Abstract

Double photonic bandgaps (PBGs) can simultaneously appear when double dark resonances in uniform cold atoms are spatially modulated by a resonance standing-wave. Theoretical calculations show that variable and efficient coherent optical control of the PBGs can be achieved by modulating the coupling field and standing-wave. The structures of double PBGs induced by the atomic coherence effect are better than those obtained in the photonic crystal heterostructures. We anticipate that this scheme has potential applications in optical networks for dual-channel all-optical switching or a dual-frequency optical Bragg reflector.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. J. D. Joannopoulos, R. D. Mcade, and J. N. Win, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  4. S. G. Romanov, H. M. Yates, M. E. Pemble, and R. M. De La Rue, “Opal-based photonic crystal with double photonic bandgap structure,” J. Phys. Condens. Matter 12, 8221–8229 (2000).
    [CrossRef]
  5. M. Egen, R. Voss, B. Griesebock, and R. Zentel, “Heterostructures of polymer crystal films,” Chem. Mater. 15, 3786–3792 (2003).
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  6. N. Gaponik, A. Eychmüller, A. L. Rogach, V. G. Solovyev, C. M. Sotomayor Torres, and S. G. Romanov, “Structure-related optical properties of luminescent hetero-opals,” J. Appl. Phys. 95, 1029–1035 (2004).
    [CrossRef]
  7. M. Bardosova, M. E. Pemble, I. M. Povey, R. H. Tredgold, and D. E. Whitehead, “Enhanced Bragg reflections from size-matched heterostructure photonic crystal thin films prepared by the Langmuir–Blodgett method,” Appl. Phys. Lett. 89, 093116 (2006).
    [CrossRef]
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    [CrossRef]
  11. M. A. G. Laso, T. Lopetegi, M. J. Erro, D. Benito, M. J. Garde, and M. Sorolla, “Multiple-frequency-tuned photonic bandgap microstrip structures,” IEEE Microw. Guid. Wave Lett. 10, 220–222 (2000).
    [CrossRef]
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  13. D. Kang, J. E. Maclennan, N. A. Clark, A. A. Zakhidov, and R. H. Baughman, “Electro-optic behavior of liquid-crystal-filled silica opal photonic crystals: Effect of liquid-crystal alignment,” Phys. Rev. Lett. 86, 4052–4055 (2001).
    [CrossRef]
  14. X. Y. Hu, Q. Zhang, Y. H. Liu, B. Y. Cheng, and D. Z. Zhang, “Ultrafast three-dimensional tunable photonic crystal,” Appl. Phys. Lett. 83, 2518–2520 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  18. Q. Y. He, J. H. Wu, T. J. Wang, and J. Y. Gao, “Dynamic control of the photonic stop bands formed by a standing wave in inhomogeneous broadening solids,” Phys. Rev. A 73, 053813 (2006).
    [CrossRef]
  19. Q. Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J. H. Xu, and J. Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
    [CrossRef]
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    [CrossRef]
  21. J. H. Wu, G. C. La Rocca, and M. Artoni, “Controlled light-pulse propagation in driven color centers in diamond,” Phys. Rev. B 77, 113106 (2008).
    [CrossRef]
  22. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
    [CrossRef]
  23. M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
    [CrossRef]
  24. D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
    [CrossRef]
  25. A. W. Brown and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. 30, 699–701 (2005).
    [CrossRef]
  26. G. Wang, Y. Xue, J. H. Wu, S. S. Liu, Z. H. Kang, and J. Y. Gao, “Dual-channel all-optical wavelength conversion switching by four-wave mixing,” Opt. Express 17, 23332–23337 (2009).
    [CrossRef]
  27. In this paper, the wavelengths of the probe, coupling, and standing fields are λp=852.335 nm, λc=852.357 nm, and λs=852.356 nm, respectively.
  28. M. Artoni, G. C. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72, 046604 (2005).
    [CrossRef]
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  31. A. André and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89, 143602 (2002).
    [CrossRef]
  32. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
    [CrossRef]

2009 (3)

2008 (2)

J. H. Wu, M. Artoni, and G. C. La Rocca, “Controlling the photonic band structure of optically driven cold atoms,” J. Opt. Soc. Am. B 25, 1840–1849 (2008).
[CrossRef]

J. H. Wu, G. C. La Rocca, and M. Artoni, “Controlled light-pulse propagation in driven color centers in diamond,” Phys. Rev. B 77, 113106 (2008).
[CrossRef]

2007 (1)

R. V. Nair and R. Vijaya, “Three-dimensionally ordered photonic crystal heterostructures with a double photonic stop band,” J. Appl. Phys. 102, 056102 (2007).
[CrossRef]

2006 (4)

M. Bardosova, M. E. Pemble, I. M. Povey, R. H. Tredgold, and D. E. Whitehead, “Enhanced Bragg reflections from size-matched heterostructure photonic crystal thin films prepared by the Langmuir–Blodgett method,” Appl. Phys. Lett. 89, 093116 (2006).
[CrossRef]

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96, 073905 (2006).
[CrossRef] [PubMed]

Q. Y. He, J. H. Wu, T. J. Wang, and J. Y. Gao, “Dynamic control of the photonic stop bands formed by a standing wave in inhomogeneous broadening solids,” Phys. Rev. A 73, 053813 (2006).
[CrossRef]

Q. Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J. H. Xu, and J. Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
[CrossRef]

2005 (2)

A. W. Brown and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. 30, 699–701 (2005).
[CrossRef]

M. Artoni, G. C. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72, 046604 (2005).
[CrossRef]

2004 (2)

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gösele, “Nonlinear optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B 70, 205110 (2004).
[CrossRef]

N. Gaponik, A. Eychmüller, A. L. Rogach, V. G. Solovyev, C. M. Sotomayor Torres, and S. G. Romanov, “Structure-related optical properties of luminescent hetero-opals,” J. Appl. Phys. 95, 1029–1035 (2004).
[CrossRef]

2003 (3)

X. Y. Hu, Q. Zhang, Y. H. Liu, B. Y. Cheng, and D. Z. Zhang, “Ultrafast three-dimensional tunable photonic crystal,” Appl. Phys. Lett. 83, 2518–2520 (2003).
[CrossRef]

M. Egen, R. Voss, B. Griesebock, and R. Zentel, “Heterostructures of polymer crystal films,” Chem. Mater. 15, 3786–3792 (2003).
[CrossRef]

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[CrossRef]

2002 (1)

A. André and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef]

2001 (2)

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

D. Kang, J. E. Maclennan, N. A. Clark, A. A. Zakhidov, and R. H. Baughman, “Electro-optic behavior of liquid-crystal-filled silica opal photonic crystals: Effect of liquid-crystal alignment,” Phys. Rev. Lett. 86, 4052–4055 (2001).
[CrossRef]

2000 (2)

S. G. Romanov, H. M. Yates, M. E. Pemble, and R. M. De La Rue, “Opal-based photonic crystal with double photonic bandgap structure,” J. Phys. Condens. Matter 12, 8221–8229 (2000).
[CrossRef]

M. A. G. Laso, T. Lopetegi, M. J. Erro, D. Benito, M. J. Garde, and M. Sorolla, “Multiple-frequency-tuned photonic bandgap microstrip structures,” IEEE Microw. Guid. Wave Lett. 10, 220–222 (2000).
[CrossRef]

1999 (2)

K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: The tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999).
[CrossRef]

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[CrossRef]

1997 (1)

S. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[CrossRef]

1994 (1)

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef]

1987 (2)

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

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

Akimov, A. V.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[CrossRef]

André, A.

A. André and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef]

Artoni, M.

J. H. Wu, M. Artoni, and G. C. La Rocca, “Controlling the photonic band structure of optically driven cold atoms,” J. Opt. Soc. Am. B 25, 1840–1849 (2008).
[CrossRef]

J. H. Wu, G. C. La Rocca, and M. Artoni, “Controlled light-pulse propagation in driven color centers in diamond,” Phys. Rev. B 77, 113106 (2008).
[CrossRef]

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96, 073905 (2006).
[CrossRef] [PubMed]

Q. Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J. H. Xu, and J. Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
[CrossRef]

M. Artoni, G. C. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72, 046604 (2005).
[CrossRef]

Bardosova, M.

M. Bardosova, M. E. Pemble, I. M. Povey, R. H. Tredgold, and D. E. Whitehead, “Enhanced Bragg reflections from size-matched heterostructure photonic crystal thin films prepared by the Langmuir–Blodgett method,” Appl. Phys. Lett. 89, 093116 (2006).
[CrossRef]

Bassani, F.

M. Artoni, G. C. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72, 046604 (2005).
[CrossRef]

Baughman, R. H.

D. Kang, J. E. Maclennan, N. A. Clark, A. A. Zakhidov, and R. H. Baughman, “Electro-optic behavior of liquid-crystal-filled silica opal photonic crystals: Effect of liquid-crystal alignment,” Phys. Rev. Lett. 86, 4052–4055 (2001).
[CrossRef]

Benito, D.

M. A. G. Laso, T. Lopetegi, M. J. Erro, D. Benito, M. J. Garde, and M. Sorolla, “Multiple-frequency-tuned photonic bandgap microstrip structures,” IEEE Microw. Guid. Wave Lett. 10, 220–222 (2000).
[CrossRef]

Bloemer, M. J.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef]

Bowden, C. M.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef]

Brown, A. W.

Busch, K.

K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: The tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999).
[CrossRef]

Chen, Y.

Cheng, B. Y.

X. Y. Hu, Q. Zhang, Y. H. Liu, B. Y. Cheng, and D. Z. Zhang, “Ultrafast three-dimensional tunable photonic crystal,” Appl. Phys. Lett. 83, 2518–2520 (2003).
[CrossRef]

Clark, N. A.

D. Kang, J. E. Maclennan, N. A. Clark, A. A. Zakhidov, and R. H. Baughman, “Electro-optic behavior of liquid-crystal-filled silica opal photonic crystals: Effect of liquid-crystal alignment,” Phys. Rev. Lett. 86, 4052–4055 (2001).
[CrossRef]

Dai, Q. F.

Z. Q. Liu, T. H. Feng, Q. F. Dai, L. J. Wu, and S. Lan, “Fabrication of high-quality three-dimensional photonic crystal heterostructures,” Chin. Phys. B 18, 2383–2388 (2009).
[CrossRef]

De La Rue, R. M.

S. G. Romanov, H. M. Yates, M. E. Pemble, and R. M. De La Rue, “Opal-based photonic crystal with double photonic bandgap structure,” J. Phys. Condens. Matter 12, 8221–8229 (2000).
[CrossRef]

Dijkhuis, J. I.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[CrossRef]

Dowling, J. P.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef]

Egen, M.

M. Egen, R. Voss, B. Griesebock, and R. Zentel, “Heterostructures of polymer crystal films,” Chem. Mater. 15, 3786–3792 (2003).
[CrossRef]

Erro, M. J.

M. A. G. Laso, T. Lopetegi, M. J. Erro, D. Benito, M. J. Garde, and M. Sorolla, “Multiple-frequency-tuned photonic bandgap microstrip structures,” IEEE Microw. Guid. Wave Lett. 10, 220–222 (2000).
[CrossRef]

Eychmüller, A.

N. Gaponik, A. Eychmüller, A. L. Rogach, V. G. Solovyev, C. M. Sotomayor Torres, and S. G. Romanov, “Structure-related optical properties of luminescent hetero-opals,” J. Appl. Phys. 95, 1029–1035 (2004).
[CrossRef]

Feng, T. H.

Z. Q. Liu, T. H. Feng, Q. F. Dai, L. J. Wu, and S. Lan, “Fabrication of high-quality three-dimensional photonic crystal heterostructures,” Chin. Phys. B 18, 2383–2388 (2009).
[CrossRef]

Fleischhauer, M.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[CrossRef]

Gao, J. Y.

G. Wang, Y. Xue, J. H. Wu, S. S. Liu, Z. H. Kang, and J. Y. Gao, “Dual-channel all-optical wavelength conversion switching by four-wave mixing,” Opt. Express 17, 23332–23337 (2009).
[CrossRef]

Q. Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J. H. Xu, and J. Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
[CrossRef]

Q. Y. He, J. H. Wu, T. J. Wang, and J. Y. Gao, “Dynamic control of the photonic stop bands formed by a standing wave in inhomogeneous broadening solids,” Phys. Rev. A 73, 053813 (2006).
[CrossRef]

Gaponik, N.

N. Gaponik, A. Eychmüller, A. L. Rogach, V. G. Solovyev, C. M. Sotomayor Torres, and S. G. Romanov, “Structure-related optical properties of luminescent hetero-opals,” J. Appl. Phys. 95, 1029–1035 (2004).
[CrossRef]

Garde, M. J.

M. A. G. Laso, T. Lopetegi, M. J. Erro, D. Benito, M. J. Garde, and M. Sorolla, “Multiple-frequency-tuned photonic bandgap microstrip structures,” IEEE Microw. Guid. Wave Lett. 10, 220–222 (2000).
[CrossRef]

Golubev, V. G.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[CrossRef]

Gösele, U.

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gösele, “Nonlinear optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Griesebock, B.

M. Egen, R. Voss, B. Griesebock, and R. Zentel, “Heterostructures of polymer crystal films,” Chem. Mater. 15, 3786–3792 (2003).
[CrossRef]

Ham, B. S.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Harris, S.

S. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[CrossRef]

He, Q. Y.

Q. Y. He, J. H. Wu, T. J. Wang, and J. Y. Gao, “Dynamic control of the photonic stop bands formed by a standing wave in inhomogeneous broadening solids,” Phys. Rev. A 73, 053813 (2006).
[CrossRef]

Q. Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J. H. Xu, and J. Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
[CrossRef]

Hemmer, P. R.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Hu, H. H.

Hu, X. Y.

X. Y. Hu, Q. Zhang, Y. H. Liu, B. Y. Cheng, and D. Z. Zhang, “Ultrafast three-dimensional tunable photonic crystal,” Appl. Phys. Lett. 83, 2518–2520 (2003).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Mcade, and J. N. Win, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

John, S.

K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: The tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999).
[CrossRef]

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

Kang, D.

D. Kang, J. E. Maclennan, N. A. Clark, A. A. Zakhidov, and R. H. Baughman, “Electro-optic behavior of liquid-crystal-filled silica opal photonic crystals: Effect of liquid-crystal alignment,” Phys. Rev. Lett. 86, 4052–4055 (2001).
[CrossRef]

Kang, Z. H.

Kerst, R.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[CrossRef]

Kurdyukov, D. A.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[CrossRef]

La Rocca, G. C.

J. H. Wu, G. C. La Rocca, and M. Artoni, “Controlled light-pulse propagation in driven color centers in diamond,” Phys. Rev. B 77, 113106 (2008).
[CrossRef]

J. H. Wu, M. Artoni, and G. C. La Rocca, “Controlling the photonic band structure of optically driven cold atoms,” J. Opt. Soc. Am. B 25, 1840–1849 (2008).
[CrossRef]

Q. Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J. H. Xu, and J. Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
[CrossRef]

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96, 073905 (2006).
[CrossRef] [PubMed]

M. Artoni, G. C. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72, 046604 (2005).
[CrossRef]

Lan, S.

Z. Q. Liu, T. H. Feng, Q. F. Dai, L. J. Wu, and S. Lan, “Fabrication of high-quality three-dimensional photonic crystal heterostructures,” Chin. Phys. B 18, 2383–2388 (2009).
[CrossRef]

Laso, M. A. G.

M. A. G. Laso, T. Lopetegi, M. J. Erro, D. Benito, M. J. Garde, and M. Sorolla, “Multiple-frequency-tuned photonic bandgap microstrip structures,” IEEE Microw. Guid. Wave Lett. 10, 220–222 (2000).
[CrossRef]

Liao, Y. B.

Liu, G. Q.

Liu, S. S.

Liu, Y. H.

X. Y. Hu, Q. Zhang, Y. H. Liu, B. Y. Cheng, and D. Z. Zhang, “Ultrafast three-dimensional tunable photonic crystal,” Appl. Phys. Lett. 83, 2518–2520 (2003).
[CrossRef]

Liu, Z. Q.

Z. Q. Liu, T. H. Feng, Q. F. Dai, L. J. Wu, and S. Lan, “Fabrication of high-quality three-dimensional photonic crystal heterostructures,” Chin. Phys. B 18, 2383–2388 (2009).
[CrossRef]

Lopetegi, T.

M. A. G. Laso, T. Lopetegi, M. J. Erro, D. Benito, M. J. Garde, and M. Sorolla, “Multiple-frequency-tuned photonic bandgap microstrip structures,” IEEE Microw. Guid. Wave Lett. 10, 220–222 (2000).
[CrossRef]

Lukin, M. D.

A. André and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef]

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[CrossRef]

Maclennan, J. E.

D. Kang, J. E. Maclennan, N. A. Clark, A. A. Zakhidov, and R. H. Baughman, “Electro-optic behavior of liquid-crystal-filled silica opal photonic crystals: Effect of liquid-crystal alignment,” Phys. Rev. Lett. 86, 4052–4055 (2001).
[CrossRef]

Mazurenko, D. A.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[CrossRef]

Mcade, R. D.

J. D. Joannopoulos, R. D. Mcade, and J. N. Win, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Musser, J. A.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Nair, R. V.

R. V. Nair and R. Vijaya, “Three-dimensionally ordered photonic crystal heterostructures with a double photonic stop band,” J. Appl. Phys. 102, 056102 (2007).
[CrossRef]

Pemble, M. E.

M. Bardosova, M. E. Pemble, I. M. Povey, R. H. Tredgold, and D. E. Whitehead, “Enhanced Bragg reflections from size-matched heterostructure photonic crystal thin films prepared by the Langmuir–Blodgett method,” Appl. Phys. Lett. 89, 093116 (2006).
[CrossRef]

S. G. Romanov, H. M. Yates, M. E. Pemble, and R. M. De La Rue, “Opal-based photonic crystal with double photonic bandgap structure,” J. Phys. Condens. Matter 12, 8221–8229 (2000).
[CrossRef]

Pevtsov, A. B.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[CrossRef]

Povey, I. M.

M. Bardosova, M. E. Pemble, I. M. Povey, R. H. Tredgold, and D. E. Whitehead, “Enhanced Bragg reflections from size-matched heterostructure photonic crystal thin films prepared by the Langmuir–Blodgett method,” Appl. Phys. Lett. 89, 093116 (2006).
[CrossRef]

Rogach, A. L.

N. Gaponik, A. Eychmüller, A. L. Rogach, V. G. Solovyev, C. M. Sotomayor Torres, and S. G. Romanov, “Structure-related optical properties of luminescent hetero-opals,” J. Appl. Phys. 95, 1029–1035 (2004).
[CrossRef]

Romanov, S. G.

N. Gaponik, A. Eychmüller, A. L. Rogach, V. G. Solovyev, C. M. Sotomayor Torres, and S. G. Romanov, “Structure-related optical properties of luminescent hetero-opals,” J. Appl. Phys. 95, 1029–1035 (2004).
[CrossRef]

S. G. Romanov, H. M. Yates, M. E. Pemble, and R. M. De La Rue, “Opal-based photonic crystal with double photonic bandgap structure,” J. Phys. Condens. Matter 12, 8221–8229 (2000).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

Scalora, M.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef]

Schweizer, S. L.

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gösele, “Nonlinear optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Scully, M. O.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[CrossRef]

Sel’kin, A. V.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[CrossRef]

Shahriar, M. S.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Solovyev, V. G.

N. Gaponik, A. Eychmüller, A. L. Rogach, V. G. Solovyev, C. M. Sotomayor Torres, and S. G. Romanov, “Structure-related optical properties of luminescent hetero-opals,” J. Appl. Phys. 95, 1029–1035 (2004).
[CrossRef]

Sorolla, M.

M. A. G. Laso, T. Lopetegi, M. J. Erro, D. Benito, M. J. Garde, and M. Sorolla, “Multiple-frequency-tuned photonic bandgap microstrip structures,” IEEE Microw. Guid. Wave Lett. 10, 220–222 (2000).
[CrossRef]

Sotomayor Torres, C. M.

N. Gaponik, A. Eychmüller, A. L. Rogach, V. G. Solovyev, C. M. Sotomayor Torres, and S. G. Romanov, “Structure-related optical properties of luminescent hetero-opals,” J. Appl. Phys. 95, 1029–1035 (2004).
[CrossRef]

Sudarshanam, V. S.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Tan, H. W.

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gösele, “Nonlinear optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Tredgold, R. H.

M. Bardosova, M. E. Pemble, I. M. Povey, R. H. Tredgold, and D. E. Whitehead, “Enhanced Bragg reflections from size-matched heterostructure photonic crystal thin films prepared by the Langmuir–Blodgett method,” Appl. Phys. Lett. 89, 093116 (2006).
[CrossRef]

Turukhin, A. V.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

van Driel, H. M.

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gösele, “Nonlinear optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Vijaya, R.

R. V. Nair and R. Vijaya, “Three-dimensionally ordered photonic crystal heterostructures with a double photonic stop band,” J. Appl. Phys. 102, 056102 (2007).
[CrossRef]

Voss, R.

M. Egen, R. Voss, B. Griesebock, and R. Zentel, “Heterostructures of polymer crystal films,” Chem. Mater. 15, 3786–3792 (2003).
[CrossRef]

Wang, G.

Wang, T. J.

Q. Y. He, J. H. Wu, T. J. Wang, and J. Y. Gao, “Dynamic control of the photonic stop bands formed by a standing wave in inhomogeneous broadening solids,” Phys. Rev. A 73, 053813 (2006).
[CrossRef]

Wang, Z. S.

Wehrspohn, R. B.

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gösele, “Nonlinear optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Whitehead, D. E.

M. Bardosova, M. E. Pemble, I. M. Povey, R. H. Tredgold, and D. E. Whitehead, “Enhanced Bragg reflections from size-matched heterostructure photonic crystal thin films prepared by the Langmuir–Blodgett method,” Appl. Phys. Lett. 89, 093116 (2006).
[CrossRef]

Win, J. N.

J. D. Joannopoulos, R. D. Mcade, and J. N. Win, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Wu, J. H.

G. Wang, Y. Xue, J. H. Wu, S. S. Liu, Z. H. Kang, and J. Y. Gao, “Dual-channel all-optical wavelength conversion switching by four-wave mixing,” Opt. Express 17, 23332–23337 (2009).
[CrossRef]

J. H. Wu, G. C. La Rocca, and M. Artoni, “Controlled light-pulse propagation in driven color centers in diamond,” Phys. Rev. B 77, 113106 (2008).
[CrossRef]

J. H. Wu, M. Artoni, and G. C. La Rocca, “Controlling the photonic band structure of optically driven cold atoms,” J. Opt. Soc. Am. B 25, 1840–1849 (2008).
[CrossRef]

Q. Y. He, J. H. Wu, T. J. Wang, and J. Y. Gao, “Dynamic control of the photonic stop bands formed by a standing wave in inhomogeneous broadening solids,” Phys. Rev. A 73, 053813 (2006).
[CrossRef]

Wu, L. J.

Z. Q. Liu, T. H. Feng, Q. F. Dai, L. J. Wu, and S. Lan, “Fabrication of high-quality three-dimensional photonic crystal heterostructures,” Chin. Phys. B 18, 2383–2388 (2009).
[CrossRef]

Xiao, M.

Xu, J. H.

Q. Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J. H. Xu, and J. Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
[CrossRef]

Xue, Y.

G. Wang, Y. Xue, J. H. Wu, S. S. Liu, Z. H. Kang, and J. Y. Gao, “Dual-channel all-optical wavelength conversion switching by four-wave mixing,” Opt. Express 17, 23332–23337 (2009).
[CrossRef]

Q. Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J. H. Xu, and J. Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
[CrossRef]

Yablonovitch, E.

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

Yates, H. M.

S. G. Romanov, H. M. Yates, M. E. Pemble, and R. M. De La Rue, “Opal-based photonic crystal with double photonic bandgap structure,” J. Phys. Condens. Matter 12, 8221–8229 (2000).
[CrossRef]

Yelin, S. F.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[CrossRef]

Zakhidov, A. A.

D. Kang, J. E. Maclennan, N. A. Clark, A. A. Zakhidov, and R. H. Baughman, “Electro-optic behavior of liquid-crystal-filled silica opal photonic crystals: Effect of liquid-crystal alignment,” Phys. Rev. Lett. 86, 4052–4055 (2001).
[CrossRef]

Zentel, R.

M. Egen, R. Voss, B. Griesebock, and R. Zentel, “Heterostructures of polymer crystal films,” Chem. Mater. 15, 3786–3792 (2003).
[CrossRef]

Zhang, D. Z.

X. Y. Hu, Q. Zhang, Y. H. Liu, B. Y. Cheng, and D. Z. Zhang, “Ultrafast three-dimensional tunable photonic crystal,” Appl. Phys. Lett. 83, 2518–2520 (2003).
[CrossRef]

Zhang, Q.

X. Y. Hu, Q. Zhang, Y. H. Liu, B. Y. Cheng, and D. Z. Zhang, “Ultrafast three-dimensional tunable photonic crystal,” Appl. Phys. Lett. 83, 2518–2520 (2003).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

X. Y. Hu, Q. Zhang, Y. H. Liu, B. Y. Cheng, and D. Z. Zhang, “Ultrafast three-dimensional tunable photonic crystal,” Appl. Phys. Lett. 83, 2518–2520 (2003).
[CrossRef]

M. Bardosova, M. E. Pemble, I. M. Povey, R. H. Tredgold, and D. E. Whitehead, “Enhanced Bragg reflections from size-matched heterostructure photonic crystal thin films prepared by the Langmuir–Blodgett method,” Appl. Phys. Lett. 89, 093116 (2006).
[CrossRef]

Chem. Mater. (1)

M. Egen, R. Voss, B. Griesebock, and R. Zentel, “Heterostructures of polymer crystal films,” Chem. Mater. 15, 3786–3792 (2003).
[CrossRef]

Chin. Phys. B (1)

Z. Q. Liu, T. H. Feng, Q. F. Dai, L. J. Wu, and S. Lan, “Fabrication of high-quality three-dimensional photonic crystal heterostructures,” Chin. Phys. B 18, 2383–2388 (2009).
[CrossRef]

IEEE Microw. Guid. Wave Lett. (1)

M. A. G. Laso, T. Lopetegi, M. J. Erro, D. Benito, M. J. Garde, and M. Sorolla, “Multiple-frequency-tuned photonic bandgap microstrip structures,” IEEE Microw. Guid. Wave Lett. 10, 220–222 (2000).
[CrossRef]

J. Appl. Phys. (2)

N. Gaponik, A. Eychmüller, A. L. Rogach, V. G. Solovyev, C. M. Sotomayor Torres, and S. G. Romanov, “Structure-related optical properties of luminescent hetero-opals,” J. Appl. Phys. 95, 1029–1035 (2004).
[CrossRef]

R. V. Nair and R. Vijaya, “Three-dimensionally ordered photonic crystal heterostructures with a double photonic stop band,” J. Appl. Phys. 102, 056102 (2007).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. Condens. Matter (1)

S. G. Romanov, H. M. Yates, M. E. Pemble, and R. M. De La Rue, “Opal-based photonic crystal with double photonic bandgap structure,” J. Phys. Condens. Matter 12, 8221–8229 (2000).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (2)

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[CrossRef]

Q. Y. He, J. H. Wu, T. J. Wang, and J. Y. Gao, “Dynamic control of the photonic stop bands formed by a standing wave in inhomogeneous broadening solids,” Phys. Rev. A 73, 053813 (2006).
[CrossRef]

Phys. Rev. B (3)

Q. Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J. H. Xu, and J. Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
[CrossRef]

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gösele, “Nonlinear optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B 70, 205110 (2004).
[CrossRef]

J. H. Wu, G. C. La Rocca, and M. Artoni, “Controlled light-pulse propagation in driven color centers in diamond,” Phys. Rev. B 77, 113106 (2008).
[CrossRef]

Phys. Rev. E (1)

M. Artoni, G. C. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72, 046604 (2005).
[CrossRef]

Phys. Rev. Lett. (9)

A. André and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef]

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef]

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[CrossRef]

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96, 073905 (2006).
[CrossRef] [PubMed]

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

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

K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: The tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999).
[CrossRef]

D. Kang, J. E. Maclennan, N. A. Clark, A. A. Zakhidov, and R. H. Baughman, “Electro-optic behavior of liquid-crystal-filled silica opal photonic crystals: Effect of liquid-crystal alignment,” Phys. Rev. Lett. 86, 4052–4055 (2001).
[CrossRef]

Phys. Today (1)

S. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[CrossRef]

Other (4)

J. D. Joannopoulos, R. D. Mcade, and J. N. Win, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

In this paper, the wavelengths of the probe, coupling, and standing fields are λp=852.335 nm, λc=852.357 nm, and λs=852.356 nm, respectively.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

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Figures (4)

Fig. 1
Fig. 1

A schematic diagram of the four-level N-type atomic system.

Fig. 2
Fig. 2

(a) Bandgap structure for the probe field in a homogeneous sample of ultracold Cs ( N 0 = 5 × 10 12 cm 3 ) in the presence of coupling field Ω c = 40 γ and a periodic modulation induced by a standing-wave Ω 1 = Ω 2 = 20 γ . Real and imaginary parts of Bloch wave vector of probe field are shown. (b) Real and imaginary parts of the susceptibility χ at probe frequency Δ p in the control of the coupling field Ω c = 40 γ and in different positions of the standing-wave: (i) Ω s = 0 ; (ii) Ω s = 20 γ ; (iii) Ω s = 40 γ . The insets show the corresponding intensity profile of standing-wave and a magnified part of the same part of real part of susceptibility χ.

Fig. 3
Fig. 3

Reflectivity of induced bandgap in a sample of length L = 2   mm . The inset shows the corresponding transmissivity. All other parameters are the same as in Fig. 2a.

Fig. 4
Fig. 4

Tuning of the bandgap reflectivity for different intensities of the coupling and standing fields. The sample has a length L = 2   mm . All other parameters are the same as in Fig. 2a.

Equations (15)

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

H I = Δ p ( | 2 2 | + | 3 3 | + | 4 4 | ) ( g | 1 3 | + Ω c | 2 3 | + Ω s | 2 4 | + H .c . ) ,
ρ ̇ 31 = ( i Δ p γ 31 ) ρ 31 + i Ω c ρ 21 + i g ,
ρ ̇ 21 = ( i Δ p Γ 21 ) ρ 21 + i Ω c ρ 31 + i Ω s ρ 41 ,
ρ ̇ 41 = ( i Δ p γ 41 ) ρ 41 + i Ω s ρ 21 ,
ρ 31 g = 1 Δ p + i γ + ( Δ p + i γ / 2 ) Ω c 2 / M ,
ϵ = 1 + χ = 1 + 3 π N γ ρ 31 g .
( E + ( z + a ) E ( z + a ) ) = M ( Δ p ) ( E + ( z ) E ( z ) ) = ( e i κ a E + ( z ) e i κ a E ( z ) ) ,
κ a = ± cos 1 [ Tr [ M ( Δ p ) ] 2 ] .
M ( N ) = sin ( N κ a ) sin ( κ a ) M sin [ ( N 1 ) κ a ] sin ( κ a ) I ,
R N = M N ( 12 ) M N ( 22 ) = M 12   sin ( N κ a ) M 22   sin ( N κ a ) sin [ ( N 1 ) κ a ] ,
T N = 1 M N ( 22 ) = sin ( κ a ) M 22   sin ( N κ a ) sin [ ( N 1 ) κ a ] .
H I = ( 0 Ω c Ω s Ω c 0 0 Ω s 0 0 ) .
| + = 2 2 ( | 2 + Ω c α | 3 + Ω s α | 4 ) ,
| = 2 2 ( | 2 + Ω c α + | 3 + Ω s α + | 4 ) ,
| 0 = Ω s α + | 3 + Ω c α + | 4 ,

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