Abstract

Three-dimensional coupled-wave theory is extended to model triangular-lattice photonic-crystal surface-emitting lasers with transverse-electric polarization. A generalized coupled-wave equation is derived to describe the sixfold symmetry of the eigenmodes in a triangular lattice. The extended theory includes the effects of both surface radiation and in-plane losses in a finite-size laser structure. Modal properties of interest including the band structure, radiation constant, threshold gain, field intensity profile, and far-field pattern (FFP) are calculated. The calculated band structure and FFP, as well as the predicted lasing mode, agree well with experimental observations. The effect of air-hole size on mode selection is also studied and confirmed by experiment.

© 2013 OSA

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  1. M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75, 316–318 (1999).
    [CrossRef]
  2. S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123–1125 (2001).
    [CrossRef] [PubMed]
  3. M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65, 195306 (2002).
    [CrossRef]
  4. I. Vurgaftman and J. R. Meyer, “Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers,” IEEE J. Quantum Electron. 39, 689–700 (2003).
    [CrossRef]
  5. D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express 12, 1562–1568 (2004).
    [CrossRef] [PubMed]
  6. E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature (London) 441, 946 (2006).
    [CrossRef]
  7. 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]
  8. M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett. 88, 191105 (2006).
    [CrossRef]
  9. 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 (London) 457, 174–178 (2009).
    [CrossRef]
  10. L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photonics Rev. 5, 647–658 (2011).
  11. Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics 4, 447–450 (2010).
    [CrossRef]
  12. S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, “Higher-order vector beams produced by photonic-crystal lasers,” Opt. Express 19, 11963–11968 (2011).
    [CrossRef] [PubMed]
  13. S. Iwahashi, K. Sakai, Y. Kurosaka, and S. Noda, “Centered-rectangular lattice photonic-crystal surface-emitting lasers,” Phys. Rev. B 85, 035304 (2012).
    [CrossRef]
  14. H. Y. Ryu, M. Notomi, and Y. H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B 68, 045209 (2003).
    [CrossRef]
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    [CrossRef] [PubMed]
  16. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
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    [CrossRef]
  18. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave model for square-lattice two-dimensional photonic crystal with transverse-electric-like mode,” Appl. Phys. Lett. 89, 021101 (2006).
    [CrossRef]
  19. K. Sakai, J. Yue, and S. Noda, “Coupled-wave model for triangular-lattice photonic crystal with transverse electric polarization,” Opt. Express 16, 6033–6040 (2008).
    [CrossRef] [PubMed]
  20. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. 46, 788–795 (2010).
    [CrossRef]
  21. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic-crystal lasers with transverse electric polarization: A general approach,” Phys. Rev. B 84, 195119 (2011).
    [CrossRef]
  22. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air-holes with arbitrary sidewalls,” Opt. Express 19, 24672–24686 (2011).
    [CrossRef] [PubMed]
  23. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic-crystal lasers with transverse electric polarization: Finite-size effects,” Opt. Express 20, 15945–15961 (2012).
    [CrossRef] [PubMed]
  24. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of centered-rectangular lattice photonic crystal with transverse-electric-like mode,” Phys. Rev. B 86, 035108 (2012).
    [CrossRef]
  25. K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B 52, 7982 (1995).
    [CrossRef]
  26. 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]
  27. M. Koba and P. Szczepanski, “The threshold mode structure analysis of the two-dimensional photonic crystal lasers,” Prog. Electromagn. Res. 125, 365–389 (2012).
    [CrossRef]
  28. K. Forberich, M. Diem, J. Crewett, U. Lemmer, A. Gombert, and K. Busch, “Lasing action in two-dimensional organic photonic crystal lasers with hexagonal symmetry,” Appl. Phys. B 82, 539–541 (2006).
    [CrossRef]
  29. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Norwood, 2005).

2012 (4)

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of centered-rectangular lattice photonic crystal with transverse-electric-like mode,” Phys. Rev. B 86, 035108 (2012).
[CrossRef]

M. Koba and P. Szczepanski, “The threshold mode structure analysis of the two-dimensional photonic crystal lasers,” Prog. Electromagn. Res. 125, 365–389 (2012).
[CrossRef]

S. Iwahashi, K. Sakai, Y. Kurosaka, and S. Noda, “Centered-rectangular lattice photonic-crystal surface-emitting lasers,” Phys. Rev. B 85, 035304 (2012).
[CrossRef]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic-crystal lasers with transverse electric polarization: Finite-size effects,” Opt. Express 20, 15945–15961 (2012).
[CrossRef] [PubMed]

2011 (4)

S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, “Higher-order vector beams produced by photonic-crystal lasers,” Opt. Express 19, 11963–11968 (2011).
[CrossRef] [PubMed]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air-holes with arbitrary sidewalls,” Opt. Express 19, 24672–24686 (2011).
[CrossRef] [PubMed]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic-crystal lasers with transverse electric polarization: A general approach,” Phys. Rev. B 84, 195119 (2011).
[CrossRef]

L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photonics Rev. 5, 647–658 (2011).

2010 (2)

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics 4, 447–450 (2010).
[CrossRef]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. 46, 788–795 (2010).
[CrossRef]

2009 (1)

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 (London) 457, 174–178 (2009).
[CrossRef]

2008 (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]

K. Sakai, J. Yue, and S. Noda, “Coupled-wave model for triangular-lattice photonic crystal with transverse electric polarization,” Opt. Express 16, 6033–6040 (2008).
[CrossRef] [PubMed]

2006 (4)

K. Forberich, M. Diem, J. Crewett, U. Lemmer, A. Gombert, and K. Busch, “Lasing action in two-dimensional organic photonic crystal lasers with hexagonal symmetry,” Appl. Phys. B 82, 539–541 (2006).
[CrossRef]

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett. 88, 191105 (2006).
[CrossRef]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature (London) 441, 946 (2006).
[CrossRef]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave model for square-lattice two-dimensional photonic crystal with transverse-electric-like mode,” Appl. Phys. Lett. 89, 021101 (2006).
[CrossRef]

2005 (2)

M. Yokoyama and S. Noda, “Finite-difference time-domain simulation of two-dimensional photonic crystal surface-emitting laser,” Opt. Express 13, 2869–2880 (2005).
[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]

2004 (1)

2003 (2)

H. Y. Ryu, M. Notomi, and Y. H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B 68, 045209 (2003).
[CrossRef]

I. Vurgaftman and J. R. Meyer, “Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers,” IEEE J. Quantum Electron. 39, 689–700 (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. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123–1125 (2001).
[CrossRef] [PubMed]

1999 (1)

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75, 316–318 (1999).
[CrossRef]

1995 (1)

K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B 52, 7982 (1995).
[CrossRef]

1992 (1)

M. Toda, “Proposed cross grating single-mode DFB laser,” IEEE J. Quantum Electron. 28, 1653–1662, (1992).
[CrossRef]

1972 (1)

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

Barbieri, S.

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 (London) 457, 174–178 (2009).
[CrossRef]

Beere, H. E.

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 (London) 457, 174–178 (2009).
[CrossRef]

Bewley, W. W.

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett. 88, 191105 (2006).
[CrossRef]

Busch, K.

K. Forberich, M. Diem, J. Crewett, U. Lemmer, A. Gombert, and K. Busch, “Lasing action in two-dimensional organic photonic crystal lasers with hexagonal symmetry,” Appl. Phys. B 82, 539–541 (2006).
[CrossRef]

Canedy, C. L.

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett. 88, 191105 (2006).
[CrossRef]

Chassagneux, Y.

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 (London) 457, 174–178 (2009).
[CrossRef]

Chutinan, A.

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

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123–1125 (2001).
[CrossRef] [PubMed]

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75, 316–318 (1999).
[CrossRef]

Colombelli, R.

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 (London) 457, 174–178 (2009).
[CrossRef]

Crewett, J.

K. Forberich, M. Diem, J. Crewett, U. Lemmer, A. Gombert, and K. Busch, “Lasing action in two-dimensional organic photonic crystal lasers with hexagonal symmetry,” Appl. Phys. B 82, 539–541 (2006).
[CrossRef]

Davies, A. G.

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 (London) 457, 174–178 (2009).
[CrossRef]

Diem, M.

K. Forberich, M. Diem, J. Crewett, U. Lemmer, A. Gombert, and K. Busch, “Lasing action in two-dimensional organic photonic crystal lasers with hexagonal symmetry,” Appl. Phys. B 82, 539–541 (2006).
[CrossRef]

Forberich, K.

K. Forberich, M. Diem, J. Crewett, U. Lemmer, A. Gombert, and K. Busch, “Lasing action in two-dimensional organic photonic crystal lasers with hexagonal symmetry,” Appl. Phys. B 82, 539–541 (2006).
[CrossRef]

Gombert, A.

K. Forberich, M. Diem, J. Crewett, U. Lemmer, A. Gombert, and K. Busch, “Lasing action in two-dimensional organic photonic crystal lasers with hexagonal symmetry,” Appl. Phys. B 82, 539–541 (2006).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Norwood, 2005).

Imada, M.

D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express 12, 1562–1568 (2004).
[CrossRef] [PubMed]

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

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123–1125 (2001).
[CrossRef] [PubMed]

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75, 316–318 (1999).
[CrossRef]

Iwahashi, S.

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic-crystal lasers with transverse electric polarization: Finite-size effects,” Opt. Express 20, 15945–15961 (2012).
[CrossRef] [PubMed]

S. Iwahashi, K. Sakai, Y. Kurosaka, and S. Noda, “Centered-rectangular lattice photonic-crystal surface-emitting lasers,” Phys. Rev. B 85, 035304 (2012).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of centered-rectangular lattice photonic crystal with transverse-electric-like mode,” Phys. Rev. B 86, 035108 (2012).
[CrossRef]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic-crystal lasers with transverse electric polarization: A general approach,” Phys. Rev. B 84, 195119 (2011).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air-holes with arbitrary sidewalls,” Opt. Express 19, 24672–24686 (2011).
[CrossRef] [PubMed]

S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, “Higher-order vector beams produced by photonic-crystal lasers,” Opt. Express 19, 11963–11968 (2011).
[CrossRef] [PubMed]

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics 4, 447–450 (2010).
[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]

Khanna, S. P.

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 (London) 457, 174–178 (2009).
[CrossRef]

Kim, C. S.

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett. 88, 191105 (2006).
[CrossRef]

Kim, M.

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett. 88, 191105 (2006).
[CrossRef]

Kitamura, K.

Koba, M.

M. Koba and P. Szczepanski, “The threshold mode structure analysis of the two-dimensional photonic crystal lasers,” Prog. Electromagn. Res. 125, 365–389 (2012).
[CrossRef]

Kogelnik, H.

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

Kunishi, W.

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics 4, 447–450 (2010).
[CrossRef]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature (London) 441, 946 (2006).
[CrossRef]

Kurosaka, Y.

S. Iwahashi, K. Sakai, Y. Kurosaka, and S. Noda, “Centered-rectangular lattice photonic-crystal surface-emitting lasers,” Phys. Rev. B 85, 035304 (2012).
[CrossRef]

S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, “Higher-order vector beams produced by photonic-crystal lasers,” Opt. Express 19, 11963–11968 (2011).
[CrossRef] [PubMed]

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics 4, 447–450 (2010).
[CrossRef]

Lee, Y. H.

H. Y. Ryu, M. Notomi, and Y. H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B 68, 045209 (2003).
[CrossRef]

Lemmer, U.

K. Forberich, M. Diem, J. Crewett, U. Lemmer, A. Gombert, and K. Busch, “Lasing action in two-dimensional organic photonic crystal lasers with hexagonal symmetry,” Appl. Phys. B 82, 539–541 (2006).
[CrossRef]

Liang, Y.

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of centered-rectangular lattice photonic crystal with transverse-electric-like mode,” Phys. Rev. B 86, 035108 (2012).
[CrossRef]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic-crystal lasers with transverse electric polarization: Finite-size effects,” Opt. Express 20, 15945–15961 (2012).
[CrossRef] [PubMed]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic-crystal lasers with transverse electric polarization: A general approach,” Phys. Rev. B 84, 195119 (2011).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air-holes with arbitrary sidewalls,” Opt. Express 19, 24672–24686 (2011).
[CrossRef] [PubMed]

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics 4, 447–450 (2010).
[CrossRef]

Lindle, J. R.

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett. 88, 191105 (2006).
[CrossRef]

Linfield, E. H.

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 (London) 457, 174–178 (2009).
[CrossRef]

Mahler, L.

L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photonics Rev. 5, 647–658 (2011).

Maineult, W.

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 (London) 457, 174–178 (2009).
[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]

Meyer, J. R.

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett. 88, 191105 (2006).
[CrossRef]

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

Miyai, E.

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics 4, 447–450 (2010).
[CrossRef]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. 46, 788–795 (2010).
[CrossRef]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave model for square-lattice two-dimensional photonic crystal with transverse-electric-like mode,” Appl. Phys. Lett. 89, 021101 (2006).
[CrossRef]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature (London) 441, 946 (2006).
[CrossRef]

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]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123–1125 (2001).
[CrossRef] [PubMed]

Murata, M.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75, 316–318 (1999).
[CrossRef]

Noda, S.

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic-crystal lasers with transverse electric polarization: Finite-size effects,” Opt. Express 20, 15945–15961 (2012).
[CrossRef] [PubMed]

S. Iwahashi, K. Sakai, Y. Kurosaka, and S. Noda, “Centered-rectangular lattice photonic-crystal surface-emitting lasers,” Phys. Rev. B 85, 035304 (2012).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of centered-rectangular lattice photonic crystal with transverse-electric-like mode,” Phys. Rev. B 86, 035108 (2012).
[CrossRef]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic-crystal lasers with transverse electric polarization: A general approach,” Phys. Rev. B 84, 195119 (2011).
[CrossRef]

S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, “Higher-order vector beams produced by photonic-crystal lasers,” Opt. Express 19, 11963–11968 (2011).
[CrossRef] [PubMed]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air-holes with arbitrary sidewalls,” Opt. Express 19, 24672–24686 (2011).
[CrossRef] [PubMed]

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics 4, 447–450 (2010).
[CrossRef]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. 46, 788–795 (2010).
[CrossRef]

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, J. Yue, and S. Noda, “Coupled-wave model for triangular-lattice photonic crystal with transverse electric polarization,” Opt. Express 16, 6033–6040 (2008).
[CrossRef] [PubMed]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature (London) 441, 946 (2006).
[CrossRef]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave model for square-lattice two-dimensional photonic crystal with transverse-electric-like mode,” Appl. Phys. Lett. 89, 021101 (2006).
[CrossRef]

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. Yokoyama and S. Noda, “Finite-difference time-domain simulation of two-dimensional photonic crystal surface-emitting laser,” Opt. Express 13, 2869–2880 (2005).
[CrossRef] [PubMed]

D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express 12, 1562–1568 (2004).
[CrossRef] [PubMed]

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

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123–1125 (2001).
[CrossRef] [PubMed]

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75, 316–318 (1999).
[CrossRef]

Notomi, M.

H. Y. Ryu, M. Notomi, and Y. H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B 68, 045209 (2003).
[CrossRef]

Ohnishi, D.

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics 4, 447–450 (2010).
[CrossRef]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature (London) 441, 946 (2006).
[CrossRef]

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]

D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express 12, 1562–1568 (2004).
[CrossRef] [PubMed]

Okano, T.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature (London) 441, 946 (2006).
[CrossRef]

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]

D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express 12, 1562–1568 (2004).
[CrossRef] [PubMed]

Peng, C.

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of centered-rectangular lattice photonic crystal with transverse-electric-like mode,” Phys. Rev. B 86, 035108 (2012).
[CrossRef]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic-crystal lasers with transverse electric polarization: Finite-size effects,” Opt. Express 20, 15945–15961 (2012).
[CrossRef] [PubMed]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic-crystal lasers with transverse electric polarization: A general approach,” Phys. Rev. B 84, 195119 (2011).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air-holes with arbitrary sidewalls,” Opt. Express 19, 24672–24686 (2011).
[CrossRef] [PubMed]

Ritchie, D. A.

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 (London) 457, 174–178 (2009).
[CrossRef]

Ryu, H. Y.

H. Y. Ryu, M. Notomi, and Y. H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B 68, 045209 (2003).
[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.

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of centered-rectangular lattice photonic crystal with transverse-electric-like mode,” Phys. Rev. B 86, 035108 (2012).
[CrossRef]

S. Iwahashi, K. Sakai, Y. Kurosaka, and S. Noda, “Centered-rectangular lattice photonic-crystal surface-emitting lasers,” Phys. Rev. B 85, 035304 (2012).
[CrossRef]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic-crystal lasers with transverse electric polarization: Finite-size effects,” Opt. Express 20, 15945–15961 (2012).
[CrossRef] [PubMed]

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic-crystal lasers with transverse electric polarization: A general approach,” Phys. Rev. B 84, 195119 (2011).
[CrossRef]

S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, “Higher-order vector beams produced by photonic-crystal lasers,” Opt. Express 19, 11963–11968 (2011).
[CrossRef] [PubMed]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air-holes with arbitrary sidewalls,” Opt. Express 19, 24672–24686 (2011).
[CrossRef] [PubMed]

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics 4, 447–450 (2010).
[CrossRef]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. 46, 788–795 (2010).
[CrossRef]

K. Sakai, J. Yue, and S. Noda, “Coupled-wave model for triangular-lattice photonic crystal with transverse electric polarization,” Opt. Express 16, 6033–6040 (2008).
[CrossRef] [PubMed]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature (London) 441, 946 (2006).
[CrossRef]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave model for square-lattice two-dimensional photonic crystal with transverse-electric-like mode,” Appl. Phys. Lett. 89, 021101 (2006).
[CrossRef]

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]

Sakoda, K.

K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B 52, 7982 (1995).
[CrossRef]

Sasaki, G.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75, 316–318 (1999).
[CrossRef]

Shank, C. V.

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

Szczepanski, P.

M. Koba and P. Szczepanski, “The threshold mode structure analysis of the two-dimensional photonic crystal lasers,” Prog. Electromagn. Res. 125, 365–389 (2012).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Norwood, 2005).

Takayama, N.

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]

Toda, M.

M. Toda, “Proposed cross grating single-mode DFB laser,” IEEE J. Quantum Electron. 28, 1653–1662, (1992).
[CrossRef]

Tokuda, T.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75, 316–318 (1999).
[CrossRef]

Tredicucci, A.

L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photonics Rev. 5, 647–658 (2011).

Vurgaftman, I.

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett. 88, 191105 (2006).
[CrossRef]

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

Yokoyama, M.

M. Yokoyama and S. Noda, “Finite-difference time-domain simulation of two-dimensional photonic crystal surface-emitting laser,” Opt. Express 13, 2869–2880 (2005).
[CrossRef] [PubMed]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123–1125 (2001).
[CrossRef] [PubMed]

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]

Yue, J.

Appl. Phys. B (1)

K. Forberich, M. Diem, J. Crewett, U. Lemmer, A. Gombert, and K. Busch, “Lasing action in two-dimensional organic photonic crystal lasers with hexagonal symmetry,” Appl. Phys. B 82, 539–541 (2006).
[CrossRef]

Appl. Phys. Lett. (3)

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75, 316–318 (1999).
[CrossRef]

M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett. 88, 191105 (2006).
[CrossRef]

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave model for square-lattice two-dimensional photonic crystal with transverse-electric-like mode,” Appl. Phys. Lett. 89, 021101 (2006).
[CrossRef]

IEEE J. Quantum Electron. (3)

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. 46, 788–795 (2010).
[CrossRef]

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

M. Toda, “Proposed cross grating single-mode DFB laser,” IEEE J. Quantum Electron. 28, 1653–1662, (1992).
[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. (1)

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

Laser Photonics Rev. (1)

L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photonics Rev. 5, 647–658 (2011).

Nat. Photonics (1)

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics 4, 447–450 (2010).
[CrossRef]

Nature (London) (2)

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 (London) 457, 174–178 (2009).
[CrossRef]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature (London) 441, 946 (2006).
[CrossRef]

Opt. Express (6)

Phys. Rev. B (6)

S. Iwahashi, K. Sakai, Y. Kurosaka, and S. Noda, “Centered-rectangular lattice photonic-crystal surface-emitting lasers,” Phys. Rev. B 85, 035304 (2012).
[CrossRef]

H. Y. Ryu, M. Notomi, and Y. H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B 68, 045209 (2003).
[CrossRef]

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

Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic-crystal lasers with transverse electric polarization: A general approach,” Phys. Rev. B 84, 195119 (2011).
[CrossRef]

C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of centered-rectangular lattice photonic crystal with transverse-electric-like mode,” Phys. Rev. B 86, 035108 (2012).
[CrossRef]

K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B 52, 7982 (1995).
[CrossRef]

Prog. Electromagn. Res. (1)

M. Koba and P. Szczepanski, “The threshold mode structure analysis of the two-dimensional photonic crystal lasers,” Prog. Electromagn. Res. 125, 365–389 (2012).
[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]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123–1125 (2001).
[CrossRef] [PubMed]

Other (1)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Norwood, 2005).

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Figures (8)

Fig. 1
Fig. 1

(a) Schematic structure of a photonic-crystal surface-emitting laser device with a triangular lattice. (b) Band structure of triangular-lattice PCs calculated by the 2D plane-wave expansion method for transverse-electric (TE) mode [3]. The red circle indicates the second-order Γ point. The inset shows the high-symmetry points at the corners of the irreducible Brillouin zone (shaded light blue).

Fig. 2
Fig. 2

(a) Schematic of a triangular-lattice PC in real space. The blue arrows denote the primitive translation vectors a1 and a2, and a is the lattice constant. (b) Reciprocal lattice space of a triangular-lattice PC. The colored arrows indicate the six basic waves: R1, S1, R2, S2, R3, and S3 at the second-order Γ point, whose wavenumber is equal to β 0 = 4 π / 3 a.

Fig. 3
Fig. 3

Calculated (a) band structure and (b) radiation constant of the eigenmodes in the vicinity of the second-order Γ point for a triangular-lattice PC-SEL with circular air holes. Six modes near the Γ point are referred to as modes: A, B1, B2, C, D1, and D2, in the order of increasing frequency. Modes B1 and B2, as well as D1 and D2, are doubly degenerate at the Γ point. (c) E-field vector distribution (arrows) and H-field patterns (in color) of the individual band-edge modes. The black circles indicate the air holes. Band-edge modes A and C are known as hexapole and monopole modes, respectively [3]. In the calculations, we use the structural parameters listed in Table 1, the lattice constant a = 341 nm, and the air-hole filling factor f = 0.15.

Fig. 4
Fig. 4

(a) Normalized threshold gain (αL) as a function of normalized mode frequency deviation (δL). The fundamental band-edge modes (A, B1, B2, and C) and an additional mode W are indicated by arrows. (b) Field intensity envelopes of the modes indicated by arrows in (a). The data are calculated by using the same parameters as specified in the caption of Fig. 3. Note that the field intensity envelopes are plotted on hexagonal grids with a circular-shape computational domain (dashed circle). The radius of the circular domain, L = 30 μm, is discretized to span seven grid cells for which the eigenvalues converge well (see Appendix B for details).

Fig. 5
Fig. 5

(a) Comparison of the measured (in color) and calculated (white dashed curves) band structures. (b) Lasing spectrum measured above the lasing threshold in the direction normal to the PC plane. The threshold current (Ith) at room temperature CW operation was 25 mA. Band structure and lasing spectrum were measured at CW current levels of 0.9Ith and 1.2Ith, respectively. The frequency of the lasing peak in (b) is 0.3434 (a/λ), indicating that the lasing mode is band-edge mode C (yellow dashed line).

Fig. 6
Fig. 6

(a) Calculated and (b) measured FFPs of mode C. A scanning microscope image of the fabricated PC with a = 341 nm and f = 0.15 (where the r/a ratio is 0.20 and r is the air-hole radius) is shown in the left inset of (b). Ex (Ey) displayed in the right insets represents the x(y) component of the FFP. Parameters used for the calculations are the same as those shown in the caption of Fig. 4. The yellow arrows in (b) indicate the directions of the measured beam polarization. The beam divergence angle of the FFPs for both cases is around 1°, reflecting the large area of coherent oscillation.

Fig. 7
Fig. 7

(a) Calculated and (b) measured FFPs of mode A. A scanning microscope image of the fabricated PC with a = 341 nm and f = 0.26 (where r/a = 0.27) is shown in the left inset of (b). Ex (Ey) displayed in the right insets represents the x (y) component of the FFP. Parameters used for the calculations are specified in the caption of Table 3. The yellow arrows in (b) indicate the directions of the measured beam polarization.

Fig. 8
Fig. 8

(a) Location of field components (basic waves) and coupling coefficients in the vicinity of the (j,k)th hexagonal cell of the grid. Positions of the unknown field components (colored hollow dots) are staggered from the positions of the known coupling coefficients (solid dots). The colored hollow dots correspond to the points that are updated using the finite-difference scheme, while the solid points are points that are not solved. (b) Schematic of a circular computational domain (yellow shaded region) discretized on the hexagonal grids with L = 2h (L is the radius of the circular shape and h is the distance between two adjacent grid points). The green hollow dots inside the black squares indicate the boundary of R3; these are set to be zero in the calculations.

Tables (3)

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Table 1 Structural parameters of the PC-SEL device.

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Table 2 Normalized threshold gain (αL) of the low-threshold modes indicated by arrows in Fig. 4(a).

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Table 3 Normalized threshold gain (αL) of the low-threshold modes AC. Parameters used for the calculations are the same as those shown in Fig. 4 except that a larger air-hole filling factor f = 0.26 is used.

Equations (34)

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a 1 = ( 3 a / 2 , a / 2 ) , a 2 = ( 3 a / 2 , a / 2 ) .
G m n = m b 1 + n b 2 = m + n 2 β 0 x ^ + 3 ( n m ) 2 β 0 y ^ m 0 x ( m , n ) β 0 x ^ + n 0 y ( m , n ) β 0 y ^ ,
u ( r ) = G m n a m n ( z ) e i ( Δ k + G m n ) r ,
× × E ( r ) = k 0 2 n ˜ 2 ( r ) E ( r ) ,
E j ( z ) = m , n E j , m , n ( z ) e i m x , m n β 0 x i n y , m n β 0 y , j = x , y ,
n 2 ( r ) = n 0 2 ( z ) + m 0 , n 0 ξ m , n ( z ) e i m 0 x β 0 x i n 0 y β 0 y ,
[ 2 z 2 + k 0 2 n 0 2 ( z ) + 2 i k 0 n 0 ( z ) α ˜ ( z ) n y 2 β 0 2 ] E x , m , n + m x n y β 0 2 E y , m , n 2 i n y β 0 E x , m , n y + i β 0 ( m x E y , m , n y + n y E y , m , n x ) = k 0 2 m m , n n ξ m m , n n E x , m , n ,
[ 2 z 2 + k 0 2 n 0 2 ( z ) + 2 i k 0 n 0 ( z ) α ˜ ( z ) m x 2 β 0 2 ] E y , m , n + m x n y β 0 2 E x , m , n 2 i m x β 0 E y , m , n x + i β 0 ( m x E x , m , n y + n y E x , m , n x ) = k 0 2 m m , n n ξ m m , n n E y , m , n ,
z [ E x , m , n x + E y , m , n y i β 0 ( m x E x , m , n + n y E y , m , n ) ] = 0 .
[ 2 z 2 + k 0 2 n 0 2 + 2 i k 0 n 0 ( z ) α ˜ ( z ) ( m x 2 + n y 2 ) β 0 2 ] ( n y E x , m , n m x E y , m , n ) + i β 0 [ m x n y E y , m , n y ( m x 2 + 2 n y 2 ) E x , m , n y ] + i β 0 [ m x n y E x , m , n x + ( n y 2 + 2 m x 2 ) E y , m , n x ] = k 0 2 m m , n n ξ m m , n n ( n y E x , m , n m x E y , m , n ) .
2 Θ 0 z 2 + [ k 0 2 n 0 2 β 2 ] Θ 0 ( z ) = 0 ,
E x , m , n = ρ m n A m , n ( x , y ) Θ 0 ( z ) , E y , m , n = η m n A m , n ( x , y ) Θ 0 ( z ) ,
ρ m n = n y , m n m x , m n 2 + n y , m n 2 , η m n = m x , m n m x , m n 2 + n y , m n 2 ,
[ 2 z 2 + k 0 2 n 0 2 + 2 i k 0 n 0 ( z ) α ˜ ( z ) ( m x 2 + n y 2 ) β 0 2 ] A m , n Θ 0 ( z ) 2 i β 0 ( m x x + n y y ) A m , n Θ 0 ( z ) = k 0 2 m , n ξ m m , n n ( ρ m n E x , m , n + η m n E y , m , n ) .
[ δ + i α + ( 1 m x 2 + n y 2 ) β 0 ] A m , n i ( m x x + n y y ) A m , n = | r 2 + s 2 | = 1 ; r m , s n κ m n ( r s ) A r , s k 0 2 2 β 0 | m 2 + n 2 | 1 ξ m m , n n P C ( ρ m n E x , m , n + η m n E y , m , n ) Θ 0 * ( z ) d z ,
κ m n ( r s ) = k 0 2 2 β 0 P C ξ m r , n s ( ρ m n ρ r s + η m n η r s ) | Θ 0 ( z ) | 2 d z
I ( x , y ) = k = 1 3 | R k ( x , y ) | 2 + | S k ( x , y ) | 2 .
( Δ E x Δ E y ) = k 0 2 P C G ( z , z ) Θ 0 ( z ) d z ( ξ 1 , 0 ρ 1 , 0 ξ 1 , 0 ρ 1 , 0 ξ 0 , 1 ρ 0 , 1 ξ 0 , 1 ρ 0 , 1 ξ 1 , 1 ρ 1 , 1 ξ 1 , 1 ρ 1 , 1 ξ 1 , 0 η 1 , 0 ξ 1 , 0 η 1 , 0 ξ 0 , 1 η 0 , 1 ξ 0 , 1 η 0 1 ξ 1 , 1 η 1 , 1 ξ 1 , 1 η 1 , 1 ) V ,
( P C E x , m , n ( z ) Θ 0 * ( z ) d z P C E y , m , n ( z ) Θ 0 * ( z ) d z ) = 1 m x 2 + n y 2 ( n y m x m x n y ) ( μ m , n ( 1 , 0 ) μ m , n ( 1 , 0 ) μ m , n ( 0 , 1 ) μ m , n ( 0 , 1 ) μ m , n ( 1 , 1 ) μ m , n ( 1 , 1 ) ν m , n ( 1 , 0 ) ν m , n ( 1 , 0 ) ν m , n ( 0 , 1 ) ν m , n ( 0 , 1 ) ν m , n ( 1 , 1 ) ν m , n ( 1 , 1 ) ) V ( ς x , m , n ( 1 , 0 ) ς x , m , n ( 1 , 0 ) ς x , m , n ( 0 , 1 ) ς x , m , n ( 0 , 1 ) ς x , m , n ( 1 , 1 ) ς x , m , n ( 1 , 1 ) ς y , m , n ( 1 , 0 ) ς y , m , n ( 1 , 0 ) ς y , m , n ( 0 , 1 ) ς y , m , n ( 0 , 1 ) ς y , m , n ( 1 , 1 ) ς y , m , n ( 1 , 1 ) ) V ,
μ m n ( r s ) = k 0 2 P C ξ m r , n s ( n y ρ r s m x η r s ) G m , n ( z , z ) Θ 0 ( z ) Θ 0 * ( z ) d z d z ,
ν m n ( r s ) = P C 1 n 0 2 ξ m r , n s ( m x ρ r s + n y η r s ) | Θ 0 ( z ) | 2 d z ,
G m , n ( z , z ) = 1 2 β z , m , n e β z , m , n | z z | , β z , m , n = ( m x 2 + n y 2 ) β 0 2 k 0 2 n 0 2 ( z ) .
( δ + i α ) V = CV ,
C = d k 0 , m n + C b + C r + C h ,
d k 0 , m n = ( d k 0 ; 1 , 0 0 0 0 0 0 0 d k 0 ; 1 , 0 0 0 0 0 0 0 d k 0 ; 0 , 1 0 0 0 0 0 0 d k 0 ; 0 , 1 0 0 0 0 0 0 d k 0 ; 1 , 1 0 0 0 0 0 0 d k 0 ; 1 , 1 ) ,
C b = ( 0 κ 1 , 0 ( 1 , 0 ) κ 1 , 0 ( 0 , 1 ) κ 1 , 0 ( 0 , 1 ) κ 1 , 0 ( 1 , 1 ) κ 1 , 0 ( 1 , 1 ) κ 1 , 0 ( 1 , 0 ) 0 κ 1 , 0 ( 0 , 1 ) κ 1 , 0 ( 0 , 1 ) κ 1 , 0 ( 1 , 1 ) κ 1 , 0 ( 1 , 1 ) κ 0 , 1 ( 1 , 0 ) κ 0 , 1 ( 1 , 0 ) 0 κ 0 , 1 ( 0 , 1 ) κ 0 , 1 ( 1 , 1 ) κ 0 , 1 ( 1 , 1 ) κ 0 , 1 ( 1 , 0 ) κ 0 , 1 ( 1 , 0 ) κ 0 , 1 ( 0 , 1 ) 0 κ 0 , 1 ( 1 , 1 ) κ 0 , 1 ( 1 , 1 ) κ 1 , 1 ( 1 , 0 ) κ 1 , 1 ( 1 , 0 ) κ 1 , 1 ( 0 , 1 ) κ 1 , 1 ( 0 , 1 ) 0 κ 1 , 1 ( 1 , 1 ) κ 1 , 1 ( 1 , 0 ) κ 1 , 1 ( 1 , 0 ) κ 1 , 1 ( 0 , 1 ) κ 1 , 1 ( 0 , 1 ) κ 1 , 1 ( 1 , 1 ) 0 ) ,
C r = ( ζ 1 , 0 ( 1 , 0 ) ζ 1 , 0 ( 1 , 0 ) ζ 1 , 0 ( 0 , 1 ) ζ 1 , 0 ( 0 , 1 ) ζ 1 , 0 ( 1 , 1 ) ζ 1 , 0 ( 1 , 1 ) ζ 1 , 0 ( 1 , 0 ) ζ 1 , 0 ( 1 , 0 ) ζ 1 , 0 ( 0 , 1 ) ζ 1 , 0 ( 0 , 1 ) ζ 1 , 0 ( 1 , 1 ) ζ 1 , 0 ( 1 , 1 ) ζ 0 , 1 ( 1 , 0 ) ζ 0 , 1 ( 1 , 0 ) ζ 0 , 1 ( 0 , 1 ) ζ 0 , 1 ( 0 , 1 ) ζ 0 , 1 ( 1 , 1 ) ζ 0 , 1 ( 1 , 1 ) ζ 0 , 1 ( 1 , 0 ) ζ 0 , 1 ( 1 , 0 ) ζ 0 , 1 ( 0 , 1 ) ζ 0 , 1 ( 0 , 1 ) ζ 0 , 1 ( 1 , 1 ) ζ 0 , 1 ( 1 , 1 ) ζ 1 , 1 ( 1 , 0 ) ζ 1 , 1 ( 1 , 0 ) ζ 1 , 1 ( 0 , 1 ) ζ 1 , 1 ( 0 , 1 ) ζ 1 , 1 ( 1 , 1 ) ζ 1 , 1 ( 1 , 1 ) ζ 1 , 1 ( 1 , 0 ) ζ 1 , 1 ( 1 , 0 ) ζ 1 , 1 ( 0 , 1 ) ζ 1 , 1 ( 0 , 1 ) ζ 1 , 1 ( 1 , 1 ) ζ 1 , 1 ( 1 , 1 ) ) ,
C h = ( χ 1 , 0 ( 1 , 0 ) χ 1 , 0 ( 1 , 0 ) χ 1 , 0 ( 0 , 1 ) χ 1 , 0 ( 0 , 1 ) χ 1 , 0 ( 1 , 1 ) χ 1 , 0 ( 1 , 1 ) χ 1 , 0 ( 1 , 0 ) χ 1 , 0 ( 1 , 0 ) χ 1 , 0 ( 0 , 1 ) χ 1 , 0 ( 0 , 1 ) χ 1 , 0 ( 1 , 1 ) χ 1 , 0 ( 1 , 1 ) χ 0 , 1 ( 1 , 0 ) χ 0 , 1 ( 1 , 0 ) χ 0 , 1 ( 0 , 1 ) χ 0 , 1 ( 0 , 1 ) χ 0 , 1 ( 1 , 1 ) χ 0 , 1 ( 1 , 1 ) χ 0 , 1 ( 1 , 0 ) χ 0 , 1 ( 1 , 0 ) χ 0 , 1 ( 0 , 1 ) χ 0 , 1 ( 0 , 1 ) χ 0 , 1 ( 1 , 1 ) χ 0 , 1 ( 1 , 1 ) χ 1 , 1 ( 1 , 0 ) χ 1 , 1 ( 1 , 0 ) χ 1 , 1 ( 0 , 1 ) χ 1 , 1 ( 0 , 1 ) χ 1 , 1 ( 1 , 1 ) χ 1 , 1 ( 1 , 1 ) χ 1 , 1 ( 1 , 0 ) χ 1 , 1 ( 1 , 0 ) χ 1 , 1 ( 0 , 1 ) χ 1 , 1 ( 0 , 1 ) χ 1 , 1 ( 1 , 1 ) χ 1 , 1 ( 1 , 1 ) ) ,
d k 0 ; m , n = ( m x ( m , n ) 2 + n y ( m , n ) 2 1 ) β 0 ,
ζ m n ( r s ) = k 0 4 2 β 0 P C ξ m , n ξ r , s ( ρ m n ρ r s + η m n η r s ) G ( z , z ) Θ 0 ( z ) Θ 0 * ( z ) d z d z ,
χ m n ( r s ) = k 0 2 2 β 0 m 2 + n 2 > 1 ξ m m , n n ( ρ m n ς x , m , n ( r s ) + η m n ς y , m , n ( r s ) ) .
( δ + i α ) [ R 1 S 1 R 2 S 2 R 3 S 3 ] = [ C ] [ R 1 S 1 R 2 S 2 R 3 S 3 ] + i [ 1 2 R 1 x 3 2 R 1 y 1 2 S 1 x + 3 2 S 1 y 1 2 R 2 x + 3 2 R 2 y 1 2 S 2 x 3 2 S 2 y R 3 x S 3 x ] ,
1 2 ( δ + i α ) [ R 1 j + 1 , k + R 1 j , k S 1 j + 1 , k + S 1 j , k R 2 j , k + 1 + R 2 j , k S 2 j , k + 1 + S 2 j , k R 3 j + 1 , k + 1 + R 3 j , k S 3 j + 1 , k + 1 + S 3 j , k ] = 1 2 [ C ] [ R 1 j + 1 , k + R 1 j , k S 1 j + 1 , k + S 1 j , k R 2 j , k + 1 + R 2 j , k S 2 j , k + 1 + S 2 j , k R 3 j + 1 , k + 1 + R 3 j , k S 3 j + 1 , k + 1 + S 3 j , k ] + i 1 h [ R 1 j + 1 , k R 1 j , k S 1 j + 1 , k + S 1 j , k R 2 j , k + 1 R 2 j , k S 2 j , k + 1 + S 2 j , k R 3 j + 1 , k + 1 R 3 j , k S 3 j + 1 , k + 1 + S 3 i , k ] ,
L ( δ + i α ) [ R 1 j + 1 , k + R 1 j , k S 1 j + 1 , k + S 1 j , k R 2 j , k + 1 + R 2 j , k S 2 j , k + 1 + S 2 j , k R 3 j + 1 , k + 1 + R 3 j , k S 3 j + 1 , k + 1 + S 3 j , k ] = L [ C ] [ R 1 j + 1 , k + R 1 j , k S 1 j + 1 , k + S 1 j , k R 2 j , k + 1 + R 2 j , k S 2 j , k + 1 + S 2 j , k R 3 j + 1 , k + 1 + R 3 j , k S 3 j + 1 , k + 1 + S 3 j , k ] + i 2 N [ R 1 j + 1 , k R 1 j , k S 1 j + 1 , k + S 1 j , k R 2 j , k + 1 R 2 j , k S 2 j , k + 1 + S 2 j , k R 3 j + 1 , k + 1 R 3 j , k S 3 j + 1 , k + 1 + S 3 j , k ] ,

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