRef]

Capasso, F.

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, 1374–1377 (2003).
[CrossRef] [PubMed]

Chan, C. T.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[CrossRef]

Cho, A. Y.

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, 1374–1377 (2003).
[CrossRef] [PubMed]

Chutinan, A.

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

Colombelli, R.

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, 1374–1377 (2003).
[CrossRef] [PubMed]

Dunbar, L. A.

G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101, 081726 (2007).
[CrossRef]

El-Kady, I.

I. El-Kady, M. M. Sigalas, R. Biswas, K. H. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[CrossRef]

Exter, M.

Faist, J.

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

G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101, 081726 (2007).
[CrossRef]

Fattinger, C.

Freeman, J. R.

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, 171112 (2008).
[CrossRef]

Gibson, J. W.

R. T. Holm, J. W. Gibson, and E. D. Palik, “Infrared reflectance studies of bulk and epitaxial-film n-type GaAs,” J. Appl. Phys. 48, 212–223 (1977).
[CrossRef]

Giovannini, M.

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

G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101, 081726 (2007).
[CrossRef]

Gmachl, C. F.

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, 1374–1377 (2003).
[CrossRef] [PubMed]

Gooch, C. H.

C. H. Gooch, Gallium Arsenide Lasers (Wiley-Interscience, 1969), pp. 64–65.

Grischkowsky, D.

Henry, C. H.

R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21, 144–150 (1985).
[CrossRef]

Ho, K. H.

I. El-Kady, M. M. Sigalas, R. Biswas, K. H. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[CrossRef]

Ho, K. M.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[CrossRef]

Holm, R. T.

R. T. Holm, J. W. Gibson, and E. D. Palik, “Infrared reflectance studies of bulk and epitaxial-film n-type GaAs,” J. Appl. Phys. 48, 212–223 (1977).
[CrossRef]

Houdre, R.

G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101, 081726 (2007).
[CrossRef]

Hoyler, N.

G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101, 081726 (2007).
[CrossRef]

Hu, Q.

B. S. Williams, H. Callebaut, S. Kumar, Q. Hu, and J. L. Reno, “3.4-THz quantum cascade laser based on longitudinal-optical-phonon scattering for depopulation,” Appl. Phys. Lett. 82, 1015–1017 (2003).
[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]

Jianglin, Y.

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445–447 (2008).
[CrossRef]

Kaliteevski, M. A.

S. Brand, R. A. Abram, and M. A. Kaliteevski, “Complex photonic band structure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. B 75, 035102 (2007).
[CrossRef]

Kazarinov, R. F.

R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21, 144–150 (1985).
[CrossRef]

Keiding, S.

Kumar, S.

B. S. Williams, H. Callebaut, S. Kumar, Q. Hu, and J. L. Reno, “3.4-THz quantum cascade laser based on longitudinal-optical-phonon scattering for depopulation,” Appl. Phys. Lett. 82, 1015–1017 (2003).
[CrossRef]

Kuzmiak, V.

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994).
[CrossRef]

Lee, S. -C.

S.-C. Lee and A. Wacker, “Theoretical analysis of spectral gain in a terahertz quantum-cascade laser: Prospects for gain at 1 THz,” Appl. Phys. Lett. 83, 2506–2508 (2003).
[CrossRef]

Long, L. L.

Macomber, S. H.

R. J. Noll and S. H. Macomber, “Analysis of grating surface emitting laser,” IEEE J. Quantum Electron. 26, 456–466 (1990).
[CrossRef]

Maradudin, A. A.

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994).
[CrossRef]

A. R. McGurn and A. A. Maradudin, “Photonic band structures of two- and three-dimensional periodic metal or semiconductor arrays,” Phys. Rev. B 48, 17576–17579 (1993).
[CrossRef]

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

Marshall, O. P.

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, 171112 (2008).
[CrossRef]

Matsubara, H.

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445–447 (2008).
[CrossRef]

McGurn, A. R.

A. R. McGurn and A. A. Maradudin, “Photonic band structures of two- and three-dimensional periodic metal or semiconductor arrays,” Phys. Rev. B 48, 17576–17579 (1993).
[CrossRef]

Meyer, J. R.

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

Miyai, E.

K. Sakai, E. Miyai, and S. Noda, “Two-dimensional coupling wave theory for square-lattice photonic crystal lasers with TM-polarization,” Opt. Express 15, 3981–3990 (2007).
[CrossRef] [PubMed]

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun. 23, 1335–1340 (2005).
[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]

Modinos, A.

V. Yannopapas, A. Modinos, and N. Strfanos, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B 60, 5359–5365 (1999).
[CrossRef]

Noda, S.

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445–447 (2008).
[CrossRef]

K. Sakai, E. Miyai, and S. Noda, “Two-dimensional coupling wave theory for square-lattice photonic crystal lasers with TM-polarization,” Opt. Express 15, 3981–3990 (2007).
[CrossRef] [PubMed]

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun. 23, 1335–1340 (2005).
[CrossRef]

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

Nojima, S.

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

S. Nojima, “Enhancement of optical gain in two-dimensional photonic crystals with active lattice points,” Jpn. J. Appl. Phys., Part 2 37, L565–L567 (1998).
[CrossRef]

Noll, R. J.

R. J. Noll and S. H. Macomber, “Analysis of grating surface emitting laser,” IEEE J. Quantum Electron. 26, 456–466 (1990).
[CrossRef]

Ohnishi, D.

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun. 23, 1335–1340 (2005).
[CrossRef]

Okano, T.

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun. 23, 1335–1340 (2005).
[CrossRef]

Ordal, M. A.

Painter, O.

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, 1374–1377 (2003).
[CrossRef] [PubMed]

Palik, E. D.

R. T. Holm, J. W. Gibson, and E. D. Palik, “Infrared reflectance studies of bulk and epitaxial-film n-type GaAs,” J. Appl. Phys. 48, 212–223 (1977).
[CrossRef]

Pincemin, F.

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994).
[CrossRef]

Plihal, M.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

Reno, J. L.

B. S. Williams, H. Callebaut, S. Kumar, Q. Hu, and J. L. Reno, “3.4-THz quantum cascade laser based on longitudinal-optical-phonon scattering for depopulation,” Appl. Phys. Lett. 82, 1015–1017 (2003).
[CrossRef]

Ritchie, D.

G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101, 081726 (2007).
[CrossRef]

Ritchie, D. A.

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, 171112 (2008).
[CrossRef]

Rungsawang, R.

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, 171112 (2008).
[CrossRef]

Sadowski, M. L.

G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101, 081726 (2007).
[CrossRef]

Saito, H.

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445–447 (2008).
[CrossRef]

Sakaguchi, T.

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun. 23, 1335–1340 (2005).
[CrossRef]

Sakai, K.

K. Sakai, E. Miyai, and S. Noda, “Two-dimensional coupling wave theory for square-lattice photonic crystal lasers with TM-polarization,” Opt. Express 15, 3981–3990 (2007).
[CrossRef] [PubMed]

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun. 23, 1335–1340 (2005).
[CrossRef]

Scalari, G.

G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101, 081726 (2007).
[CrossRef]

Sergent, A. M.

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, 1374–1377 (2003).
[CrossRef] [PubMed]

Sigalas, M. M.

I. El-Kady, M. M. Sigalas, R. Biswas, K. H. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[CrossRef]

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[CrossRef]

Sirigu, L.

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

G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101, 081726 (2007).
[CrossRef]

Sivco, D. L.

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, 1374–1377 (2003).
[CrossRef] [PubMed]

Soukoulis, C. M.

I. El-Kady, M. M. Sigalas, R. Biswas, K. H. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[CrossRef]

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[CrossRef]

Srinivasan, K.

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, 1374–1377 (2003).
[CrossRef] [PubMed]

Strfanos, N.

V. Yannopapas, A. Modinos, and N. Strfanos, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B 60, 5359–5365 (1999).
[CrossRef]

Tanaka, Y.

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445–447 (2008).
[CrossRef]

Tennant, D. M.

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, 1374–1377 (2003).
[CrossRef] [PubMed]

Terazzi, R.

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

G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101, 081726 (2007).
[CrossRef]

Troccoli, M.

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, 1374–1377 (2003).
[CrossRef] [PubMed]

Vurgaftman, I.

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

Wacker, A.

S.-C. Lee and A. Wacker, “Theoretical analysis of spectral gain in a terahertz quantum-cascade laser: Prospects for gain at 1 THz,” Appl. Phys. Lett. 83, 2506–2508 (2003).
[CrossRef]

Walther, C.

G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101, 081726 (2007).
[CrossRef]

Ward, C. A.

Williams, B. S.

B. S. Williams, H. Callebaut, S. Kumar, Q. Hu, and J. L. Reno, “3.4-THz quantum cascade laser based on longitudinal-optical-phonon scattering for depopulation,” Appl. Phys. Lett. 82, 1015–1017 (2003).
[CrossRef]

Yannopapas, V.

V. Yannopapas, A. Modinos, and N. Strfanos, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B 60, 5359–5365 (1999).
[CrossRef]

Yoshimoto, S.

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445–447 (2008).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

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, 171112 (2008).
[CrossRef]

B. S. Williams, H. Callebaut, S. Kumar, Q. Hu, and J. L. Reno, “3.4-THz quantum cascade laser based on longitudinal-optical-phonon scattering for depopulation,” Appl. Phys. Lett. 82, 1015–1017 (2003).
[CrossRef]

S.-C. Lee and A. Wacker, “Theoretical analysis of spectral gain in a terahertz quantum-cascade laser: Prospects for gain at 1 THz,” Appl. Phys. Lett. 83, 2506–2508 (2003).
[CrossRef]

IEEE J. Quantum Electron. (3)

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

R. J. Noll and S. H. Macomber, “Analysis of grating surface emitting laser,” IEEE J. Quantum Electron. 26, 456–466 (1990).
[CrossRef]

R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21, 144–150 (1985).
[CrossRef]

IEEE J. Sel. Areas Commun. (1)

K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun. 23, 1335–1340 (2005).
[CrossRef]

J. Appl. Phys. (4)

G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101, 081726 (2007).
[CrossRef]

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

R. T. Holm, J. W. Gibson, and E. D. Palik, “Infrared reflectance studies of bulk and epitaxial-film n-type GaAs,” J. Appl. Phys. 48, 212–223 (1977).
[CrossRef]

S. Adachi, “GaAs, AlAs, and AlxGa1−xAs: Material parameters for use in research and device applications,” J. Appl. Phys. 58, R1–R29 (1985).
[CrossRef]

J. Opt. Soc. Am. B (1)

Jpn. J. Appl. Phys., Part 2 (1)

S. Nojima, “Enhancement of optical gain in two-dimensional photonic crystals with active lattice points,” Jpn. J. Appl. Phys., Part 2 37, L565–L567 (1998).
[CrossRef]

Opt. Express (2)

Phys. Rev. B (8)

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[CrossRef]

V. Yannopapas, A. Modinos, and N. Strfanos, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B 60, 5359–5365 (1999).
[CrossRef]

I. El-Kady, M. M. Sigalas, R. Biswas, K. H. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62, 15299–15302 (2000).
[CrossRef]

S. Brand, R. A. Abram, and M. A. Kaliteevski, “Complex photonic band structure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. B 75, 035102 (2007).
[CrossRef]

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

A. R. McGurn and A. A. Maradudin, “Photonic band structures of two- and three-dimensional periodic metal or semiconductor arrays,” Phys. Rev. B 48, 17576–17579 (1993).
[CrossRef]

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994).
[CrossRef]

Science (2)

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445–447 (2008).
[CrossRef]

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, 1374–1377 (2003).
[CrossRef] [PubMed]

Other (1)

C. H. Gooch, Gallium Arsenide Lasers (Wiley-Interscience, 1969), pp. 64–65.

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Figures (14)

Fig. 1
Fig. 1

(a) Schematic view of the calculation model for PC surface-emitting laser consisting of triangle-aligned metal holes. (b) Cross-section of the laser.

Fig. 2
Fig. 2

Real ( ε ) and imaginary ( ε ) parts of effective relative dielectric constants of EM wave guided in (a) hole and (b) metal regions. The calculations are done with or without gain in the semiconductor layer.

Fig. 3
Fig. 3

PBS with a period of 32 μ m and an r / a ratio of 0.3. Color of each line indicates the absorption loss α. Positive and negative α’s indicate loss and gain, respectively. Γ 2 indicates the second symmetric point at the Γ point. Dashed lines indicate the light cone.

Fig. 4
Fig. 4

EM-field distribution of hexapole, quadrupole, dipole, and monopole modes around Γ 2 point. The intensity includes the amplitude and phase. The calculations for X and J directions are done with wavevectors ( k x , k y ) = ( 0.003 12 , 0.001 80 ) π / a and ( 0.00417 , 0 ) π / a , respectively.

Fig. 5
Fig. 5

(a) PBS and (b) gain/loss dispersion characteristics of each photonic mode with a = 32 μ m and r / a = 0.3 . “H,” “Q,” “AQ,” “D,” “AD,” and “M” stand for hexapole, (symmetric) quadrupole, antisymmetric quadrupole, (symmetric) dipole, antisymmetric dipole, and monopole, respectively. Dotted and dashed lines in (a) show the QCL gain peak frequency and the light cone, respectively.

Fig. 6
Fig. 6

(a) PBS and (b) gain/loss dispersion characteristics of each photonic mode with a = 34 μ m and r / a = 0.3 . H, Q, AQ, D, AD, and M represent the same as in Fig. 5. Dotted and dashed lines in (a) show the QCL gain peak frequency and the light cone, respectively.

Fig. 7
Fig. 7

(a),(c) PBS and (b),(d) gain/loss dispersion characteristics around Γ 2 . (a) and (b) are with surface-emission loss, while (c) and (d) without surface-emission loss. Gain is only in the metal region. Period and r / a ratio are 32 μ m and of 0.3, respectively. H, Q, AQ, D, AD, and M represent the same as in Fig. 5. Dotted lines in (a) and (c) indicate QCL gain peak frequency.

Fig. 8
Fig. 8

(a) PBS and (b) gain/loss dispersion characteristics around Γ 2 with surface-emission loss. Gain exits in both metal and hole regions. Period and r / a ratio are 32 μ m and 0.3, respectively. H, Q, AQ, D, AD, and M represent the same as in Fig. 5. Dotted horizontal line in (a) indicates QCL gain peak frequency.

Fig. 9
Fig. 9

Radiation factor of each photonic mode as a function of PC period and r / a ratio. (a), (b), (c), and (d) show the hexapole, quadrupole, dipole, and monopole modes, respectively. Only metal region possesses gain.

Fig. 10
Fig. 10

Resonant frequency of each photonic mode as a function of period and r / a ratio of PC structure. (a), (b), (c), and (d) show the hexapole, quadrupole, dipole, and monopole modes, respectively. Red zone includes the QCL gain peak frequency of 3.03 THz.

Fig. 11
Fig. 11

Gain/loss factor of each photonic mode at Γ 2 point as a function of PC period and r / a ratio with surface-emission loss. (a), (b), (c), and (d) show the hexapole, quadrupole, dipole, and monopole modes, respectively. Only metal region possesses gain.

Fig. 12
Fig. 12

Gain/loss factor of each photonic mode at Γ 2 point as a function of PC period and r / a ratio with surface-emission loss. (a), (b), (c), and (d) show the hexapole, quadrupole, dipole, and monopole modes, respectively. Both metal and hole regions possess gain.

Fig. 13
Fig. 13

(a) PBS and (b) gain/loss dispersion characteristics around Γ 2 with surface-emission loss for an optimized PC structure. Gain exits in both metal and hole regions. Period and r / a ratio are 34.6 μ m and 0.2, respectively. H, Q, AQ, D, AD, and M represent the same as in Fig. 5. Dotted horizontal line in (a) indicates QCL gain peak frequency.

Fig. 14
Fig. 14

Normalized frequency ( ω a / 2 π c ) of experimental results compared with that of our calculated results. Squares represent experimental results [7]. Solid lines represent the case with surface-emission loss and gain under metal and hole regions.

Equations (24)

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E 3 ( x ̂ ) = G B ( k , G , ω ) exp [ i ( k + G ) x ¯ ] ϕ ( x 3 ) + Δ E ( x ̂ ) ,
k = k r + i k i = k r + i k i ( k r + G ) / | k r + G | ,
ε ( x ¯ , ω ) = G κ ¯ ( G , ω ) exp ( i G x ¯ ) ,
κ ¯ ( G , ω ) = { ε a ( ω ) f + ε b ( ω ) ( 1 f ) ( G = 0 ) 2 [ ε a ( ω ) ε b ( ω ) ] J 1 ( | G r | ) / | G r | ( G 0 ) , }
( k + G ) 2 B ( k , G , ω ) = ω 2 c 2 [ Γ g G G κ ¯ ( G G , ω ) B ( k , G , ω ) + ε e f f B ( k , G , ω ) ] + i G 0 2 | G | h ( G , G ) B ( k , G , ω ) ,
h ( G , G ) = Γ g ω 4 2 d g | G | K 3 3 c 4 κ ¯ ( G , ω ) κ ¯ ( G , ω ) [ ( 1 cos   K 3 d g ) + i ( K 3 d g sin   K 3 d g ) ] ,
g ( ω ) = ω 2 c n B ε 0 L p i , j e 2 | z i j | 2 ( n i n j ) ( Γ i + Γ j ) ( ω Δ E i j ) 2 + [ ( Γ i + Γ j ) / 2 ] 2 ,
( 2 x 1 2 + 2 x 2 2 + 2 x 3 2 ) E ( x ̂ ) = ω 2 c 2 ε ( x ̂ , ω ) E ( x ̂ ) .
G ( | k + G | 2 + 2 x 3 2 ) B ( k , G , ω ) exp { i ( k + G ) x ¯ } ϕ ( x 3 ) + ω 2 c 2 { G κ ̂ ( G , ω , x 3 ) G B ( k , G , ω ) exp [ i ( k + G + G ) x ¯ ] } ϕ ( x 3 ) + 2 x 3 2 Δ E ( x ̂ ) + ω 2 c 2 G κ ̂ ( G , ω , x 3 ) exp ( i G x ¯ ) Δ E ( x ̂ ) = 0 .
[ 2 x 3 2 + ω 2 c 2 κ ̂ ( 0 , ω , x 3 ) ] Δ E ( x ̂ ) = ω 2 c 2 G 0 κ ̂ ( G , ω , x 3 ) B ( k , G , ω ) ϕ ( x 3 ) .
Δ E ( x ̂ ) = ω 2 c 2 G 0 G ̃ ( x 3 , t ) κ ̂ ( G , ω , t ) B ( k , G , ω ) ϕ ( t ) d t ,
G ̃ ( x 3 , t ) = exp [ ± i K 3 ( x 3 t ) ] / 2 i K 3     ( positive   sign   for   x 3 > t ,   negative   for   x 3 < t ) .
{ 2 x 3 2 + ω 2 c 2 [ κ ̂ ( 0 , ω , x 3 ) ε e f f ] } ϕ ( x 3 ) = 0 ,
ϕ ( x 3 ) 2 d x 3 = 1.
G { [ ( k + G ) 2 + ω 2 c 2 ε e f f ] B ( k , G , ω ) + Γ g ω 2 c 2 G G κ ¯ ( G G , ω ) B ( k , G , ω ) } exp [ i ( k + G ) x ¯ ] + 2 Δ E ( x ̂ ) x 3 2 ϕ ( x 3 ) d x 3 + i G G 0 2 | G | h ( G , G ) B ( k , G , ω ) exp ( i G x ¯ ) = 0 ,
h ( G , G ) = i 1 2 | G | ω 4 c 4 G ̃ ( x 3 , t ) κ ̂ ( G , ω , x 3 ) κ ̂ ( G , ω , t ) ϕ ( t ) ϕ ( x 3 ) d t d x 3 ,
Γ g = P C L ϕ ( x 3 ) 2 d x 3 .
h ( G , G ) = ω 4 4 | G | K 3 c 4 d x 3 d t κ ̂ ( G , ω , x 3 ) κ ̂ ( G , ω , t ) ϕ ( x 3 ) ϕ ( t ) exp [ ± i K 3 ( x 3 t ) ] = ω 4 4 | G | K 3 c 4 κ ¯ ( G , ω ) κ ¯ ( G , ω ) ϕ c 2 d g 1 d g 2 d x 3 d g 1 d g 2 d t   exp [ ± i K 3 ( x 3 t ) ] ,
d g 1 d g 2 d x 3 d g 1 d g 2 d t   exp [ ± i K 3 ( x 3 t ) ] = d g 1 d g 2 { d g 1 x 3 exp [ i K 3 ( x 3 t ) ] d t + x 3 d g 2 exp [ i K 3 ( x 3 t ) ] d t } d x 3 = 2 K 3 2 [ ( 1 cos   K 3 d g ) + i ( K 3 d g sin   K 3 d g ) ] ,
Γ g = ϕ c 2 d g 1 d g 2 d x 3 = ϕ c 2 d g .
h Γ g d g [ 2 π Δ n λ 1 a c R d 2 x ¯   exp ( i G x ¯ ) ] 2 ,
[ n ( ω ) + i κ ( ω ) ] 2 = ε [ 1 + ω L 2 ω T 2 ω T 2 ω 2 i ω γ p h ω p 2 ω ( ω + i γ p l ) ] ,
ω p 2 = N e 2 / m ε ,
Δ n ( ω ) = ( 2 / π ) 0 ω κ g ( ω ) / ( ω 2 ω 2 ) d ω ,

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