Abstract

We present a useful framework based on the coupled-wave theory, through which we can survey the resonant modes of TM polarization in 2D photonic-crystal lasers and understand their properties in detail. Through numerical calculations, we clarify their threshold gains, deviations from the Bragg frequency and field distributions. We find that the lasing mode can be selected by manipulating the hole-filling factor or the boundary reflection.

© 2007 Optical Society of America

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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. M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, "Laser action from two-dimensional distributed feedback in photonic crystals," Appl. Phys. Lett. 74, 7-9 (1999).
    [CrossRef]
  3. 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]
  4. E. Miyai, K. Sakai, T. Okano,W. Kunishi, D. Ohnishi, S. Noda, "Lasers producing tailored beams," Nature 441, 946 (2006).
    [CrossRef] [PubMed]
  5. G. A. Turnbull, P. Andrew, W. L. Barnes, and I. D. W. Samuel," Operating characteristics of a emiconducting polymer laser pumped by a microchip laser," Appl. Phys. Lett. 82, 313-315 (2003).
    [CrossRef]
  6. 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]
  7. M. Imada, A. Chutinan, S. Noda and M. Mochizuki, "Multidirectionally distributed feedback photonic rystal lasers," Phys. Rev. B 65, 195306 1-8 (2002).
    [CrossRef]
  8. 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]
  9. 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]
  10. H. Kogelnik and C. V. Shank, "Coupled-Wave Theory of Distributed Feedback Lasers," J. Appl. Phys. 43, 2327-2335 (1972).
    [CrossRef]
  11. G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A. L. Hutchinson, D. L. Sivco, A. Y. Cho, "High-Power infrared (8-Micrometer Wavelength) Superlattice Lasers," Science 276, 773-776 (1997).
    [CrossRef] [PubMed]
  12. R. Dorn, S. Quabis and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 1-4 (2003).
    [CrossRef]
  13. M. Plihal, and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991).
    [CrossRef]
  14. H. Kogelnik, "Coupled Wave Theory for Thick Hologram Gratings," Bell System Tech. J. 48, 2909-2947 (1969).
  15. 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 1-3 (2006).
    [CrossRef]
  16. R. F. Kazarinov, and C. H. Henry, "Second-order distributed-feedback lasers with mode selection provided by first-order radiation losses," IEEE J. Quantum Electronics 21, 144-150 (1985).
    [CrossRef]
  17. S. R. Chinn "Effects of Mirror Reflectivity in a Distributed-Feedback Laser," IEEE J. Quantum Electron. 9, 574-580 (1973).
    [CrossRef]
  18. W. Streifer, R. D. Burnham, and D. R. Scifres, "Effect of External Reflectors on Longitudinal Modes of Distributed Feedback Lasers," IEEE J. Quantum Electron. 11, 154-161 (1975).
    [CrossRef]

2006 (1)

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

2005 (2)

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]

2003 (2)

G. A. Turnbull, P. Andrew, W. L. Barnes, and I. D. W. Samuel," Operating characteristics of a emiconducting polymer laser pumped by a microchip laser," Appl. Phys. Lett. 82, 313-315 (2003).
[CrossRef]

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]

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 (2)

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. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, "Laser action from two-dimensional distributed feedback in photonic crystals," Appl. Phys. Lett. 74, 7-9 (1999).
[CrossRef]

1997 (1)

G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A. L. Hutchinson, D. L. Sivco, A. Y. Cho, "High-Power infrared (8-Micrometer Wavelength) Superlattice Lasers," Science 276, 773-776 (1997).
[CrossRef] [PubMed]

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]

1985 (1)

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

1975 (1)

W. Streifer, R. D. Burnham, and D. R. Scifres, "Effect of External Reflectors on Longitudinal Modes of Distributed Feedback Lasers," IEEE J. Quantum Electron. 11, 154-161 (1975).
[CrossRef]

1973 (1)

S. R. Chinn "Effects of Mirror Reflectivity in a Distributed-Feedback Laser," IEEE J. Quantum Electron. 9, 574-580 (1973).
[CrossRef]

1972 (1)

H. Kogelnik and C. V. Shank, "Coupled-Wave Theory of Distributed Feedback Lasers," J. Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

1969 (1)

H. Kogelnik, "Coupled Wave Theory for Thick Hologram Gratings," Bell System Tech. J. 48, 2909-2947 (1969).

Andrew, P.

G. A. Turnbull, P. Andrew, W. L. Barnes, and I. D. W. Samuel," Operating characteristics of a emiconducting polymer laser pumped by a microchip laser," Appl. Phys. Lett. 82, 313-315 (2003).
[CrossRef]

Barnes, W.L.

G. A. Turnbull, P. Andrew, W. L. Barnes, and I. D. W. Samuel," Operating characteristics of a emiconducting polymer laser pumped by a microchip laser," Appl. Phys. Lett. 82, 313-315 (2003).
[CrossRef]

Burnham, R. D.

W. Streifer, R. D. Burnham, and D. R. Scifres, "Effect of External Reflectors on Longitudinal Modes of Distributed Feedback Lasers," IEEE J. Quantum Electron. 11, 154-161 (1975).
[CrossRef]

Capasso, F.

G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A. L. Hutchinson, D. L. Sivco, A. Y. Cho, "High-Power infrared (8-Micrometer Wavelength) Superlattice Lasers," Science 276, 773-776 (1997).
[CrossRef] [PubMed]

Chinn, S. R.

S. R. Chinn "Effects of Mirror Reflectivity in a Distributed-Feedback Laser," IEEE J. Quantum Electron. 9, 574-580 (1973).
[CrossRef]

Cho, A. Y.

G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A. L. Hutchinson, D. L. Sivco, A. Y. Cho, "High-Power infrared (8-Micrometer Wavelength) Superlattice Lasers," Science 276, 773-776 (1997).
[CrossRef] [PubMed]

Chutinan, A.

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]

Dodabalapur, A.

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, "Laser action from two-dimensional distributed feedback in photonic crystals," Appl. Phys. Lett. 74, 7-9 (1999).
[CrossRef]

Faist, J.

G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A. L. Hutchinson, D. L. Sivco, A. Y. Cho, "High-Power infrared (8-Micrometer Wavelength) Superlattice Lasers," Science 276, 773-776 (1997).
[CrossRef] [PubMed]

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 Electronics 21, 144-150 (1985).
[CrossRef]

Hutchinson, A. L.

G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A. L. Hutchinson, D. L. Sivco, A. Y. Cho, "High-Power infrared (8-Micrometer Wavelength) Superlattice Lasers," Science 276, 773-776 (1997).
[CrossRef] [PubMed]

Imada, M.

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]

Joannopoulos, J. D.

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, "Laser action from two-dimensional distributed feedback in photonic crystals," Appl. Phys. Lett. 74, 7-9 (1999).
[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 Electronics 21, 144-150 (1985).
[CrossRef]

Kogelnik, H.

H. Kogelnik and C. V. Shank, "Coupled-Wave Theory of Distributed Feedback Lasers," J. Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

H. Kogelnik, "Coupled Wave Theory for Thick Hologram Gratings," Bell System Tech. J. 48, 2909-2947 (1969).

Kunishi, W.

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

Maradudin, A. A.

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

Meier, M.

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, "Laser action from two-dimensional distributed feedback in photonic crystals," Appl. Phys. Lett. 74, 7-9 (1999).
[CrossRef]

Mekis, A.

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, "Laser action from two-dimensional distributed feedback in photonic crystals," Appl. Phys. Lett. 74, 7-9 (1999).
[CrossRef]

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]

Miyai, E.

E. Miyai, K. Sakai, T. Okano,W. Kunishi, D. Ohnishi, S. Noda, "Lasers producing tailored beams," Nature 441, 946 (2006).
[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.

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]

Nalamasu, O.

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, "Laser action from two-dimensional distributed feedback in photonic crystals," Appl. Phys. Lett. 74, 7-9 (1999).
[CrossRef]

Noda, S.

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

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]

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]

Ohnishi, D.

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

Okano, T.

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

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]

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.

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

Samuel, I.D.W.

G. A. Turnbull, P. Andrew, W. L. Barnes, and I. D. W. Samuel," Operating characteristics of a emiconducting polymer laser pumped by a microchip laser," Appl. Phys. Lett. 82, 313-315 (2003).
[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]

Scamarcio, G.

G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A. L. Hutchinson, D. L. Sivco, A. Y. Cho, "High-Power infrared (8-Micrometer Wavelength) Superlattice Lasers," Science 276, 773-776 (1997).
[CrossRef] [PubMed]

Scifres, D. R.

W. Streifer, R. D. Burnham, and D. R. Scifres, "Effect of External Reflectors on Longitudinal Modes of Distributed Feedback Lasers," IEEE J. Quantum Electron. 11, 154-161 (1975).
[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]

Sirtori, C.

G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A. L. Hutchinson, D. L. Sivco, A. Y. Cho, "High-Power infrared (8-Micrometer Wavelength) Superlattice Lasers," Science 276, 773-776 (1997).
[CrossRef] [PubMed]

Sivco, D. L.

G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A. L. Hutchinson, D. L. Sivco, A. Y. Cho, "High-Power infrared (8-Micrometer Wavelength) Superlattice Lasers," Science 276, 773-776 (1997).
[CrossRef] [PubMed]

Slusher, R. E.

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, "Laser action from two-dimensional distributed feedback in photonic crystals," Appl. Phys. Lett. 74, 7-9 (1999).
[CrossRef]

Streifer, W.

W. Streifer, R. D. Burnham, and D. R. Scifres, "Effect of External Reflectors on Longitudinal Modes of Distributed Feedback Lasers," IEEE J. Quantum Electron. 11, 154-161 (1975).
[CrossRef]

Timko, A.

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, "Laser action from two-dimensional distributed feedback in photonic crystals," Appl. Phys. Lett. 74, 7-9 (1999).
[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]

Turnbull, G.A.

G. A. Turnbull, P. Andrew, W. L. Barnes, and I. D. W. Samuel," Operating characteristics of a emiconducting polymer laser pumped by a microchip laser," Appl. Phys. Lett. 82, 313-315 (2003).
[CrossRef]

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]

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]

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. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, "Laser action from two-dimensional distributed feedback in photonic crystals," Appl. Phys. Lett. 74, 7-9 (1999).
[CrossRef]

G. A. Turnbull, P. Andrew, W. L. Barnes, and I. D. W. Samuel," Operating characteristics of a emiconducting polymer laser pumped by a microchip laser," Appl. Phys. Lett. 82, 313-315 (2003).
[CrossRef]

Bell System Tech. J. (1)

H. Kogelnik, "Coupled Wave Theory for Thick Hologram Gratings," Bell System Tech. J. 48, 2909-2947 (1969).

IEEE J. Quantum Electron. (2)

S. R. Chinn "Effects of Mirror Reflectivity in a Distributed-Feedback Laser," IEEE J. Quantum Electron. 9, 574-580 (1973).
[CrossRef]

W. Streifer, R. D. Burnham, and D. R. Scifres, "Effect of External Reflectors on Longitudinal Modes of Distributed Feedback Lasers," IEEE J. Quantum Electron. 11, 154-161 (1975).
[CrossRef]

IEEE J. Quantum Electronics (1)

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

IEEE J. Quantum. Electron. (1)

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]

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]

Nature (1)

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

Opt. Express (1)

Phys. Rev. B (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]

Science (2)

G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A. L. Hutchinson, D. L. Sivco, A. Y. Cho, "High-Power infrared (8-Micrometer Wavelength) Superlattice Lasers," Science 276, 773-776 (1997).
[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]

Other (3)

M. Imada, A. Chutinan, S. Noda and M. Mochizuki, "Multidirectionally distributed feedback photonic rystal lasers," Phys. Rev. B 65, 195306 1-8 (2002).
[CrossRef]

R. Dorn, S. Quabis and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 1-4 (2003).
[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 1-3 (2006).
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

Schematic diagram of the four basic waves in a square-lattice PC with circular holes.

Fig. 2.
Fig. 2.

(a) Plot of the threshold gain (αL) vs the wavenumber deviation from the Bragg condition (δL). (b) The longitudinal mode pattern for the three orders.

Fig. 3.
Fig. 3.

(a) Plot of aL vs dL for the 1st order modes. (b) The amplitude sum, Rx +Sx , for the 1st order modes.

Fig. 4.
Fig. 4.

(a) The intensity distribution for the four waves, |Rx |2+|Sx |2+|Ry |2+|Sy |2, for the basic modes. (b) The electric field distribution around the center of the cavity.

Fig.5.
Fig.5.

The product of the coupling constants κ1, κ2, κ3 and the cavity length L as functions of the hole-filling factor f.

Fig. 6.
Fig. 6.

(a) Plot of αL vs δL of the two modes A and O for several magnitudes of the coupling constants under the same ratio of κ2L/κ3L=2. (b) The intensity distribution of the four waves, |Rx |2+|Sx |2+|Ry |2+|Sy |2, for mode A, along the x axis at the center of the cavity.

Fig. 7.
Fig. 7.

The threshold gain for the basic modes as functions of the hole-filling factor f.

Fig. 8.
Fig. 8.

Phase ϕ as a function of the edge position for a typical cleaved facet in the case of the second order grating.

Fig. 9
Fig. 9

Threshold gain of mode A (a) as a functions of reflectivity ρ for the phase ϕ=0 and ϕ=π, (b) as a function of the phase, ϕ, for the reflectivity of ρ=0.2.

Equations (17)

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2 E z x 2 + 2 E z y 2 + k 2 E z = 0 ,
k 2 = β 2 + 2 iαβ + 2 β G 0 κ ( G ) exp [ i ( G r ) ]
κ ( G ) = π λ n ( G ) + i 1 2 α ( G )
E z ( r ) = G h G exp [ i ( k + G ) r ] ,
E z = R x ( x , y ) e i β 0 x + S x ( x , y ) e i β 0 x + R y ( x , y ) e i β 0 y + S y ( x , y ) e i β 0 y
x R x + ( α κ 0 i δ ) R x = ( i κ 3 κ 0 ) S x + i κ 2 ( S y + R y ) ,
x S x + ( α κ 0 i δ ) S x = ( i κ 3 κ 0 ) R x + i κ 2 ( S y + R y ) ,
y R y + ( α κ 0 i δ ) R y = ( i κ 3 κ 0 ) S y + i κ 2 ( S x + R x ) ,
y S y + ( α κ 0 i δ ) S y = ( i κ 3 κ 0 ) R y + i κ 2 ( S x + R x ) .
κ 1 = κ ( G ) | G = β 0 , κ 2 = κ ( G ) | G = 2 β 0 , κ 3 = κ ( G ) | G = 2 β 0 .
n ( G ) = ( n a n b ) f 2 J 1 ( G 2 π f ) ( G 2 π f )
α ( G ) = ( α a α b ) f 2 J 1 ( G 2 π f ) ( G 2 π f )
r = ρ exp ( i ϕ ) ,
R x ( L 2 , y ) = S x ( L 2 , y ) ρ x exp ( i ϕ x ) ,
S x ( L 2 , y ) = R x ( L 2 , y ) ρ x + exp ( i ϕ x + ) ,
R y ( x , L 2 ) = S y ( x , L 2 ) ρ y exp ( i ϕ y ) ,
S y ( x , L 2 ) = R y ( x , L 2 ) ρ y + exp ( i ϕ y + ) .

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