Abstract

We present a theoretical analysis of laser action within the bands of propagating modes of a photonic crystal. Using Bloch functions as carrier waves in conjunction with a multiscale analysis, we derive the generalized Maxwell–Bloch equations for an incoherently pumped atomic system in interaction with the electromagnetic reservoir of a photonic crystal. These general Maxwell–Bloch equations are similar to the conventional semiclassical laser equations but contain effective parameters that depend on the band structure of the linear photonic crystal. Through an investigation of steady-state laser behavior, we show that, near a photonic band edge, the rate of stimulated emission may be enhanced and the internal losses are reduced, which leads to an important lowering of the laser threshold. In addition, we find an increase of the laser output along with an additional narrowing of the linewidth at a photonic band edge.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2001 (5)

M. Florescu and S. John, “Single-atom switching in photonic crystals,” Phys. Rev. A 64, 033801–1–033801–21 (2001).
[CrossRef]

S. John and M. Florescu, “Photonic band gap materials: toward an all-optical micro-transistor,” J. Opt. A, Pure Appl. Opt. 3, S103–S120 (2001).
[CrossRef]

G. A. Turnbull, P. Andrew, M. J. Jory, W. L. Barnes, and I. D. W. Samuel, “Relationship between photonic band structure and emission characteristics of a polymer distributed feedback laser,” Phys. Rev. B 64, 125122–1–125122–6 (2001).
[CrossRef]

N. Susa, “Threshold gain and gain-enhancement due to distributed-feedback in two-dimensional photonic-crystal lasers,” J. Appl. Phys. 89, 815–823 (2001).
[CrossRef]

D. Hermann, M. Frank, K. Busch, and P. Wölfle, “Photonic band structure computation,” Opt. Express 8, 167–172 (2001), http://epubs.osa.org/opticsexpress.
[CrossRef] [PubMed]

2000 (3)

J. E. Sipe, “Vector k⋅p approach for photonic band structures,” Phys. Rev. E 62, 5672–5677 (2000).
[CrossRef]

J.-K. Hwang, H.-Y. Ryu, D.-S. Song, I.-Y. Han, H.-W. Song, H.-K. Park, Y.-H. Lee, and D.-H. Jang, “Room-temperature triangular-lattice two-dimensional photonic band gap lasers operating at 1.54 μm,” Appl. Phys. Lett. 76, 2982–2984 (2000).
[CrossRef]

S. Riechel, C. Kallinger, U. Lemmer, J. Feldmann, A. Gombert, V. Wittwer, and U. Scherf, “A nearly diffraction limited surface emitting conjugated polymer laser utilizing a two-dimensional photonic band structure,” Appl. Phys. Lett. 77, 2310–2312 (2000).
[CrossRef]

1999 (4)

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Mekis, A. Timko, C. A. Murray, R. Ruel, and O. Nalamasu, “Emission characteristics of two-dimensional organic photonic crystal lasers fabricated by replica molding,” J. Appl. Phys. 86, 3502–3507 (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]

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]

K. Sakoda, K. Ohtaka, and T. Ueta, “Low-threshold laser oscillation due to group-velocity anomaly pelicular to two- and three-dimensional photonic crystals,” Opt. Express 4, 481–489 (1999), http://epubs.osa.org/opticsexpress.
[CrossRef] [PubMed]

1998 (3)

V. I. Kopp, B. Fan, H. K. M. Vithana, and A. Z. Genack, “Low-threshold lasing at the edge of a photonic stop band in cholesteric liquid crystals,” Opt. Lett. 23, 1707–1709 (1998).
[CrossRef]

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

N. Vats and S. John, “Non-Markovian quantum fluctuations and superradiance near a photonic band edge,” Phys. Rev. A 58, 4168–4184 (1998).
[CrossRef]

1997 (1)

S. John and T. Quang, “Collective switching and inversion without fluctuation of two-level atoms in confined photonic systems,” Phys. Rev. Lett. 78, 1888–1891 (1997).
[CrossRef]

1994 (1)

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

1991 (1)

G. Bjork and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. 27, 2386–2396 (1991).
[CrossRef]

1990 (1)

S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[CrossRef] [PubMed]

1989 (1)

H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. 86, 4801–4805 (1989).
[CrossRef]

1988 (1)

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149–5165 (1988).
[CrossRef] [PubMed]

1987 (2)

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

1984 (1)

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984).
[CrossRef]

Andrew, P.

G. A. Turnbull, P. Andrew, M. J. Jory, W. L. Barnes, and I. D. W. Samuel, “Relationship between photonic band structure and emission characteristics of a polymer distributed feedback laser,” Phys. Rev. B 64, 125122–1–125122–6 (2001).
[CrossRef]

Barnes, W. L.

G. A. Turnbull, P. Andrew, M. J. Jory, W. L. Barnes, and I. D. W. Samuel, “Relationship between photonic band structure and emission characteristics of a polymer distributed feedback laser,” Phys. Rev. B 64, 125122–1–125122–6 (2001).
[CrossRef]

Bjork, G.

G. Bjork and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. 27, 2386–2396 (1991).
[CrossRef]

Bloemer, M. J.

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

Bowden, C. M.

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

Brorson, S. D.

H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. 86, 4801–4805 (1989).
[CrossRef]

Busch, K.

Chutinan, A.

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]

de Sterke, C. M.

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149–5165 (1988).
[CrossRef] [PubMed]

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]

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Mekis, A. Timko, C. A. Murray, R. Ruel, and O. Nalamasu, “Emission characteristics of two-dimensional organic photonic crystal lasers fabricated by replica molding,” J. Appl. Phys. 86, 3502–3507 (1999).
[CrossRef]

Dowling, J. P.

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

Fan, B.

Feldmann, J.

S. Riechel, C. Kallinger, U. Lemmer, J. Feldmann, A. Gombert, V. Wittwer, and U. Scherf, “A nearly diffraction limited surface emitting conjugated polymer laser utilizing a two-dimensional photonic band structure,” Appl. Phys. Lett. 77, 2310–2312 (2000).
[CrossRef]

Florescu, M.

S. John and M. Florescu, “Photonic band gap materials: toward an all-optical micro-transistor,” J. Opt. A, Pure Appl. Opt. 3, S103–S120 (2001).
[CrossRef]

M. Florescu and S. John, “Single-atom switching in photonic crystals,” Phys. Rev. A 64, 033801–1–033801–21 (2001).
[CrossRef]

Frank, M.

Genack, A. Z.

Gombert, A.

S. Riechel, C. Kallinger, U. Lemmer, J. Feldmann, A. Gombert, V. Wittwer, and U. Scherf, “A nearly diffraction limited surface emitting conjugated polymer laser utilizing a two-dimensional photonic band structure,” Appl. Phys. Lett. 77, 2310–2312 (2000).
[CrossRef]

Han, I.-Y.

J.-K. Hwang, H.-Y. Ryu, D.-S. Song, I.-Y. Han, H.-W. Song, H.-K. Park, Y.-H. Lee, and D.-H. Jang, “Room-temperature triangular-lattice two-dimensional photonic band gap lasers operating at 1.54 μm,” Appl. Phys. Lett. 76, 2982–2984 (2000).
[CrossRef]

Hermann, D.

Hwang, J.-K.

J.-K. Hwang, H.-Y. Ryu, D.-S. Song, I.-Y. Han, H.-W. Song, H.-K. Park, Y.-H. Lee, and D.-H. Jang, “Room-temperature triangular-lattice two-dimensional photonic band gap lasers operating at 1.54 μm,” Appl. Phys. Lett. 76, 2982–2984 (2000).
[CrossRef]

Imada, 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]

Jang, D.-H.

J.-K. Hwang, H.-Y. Ryu, D.-S. Song, I.-Y. Han, H.-W. Song, H.-K. Park, Y.-H. Lee, and D.-H. Jang, “Room-temperature triangular-lattice two-dimensional photonic band gap lasers operating at 1.54 μm,” Appl. Phys. Lett. 76, 2982–2984 (2000).
[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]

John, S.

M. Florescu and S. John, “Single-atom switching in photonic crystals,” Phys. Rev. A 64, 033801–1–033801–21 (2001).
[CrossRef]

S. John and M. Florescu, “Photonic band gap materials: toward an all-optical micro-transistor,” J. Opt. A, Pure Appl. Opt. 3, S103–S120 (2001).
[CrossRef]

N. Vats and S. John, “Non-Markovian quantum fluctuations and superradiance near a photonic band edge,” Phys. Rev. A 58, 4168–4184 (1998).
[CrossRef]

S. John and T. Quang, “Collective switching and inversion without fluctuation of two-level atoms in confined photonic systems,” Phys. Rev. Lett. 78, 1888–1891 (1997).
[CrossRef]

S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984).
[CrossRef]

Jory, M. J.

G. A. Turnbull, P. Andrew, M. J. Jory, W. L. Barnes, and I. D. W. Samuel, “Relationship between photonic band structure and emission characteristics of a polymer distributed feedback laser,” Phys. Rev. B 64, 125122–1–125122–6 (2001).
[CrossRef]

Kallinger, C.

S. Riechel, C. Kallinger, U. Lemmer, J. Feldmann, A. Gombert, V. Wittwer, and U. Scherf, “A nearly diffraction limited surface emitting conjugated polymer laser utilizing a two-dimensional photonic band structure,” Appl. Phys. Lett. 77, 2310–2312 (2000).
[CrossRef]

Kopp, V. I.

Lee, Y.-H.

J.-K. Hwang, H.-Y. Ryu, D.-S. Song, I.-Y. Han, H.-W. Song, H.-K. Park, Y.-H. Lee, and D.-H. Jang, “Room-temperature triangular-lattice two-dimensional photonic band gap lasers operating at 1.54 μm,” Appl. Phys. Lett. 76, 2982–2984 (2000).
[CrossRef]

Lemmer, U.

S. Riechel, C. Kallinger, U. Lemmer, J. Feldmann, A. Gombert, V. Wittwer, and U. Scherf, “A nearly diffraction limited surface emitting conjugated polymer laser utilizing a two-dimensional photonic band structure,” Appl. Phys. Lett. 77, 2310–2312 (2000).
[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]

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Mekis, A. Timko, C. A. Murray, R. Ruel, and O. Nalamasu, “Emission characteristics of two-dimensional organic photonic crystal lasers fabricated by replica molding,” J. Appl. Phys. 86, 3502–3507 (1999).
[CrossRef]

Mekis, A.

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Mekis, A. Timko, C. A. Murray, R. Ruel, and O. Nalamasu, “Emission characteristics of two-dimensional organic photonic crystal lasers fabricated by replica molding,” J. Appl. Phys. 86, 3502–3507 (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]

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]

Murray, C. A.

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Mekis, A. Timko, C. A. Murray, R. Ruel, and O. Nalamasu, “Emission characteristics of two-dimensional organic photonic crystal lasers fabricated by replica molding,” J. Appl. Phys. 86, 3502–3507 (1999).
[CrossRef]

Nalamasu, O.

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Mekis, A. Timko, C. A. Murray, R. Ruel, and O. Nalamasu, “Emission characteristics of two-dimensional organic photonic crystal lasers fabricated by replica molding,” J. Appl. Phys. 86, 3502–3507 (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]

Noda, S.

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]

Nojima, S.

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

Ohtaka, K.

Park, H.-K.

J.-K. Hwang, H.-Y. Ryu, D.-S. Song, I.-Y. Han, H.-W. Song, H.-K. Park, Y.-H. Lee, and D.-H. Jang, “Room-temperature triangular-lattice two-dimensional photonic band gap lasers operating at 1.54 μm,” Appl. Phys. Lett. 76, 2982–2984 (2000).
[CrossRef]

Quang, T.

S. John and T. Quang, “Collective switching and inversion without fluctuation of two-level atoms in confined photonic systems,” Phys. Rev. Lett. 78, 1888–1891 (1997).
[CrossRef]

Riechel, S.

S. Riechel, C. Kallinger, U. Lemmer, J. Feldmann, A. Gombert, V. Wittwer, and U. Scherf, “A nearly diffraction limited surface emitting conjugated polymer laser utilizing a two-dimensional photonic band structure,” Appl. Phys. Lett. 77, 2310–2312 (2000).
[CrossRef]

Rogers, J. A.

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Mekis, A. Timko, C. A. Murray, R. Ruel, and O. Nalamasu, “Emission characteristics of two-dimensional organic photonic crystal lasers fabricated by replica molding,” J. Appl. Phys. 86, 3502–3507 (1999).
[CrossRef]

Ruel, R.

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Mekis, A. Timko, C. A. Murray, R. Ruel, and O. Nalamasu, “Emission characteristics of two-dimensional organic photonic crystal lasers fabricated by replica molding,” J. Appl. Phys. 86, 3502–3507 (1999).
[CrossRef]

Ryu, H.-Y.

J.-K. Hwang, H.-Y. Ryu, D.-S. Song, I.-Y. Han, H.-W. Song, H.-K. Park, Y.-H. Lee, and D.-H. Jang, “Room-temperature triangular-lattice two-dimensional photonic band gap lasers operating at 1.54 μm,” Appl. Phys. Lett. 76, 2982–2984 (2000).
[CrossRef]

Sakoda, K.

Samuel, I. D. W.

G. A. Turnbull, P. Andrew, M. J. Jory, W. L. Barnes, and I. D. W. Samuel, “Relationship between photonic band structure and emission characteristics of a polymer distributed feedback laser,” Phys. Rev. B 64, 125122–1–125122–6 (2001).
[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]

Scalora, M.

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

Scherf, U.

S. Riechel, C. Kallinger, U. Lemmer, J. Feldmann, A. Gombert, V. Wittwer, and U. Scherf, “A nearly diffraction limited surface emitting conjugated polymer laser utilizing a two-dimensional photonic band structure,” Appl. Phys. Lett. 77, 2310–2312 (2000).
[CrossRef]

Sipe, J. E.

J. E. Sipe, “Vector k⋅p approach for photonic band structures,” Phys. Rev. E 62, 5672–5677 (2000).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149–5165 (1988).
[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]

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Mekis, A. Timko, C. A. Murray, R. Ruel, and O. Nalamasu, “Emission characteristics of two-dimensional organic photonic crystal lasers fabricated by replica molding,” J. Appl. Phys. 86, 3502–3507 (1999).
[CrossRef]

Song, D.-S.

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[CrossRef]

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J.-K. Hwang, H.-Y. Ryu, D.-S. Song, I.-Y. Han, H.-W. Song, H.-K. Park, Y.-H. Lee, and D.-H. Jang, “Room-temperature triangular-lattice two-dimensional photonic band gap lasers operating at 1.54 μm,” Appl. Phys. Lett. 76, 2982–2984 (2000).
[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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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]

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[CrossRef]

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[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Photonic band structure for E-polarized radiation in a 2-D photonic crystal consisting of a square array of dielectric cylinders (r/a=0.4 and c=12) in air.

Fig. 2
Fig. 2

Group velocities q·vm for the three lowest bands for E-polarized radiation in a 2-D photonic crystal consisting of a square array of dielectric cylinders. The photonic-crystal parameters are the same as for Fig. 1.

Fig. 3
Fig. 3

Dimensionless gain-enhancement factor α¯m for the three lowest bands for E-polarized radiation in a 2-D photonic crystal consisting of a square array of dielectric cylinders. The active medium occupies the space between the cylinders, and the photonic-crystal parameters are the same as for Fig. 1.

Fig. 4
Fig. 4

Dimensionless saturation-enhancement factor β¯m for the three lowest bands for E-polarized radiation in a 2-D photonic crystal consisting of a square array of dielectric cylinders. The active medium occupies the space between the cylinders, and the photonic-crystal parameters are the same as for Fig. 1.

Equations (65)

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2E(x, t)-(x)c2 2E(x, t)t2-4πσ˜(x)c2 E(x, t)t
=4πc2 2Pnlatoms(x, t)t2,
Pnlatoms(x, t)=aδ(x-xa)Pa(t)=n(x)P˜(x, t),
σ˜(x)μσ(x),
P˜(x, t)μP(x, t),
2E(x, t)-(x)c2 2E(x, t)t2-μ 4πσ(x)c2 E(x, t)t
=μ 4πc2n(x) 2P(x, t)t2.
d2Pa(t)dt2+2γ˜ dPa(t)dt+ωa2Pa(t)=-2ωaΩ˜|d12|ΔNa(t).
Ω˜|d12|E(xa, t)
dΔNa(t)dt=γ˜[ΔNeq,a-ΔNa(t)]+2 Ω˜ωa 1|d12| dPa(t)dt,
γ˜μγ,
γ˜μγ,
ωaΩ˜|d12|ΔNa(t)ωaΩ˜Pa(t)ωa2Pa(t),
Ω˜ωa 1|d12| dPa(t)dtΩ˜ωa dΔNa(t)dtdΔNa(t)dt.
Ω˜μΩ,
d2P(x, t)dt2+μ2γ dP(x, t)dt+ωa2P(x, t)
=μ-2ωa |d12|2E(x, t)ΔN(x, t),
dΔN(x, t)dt=μR-γΔN(x, t)+2ωaE(x, t) dP(x, t)dt.
x=x0+μ x1+μ2 x2+,
t=t0+μ t1+μ2 t2+,
E(x, t)=E(0)+μE(1)+μ2E(2)+,
P(x, t)=P(0)+μP(1)+μ2P(2)+,
ΔN(x, t)=ΔN(0)+μΔN(1)+μ2ΔN(2)+,
2x02-(x0)c2 2t02E(0)=0,
2P(0)t02+ωa2P(0)=0.
E(0)(x0, x1 ,; t0, t1 ,)
=E(x1, x2 ,;t1, t2 ,)Φm(x0)exp(-iωmt0)+c.c.,
P(0)(x0, x1 ,; t0, t1 ,)
=P(x1, x2 ,; t1, t2 ,)Φm(x0)exp(-iωmt0)+c.c..
2x02+ωm2c2(x0)Φm(x0)=0
cellΦm*(x0)(x0)Φm(x0)dx0=δm,m.
ωm=ωa.
-c2 2x02+(x0) 2t02E(1)
=2c2 x0 x1-2(x0) t0 t1+4πσ(x0) t0E(0)+4πn(x0) 2t02P(0),
2t02+ωa2P(1)=-22t0t1+γ t0P(0)-2ωa |d12|2E(0)ΔN(0),
t1ΔN(0)=R-γΔN(0)+2ωaE(0) P(0)t0.
E(1)(x0, x1 ,; t0, t1 ,)
=lme(x1, x2 ,; t1, t2 ,)Φl(x0)exp(-iωmt0)+c.c.,
P(1)(x0, x1 ,; t0, t1 ,)
=lmp(x1, x2 ,; t1, t2 ,)Φl(x0)exp(-iωmt0)+c.c.
vm·E(x, t)+E(x, t)t+2πσmE(x, t)
=2πiαmωmP(x, t),
P(x, t)t+γP(x, t)
=-i |d12|2 E(x, t)ΔN˜(x, t),
ΔN˜(x, t)t=R-γΔN˜(x, t)+2iβm[E(x, t)P*(x, t)-P(x, t)E*(x, t)],
vm=c2ωm cellϕm*(x0)-i ddx0Φm(x0)dx0.
ΔN˜(x, t)cellΔN(x0, x1; t1)x|Φm(x0)|2n(x0)dx0cell|Φm(x0)|2n(x0)dx0.
αm=cell|Φm(x0)|2n(x0)dx0,
βm=cell|Φm(x0)|4n(x0)dx0cell|Φm(x0)|2n(x0)dx0.
σm=cell|Φm(x0)|2σ(x0)dx0.
P(x, t)=-i |d12|2γ E(x, t)ΔN˜(x, t),
vm·E(x, t)+E(x, t)t
=12 -γm+αm 4π|d12|2ωmω ΔN˜(x, t)E(x, t),
ΔN˜(x, t)t=R-γΔN˜(x, t)-βm 4|d12|22γ|E(x, t)|2ΔN˜(x, t),
ΔN¯(x)=R/γ1+βmI(x),
I(x)=|E(x, t)|2Isat,
Isat2γγ4|d12|2
vm·E(x)=12 -γm+Gm1+βmI(x)E(x),
Gm=παm(ωm)RIsat.
Iss=1βm Gmγm-1.
Rthr=I¯satγmγ0 1α¯mRthr0,
αmcelldx0|Φm(x0)|2n(x0)/celldx0n(x0).
Ioutput=0forR<Rthrπβmγm αm(ωm)Isat(R-Rthr)forR>Rthr,
Δν=Δνcnss,
nss=nsat0βm RRthr-1,

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