Abstract

We analyze in depth the eigenmodes symmetry of the vectorial electromagnetic wave equation with discrete symmetry, using a recently developed maximal symmetrization and reduction scheme leading to an automatic technique which decomposes every mode into its most fundamental internal geometrical components carrying independent symmetries, the ultimately reduced component functions (URCFs). Using URCFs, geometrical properties of photonic crystal defect modes can be analyzed in great details. In particular we analytically identify the kind of modes that display non-vanishing transverse electric or transverse magnetic amplitude at the cavity center in C2v, C3v, C4v, and C6v symmetries, and their degeneracies. We also build a postprocessing tool able to extract and identify URCFs out of the modes whether from experimental or numerical origin. In the latter case it is independent of the eigenmode computation method. In another variant the whole eigenmode computation can be systematically reduced to a minimal domain, without any need for applying specific non-trivial boundary conditions. The approach leads to strong analytical predictions which are illustrated for specific H1 and L3 cavities using the postprocessing tool on full three-dimensional computed modes. It not only constitutes an unprecedented check of the symmetry of the computational results, but it is shown to also deliver a deep geometrical and physical insight into the structure of the modes of photonic bandgap microcavities, which is of direct use for most modern applications in quantum photonics.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]

2010

D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vučković, “Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity,” Phys. Rev. Lett. 104, 073904 (2010).
[CrossRef] [PubMed]

S. Dalessi and M.-A. Dupertuis, “Maximal symmetrization and reduction of fields: application to wave functions in solid state nanostructures,” Phys. Rev. B 81, 125106 (2010).
[CrossRef]

K. F. Karlsson, M. A. Dupertuis, D. Y. Oberli, E. Pelucchi, A. Rudra, P. O. Holtz, and E. Kapon, “Fine structure of exciton complexes in high symmetry quantum dots: symmetry breaking and symmetry elevation” Phys. Rev. B 81161307 (2010).
[CrossRef]

2009

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79, 013814 (2009).
[CrossRef]

M. Larqué, T. Karle, I. Robert-Philip, and A. Beveratos, “Optimizing h1 cavities for the generation of entangled photon pairs,” New J. Phys. 11, 033022 (2009).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef] [PubMed]

2008

K. Aoki, D. Guimard, M. Nishioka, M. Nomura, S. Iwamoto, and Y. Arakawa, “Coupling of quantum-dot light emission with a three-dimensional photonic-crystal nanocavity,” Nat. Photonics 2, 688–692 (2008).
[CrossRef]

P. Boucaud, M. El Kurdi, S. David, X. Checoury, X. Li, T. P. Ngo, S. Sauvage, D. Bouchier, G. Fishman, O. Kermarrec, Y. Campidelli, D. Bensahel, T. Akatsu, C. Richtarch, and B. Ghyselen, “Germanium-based nanophotonic devices: two-dimensional photonic crystals and cavities,” Thin Solid Films 517, 121–124 (2008).
[CrossRef]

M. El Kurdi, S. David, X. Checoury, G. Fishman, P. Boucaud, O. Kermarrec, D. Bensahel, and B. Ghyselen, “Two-dimensional photonic crystals with pure germanium-on-insulator,” Opt. Commun. 281, 846–850 (2008).
[CrossRef]

R. Johne, N. A. Gippius, G. Pavlovic, D. D. Solnyshkov, I. A. Shelykh, and G. Malpuech, “Entangled photon pairs produced by a quantum dot strongly coupled to a microcavity,” Phys. Rev. Lett. 100, 240404 (2008).
[CrossRef] [PubMed]

I. Fushman, D. Englund, A. Faraon, N. Stolz, P. Petroff, and J. Vuckovic, “Controlled phase shifts with a single quantum dot,” Science 320, 769–772 (2008).
[CrossRef] [PubMed]

F. Römer and B. Witzigmann, “Spectral and spatial properties of the spontaneous emission enhancement in photonic crystal cavities,” J. Opt. Soc. Am. B 25, 31–39 (2008).
[CrossRef]

M. Belotti, J. F. Galisteo-Lopez, S. De Angelis, M. Galli, I. Maksymov, L. C. Andreani, D. Peyrade, and Y. Chen, “All-optical switching in 2D silicon photonic crystals with low loss waveguides and optical cavities,” Opt. Express 16, 11624–11636 (2008).
[PubMed]

2007

F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, “Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver,” Opt. Quantum Electron. 39, 341–352 (2007).
[CrossRef]

D. Englund, A. Faraon, B. Y. Zhang, Y. Yamamoto, and J. Vuckovic, “Generation and transfer of single photons on a photonic crystal chip,” Opt. Express 15, 5550–5558 (2007).
[CrossRef] [PubMed]

2006

S. H. Kim, S. K. Kim, and Y. H. Lee, “Vertical beaming of wavelength-scale photonic crystal resonators,” Phys. Rev. B 73, 235117 (2006).
[CrossRef]

S. K. Kim, G. H. Kim, S. H. Kim, Y. H. Lee, S. B. Kim, and I. Kim, “Loss management using parity-selective barriers for single-mode, single-cell photonic crystal resonators,” Appl. Phys. Lett. 88, 161119 (2006).
[CrossRef]

J. Vuckovic, D. Englund, D. Fattal, E. Waks, and Y. Yamamoto, “Generation and manipulation of nonclassical light using photonic crystals,” Physica E (Amsterdam) 32, 466–470 (2006).
[CrossRef]

W. H. Chang, W. Y. Chen, H. S. Chang, T. P. Hsieh, J. I. Chyi, and T. M. Hsu, “Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities,” Phys. Rev. Lett. 96, 117401 (2006).
[CrossRef] [PubMed]

2005

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005).
[CrossRef]

N. Mattiucci, G. D’Aguanno, M. Scalora, and M. Bloemer, “Cross-phase modulation in one-dimensional photonic crystals: applications to all-optical devices,” Appl. Phys. B 81, 389–391 (2005).
[CrossRef]

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]

Y. Akahane, T. Asano, H. Takano, B. S. Song, Y. Takana, and S. Noda, “Two-dimensional photonic-crystal-slab channel-drop filter with flat-top response,” Opt. Express 13, 2512–2530 (2005).
[CrossRef] [PubMed]

W. Kuang, J. R. Cao, T. Yang, S. J. Choi, P. T. Lee, J. D. O’Brien, and P. D. Dapkus, “Classification of modes in suspended-membrane, 19-missing-hole photonic-crystal microcavities,” J. Opt. Soc. Am. B 22, 1092–1099 (2005).
[CrossRef]

A. Badolato, K. Hennessy, M. Atature, J. Dreiser, E. Hu, P. Petroff, and A. Imamoglu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158–1161 (2005).
[CrossRef] [PubMed]

2004

B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two-dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59–63 (2004).
[CrossRef]

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[CrossRef] [PubMed]

M. Okano and S. Noda, “Analysis of multimode point-defect cavities in three-dimensional photonic crystals using group theory in frequency and time domains,” Phys. Rev. B 70, 125105 (2004).
[CrossRef]

2003

O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” Phys. Rev. B 68, 035110 (2003).
[CrossRef]

S. H. Kim and Y. H. Lee, “Symmetry relations of two-dimensional photonic crystal cavity modes,” IEEE J. Quantum Electron. 39, 1081–1085 (2003).
[CrossRef]

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302, 1374–1377 (2003).
[CrossRef] [PubMed]

1999

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

1998

1997

J. D. Joannopoulos, P. R. Villeneuve, and S. H. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

1993

B. Acklin, M. Cada, J. He, and M.-A. Dupertuis, “Bistable switching in a nonlinear Bragg reflector,” Appl. Phys. Lett. 63, 2177–2179 (1993).
[CrossRef]

Acklin, B.

B. Acklin, M. Cada, J. He, and M.-A. Dupertuis, “Bistable switching in a nonlinear Bragg reflector,” Appl. Phys. Lett. 63, 2177–2179 (1993).
[CrossRef]

Akahane, Y.

Akatsu, T.

P. Boucaud, M. El Kurdi, S. David, X. Checoury, X. Li, T. P. Ngo, S. Sauvage, D. Bouchier, G. Fishman, O. Kermarrec, Y. Campidelli, D. Bensahel, T. Akatsu, C. Richtarch, and B. Ghyselen, “Germanium-based nanophotonic devices: two-dimensional photonic crystals and cavities,” Thin Solid Films 517, 121–124 (2008).
[CrossRef]

Altmann, S. L.

S. L. Altmann and P. Herzig, Point-Group Theory Tables (Clarendon, 1994).

Andreani, L. C.

Aoki, K.

K. Aoki, D. Guimard, M. Nishioka, M. Nomura, S. Iwamoto, and Y. Arakawa, “Coupling of quantum-dot light emission with a three-dimensional photonic-crystal nanocavity,” Nat. Photonics 2, 688–692 (2008).
[CrossRef]

Arakawa, Y.

K. Aoki, D. Guimard, M. Nishioka, M. Nomura, S. Iwamoto, and Y. Arakawa, “Coupling of quantum-dot light emission with a three-dimensional photonic-crystal nanocavity,” Nat. Photonics 2, 688–692 (2008).
[CrossRef]

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]

Arbenz, P.

F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, “Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver,” Opt. Quantum Electron. 39, 341–352 (2007).
[CrossRef]

Asano, T.

Atature, M.

A. Badolato, K. Hennessy, M. Atature, J. Dreiser, E. Hu, P. Petroff, and A. Imamoglu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158–1161 (2005).
[CrossRef] [PubMed]

Badolato, A.

A. Badolato, K. Hennessy, M. Atature, J. Dreiser, E. Hu, P. Petroff, and A. Imamoglu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158–1161 (2005).
[CrossRef] [PubMed]

Baek, J. H.

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[CrossRef] [PubMed]

Belotti, M.

Bensahel, D.

M. El Kurdi, S. David, X. Checoury, G. Fishman, P. Boucaud, O. Kermarrec, D. Bensahel, and B. Ghyselen, “Two-dimensional photonic crystals with pure germanium-on-insulator,” Opt. Commun. 281, 846–850 (2008).
[CrossRef]

P. Boucaud, M. El Kurdi, S. David, X. Checoury, X. Li, T. P. Ngo, S. Sauvage, D. Bouchier, G. Fishman, O. Kermarrec, Y. Campidelli, D. Bensahel, T. Akatsu, C. Richtarch, and B. Ghyselen, “Germanium-based nanophotonic devices: two-dimensional photonic crystals and cavities,” Thin Solid Films 517, 121–124 (2008).
[CrossRef]

Beveratos, A.

M. Larqué, T. Karle, I. Robert-Philip, and A. Beveratos, “Optimizing h1 cavities for the generation of entangled photon pairs,” New J. Phys. 11, 033022 (2009).
[CrossRef]

Bloemer, M.

N. Mattiucci, G. D’Aguanno, M. Scalora, and M. Bloemer, “Cross-phase modulation in one-dimensional photonic crystals: applications to all-optical devices,” Appl. Phys. B 81, 389–391 (2005).
[CrossRef]

Botten, L. C.

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79, 013814 (2009).
[CrossRef]

Boucaud, P.

P. Boucaud, M. El Kurdi, S. David, X. Checoury, X. Li, T. P. Ngo, S. Sauvage, D. Bouchier, G. Fishman, O. Kermarrec, Y. Campidelli, D. Bensahel, T. Akatsu, C. Richtarch, and B. Ghyselen, “Germanium-based nanophotonic devices: two-dimensional photonic crystals and cavities,” Thin Solid Films 517, 121–124 (2008).
[CrossRef]

M. El Kurdi, S. David, X. Checoury, G. Fishman, P. Boucaud, O. Kermarrec, D. Bensahel, and B. Ghyselen, “Two-dimensional photonic crystals with pure germanium-on-insulator,” Opt. Commun. 281, 846–850 (2008).
[CrossRef]

Bouchier, D.

P. Boucaud, M. El Kurdi, S. David, X. Checoury, X. Li, T. P. Ngo, S. Sauvage, D. Bouchier, G. Fishman, O. Kermarrec, Y. Campidelli, D. Bensahel, T. Akatsu, C. Richtarch, and B. Ghyselen, “Germanium-based nanophotonic devices: two-dimensional photonic crystals and cavities,” Thin Solid Films 517, 121–124 (2008).
[CrossRef]

Cada, M.

B. Acklin, M. Cada, J. He, and M.-A. Dupertuis, “Bistable switching in a nonlinear Bragg reflector,” Appl. Phys. Lett. 63, 2177–2179 (1993).
[CrossRef]

Camacho, R.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef] [PubMed]

Campidelli, Y.

P. Boucaud, M. El Kurdi, S. David, X. Checoury, X. Li, T. P. Ngo, S. Sauvage, D. Bouchier, G. Fishman, O. Kermarrec, Y. Campidelli, D. Bensahel, T. Akatsu, C. Richtarch, and B. Ghyselen, “Germanium-based nanophotonic devices: two-dimensional photonic crystals and cavities,” Thin Solid Films 517, 121–124 (2008).
[CrossRef]

Cao, J. R.

Capasso, F.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302, 1374–1377 (2003).
[CrossRef] [PubMed]

Chan, J.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef] [PubMed]

Chang, H. S.

W. H. Chang, W. Y. Chen, H. S. Chang, T. P. Hsieh, J. I. Chyi, and T. M. Hsu, “Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities,” Phys. Rev. Lett. 96, 117401 (2006).
[CrossRef] [PubMed]

Chang, W. H.

W. H. Chang, W. Y. Chen, H. S. Chang, T. P. Hsieh, J. I. Chyi, and T. M. Hsu, “Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities,” Phys. Rev. Lett. 96, 117401 (2006).
[CrossRef] [PubMed]

Checoury, X.

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P. Boucaud, M. El Kurdi, S. David, X. Checoury, X. Li, T. P. Ngo, S. Sauvage, D. Bouchier, G. Fishman, O. Kermarrec, Y. Campidelli, D. Bensahel, T. Akatsu, C. Richtarch, and B. Ghyselen, “Germanium-based nanophotonic devices: two-dimensional photonic crystals and cavities,” Thin Solid Films 517, 121–124 (2008).
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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).
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[CrossRef]

P. Boucaud, M. El Kurdi, S. David, X. Checoury, X. Li, T. P. Ngo, S. Sauvage, D. Bouchier, G. Fishman, O. Kermarrec, Y. Campidelli, D. Bensahel, T. Akatsu, C. Richtarch, and B. Ghyselen, “Germanium-based nanophotonic devices: two-dimensional photonic crystals and cavities,” Thin Solid Films 517, 121–124 (2008).
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M. El Kurdi, S. David, X. Checoury, G. Fishman, P. Boucaud, O. Kermarrec, D. Bensahel, and B. Ghyselen, “Two-dimensional photonic crystals with pure germanium-on-insulator,” Opt. Commun. 281, 846–850 (2008).
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W. H. Chang, W. Y. Chen, H. S. Chang, T. P. Hsieh, J. I. Chyi, and T. M. Hsu, “Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities,” Phys. Rev. Lett. 96, 117401 (2006).
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A. Badolato, K. Hennessy, M. Atature, J. Dreiser, E. Hu, P. Petroff, and A. Imamoglu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158–1161 (2005).
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M. El Kurdi, S. David, X. Checoury, G. Fishman, P. Boucaud, O. Kermarrec, D. Bensahel, and B. Ghyselen, “Two-dimensional photonic crystals with pure germanium-on-insulator,” Opt. Commun. 281, 846–850 (2008).
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O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
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B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two-dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59–63 (2004).
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H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
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H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
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S. H. Kim and Y. H. Lee, “Symmetry relations of two-dimensional photonic crystal cavity modes,” IEEE J. Quantum Electron. 39, 1081–1085 (2003).
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S. H. Kim, S. K. Kim, and Y. H. Lee, “Vertical beaming of wavelength-scale photonic crystal resonators,” Phys. Rev. B 73, 235117 (2006).
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Kwon, S. H.

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M. Larqué, T. Karle, I. Robert-Philip, and A. Beveratos, “Optimizing h1 cavities for the generation of entangled photon pairs,” New J. Phys. 11, 033022 (2009).
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Lee, R. K.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
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S. H. Kim, S. K. Kim, and Y. H. Lee, “Vertical beaming of wavelength-scale photonic crystal resonators,” Phys. Rev. B 73, 235117 (2006).
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S. K. Kim, G. H. Kim, S. H. Kim, Y. H. Lee, S. B. Kim, and I. Kim, “Loss management using parity-selective barriers for single-mode, single-cell photonic crystal resonators,” Appl. Phys. Lett. 88, 161119 (2006).
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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).
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[CrossRef]

Nomura, M.

K. Aoki, D. Guimard, M. Nishioka, M. Nomura, S. Iwamoto, and Y. Arakawa, “Coupling of quantum-dot light emission with a three-dimensional photonic-crystal nanocavity,” Nat. Photonics 2, 688–692 (2008).
[CrossRef]

O’Brien, J. D.

W. Kuang, J. R. Cao, T. Yang, S. J. Choi, P. T. Lee, J. D. O’Brien, and P. D. Dapkus, “Classification of modes in suspended-membrane, 19-missing-hole photonic-crystal microcavities,” J. Opt. Soc. Am. B 22, 1092–1099 (2005).
[CrossRef]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Oberli, D. Y.

K. F. Karlsson, M. A. Dupertuis, D. Y. Oberli, E. Pelucchi, A. Rudra, P. O. Holtz, and E. Kapon, “Fine structure of exciton complexes in high symmetry quantum dots: symmetry breaking and symmetry elevation” Phys. Rev. B 81161307 (2010).
[CrossRef]

Okano, M.

M. Okano and S. Noda, “Analysis of multimode point-defect cavities in three-dimensional photonic crystals using group theory in frequency and time domains,” Phys. Rev. B 70, 125105 (2004).
[CrossRef]

Painter, O.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef] [PubMed]

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302, 1374–1377 (2003).
[CrossRef] [PubMed]

O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” Phys. Rev. B 68, 035110 (2003).
[CrossRef]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Park, H. G.

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[CrossRef] [PubMed]

Park, H. Y.

B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two-dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59–63 (2004).
[CrossRef]

Pavlovic, G.

R. Johne, N. A. Gippius, G. Pavlovic, D. D. Solnyshkov, I. A. Shelykh, and G. Malpuech, “Entangled photon pairs produced by a quantum dot strongly coupled to a microcavity,” Phys. Rev. Lett. 100, 240404 (2008).
[CrossRef] [PubMed]

Pelucchi, E.

K. F. Karlsson, M. A. Dupertuis, D. Y. Oberli, E. Pelucchi, A. Rudra, P. O. Holtz, and E. Kapon, “Fine structure of exciton complexes in high symmetry quantum dots: symmetry breaking and symmetry elevation” Phys. Rev. B 81161307 (2010).
[CrossRef]

Petroff, P.

D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vučković, “Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity,” Phys. Rev. Lett. 104, 073904 (2010).
[CrossRef] [PubMed]

I. Fushman, D. Englund, A. Faraon, N. Stolz, P. Petroff, and J. Vuckovic, “Controlled phase shifts with a single quantum dot,” Science 320, 769–772 (2008).
[CrossRef] [PubMed]

A. Badolato, K. Hennessy, M. Atature, J. Dreiser, E. Hu, P. Petroff, and A. Imamoglu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158–1161 (2005).
[CrossRef] [PubMed]

Peyrade, D.

Poulton, C. G.

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79, 013814 (2009).
[CrossRef]

Reuse, F.

B. Gallinet, M.-A. Dupertuis, and F. Reuse are preparing a manuscript to be called “Spatial domain reduction: a systematic approach of symmetry effects in nanostructures and photonic band-gap microcavities.”

Richtarch, C.

P. Boucaud, M. El Kurdi, S. David, X. Checoury, X. Li, T. P. Ngo, S. Sauvage, D. Bouchier, G. Fishman, O. Kermarrec, Y. Campidelli, D. Bensahel, T. Akatsu, C. Richtarch, and B. Ghyselen, “Germanium-based nanophotonic devices: two-dimensional photonic crystals and cavities,” Thin Solid Films 517, 121–124 (2008).
[CrossRef]

Robert-Philip, I.

M. Larqué, T. Karle, I. Robert-Philip, and A. Beveratos, “Optimizing h1 cavities for the generation of entangled photon pairs,” New J. Phys. 11, 033022 (2009).
[CrossRef]

Römer, F.

F. Römer and B. Witzigmann, “Spectral and spatial properties of the spontaneous emission enhancement in photonic crystal cavities,” J. Opt. Soc. Am. B 25, 31–39 (2008).
[CrossRef]

F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, “Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver,” Opt. Quantum Electron. 39, 341–352 (2007).
[CrossRef]

Rudra, A.

K. F. Karlsson, M. A. Dupertuis, D. Y. Oberli, E. Pelucchi, A. Rudra, P. O. Holtz, and E. Kapon, “Fine structure of exciton complexes in high symmetry quantum dots: symmetry breaking and symmetry elevation” Phys. Rev. B 81161307 (2010).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2005).

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994).

Sauvage, S.

P. Boucaud, M. El Kurdi, S. David, X. Checoury, X. Li, T. P. Ngo, S. Sauvage, D. Bouchier, G. Fishman, O. Kermarrec, Y. Campidelli, D. Bensahel, T. Akatsu, C. Richtarch, and B. Ghyselen, “Germanium-based nanophotonic devices: two-dimensional photonic crystals and cavities,” Thin Solid Films 517, 121–124 (2008).
[CrossRef]

Scalora, M.

N. Mattiucci, G. D’Aguanno, M. Scalora, and M. Bloemer, “Cross-phase modulation in one-dimensional photonic crystals: applications to all-optical devices,” Appl. Phys. B 81, 389–391 (2005).
[CrossRef]

Scherer, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Sergent, A. M.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302, 1374–1377 (2003).
[CrossRef] [PubMed]

Shelykh, I. A.

R. Johne, N. A. Gippius, G. Pavlovic, D. D. Solnyshkov, I. A. Shelykh, and G. Malpuech, “Entangled photon pairs produced by a quantum dot strongly coupled to a microcavity,” Phys. Rev. Lett. 100, 240404 (2008).
[CrossRef] [PubMed]

Sivco, D. L.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302, 1374–1377 (2003).
[CrossRef] [PubMed]

Solnyshkov, D. D.

R. Johne, N. A. Gippius, G. Pavlovic, D. D. Solnyshkov, I. A. Shelykh, and G. Malpuech, “Entangled photon pairs produced by a quantum dot strongly coupled to a microcavity,” Phys. Rev. Lett. 100, 240404 (2008).
[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]

Song, B. S.

Srinivasan, K.

O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” Phys. Rev. B 68, 035110 (2003).
[CrossRef]

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302, 1374–1377 (2003).
[CrossRef] [PubMed]

Stoltz, N.

D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vučković, “Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity,” Phys. Rev. Lett. 104, 073904 (2010).
[CrossRef] [PubMed]

Stolz, N.

I. Fushman, D. Englund, A. Faraon, N. Stolz, P. Petroff, and J. Vuckovic, “Controlled phase shifts with a single quantum dot,” Science 320, 769–772 (2008).
[CrossRef] [PubMed]

Takana, Y.

Takano, H.

Tennant, D. M.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302, 1374–1377 (2003).
[CrossRef] [PubMed]

Toishi, M.

D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vučković, “Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity,” Phys. Rev. Lett. 104, 073904 (2010).
[CrossRef] [PubMed]

Troccoli, M.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302, 1374–1377 (2003).
[CrossRef] [PubMed]

Vahala, K. J.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef] [PubMed]

Villeneuve, P. R.

S. H. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express 3, 4–11 (1998).
[CrossRef] [PubMed]

J. D. Joannopoulos, P. R. Villeneuve, and S. H. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

Vuckovic, J.

D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vučković, “Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity,” Phys. Rev. Lett. 104, 073904 (2010).
[CrossRef] [PubMed]

I. Fushman, D. Englund, A. Faraon, N. Stolz, P. Petroff, and J. Vuckovic, “Controlled phase shifts with a single quantum dot,” Science 320, 769–772 (2008).
[CrossRef] [PubMed]

D. Englund, A. Faraon, B. Y. Zhang, Y. Yamamoto, and J. Vuckovic, “Generation and transfer of single photons on a photonic crystal chip,” Opt. Express 15, 5550–5558 (2007).
[CrossRef] [PubMed]

J. Vuckovic, D. Englund, D. Fattal, E. Waks, and Y. Yamamoto, “Generation and manipulation of nonclassical light using photonic crystals,” Physica E (Amsterdam) 32, 466–470 (2006).
[CrossRef]

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.

J. Vuckovic, D. Englund, D. Fattal, E. Waks, and Y. Yamamoto, “Generation and manipulation of nonclassical light using photonic crystals,” Physica E (Amsterdam) 32, 466–470 (2006).
[CrossRef]

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]

Witzigmann, B.

F. Römer and B. Witzigmann, “Spectral and spatial properties of the spontaneous emission enhancement in photonic crystal cavities,” J. Opt. Soc. Am. B 25, 31–39 (2008).
[CrossRef]

F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, “Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver,” Opt. Quantum Electron. 39, 341–352 (2007).
[CrossRef]

Yamamoto, Y.

D. Englund, A. Faraon, B. Y. Zhang, Y. Yamamoto, and J. Vuckovic, “Generation and transfer of single photons on a photonic crystal chip,” Opt. Express 15, 5550–5558 (2007).
[CrossRef] [PubMed]

J. Vuckovic, D. Englund, D. Fattal, E. Waks, and Y. Yamamoto, “Generation and manipulation of nonclassical light using photonic crystals,” Physica E (Amsterdam) 32, 466–470 (2006).
[CrossRef]

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]

Yang, J. K.

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[CrossRef] [PubMed]

Yang, T.

Yariv, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[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]

Zhang, B. Y.

Appl. Phys. B

N. Mattiucci, G. D’Aguanno, M. Scalora, and M. Bloemer, “Cross-phase modulation in one-dimensional photonic crystals: applications to all-optical devices,” Appl. Phys. B 81, 389–391 (2005).
[CrossRef]

Appl. Phys. Lett.

B. Acklin, M. Cada, J. He, and M.-A. Dupertuis, “Bistable switching in a nonlinear Bragg reflector,” Appl. Phys. Lett. 63, 2177–2179 (1993).
[CrossRef]

S. K. Kim, G. H. Kim, S. H. Kim, Y. H. Lee, S. B. Kim, and I. Kim, “Loss management using parity-selective barriers for single-mode, single-cell photonic crystal resonators,” Appl. Phys. Lett. 88, 161119 (2006).
[CrossRef]

IEEE J. Quantum Electron.

S. H. Kim and Y. H. Lee, “Symmetry relations of two-dimensional photonic crystal cavity modes,” IEEE J. Quantum Electron. 39, 1081–1085 (2003).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Photonics

K. Aoki, D. Guimard, M. Nishioka, M. Nomura, S. Iwamoto, and Y. Arakawa, “Coupling of quantum-dot light emission with a three-dimensional photonic-crystal nanocavity,” Nat. Photonics 2, 688–692 (2008).
[CrossRef]

Nature

J. D. Joannopoulos, P. R. Villeneuve, and S. H. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef] [PubMed]

Nature Mater.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005).
[CrossRef]

New J. Phys.

M. Larqué, T. Karle, I. Robert-Philip, and A. Beveratos, “Optimizing h1 cavities for the generation of entangled photon pairs,” New J. Phys. 11, 033022 (2009).
[CrossRef]

Opt. Commun.

B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two-dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59–63 (2004).
[CrossRef]

M. El Kurdi, S. David, X. Checoury, G. Fishman, P. Boucaud, O. Kermarrec, D. Bensahel, and B. Ghyselen, “Two-dimensional photonic crystals with pure germanium-on-insulator,” Opt. Commun. 281, 846–850 (2008).
[CrossRef]

Opt. Express

Opt. Quantum Electron.

F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, “Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver,” Opt. Quantum Electron. 39, 341–352 (2007).
[CrossRef]

Phys. Rev. A

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79, 013814 (2009).
[CrossRef]

Phys. Rev. B

S. H. Kim, S. K. Kim, and Y. H. Lee, “Vertical beaming of wavelength-scale photonic crystal resonators,” Phys. Rev. B 73, 235117 (2006).
[CrossRef]

K. F. Karlsson, M. A. Dupertuis, D. Y. Oberli, E. Pelucchi, A. Rudra, P. O. Holtz, and E. Kapon, “Fine structure of exciton complexes in high symmetry quantum dots: symmetry breaking and symmetry elevation” Phys. Rev. B 81161307 (2010).
[CrossRef]

M. Okano and S. Noda, “Analysis of multimode point-defect cavities in three-dimensional photonic crystals using group theory in frequency and time domains,” Phys. Rev. B 70, 125105 (2004).
[CrossRef]

O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” Phys. Rev. B 68, 035110 (2003).
[CrossRef]

S. Dalessi and M.-A. Dupertuis, “Maximal symmetrization and reduction of fields: application to wave functions in solid state nanostructures,” Phys. Rev. B 81, 125106 (2010).
[CrossRef]

Phys. Rev. Lett.

R. Johne, N. A. Gippius, G. Pavlovic, D. D. Solnyshkov, I. A. Shelykh, and G. Malpuech, “Entangled photon pairs produced by a quantum dot strongly coupled to a microcavity,” Phys. Rev. Lett. 100, 240404 (2008).
[CrossRef] [PubMed]

D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vučković, “Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity,” Phys. Rev. Lett. 104, 073904 (2010).
[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]

W. H. Chang, W. Y. Chen, H. S. Chang, T. P. Hsieh, J. I. Chyi, and T. M. Hsu, “Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities,” Phys. Rev. Lett. 96, 117401 (2006).
[CrossRef] [PubMed]

Physica E (Amsterdam)

J. Vuckovic, D. Englund, D. Fattal, E. Waks, and Y. Yamamoto, “Generation and manipulation of nonclassical light using photonic crystals,” Physica E (Amsterdam) 32, 466–470 (2006).
[CrossRef]

Science

I. Fushman, D. Englund, A. Faraon, N. Stolz, P. Petroff, and J. Vuckovic, “Controlled phase shifts with a single quantum dot,” Science 320, 769–772 (2008).
[CrossRef] [PubMed]

A. Badolato, K. Hennessy, M. Atature, J. Dreiser, E. Hu, P. Petroff, and A. Imamoglu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158–1161 (2005).
[CrossRef] [PubMed]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science 302, 1374–1377 (2003).
[CrossRef] [PubMed]

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[CrossRef] [PubMed]

Thin Solid Films

P. Boucaud, M. El Kurdi, S. David, X. Checoury, X. Li, T. P. Ngo, S. Sauvage, D. Bouchier, G. Fishman, O. Kermarrec, Y. Campidelli, D. Bensahel, T. Akatsu, C. Richtarch, and B. Ghyselen, “Germanium-based nanophotonic devices: two-dimensional photonic crystals and cavities,” Thin Solid Films 517, 121–124 (2008).
[CrossRef]

Other

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2005).

B. Gallinet, M.-A. Dupertuis, and F. Reuse are preparing a manuscript to be called “Spatial domain reduction: a systematic approach of symmetry effects in nanostructures and photonic band-gap microcavities.”

J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994).

S. L. Altmann and P. Herzig, Point-Group Theory Tables (Clarendon, 1994).

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Figures (7)

Fig. 1
Fig. 1

Group symmetry and axis choices: (a) C 2 v , (b) C 3 v and C 6 v , (c) C 4 v (see Appendix C for the mode decomposition in URCFs for C 3 v and C 4 v symmetries).

Fig. 2
Fig. 2

Hexagonal grid of coordinates for sampling: every point is the image of one another by symmetry operations. It can be decomposed into separate sub-domains with respect to the C 6 v symmetry: 12 interiors ( S s , s = 1 , , 12 ) , six σ v -type borders ( B s , s = 1 , , 6 ) , six σ d -type borders ( b s , s = 1 , , 6 ) , and the central point ( C ) .

Fig. 3
Fig. 3

(a) Cavity H 1 , D 6 h symmetric. The holes surrounding the defect have been scaled down by a factor of 0.7 and displaced by 0.09 a in radial direction. Basic geometric data: slab thickness d = 320   nm , hole distance a = 340   nm , hole radius r = 105   nm . (b) Cavity L 3 , D 2 h symmetric. The basic geometry data are slab thickness d = 320   nm , hole distance a = 310   nm , hole radius r = 90   nm . The holes defining the cavity on the long axis have been dislocated by 0.17 a and scaled to 0.465 r .

Fig. 4
Fig. 4

Fundamental y-polarized B 2 mode (arbitrary units, see color (gray) scale bars for intensities or field amplitudes; for each component mapped: irrep on top-left, energy weight on top-right) with the group representation label of the mode and its components on top.

Fig. 5
Fig. 5

Fundamental E 1 modes (arbitrary units, see color (gray) scale bars for intensities or field amplitudes; for each function mapped: irrep on top-left, energy weight on top-right). Each component of these degenerate modes carries a combination of scalar URCFs associated with an irrep of C 6 v . These four independent URCFs are found in both partners but combined differently, and make the explicit link between the two partners.

Fig. 6
Fig. 6

E 2 modes (arbitrary units, see color (gray) scale bars for intensities or field amplitudes; for each function mapped: irrep on top-left, energy weight on top-right). Each component of these degenerate modes carries a combination of scalar URCFs associated with an irrep of C 6 v . These four independent URCFs are found in both partners but combined differently, and make the explicit link between the two partners.

Fig. 7
Fig. 7

Non-degenerate (a) B 2 and (b) A 2 modes (arbitrary units, see color (gray) scale bars for intensities or field amplitudes; for each function mapped: irrep on top-left, energy weight on top-right). Both x and y components have exactly the same weight in the energy due to symmetry effects.

Tables (7)

Tables Icon

Table 1 Eigenmodes of the L 3 Cavity, with Their Normalized Frequency ω n = a k / 2 π

Tables Icon

Table 2 Eigenmodes of the H 1 Cavity, with Their Normalized Frequency ω n = a k / 2 π

Tables Icon

Table 3 C 2 v Character Table

Tables Icon

Table 4 C 6 v Character Table

Tables Icon

Table 5 C 3 v Character Table

Tables Icon

Table 6 C 4 v Character Table

Tables Icon

Table 7 Correspondence between Irreps of the Magnetic and Electric Fields

Equations (92)

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1 ϵ ( r ) × { × E ( r ) } = ω 2 c 2 E ( r ) ,
× { 1 ϵ ( r ) × H ( r ) } = ω 2 c 2 H ( r ) .
( ϵ ( r ) E ( r ) ) = 0 ,
H ( r ) = 0 ,
[ M E E ] ( r ) 1 ϵ ( r ) × { × E ( r ) } ,
[ M H H ] ( r ) × { 1 ϵ ( r ) × H ( r ) } .
H 1 | H 2 H = d 3 r H 1 ( r ) H 2 ( r ) ,
H 1 | M H H 2 H = M H H 1 | H 2 H .
E 1 | E 2 E = d 3 r ϵ ( r ) E 1 ( r ) E 2 ( r ) ,
E 1 | M E E 2 E = M E E 1 | E 2 E .
ϵ ( R ( g ) r ) ϵ ( r ) ,
[ T g V ] ( r ) = R ( g ) V ( R 1 ( g ) r ) ,
T g M E T g 1 = M E ,
T g M H T g 1 = M H ,
[ T g V μ Γ ] ( r ) = ν = 1 d Γ [ D Γ ( g ) ] ν μ V ν Γ ( r ) ,
V μ , m Γ | V ν , n Γ = δ Γ Γ δ μ ν δ m n ,
× E ( r ) = i ω μ 0 H ( r ) ,
χ Γ ( g ) = [ Det   R ( g ) ] χ Γ ( g ) ,
ν = 1 d Γ [ D Γ ( g ) ] ν μ V ν Γ ( r ) = R ( g ) V μ Γ ( R 1 ( g ) r ) .
e ̂ i = Γ b , j U Γ b , j ; i e ̂ j Γ b ,
e ̂ j Γ b = i U i ; Γ b , j 1 e ̂ i ,
g e ̂ i Γ b = j = 1 d Γ b [ D Γ b ( g 1 ) ] j i e ̂ j Γ b ,
[ T g V B ] ( r ) = R B ( g ) V B ( R 1 ( g ) r ) .
ν = 1 d Γ [ D Γ ( g ) ] ν μ V ν Γ , Γ b ( r ) = D Γ b ( g ) V μ Γ , Γ b ( R 1 ( g ) r ) ,
V μ B , Γ = Γ b V μ Γ , Γ b ,
R B ( g ) = Γ b D Γ b ( g ) .
ν = 1 d Γ [ D Γ ( g ) ] ν μ [ D Γ b ( g ) ] 1 V ν Γ , Γ b ( r ) = V μ Γ , Γ b ( R 1 ( g ) r ) .
ν = 1 d Γ j = 1 d Γ b [ D Γ ( g ) ] ν μ [ D Γ b ( g 1 ) ] i j V ν j Γ , Γ b ( r ) = V μ i Γ , Γ b ( R 1 ( g ) r ) .
Γ Γ b = a n Γ , Γ b ; Γ a Γ a ,
D α α Γ a ( g ) = i , j = 1 d Γ b μ , ν = 1 d Γ [ C ν , j ; α Γ , Γ b ; Γ a ] [ D Γ ( g ) ] ν μ [ D Γ b ( g ) ] j i C μ , i ; α Γ , Γ b ; Γ a .
f α Γ , Γ b , Γ a = μ = 1 d Γ i = 1 d Γ b [ C μ , i ; α Γ , Γ b ; Γ a ] V μ i Γ , Γ b ,
V μ i Γ , Γ b = Γ a α = 1 d Γ a C μ , i ; α Γ , Γ b ; Γ a f α Γ , Γ b , Γ a ,
μ , i [ C μ , i ; α Γ , Γ b ; Γ a ] C μ , i ; α Γ , Γ b ; Γ a = δ α α δ Γ a Γ a ,
Γ a , α [ C μ , i ; α Γ , Γ b ; Γ a ] C ν , j ; α Γ , Γ b ; Γ a = δ i j δ μ ν .
f α Γ , Γ b , Γ a ( R 1 ( g ) r ) = α = 1 d Γ a [ D Γ a ( g ) ] α α f α Γ , Γ b , Γ a ( r ) .
V μ l Γ l | V μ r Γ r X = δ Γ l , Γ r δ μ l , μ r = Γ b Γ a ( i , a C μ l , i ; α Γ l , Γ b ; Γ a [ C μ r , i ; α Γ r , Γ b ; Γ a ] ) f Γ l , Γ b , Γ a f Γ , Γ b , Γ a X ,
f α Γ l , Γ b , Γ a | f α Γ r , Γ b , Γ a X = f Γ l , Γ b , Γ a f Γ , Γ b , Γ a X δ Γ a , Γ a δ α , α .
d Γ = Γ b Γ a d Γ a f Γ l , Γ b , Γ a f Γ , Γ b , Γ a X .
w ( Γ , Γ b , Γ a ) = f Γ , Γ b , Γ a f Γ , Γ b , Γ a X / d Γ ,
V μ Γ | V μ Γ X = 1 = Γ b , Γ a d Γ a w ( Γ , Γ b , Γ a ) .
R ( g ) = ( χ B 1 ( g ) 0 0 0 χ B 2 ( g ) 0 0 0 χ A 1 ( g ) ) ,
V A 1 = ( f A 1 , B 1 f A 1 , B 2 f A 1 , A 1 ) ,     V A 2 = ( f A 2 , B 2 f A 2 , B 1 f A 2 , A 2 ) ,
V B 1 = ( f B 1 , A 1 f B 1 , A 2 f B 1 , B 1 ) ,     V B 2 = ( f B 2 , A 2 f B 2 , A 1 f B 2 , B 2 ) .
R ( g ) = ( D E 1 ( g ) 0 0 χ A 1 ( g ) ) ,
V A 1 = ( f 1 A 1 , E 1 f 2 A 1 , E 1 f A 1 , A 1 ) ,     V A 2 = ( f 2 A 2 , E 1 f 1 A 2 , E 1 f A 2 , A 2 ) ,
V B 1 = ( f 2 B 1 , E 2 f 1 B 1 , E 2 f B 1 , B 1 ) ,     V B 2 = ( f 1 B 2 , E 2 f 2 B 2 , E 2 f B 2 , B 2 ) .
V 1 E 1 = ( 1 2 [ f E 1 , A 1 f 2 E 1 , E 2 ] 1 2 [ f E 1 , A 2 + f 1 E 1 , E 2 ] f 1 E 1 , E 1 ) ,
V 2 E 1 = ( 1 2 [ f E 1 , A 2 + f 1 E 1 , E 2 ] 1 2 [ f E 1 , A 1 + f 2 E 1 , E 2 ] f 2 E 1 , E 1 ) ,
V 1 E 2 = ( 1 2 [ f E 2 , B 2 f 2 E 2 , E 1 ] 1 2 [ f E 2 , B 1 f 1 E 2 , E 1 ] f 1 E 2 , E 2 ) ,
V 2 E 2 = ( 1 2 [ f E 2 , B 1 + f 1 E 2 , E 1 ] 1 2 [ f E 2 , B 2 f 2 E 2 , E 1 ] f 2 E 2 , E 2 ) .
M = Γ b , i ; Γ b , j M i , j Γ b , Γ b e ̂ i Γ b ( e ̂ j Γ b ) ,
Γ b , Γ a α = 1 d Γ a M Γ ( Γ b , Γ a , α ; Γ b , Γ a , α ) f α Γ , Γ b , Γ a = ω 2 c 2 f α Γ , Γ b , Γ a .
M Γ ( Γ b , Γ a , α ; Γ b , Γ a , α ) = μ = 1 d Γ i = 1 d Γ b j = 1 d Γ b C μ , i ; α Γ , Γ b ; Γ a M i , j Γ b , Γ b [ C μ , j ; α Γ , Γ b ; Γ a ] .
R ( g ) r s , i = r π g ( s ) , i ,     s = 1 , , N S ;     i = 1 , , n s ,
[ T g ψ ] s , i = ψ π g 1 ( s ) , i = m P n m ( g ) ψ m .
[ T g V ] s , i = R ( g ) V π g 1 ( s ) , i = R ( g ) P n m ( g ) V m .
P σ d 1 T ( C B 1 S 1 B 2 S 2 B 3 S 3 B 4 S 4 B 5 S 5 B 6 S 6 B 7 S 7 B 8 S 8 B 9 S 9 B 10 S 10 B 11 S 11 B 12 S 12 ) = ( C B 7 S 6 B 6 S 5 B 5 S 4 B 4 S 3 B 3 S 2 B 2 S 1 B 1 S 12 B 12 S 11 B 11 S 10 B 10 S 9 B 9 S 8 B 8 S 7 ) = ( 1 0 . . . . . . . . . . . 0 0 0 . . . . . . . . . 0 . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 0 . . . . . . . . . ) ( C B 1 S 1 B 2 S 2 B 3 S 3 B 4 S 4 B 5 S 5 B 6 S 6 B 7 S 7 B 8 S 8 B 9 S 9 B 10 S 10 B 11 S 11 B 12 S 12 ) ,
E | E E = n ϵ n E n E n .
Π μ Γ = d Γ | G | g G [ D μ μ Γ ( g ) ] T g ,
V ( n ) = Γ , μ c μ Γ , ( n ) V μ Γ , ( n ) ,
V ( n ) | Π μ Γ V ( n ) = | c μ Γ , ( n ) | 2 .
[ D ( g ) ] l r = V l | T g V r .
V μ Γ ( r n ) = r S r μ Γ V r ( r n ) .
( ω c ) 2 = E μ Γ | M E | E μ Γ E μ Γ | E μ Γ .
D E 1 ( E ) = ( 1 0 0 1 ) ,     D E 1 ( C 6 + ) = ( 1 2 3 2 3 2 1 2 ) ,
D E 1 ( C 6 ) = ( 1 2 3 2 3 2 1 2 ) ,
D E 1 ( C 3 + ) = ( 1 2 3 2 3 2 1 2 ) ,
D E 1 ( C 3 ) = ( 1 2 3 2 3 2 1 2 ) ,     D E 1 ( C 2 ) = ( 1 0 0 1 ) ,
D E 1 ( σ d 1 ) = ( 1 0 0 1 ) ,     D E 1 ( σ d 2 ) = ( 1 2 3 2 3 2 1 2 ) ,
D E 1 ( σ d 3 ) = ( 1 2 3 2 3 2 1 2 ) ,
D E 1 ( σ v 1 ) = ( 1 0 0 1 ) ,
D E 1 ( σ v 2 ) = ( 1 2 3 2 3 2 1 2 ) ,     D E 1 ( σ v 3 ) = ( 1 2 3 2 3 2 1 2 ) ,
D E 2 ( E ) = D E 2 ( C 2 ) = ( 1 0 0 1 ) ,
D E 2 ( σ d 1 ) = D E 2 ( σ v 1 ) = ( 1 0 0 1 ) ,
D E 2 ( C 6 + ) = D E 2 ( C 3 ) = ( 1 2 3 2 3 2 1 2 ) ,
D E 2 ( C 6 ) = D E 2 ( C 3 + ) = ( 1 2 3 2 3 2 1 2 ) ,
D E 2 ( σ d 2 ) = D E 2 ( σ v 2 ) = ( 1 2 3 2 3 2 1 2 ) ,
D E 2 ( σ d 3 ) = D E 2 ( σ v 3 ) = ( 1 2 3 2 3 2 1 2 ) .
D E ( E ) = ( 1 0 0 1 ) ,     D E ( C 3 + ) = ( 1 2 3 2 3 2 1 2 ) ,
D E ( C 3 ) = ( 1 2 3 2 3 2 1 2 ) ,
D E ( σ v 1 ) = ( 1 0 0 1 ) ,     D E ( σ v 2 ) = ( 1 2 3 2 3 2 1 2 ) ,
D E ( σ v 3 ) = ( 1 2 3 2 3 2 1 2 ) .
D E ( E ) = ( 1 0 0 1 ) ,     D E ( C 4 + ) = ( 0 1 1 0 ) ,
D E ( C 4 ) = ( 0 1 1 0 ) ,     D E ( C 2 ) = ( 1 0 0 1 ) ,
D E ( σ v 1 ) = ( 1 0 0 1 ) ,     D E ( σ v 2 ) = ( 1 0 0 1 ) ,
D E ( σ d 1 ) = ( 0 1 1 0 ) ,     D E ( σ d 2 ) = ( 0 1 1 0 ) .
R ( g ) = ( D E ( g ) 0 0 χ A 1 ( g ) ) .
V A 1 = ( f 1 A 1 , E f 2 A 1 , E f A 1 , A 1 ) ,     V A 2 = ( f 2 A 2 , E f 1 A 2 , E f A 2 , A 2 ) ,
V 1 E = ( 1 2 [ f E , A 2 f 1 E , E ] 1 2 [ f E , A 1 + f 2 E , E ] f 1 E , E ) ,     V 2 E = ( 1 2 [ f E , A 1 + f 2 E , E ] 1 2 [ f E , A 2 + f 1 E , E ] f 2 E , E ) .
V A 1 = ( f 1 A 1 , E f 2 A 1 , E f A 1 , A 1 ) ,     V A 2 = ( f 2 A 2 , E f 1 A 2 , E f A 2 , A 2 ) ,
V B 1 = ( f 1 B 1 , E f 2 B 1 , E f B 1 , B 1 ) ,     V B 2 = ( f 2 B 2 , E f 1 B 2 , E f B 2 , B 2 ) ,
V 1 E = ( 1 2 [ f E , A 1 f E , B 1 ] 1 2 [ f E , A 2 + f E , B 2 ] f 1 E , E ) ,     V 2 E = ( 1 2 [ f E , A 2 + f E , B 2 ] 1 2 [ f E , A 1 + f E , B 1 ] f 2 E , E ) .

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