Abstract

We investigate the physics of spontaneous emission in a photonic crystal (PhC) made of GaN rods with embedded InGaN quantum wells, formed on a thick GaN layer. Although the PhC lies on a higher-index medium, we evidence the existence of unexpected quasi-guided Bloch modes which are strongly localized in the PhC region and possess a long lifetime. These modes determine the behavior of spontaneous emission such as the emission diagram and Purcell effect, as would happen in the usual case of emission in a PhC membrane.

©2007 Optical Society of America

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  1. P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. L. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657, 2004.
    [Crossref] [PubMed]
  2. M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296–1298, 2005.
    [Crossref] [PubMed]
  3. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904, 2005.
    [Crossref] [PubMed]
  4. E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, S. Olivier, S. Varoutsis, I. Robert-Philip, R. Houdre, and C. J. M. Smith, “Spontaneous emission enhancement of quantum dots in a photonic crystal wire,” Phys. Rev. Lett 95, 183901, 2005.
    [Crossref] [PubMed]
  5. M. Kitamura, S. Iwamoto, and Y. Arakawa, “Enhanced light emission from an organic photonic crystal with a nanocavity,” Appl. Phys. Lett. 87, 151119, 2005.
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  6. J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887, 2004.
    [Crossref]
  7. T. N. Oder, K. H. Kim, J. Y. Lin, and H. X. Jiang, “III-nitride blue and ultraviolet photonic crystal light emitting diodes,” Appl. Phys. Lett. 84, 466–468, 2004.
    [Crossref]
  8. A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124, 2006.
    [Crossref]
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    [Crossref]
  22. S. Mathias, M. Wiesenmayer, M. Aeschlimann, and M. Bauer, “Quantum-well wave-function localization and the electron-phonon interaction in thin Ag nanofilms,” Phys. Rev. Lett. 9, 7236809, 2006.
    [Crossref]
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    [Crossref]

2007 (1)

D. Gerace and L. C. Andreani, “Quantum theory of exciton-photon coupling in photonic crystal slabs with embedded quantum wells,” Phys. Rev. B 75, 235325, 2007.
[Crossref]

2006 (3)

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124, 2006.
[Crossref]

S. Keller, C. Schaake, N. A. Fichtenbaum, C. J. Neufeld, Y. Wu, K. McGroddy, A. David, S. P. DenBaars, C. Weisbuch, J. S. Speck, and U. K. Mishra, “Optical and structural properties of GaN nanopillar and nanostripe arrays with embedded InGaN/GaN multi-quantum wells,” J. Appl. Phys. 10 0,054314, 2006.
[Crossref]

S. Mathias, M. Wiesenmayer, M. Aeschlimann, and M. Bauer, “Quantum-well wave-function localization and the electron-phonon interaction in thin Ag nanofilms,” Phys. Rev. Lett. 9, 7236809, 2006.
[Crossref]

2005 (5)

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107, 2005.
[Crossref]

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296–1298, 2005.
[Crossref] [PubMed]

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904, 2005.
[Crossref] [PubMed]

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, S. Olivier, S. Varoutsis, I. Robert-Philip, R. Houdre, and C. J. M. Smith, “Spontaneous emission enhancement of quantum dots in a photonic crystal wire,” Phys. Rev. Lett 95, 183901, 2005.
[Crossref] [PubMed]

M. Kitamura, S. Iwamoto, and Y. Arakawa, “Enhanced light emission from an organic photonic crystal with a nanocavity,” Appl. Phys. Lett. 87, 151119, 2005.
[Crossref]

2004 (3)

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887, 2004.
[Crossref]

T. N. Oder, K. H. Kim, J. Y. Lin, and H. X. Jiang, “III-nitride blue and ultraviolet photonic crystal light emitting diodes,” Appl. Phys. Lett. 84, 466–468, 2004.
[Crossref]

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. L. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657, 2004.
[Crossref] [PubMed]

2002 (2)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102, 2002.
[Crossref]

D. Delbeke, P. Bienstman, R. Bockstaele, and R. Baets, “Rigorous electromagnetic analysis of dipole emission in periodically corrugated layers: the grating-assisted resonant-cavity light-emitting diode,” J. Opt. Soc. Am. A 19, 871–880, 2002.
[Crossref]

2001 (1)

2000 (1)

1999 (2)

1994 (1)

N. Koide and K. Ujihara, “Analysis of spontaneous emission from a planar microcavity dependence on the refractive-index of the cavity region and atomic location,” Opt. Commun. 111, 381–393, 1994.
[Crossref]

1992 (1)

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous-media with applications to photonic band structures,” Phys. Rev. A 46, 612–622, 1992.
[Crossref] [PubMed]

1975 (1)

J. P. Wittke, “Spontaneous emission-rate alteration by dielectric and other waveguiding structures,” RCA Review 36, 655–666, 1975.

1974 (1)

R. R. Chance, A. Prock, and R. Silbey, “Lifetime of an emitting molecule near a partially reflecting surface,” J. Chemical Phys. 60, 2744–2748, 1974.
[Crossref]

Aeschlimann, M.

S. Mathias, M. Wiesenmayer, M. Aeschlimann, and M. Bauer, “Quantum-well wave-function localization and the electron-phonon interaction in thin Ag nanofilms,” Phys. Rev. Lett. 9, 7236809, 2006.
[Crossref]

Andreani, L. C.

D. Gerace and L. C. Andreani, “Quantum theory of exciton-photon coupling in photonic crystal slabs with embedded quantum wells,” Phys. Rev. B 75, 235325, 2007.
[Crossref]

Arakawa, Y.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904, 2005.
[Crossref] [PubMed]

M. Kitamura, S. Iwamoto, and Y. Arakawa, “Enhanced light emission from an organic photonic crystal with a nanocavity,” Appl. Phys. Lett. 87, 151119, 2005.
[Crossref]

Asano, T.

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296–1298, 2005.
[Crossref] [PubMed]

Baets, R.

Bauer, M.

S. Mathias, M. Wiesenmayer, M. Aeschlimann, and M. Bauer, “Quantum-well wave-function localization and the electron-phonon interaction in thin Ag nanofilms,” Phys. Rev. Lett. 9, 7236809, 2006.
[Crossref]

Benisty, H.

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124, 2006.
[Crossref]

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, S. Olivier, S. Varoutsis, I. Robert-Philip, R. Houdre, and C. J. M. Smith, “Spontaneous emission enhancement of quantum dots in a photonic crystal wire,” Phys. Rev. Lett 95, 183901, 2005.
[Crossref] [PubMed]

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107, 2005.
[Crossref]

Bhat, R.

Bienstman, P.

Bockstaele, R.

Boroditsky, M.

Bowden, C. M.

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous-media with applications to photonic band structures,” Phys. Rev. A 46, 612–622, 1992.
[Crossref] [PubMed]

Chance, R. R.

R. R. Chance, A. Prock, and R. Silbey, “Lifetime of an emitting molecule near a partially reflecting surface,” J. Chemical Phys. 60, 2744–2748, 1974.
[Crossref]

Coccioli, R.

Craford, M. G.

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887, 2004.
[Crossref]

Culshaw, I. S.

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618, 1999.
[Crossref]

David, A.

S. Keller, C. Schaake, N. A. Fichtenbaum, C. J. Neufeld, Y. Wu, K. McGroddy, A. David, S. P. DenBaars, C. Weisbuch, J. S. Speck, and U. K. Mishra, “Optical and structural properties of GaN nanopillar and nanostripe arrays with embedded InGaN/GaN multi-quantum wells,” J. Appl. Phys. 10 0,054314, 2006.
[Crossref]

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124, 2006.
[Crossref]

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107, 2005.
[Crossref]

Delbeke, D.

DenBaars, S. P.

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124, 2006.
[Crossref]

S. Keller, C. Schaake, N. A. Fichtenbaum, C. J. Neufeld, Y. Wu, K. McGroddy, A. David, S. P. DenBaars, C. Weisbuch, J. S. Speck, and U. K. Mishra, “Optical and structural properties of GaN nanopillar and nanostripe arrays with embedded InGaN/GaN multi-quantum wells,” J. Appl. Phys. 10 0,054314, 2006.
[Crossref]

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107, 2005.
[Crossref]

Diana, F. S.

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107, 2005.
[Crossref]

Dowling, J. P.

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous-media with applications to photonic band structures,” Phys. Rev. A 46, 612–622, 1992.
[Crossref] [PubMed]

Englund, D.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904, 2005.
[Crossref] [PubMed]

Epler, J. E.

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887, 2004.
[Crossref]

Fattal, D.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904, 2005.
[Crossref] [PubMed]

Fichtenbaum, N. A.

S. Keller, C. Schaake, N. A. Fichtenbaum, C. J. Neufeld, Y. Wu, K. McGroddy, A. David, S. P. DenBaars, C. Weisbuch, J. S. Speck, and U. K. Mishra, “Optical and structural properties of GaN nanopillar and nanostripe arrays with embedded InGaN/GaN multi-quantum wells,” J. Appl. Phys. 10 0,054314, 2006.
[Crossref]

Fujii, T.

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124, 2006.
[Crossref]

Fujita, M.

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296–1298, 2005.
[Crossref] [PubMed]

Gardner, N. F.

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887, 2004.
[Crossref]

Gerace, D.

D. Gerace and L. C. Andreani, “Quantum theory of exciton-photon coupling in photonic crystal slabs with embedded quantum wells,” Phys. Rev. B 75, 235325, 2007.
[Crossref]

Gippius, N. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102, 2002.
[Crossref]

Houdre, R.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, S. Olivier, S. Varoutsis, I. Robert-Philip, R. Houdre, and C. J. M. Smith, “Spontaneous emission enhancement of quantum dots in a photonic crystal wire,” Phys. Rev. Lett 95, 183901, 2005.
[Crossref] [PubMed]

Hu, E.

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107, 2005.
[Crossref]

Hu, E. L.

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124, 2006.
[Crossref]

Irman, A.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. L. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657, 2004.
[Crossref] [PubMed]

Ishihara, T.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102, 2002.
[Crossref]

Iwamoto, S.

M. Kitamura, S. Iwamoto, and Y. Arakawa, “Enhanced light emission from an organic photonic crystal with a nanocavity,” Appl. Phys. Lett. 87, 151119, 2005.
[Crossref]

Jiang, H. X.

T. N. Oder, K. H. Kim, J. Y. Lin, and H. X. Jiang, “III-nitride blue and ultraviolet photonic crystal light emitting diodes,” Appl. Phys. Lett. 84, 466–468, 2004.
[Crossref]

Keller, S.

S. Keller, C. Schaake, N. A. Fichtenbaum, C. J. Neufeld, Y. Wu, K. McGroddy, A. David, S. P. DenBaars, C. Weisbuch, J. S. Speck, and U. K. Mishra, “Optical and structural properties of GaN nanopillar and nanostripe arrays with embedded InGaN/GaN multi-quantum wells,” J. Appl. Phys. 10 0,054314, 2006.
[Crossref]

Kim, K. H.

T. N. Oder, K. H. Kim, J. Y. Lin, and H. X. Jiang, “III-nitride blue and ultraviolet photonic crystal light emitting diodes,” Appl. Phys. Lett. 84, 466–468, 2004.
[Crossref]

Kitamura, M.

M. Kitamura, S. Iwamoto, and Y. Arakawa, “Enhanced light emission from an organic photonic crystal with a nanocavity,” Appl. Phys. Lett. 87, 151119, 2005.
[Crossref]

Koide, N.

N. Koide and K. Ujihara, “Analysis of spontaneous emission from a planar microcavity dependence on the refractive-index of the cavity region and atomic location,” Opt. Commun. 111, 381–393, 1994.
[Crossref]

Krames, M. R.

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887, 2004.
[Crossref]

Krauss, T. F.

Lemarchand, F.

Lin, J. Y.

T. N. Oder, K. H. Kim, J. Y. Lin, and H. X. Jiang, “III-nitride blue and ultraviolet photonic crystal light emitting diodes,” Appl. Phys. Lett. 84, 466–468, 2004.
[Crossref]

Lodahl, P.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. L. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657, 2004.
[Crossref] [PubMed]

Mathias, S.

S. Mathias, M. Wiesenmayer, M. Aeschlimann, and M. Bauer, “Quantum-well wave-function localization and the electron-phonon interaction in thin Ag nanofilms,” Phys. Rev. Lett. 9, 7236809, 2006.
[Crossref]

McGroddy, K.

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124, 2006.
[Crossref]

S. Keller, C. Schaake, N. A. Fichtenbaum, C. J. Neufeld, Y. Wu, K. McGroddy, A. David, S. P. DenBaars, C. Weisbuch, J. S. Speck, and U. K. Mishra, “Optical and structural properties of GaN nanopillar and nanostripe arrays with embedded InGaN/GaN multi-quantum wells,” J. Appl. Phys. 10 0,054314, 2006.
[Crossref]

Meier, C.

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107, 2005.
[Crossref]

Mishra, U. K.

S. Keller, C. Schaake, N. A. Fichtenbaum, C. J. Neufeld, Y. Wu, K. McGroddy, A. David, S. P. DenBaars, C. Weisbuch, J. S. Speck, and U. K. Mishra, “Optical and structural properties of GaN nanopillar and nanostripe arrays with embedded InGaN/GaN multi-quantum wells,” J. Appl. Phys. 10 0,054314, 2006.
[Crossref]

Muljarov, E. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102, 2002.
[Crossref]

Nakamura, S.

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124, 2006.
[Crossref]

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107, 2005.
[Crossref]

Nakaoka, T.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904, 2005.
[Crossref] [PubMed]

Neufeld, C. J.

S. Keller, C. Schaake, N. A. Fichtenbaum, C. J. Neufeld, Y. Wu, K. McGroddy, A. David, S. P. DenBaars, C. Weisbuch, J. S. Speck, and U. K. Mishra, “Optical and structural properties of GaN nanopillar and nanostripe arrays with embedded InGaN/GaN multi-quantum wells,” J. Appl. Phys. 10 0,054314, 2006.
[Crossref]

Nikolaev, I. S.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. L. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657, 2004.
[Crossref] [PubMed]

Noda, S.

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296–1298, 2005.
[Crossref] [PubMed]

Oder, T. N.

T. N. Oder, K. H. Kim, J. Y. Lin, and H. X. Jiang, “III-nitride blue and ultraviolet photonic crystal light emitting diodes,” Appl. Phys. Lett. 84, 466–468, 2004.
[Crossref]

Olivier, S.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, S. Olivier, S. Varoutsis, I. Robert-Philip, R. Houdre, and C. J. M. Smith, “Spontaneous emission enhancement of quantum dots in a photonic crystal wire,” Phys. Rev. Lett 95, 183901, 2005.
[Crossref] [PubMed]

Overgaag, K.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. L. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657, 2004.
[Crossref] [PubMed]

Pottage, J. M.

Prock, A.

R. R. Chance, A. Prock, and R. Silbey, “Lifetime of an emitting molecule near a partially reflecting surface,” J. Chemical Phys. 60, 2744–2748, 1974.
[Crossref]

Rigneault, H.

Robert-Philip, I.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, S. Olivier, S. Varoutsis, I. Robert-Philip, R. Houdre, and C. J. M. Smith, “Spontaneous emission enhancement of quantum dots in a photonic crystal wire,” Phys. Rev. Lett 95, 183901, 2005.
[Crossref] [PubMed]

Roberts, P. J.

P. S. J. Russell, S. Tredwell, and P. J. Roberts, Full photonic bandgaps and spontaneous emission control in 1d multilayer dielectric structures,” Opt. Commun.160, 66–71, 1999.
[Crossref]

Russell, P. S.

Russell, P. S. J.

P. S. J. Russell, S. Tredwell, and P. J. Roberts, Full photonic bandgaps and spontaneous emission control in 1d multilayer dielectric structures,” Opt. Commun.160, 66–71, 1999.
[Crossref]

Schaake, C.

S. Keller, C. Schaake, N. A. Fichtenbaum, C. J. Neufeld, Y. Wu, K. McGroddy, A. David, S. P. DenBaars, C. Weisbuch, J. S. Speck, and U. K. Mishra, “Optical and structural properties of GaN nanopillar and nanostripe arrays with embedded InGaN/GaN multi-quantum wells,” J. Appl. Phys. 10 0,054314, 2006.
[Crossref]

Sentenac, A.

Sharma, R.

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124, 2006.
[Crossref]

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107, 2005.
[Crossref]

Sigalas, M. M.

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887, 2004.
[Crossref]

Silbey, R.

R. R. Chance, A. Prock, and R. Silbey, “Lifetime of an emitting molecule near a partially reflecting surface,” J. Chemical Phys. 60, 2744–2748, 1974.
[Crossref]

Silvestre, E.

Simmons, J. A.

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887, 2004.
[Crossref]

Smith, C. J. M.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, S. Olivier, S. Varoutsis, I. Robert-Philip, R. Houdre, and C. J. M. Smith, “Spontaneous emission enhancement of quantum dots in a photonic crystal wire,” Phys. Rev. Lett 95, 183901, 2005.
[Crossref] [PubMed]

Solomon, G.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904, 2005.
[Crossref] [PubMed]

Speck, J. S.

S. Keller, C. Schaake, N. A. Fichtenbaum, C. J. Neufeld, Y. Wu, K. McGroddy, A. David, S. P. DenBaars, C. Weisbuch, J. S. Speck, and U. K. Mishra, “Optical and structural properties of GaN nanopillar and nanostripe arrays with embedded InGaN/GaN multi-quantum wells,” J. Appl. Phys. 10 0,054314, 2006.
[Crossref]

Takahashi, S.

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296–1298, 2005.
[Crossref] [PubMed]

Tanaka, Y.

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296–1298, 2005.
[Crossref] [PubMed]

Tikhodeev, S. G.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102, 2002.
[Crossref]

Tredwell, S.

P. S. J. Russell, S. Tredwell, and P. J. Roberts, Full photonic bandgaps and spontaneous emission control in 1d multilayer dielectric structures,” Opt. Commun.160, 66–71, 1999.
[Crossref]

Ujihara, K.

N. Koide and K. Ujihara, “Analysis of spontaneous emission from a planar microcavity dependence on the refractive-index of the cavity region and atomic location,” Opt. Commun. 111, 381–393, 1994.
[Crossref]

van Driel, A. F.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. L. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657, 2004.
[Crossref] [PubMed]

Vanmaekelbergh, D. L.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. L. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657, 2004.
[Crossref] [PubMed]

Varoutsis, S.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, S. Olivier, S. Varoutsis, I. Robert-Philip, R. Houdre, and C. J. M. Smith, “Spontaneous emission enhancement of quantum dots in a photonic crystal wire,” Phys. Rev. Lett 95, 183901, 2005.
[Crossref] [PubMed]

Viasnoff-Schwoob, E.

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, S. Olivier, S. Varoutsis, I. Robert-Philip, R. Houdre, and C. J. M. Smith, “Spontaneous emission enhancement of quantum dots in a photonic crystal wire,” Phys. Rev. Lett 95, 183901, 2005.
[Crossref] [PubMed]

Vos, W. L.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. L. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657, 2004.
[Crossref] [PubMed]

Vrijen, R.

Vuckovic, J.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904, 2005.
[Crossref] [PubMed]

Waks, E.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904, 2005.
[Crossref] [PubMed]

Weisbuch, C.

S. Keller, C. Schaake, N. A. Fichtenbaum, C. J. Neufeld, Y. Wu, K. McGroddy, A. David, S. P. DenBaars, C. Weisbuch, J. S. Speck, and U. K. Mishra, “Optical and structural properties of GaN nanopillar and nanostripe arrays with embedded InGaN/GaN multi-quantum wells,” J. Appl. Phys. 10 0,054314, 2006.
[Crossref]

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124, 2006.
[Crossref]

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, S. Olivier, S. Varoutsis, I. Robert-Philip, R. Houdre, and C. J. M. Smith, “Spontaneous emission enhancement of quantum dots in a photonic crystal wire,” Phys. Rev. Lett 95, 183901, 2005.
[Crossref] [PubMed]

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107, 2005.
[Crossref]

Wendt, J. R.

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887, 2004.
[Crossref]

Whittaker, D. M.

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618, 1999.
[Crossref]

Wierer, J. J.

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887, 2004.
[Crossref]

Wiesenmayer, M.

S. Mathias, M. Wiesenmayer, M. Aeschlimann, and M. Bauer, “Quantum-well wave-function localization and the electron-phonon interaction in thin Ag nanofilms,” Phys. Rev. Lett. 9, 7236809, 2006.
[Crossref]

Wittke, J. P.

J. P. Wittke, “Spontaneous emission-rate alteration by dielectric and other waveguiding structures,” RCA Review 36, 655–666, 1975.

Wu, Y.

S. Keller, C. Schaake, N. A. Fichtenbaum, C. J. Neufeld, Y. Wu, K. McGroddy, A. David, S. P. DenBaars, C. Weisbuch, J. S. Speck, and U. K. Mishra, “Optical and structural properties of GaN nanopillar and nanostripe arrays with embedded InGaN/GaN multi-quantum wells,” J. Appl. Phys. 10 0,054314, 2006.
[Crossref]

Yablonovitch, E.

Yablonskii, A. L.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102, 2002.
[Crossref]

Yamamoto, Y.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904, 2005.
[Crossref] [PubMed]

Zhang, B.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904, 2005.
[Crossref] [PubMed]

Appl. Phys. Lett. (5)

M. Kitamura, S. Iwamoto, and Y. Arakawa, “Enhanced light emission from an organic photonic crystal with a nanocavity,” Appl. Phys. Lett. 87, 151119, 2005.
[Crossref]

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887, 2004.
[Crossref]

T. N. Oder, K. H. Kim, J. Y. Lin, and H. X. Jiang, “III-nitride blue and ultraviolet photonic crystal light emitting diodes,” Appl. Phys. Lett. 84, 466–468, 2004.
[Crossref]

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124, 2006.
[Crossref]

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107, 2005.
[Crossref]

J. Appl. Phys. (1)

S. Keller, C. Schaake, N. A. Fichtenbaum, C. J. Neufeld, Y. Wu, K. McGroddy, A. David, S. P. DenBaars, C. Weisbuch, J. S. Speck, and U. K. Mishra, “Optical and structural properties of GaN nanopillar and nanostripe arrays with embedded InGaN/GaN multi-quantum wells,” J. Appl. Phys. 10 0,054314, 2006.
[Crossref]

J. Chemical Phys. (1)

R. R. Chance, A. Prock, and R. Silbey, “Lifetime of an emitting molecule near a partially reflecting surface,” J. Chemical Phys. 60, 2744–2748, 1974.
[Crossref]

J. Lightwave Technol. (1)

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

Nature (1)

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. L. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657, 2004.
[Crossref] [PubMed]

Opt. Commun. (1)

N. Koide and K. Ujihara, “Analysis of spontaneous emission from a planar microcavity dependence on the refractive-index of the cavity region and atomic location,” Opt. Commun. 111, 381–393, 1994.
[Crossref]

Phys. Rev. A (1)

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous-media with applications to photonic band structures,” Phys. Rev. A 46, 612–622, 1992.
[Crossref] [PubMed]

Phys. Rev. B (3)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102, 2002.
[Crossref]

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618, 1999.
[Crossref]

D. Gerace and L. C. Andreani, “Quantum theory of exciton-photon coupling in photonic crystal slabs with embedded quantum wells,” Phys. Rev. B 75, 235325, 2007.
[Crossref]

Phys. Rev. Lett (1)

E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, S. Olivier, S. Varoutsis, I. Robert-Philip, R. Houdre, and C. J. M. Smith, “Spontaneous emission enhancement of quantum dots in a photonic crystal wire,” Phys. Rev. Lett 95, 183901, 2005.
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904, 2005.
[Crossref] [PubMed]

S. Mathias, M. Wiesenmayer, M. Aeschlimann, and M. Bauer, “Quantum-well wave-function localization and the electron-phonon interaction in thin Ag nanofilms,” Phys. Rev. Lett. 9, 7236809, 2006.
[Crossref]

RCA Review (1)

J. P. Wittke, “Spontaneous emission-rate alteration by dielectric and other waveguiding structures,” RCA Review 36, 655–666, 1975.

Science (1)

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296–1298, 2005.
[Crossref] [PubMed]

Other (1)

P. S. J. Russell, S. Tredwell, and P. J. Roberts, Full photonic bandgaps and spontaneous emission control in 1d multilayer dielectric structures,” Opt. Commun.160, 66–71, 1999.
[Crossref]

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Figures (10)

Fig. 1.
Fig. 1. (Left) Band structure of a GaN PhC membrane. Blue: band structure of the fundamental TE Bloch mode. Red: Corresponding DOS ρ guided . Magenta: air cone. (Right) Corresponding Fermi surfaces at frequencies a/λ=0.15, 0.2, 0.25, 0.3, 0.33, 0.34, 0.35, 0.36, 0.37 (from center to edge of the reduced Brillouin zone). Magenta dashed circle: air cone.
Fig. 2.
Fig. 2. (Left) Red dots: Purcell factor F guided for the guided Bloch mode. Blue line: ρ guided /ω 2 (the guided DOS normalized by the frequency) is seen to coincide with the Purcell factor in this simple case. Black dots: Purcell factor F air for emission to air - emission in the air continuum is nearly constant and not affected by the PhC (the dashed black line is a guide to the eye). Note that the total Purcell factor F guided +F air is usually smaller than one (except close enough to the Van Hove singularity). This is because the calculation is normalized to F p =1 for emission in bulk GaN, and on average the dipole in the membrane “feels” a dielectric with an index lower than that of bulk GaN. (Right) Emission diagram of a quantum well in a GaN PhC membrane, at a frequency u=0.35. This diagram displays the power emitted (in log scale) as a function of direction. k x and k y are the components of the in-plane wavevector of photons in units of k 0. The white circle is the air cone, and the white hexagon the first Brillouin Zone. The emission diagram is composed of direct emission into the air continuum, and of emission into the Bloch mode following the mode’s isofrequencies.
Fig. 3.
Fig. 3. (a) Generic quasi-free photon band structure along ΓM and ΓKM for photons of effective index n eff =1.6. Diffraction to air first becomes possible along ΓM around a frequency u=0.42 (red dashed line). At higher frequencies u=0.54 and u=0.58 (green and blue dashed lines respectively) a larger part of the band structure is located above the light line. (b) Corresponding isofrequencies and air cones for these three frequencies (red u=0.42, green u=0.54, blue u=0.58). The n=1.6” circles are the isofrequencies (i.e. wavevector locus) for n eff =1.6, and their foldings by the reciprocal lattice. The air circles correspond to n eff =1. Diffraction is possible when an isofrequency is folded inside the corresponding air circle. (c) Zoom on (b) detailing the fate of the mode: the red frequency is never extracted, the green frequency is partly unextracted. The blue frequency is almost completely extracted, while keeping the air-circle at the smallest possible diameter.
Fig. 4.
Fig. 4. PhC membrane emission diagram. (Left) Emission diagram of a quantum well in a GaN PhC membrane (the emitted power is in log scale, k x and k y in units of k 0). The small full circle is the air cone (n=1), and the large dashed circle corresponds to the average index of the membrane (n eff =1.9): emission is strongly suppressed beyond this circle. The sharp features correspond to emission in Bloch modes. (Right) Emission diagram with the isofrequency contours of a free-photon mode (n eff =1.6) superimposed as red lines. The full red circle is the dispersion of a guided mode with n eff =1.6, and the dashed circles correspond to periodization by the reciprocal lattice, e.g. to the harmonics of the Bloch mode.
Fig. 5.
Fig. 5. Radiative channels in the structure. Dashed arrows: corresponding in-plane wavevectors k . (a) Continuum of air modes, with k <k 0(b) Continuum of sapphire modes with k <n sapphire k 0 (c) Bloch mode propagative in the GaN and PhC region (d) Bloch mode of larger k , evanescent in the PhC region.
Fig. 6.
Fig. 6. (Left) Emission diagram for z s =1.45. The green lines correspond to the boundary of the first Brillouin zone. The isofrequency of the Bloch mode localized in the PhC region is superimposed as a blue dashed line: it follows exactly the emission peak of the diagram. (Right) Modulus of the electric field (|E|) along z for the localized Bloch mode, in the ΓK direction. Blue line: fundamental harmonic. Magenta line: second-largest harmonic. The average index profile is represented by the black line.
Fig. 7.
Fig. 7. (Left) Geometry of the structure. (Right) Blue line: profile of the electric field |E| of the 1D QGM (with n eff =1.86), obtained by solving numerically the Helmholtz equation. Red line: |E| for the fundamental harmonic of corresponding Bloch QGM (taking scattering by the PhC into account), propagating with n eff =1.6. The profile of both modes is nearly identical despite the difference in effective index, evidencing that the mode’s profile is imposed by phase matching in the vertical direction.
Fig. 8.
Fig. 8. Propagation properties of the uncoupled 1D QGM (thin blue line), and the coupled QGBM (red thick line: numerical result, green dashed line: analytical solution of Eq 10). As can be seen, the analytical solution is in general very close to the numerical solution. (Top left) Dispersion relation. The coupling causes a gap opening at G/2 (dashed vertical line). (Top right) Loss coefficient k . In the presence of coupling, k is high in the gap due to reflection. Outside of the gap, losses are dictated by the leakage of the fundamental harmonic and barely modified by the coupling. (Bottom left) Effective index n eff . Below the gap, the mode localizes in the dielectric and n eff increases with respect to the 1D value. Above the gap on the other hand, the mode localizes in air and n eff decreases. Notice the rigid shift of n eff caused by the coupling at high frequency: its asymptotic value is not 〈n〉 but n 1 ε 1 n 2 . (Bottom right) Corresponding angle of propagation.
Fig. 9.
Fig. 9. Purcell factor as a function of the source’s position z s in the structure. Magenta line with dots: actual Purcell factor F p . The strong oscillations of F p are due to coupling to the QGBM. Dashed line: ‘average index’ result. Oscillations of F p are weak in this approximation, because spontaneous emission is dominated by emission to the sapphire and air continua. Red line: fit by Eq. 13.
Fig. 10.
Fig. 10. (Left) Generalization of the QGM to higher-order quasi-guided resonances in the PhC region. Here, the first and second resonances are depicted. (Right) QGM in a GaN/PhC/GaN structure: the mode is trapped in the low-index region, and constitutes an easily injectable photonic crystal membrane.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

F p = F air + F guided
F guided = ρ guided ρ 0 = ρ guided ω 2 n 2 8 π /3
ρ guided ( ω ) = n Sn d k ω d . A n 2
F p = 3 8 π ω 2 n 2 ( ρ air + ρ guided ( ω ) )
k z = π L
k 1 D = n 1 2 k 0 2 ( π L ) 2 , cos θ = 1 2 L n 1 u
k " = ln ( r ) 2 L tan θ
ω 2 ( ε 0 ε 1 ε 1 ε 0 ) E + E = k 2 + k z 2 ( k G ) 2 + k z 2 . E + E
k z 2 = ε 0 ω 2 k 1 D 2
K 2 = G 2 + 2 ( k 1 D 2 G 2 4 ) ( G 2 + 2 ( k 1 D 2 G 2 4 ) ) 2 + 4 ω 4 ε 1 2 4 ( k 1 D 2 G 2 4 ) 2 2
k 1 D 2 G 2 4 < ω 2 ε 1 K 2 > 0 , K , K < 0 ( dielectric band ) ω 2 ε 1 < k 1 D 2 G 2 4 < ω 2 ε 1 K 2 < 0 , K i ( gap ) ω 2 ε 1 < k 1 D 2 G 2 4 K 2 > 0 , K , K > 0 ( air band )
k z p = p π L , E p cos ( k z p z )
F p ( z s ) p sin ( k z p z s ) 2

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