Abstract

Using the transfer-matrix method, the effects of absorption and inhomogeneous broadening, in one-dimensional optical lattice constructed from inhomogeneously broadened spin transitions of nitrogen-vacancy color centers in single crystal diamond (NV diamond), on the reflection and absorption spectrum are presented. Further analysis show that, in realistic periodic stacks of the NV diamond, modulating the geometrical configuration of the external optical potential, the absorption lineshape scale, and the inhomogeneous broadening, one could easily access the diverse gap structures and a high band-gap reflectivity. These pretty useful calculations hold more potential for effective control of the light-matter interaction and realization in practice.

© 2006 Optical Society of America

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  1. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58,2059-2062 (1987).
    [CrossRef] [PubMed]
  2. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58,2486-2489 (1987).
    [CrossRef] [PubMed]
  3. S. Harris, "Electromagnetically induced transparency," Phys. Today 50,36-42 (1997).
    [CrossRef]
  4. A. Andre and M. D. Lukin, "Manipulating light pulses via dynamically controlled photonic band gap," Phys. Rev. Lett. 89,143602 (2002).
    [CrossRef] [PubMed]
  5. H. Kang, G. Hernandez, and Y. Zhu, "Slow-light six-wave mixing at low light intensities," Phys. Rev. Lett. 93,073601 (2004).
    [CrossRef] [PubMed]
  6. M. Artoni and G. La Rocca, "Optically tunable photonic stop bands in homogeneous absorbing media," Phys. Rev. Lett. 96,073905 (2006).
    [CrossRef] [PubMed]
  7. 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]
  8. 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]
  9. X. M. Su and B. S. Ham, "Dynamic control of the photonic band gap using quantum coherence," Phys. Rev. A 71,013821 (2005).
    [CrossRef]
  10. M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, "Photonic band gaps and defects in two dimensions: studies of the transmission coefficient," Phys. Rev. B 48,14121-14126 (1993).
    [CrossRef]
  11. M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, "Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials," Phys. Rev. B 49,11080-11087 (1994).
    [CrossRef]
  12. A. A. Krokhin and P. Halevi, "Influence of weak dissipation on the photonic band structure of periodic composites," Phys. Rev. B 53,1205-1214 (1996).
    [CrossRef]
  13. A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, "Optical properties of polaritonic crystal slab," Phys. Stat. Solidi A 190,413-419 (2002).
    [CrossRef]
  14. M. Hubner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Optical lattices achieved by excitons in periodic quantum well structures," Phys. Rev. Lett. 83,2841-2844 (1999).
    [CrossRef]
  15. J. P. Prineas, C. Ell, E. S. L EE, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Exciton-polariton eigenmodes in light-coupled in 0.04Ga0.96As/GaAs semiconductor multiple-quantum-well periodic structures," Phys. Rev. B 61,13863-13872 (2000).
    [CrossRef]
  16. L. I. Deych, M. V. Erementchouk, and A. A. Lisyansky, "Effects of inhomogeneous broadening on reflection spectra of Bragg multiple quantum well structures with a defect," Phys. Rev. B 69,075308 (2004).
    [CrossRef]
  17. E. L. Ivchenko, M. M. Voronov, M. V. Eremetchouk, L. I. Deych, and A. A. Lisyansky, "Multiple-quantum-wellbased photonic crystals with simple and compound elementary supercells," Phys. Rev. B 70,195106 (2004).
    [CrossRef]
  18. M. Artoni, G. La Rocca, and F. Bassani, "Resonantly absorbing one-dimensional photonic crystals," Phys. Rev. E 72,046604 (2005).
    [CrossRef]
  19. X. F. He, N. B. Manson, and P. T. H. Fisk, "Paramagnetic resonance of photoexcited N-V defects in diamond. I. Level anticrossing in the 3A ground state," Phys. Rev. B 47,8809 (1993).
    [CrossRef]
  20. P. R. Hemmer, A. V. Turukhin, M. S. Shahriar, and J. Musser, "Raman excited spin coherence in NV-Diamond," Opt. Lett. 26,361-363 (2001).
    [CrossRef]
  21. M. Born and E. Wolf, Principles of Optics, 6th Edition (Cambridge University Press, Cambridge, 1980).
  22. E. Kuznetsova, O. Kocharovskaya, P. Hemmer, and M. O. Scully, "Atomic interference phenomena in solids with a long-lived spin coherence," Phys. Rev. A 66,063802 (2002).
    [CrossRef]
  23. A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, "Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium," Phys. Rev. A 66,013805 (2002).
    [CrossRef]
  24. The use of Lorentzian line shapes allows us to obtain analytical results for χ as described in detail in Ref. [22].
  25. Here, χ (ω,ωab(cb)_ = Nμ2 abρab/(2¯hΩp) the density equations and all parameters are shown in detail in our earlier work [7, 8].
  26. F. Bassani and G. Pastori Parravicini, Electronic States and Optical Transitions in Solids (Pergamon Press, Oxford, 1975).
  27. In typical experimental configurations a is set by the periodicity of the standing wave and it is just half the wavelength of the two counter-propagating laser beams creating the optical potential as described in following Ref. [28, 29]. Each slab has a thickness d sufficiently smaller than the periodicity.
  28. I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, and W. D. Phillips, "Photonic band gaps in optical lattices," Phys. Rev. A 52,1394-1410 (1995).
    [CrossRef] [PubMed]
  29. M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Stationary pulses of light in an atomic medium," Nature 426,638-641 (2003).
    [CrossRef] [PubMed]
  30. The scaling affects both the resonant absorption (κ) and the refractive index (η) as shown in Fig. 2(c) for NV diamond. α → 1 corresponds to the actual linewidth profile, smaller α yield a linewidth narrowing with a concomitant peak absorption increase.
  31. It could be realized by using a rather large misalignment between the two beams. While in Fig. 2(b) the periodicity a ≃ 318.5 nm is the situation in which the two beams are exactly counter propagating, because it is just equal to the half-wavelength of the resonant transition from the excited state |ai to the ground-state spin sublevel |ci in NV diamond.
  32. B. S. Ham, P. R. Hemmer, and M. S. Shahriar, "Efficient electromagnetically induced transparency in a rare-earth doped crystal," Opt. Commun. 144,227-230 (1997).
    [CrossRef]
  33. According to the experiment done by Ham in Ref. [32], we know that inhomogeneous line broadening can be effectively reduced up to the magnitude of the laser beam jitter using an optical repump scheme, corresponding reduction in the effective atomic density. As for the use of a repumper in Ref. [20], the authors explicitly say that for NV diamond this procedure is not frequency selective, however in Ref. [22, 23] the discrepancy in the transparency value (bigger in experiment than in theory) is attributed to possible effects of the repumper. Maybe that the use of the repumper in NV diamond leads to a minor correction of the broadening compared to the case of Pr:YSO.
  34. M. Artoni, G. La Rocca, and F. Bassani have described the detail about labs and lext for atom stacks in Ref. [18].
  35. T. Shibata, "Micromachining of diamond thin film," New Diamond Front. Carbon Technol. 10,161-175 (2000).
  36. C. Tavares, F. Omnes, J Pernot, and E. Bustarret, "Electronic properties of boron-doped 111-oriented homoepitaxial diamond layers," Diamond Relat. Mater. 15,582-585 (2006).
    [CrossRef]
  37. S. Tomljenovic-Hanic, M. J. Steel, and C. Martijn de Sterke, "Diamond based photonic crystal microcavities," Opt. Express 14,3556 (2006).
    [CrossRef] [PubMed]
  38. M. Lukin, "Colloquium: trapping and manipulating photon states in atomic ensembles," Rev. Mod. Phys. 75,457 (2003).
    [CrossRef]
  39. A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Nonlinear optics with stationary pulses of light," Phys. Rev. Lett. 94,063902 (2005).
    [CrossRef] [PubMed]

2006 (5)

M. Artoni and G. 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]

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]

C. Tavares, F. Omnes, J Pernot, and E. Bustarret, "Electronic properties of boron-doped 111-oriented homoepitaxial diamond layers," Diamond Relat. Mater. 15,582-585 (2006).
[CrossRef]

S. Tomljenovic-Hanic, M. J. Steel, and C. Martijn de Sterke, "Diamond based photonic crystal microcavities," Opt. Express 14,3556 (2006).
[CrossRef] [PubMed]

2005 (3)

A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Nonlinear optics with stationary pulses of light," Phys. Rev. Lett. 94,063902 (2005).
[CrossRef] [PubMed]

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

X. M. Su and B. S. Ham, "Dynamic control of the photonic band gap using quantum coherence," Phys. Rev. A 71,013821 (2005).
[CrossRef]

2004 (3)

H. Kang, G. Hernandez, and Y. Zhu, "Slow-light six-wave mixing at low light intensities," Phys. Rev. Lett. 93,073601 (2004).
[CrossRef] [PubMed]

L. I. Deych, M. V. Erementchouk, and A. A. Lisyansky, "Effects of inhomogeneous broadening on reflection spectra of Bragg multiple quantum well structures with a defect," Phys. Rev. B 69,075308 (2004).
[CrossRef]

E. L. Ivchenko, M. M. Voronov, M. V. Eremetchouk, L. I. Deych, and A. A. Lisyansky, "Multiple-quantum-wellbased photonic crystals with simple and compound elementary supercells," Phys. Rev. B 70,195106 (2004).
[CrossRef]

2003 (2)

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Stationary pulses of light in an atomic medium," Nature 426,638-641 (2003).
[CrossRef] [PubMed]

M. Lukin, "Colloquium: trapping and manipulating photon states in atomic ensembles," Rev. Mod. Phys. 75,457 (2003).
[CrossRef]

2002 (4)

E. Kuznetsova, O. Kocharovskaya, P. Hemmer, and M. O. Scully, "Atomic interference phenomena in solids with a long-lived spin coherence," Phys. Rev. A 66,063802 (2002).
[CrossRef]

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, "Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium," Phys. Rev. A 66,013805 (2002).
[CrossRef]

A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, "Optical properties of polaritonic crystal slab," Phys. Stat. Solidi A 190,413-419 (2002).
[CrossRef]

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

2001 (1)

2000 (2)

T. Shibata, "Micromachining of diamond thin film," New Diamond Front. Carbon Technol. 10,161-175 (2000).

J. P. Prineas, C. Ell, E. S. L EE, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Exciton-polariton eigenmodes in light-coupled in 0.04Ga0.96As/GaAs semiconductor multiple-quantum-well periodic structures," Phys. Rev. B 61,13863-13872 (2000).
[CrossRef]

1999 (1)

M. Hubner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Optical lattices achieved by excitons in periodic quantum well structures," Phys. Rev. Lett. 83,2841-2844 (1999).
[CrossRef]

1997 (2)

S. Harris, "Electromagnetically induced transparency," Phys. Today 50,36-42 (1997).
[CrossRef]

B. S. Ham, P. R. Hemmer, and M. S. Shahriar, "Efficient electromagnetically induced transparency in a rare-earth doped crystal," Opt. Commun. 144,227-230 (1997).
[CrossRef]

1996 (1)

A. A. Krokhin and P. Halevi, "Influence of weak dissipation on the photonic band structure of periodic composites," Phys. Rev. B 53,1205-1214 (1996).
[CrossRef]

1995 (1)

I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, and W. D. Phillips, "Photonic band gaps in optical lattices," Phys. Rev. A 52,1394-1410 (1995).
[CrossRef] [PubMed]

1994 (1)

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, "Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials," Phys. Rev. B 49,11080-11087 (1994).
[CrossRef]

1993 (2)

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, "Photonic band gaps and defects in two dimensions: studies of the transmission coefficient," Phys. Rev. B 48,14121-14126 (1993).
[CrossRef]

X. F. He, N. B. Manson, and P. T. H. Fisk, "Paramagnetic resonance of photoexcited N-V defects in diamond. I. Level anticrossing in the 3A ground state," Phys. Rev. B 47,8809 (1993).
[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]

Andre, A.

A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Nonlinear optics with stationary pulses of light," Phys. Rev. Lett. 94,063902 (2005).
[CrossRef] [PubMed]

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

Artoni, M.

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

Bajcsy, M.

A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Nonlinear optics with stationary pulses of light," Phys. Rev. Lett. 94,063902 (2005).
[CrossRef] [PubMed]

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Stationary pulses of light in an atomic medium," Nature 426,638-641 (2003).
[CrossRef] [PubMed]

Bassani, F.

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

Brick, P.

M. Hubner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Optical lattices achieved by excitons in periodic quantum well structures," Phys. Rev. Lett. 83,2841-2844 (1999).
[CrossRef]

Bustarret, E.

C. Tavares, F. Omnes, J Pernot, and E. Bustarret, "Electronic properties of boron-doped 111-oriented homoepitaxial diamond layers," Diamond Relat. Mater. 15,582-585 (2006).
[CrossRef]

Chan, C. T.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, "Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials," Phys. Rev. B 49,11080-11087 (1994).
[CrossRef]

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, "Photonic band gaps and defects in two dimensions: studies of the transmission coefficient," Phys. Rev. B 48,14121-14126 (1993).
[CrossRef]

Deutsch, I. H.

I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, and W. D. Phillips, "Photonic band gaps in optical lattices," Phys. Rev. A 52,1394-1410 (1995).
[CrossRef] [PubMed]

Deych, L. I.

L. I. Deych, M. V. Erementchouk, and A. A. Lisyansky, "Effects of inhomogeneous broadening on reflection spectra of Bragg multiple quantum well structures with a defect," Phys. Rev. B 69,075308 (2004).
[CrossRef]

E. L. Ivchenko, M. M. Voronov, M. V. Eremetchouk, L. I. Deych, and A. A. Lisyansky, "Multiple-quantum-wellbased photonic crystals with simple and compound elementary supercells," Phys. Rev. B 70,195106 (2004).
[CrossRef]

Economou, E. N.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, "Photonic band gaps and defects in two dimensions: studies of the transmission coefficient," Phys. Rev. B 48,14121-14126 (1993).
[CrossRef]

Ell, C.

J. P. Prineas, C. Ell, E. S. L EE, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Exciton-polariton eigenmodes in light-coupled in 0.04Ga0.96As/GaAs semiconductor multiple-quantum-well periodic structures," Phys. Rev. B 61,13863-13872 (2000).
[CrossRef]

M. Hubner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Optical lattices achieved by excitons in periodic quantum well structures," Phys. Rev. Lett. 83,2841-2844 (1999).
[CrossRef]

Erementchouk, M. V.

L. I. Deych, M. V. Erementchouk, and A. A. Lisyansky, "Effects of inhomogeneous broadening on reflection spectra of Bragg multiple quantum well structures with a defect," Phys. Rev. B 69,075308 (2004).
[CrossRef]

Eremetchouk, M. V.

E. L. Ivchenko, M. M. Voronov, M. V. Eremetchouk, L. I. Deych, and A. A. Lisyansky, "Multiple-quantum-wellbased photonic crystals with simple and compound elementary supercells," Phys. Rev. B 70,195106 (2004).
[CrossRef]

Fisk, P. T. H.

X. F. He, N. B. Manson, and P. T. H. Fisk, "Paramagnetic resonance of photoexcited N-V defects in diamond. I. Level anticrossing in the 3A ground state," Phys. Rev. B 47,8809 (1993).
[CrossRef]

Gao, J. 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]

Gibbs, H. M.

M. Hubner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Optical lattices achieved by excitons in periodic quantum well structures," Phys. Rev. Lett. 83,2841-2844 (1999).
[CrossRef]

Gippius, N. A.

A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, "Optical properties of polaritonic crystal slab," Phys. Stat. Solidi A 190,413-419 (2002).
[CrossRef]

Halevi, P.

A. A. Krokhin and P. Halevi, "Influence of weak dissipation on the photonic band structure of periodic composites," Phys. Rev. B 53,1205-1214 (1996).
[CrossRef]

Ham, B. S.

X. M. Su and B. S. Ham, "Dynamic control of the photonic band gap using quantum coherence," Phys. Rev. A 71,013821 (2005).
[CrossRef]

B. S. Ham, P. R. Hemmer, and M. S. Shahriar, "Efficient electromagnetically induced transparency in a rare-earth doped crystal," Opt. Commun. 144,227-230 (1997).
[CrossRef]

Harris, S.

S. Harris, "Electromagnetically induced transparency," Phys. Today 50,36-42 (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]

He, X. F.

X. F. He, N. B. Manson, and P. T. H. Fisk, "Paramagnetic resonance of photoexcited N-V defects in diamond. I. Level anticrossing in the 3A ground state," Phys. Rev. B 47,8809 (1993).
[CrossRef]

Hemmer, P.

E. Kuznetsova, O. Kocharovskaya, P. Hemmer, and M. O. Scully, "Atomic interference phenomena in solids with a long-lived spin coherence," Phys. Rev. A 66,063802 (2002).
[CrossRef]

Hemmer, P. R.

P. R. Hemmer, A. V. Turukhin, M. S. Shahriar, and J. Musser, "Raman excited spin coherence in NV-Diamond," Opt. Lett. 26,361-363 (2001).
[CrossRef]

B. S. Ham, P. R. Hemmer, and M. S. Shahriar, "Efficient electromagnetically induced transparency in a rare-earth doped crystal," Opt. Commun. 144,227-230 (1997).
[CrossRef]

Hernandez, G.

H. Kang, G. Hernandez, and Y. Zhu, "Slow-light six-wave mixing at low light intensities," Phys. Rev. Lett. 93,073601 (2004).
[CrossRef] [PubMed]

Ho, K. M.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, "Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials," Phys. Rev. B 49,11080-11087 (1994).
[CrossRef]

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, "Photonic band gaps and defects in two dimensions: studies of the transmission coefficient," Phys. Rev. B 48,14121-14126 (1993).
[CrossRef]

Hubner, M.

M. Hubner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Optical lattices achieved by excitons in periodic quantum well structures," Phys. Rev. Lett. 83,2841-2844 (1999).
[CrossRef]

Ishihara, T.

A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, "Optical properties of polaritonic crystal slab," Phys. Stat. Solidi A 190,413-419 (2002).
[CrossRef]

Ivchenko, E. L.

E. L. Ivchenko, M. M. Voronov, M. V. Eremetchouk, L. I. Deych, and A. A. Lisyansky, "Multiple-quantum-wellbased photonic crystals with simple and compound elementary supercells," Phys. Rev. B 70,195106 (2004).
[CrossRef]

Javan, A.

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, "Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium," Phys. Rev. A 66,013805 (2002).
[CrossRef]

John, S.

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

Kang, H.

H. Kang, G. Hernandez, and Y. Zhu, "Slow-light six-wave mixing at low light intensities," Phys. Rev. Lett. 93,073601 (2004).
[CrossRef] [PubMed]

Khitrova, G.

M. Hubner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Optical lattices achieved by excitons in periodic quantum well structures," Phys. Rev. Lett. 83,2841-2844 (1999).
[CrossRef]

Koch, S. W.

M. Hubner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Optical lattices achieved by excitons in periodic quantum well structures," Phys. Rev. Lett. 83,2841-2844 (1999).
[CrossRef]

Kocharovskaya, O.

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, "Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium," Phys. Rev. A 66,013805 (2002).
[CrossRef]

E. Kuznetsova, O. Kocharovskaya, P. Hemmer, and M. O. Scully, "Atomic interference phenomena in solids with a long-lived spin coherence," Phys. Rev. A 66,063802 (2002).
[CrossRef]

Krokhin, A. A.

A. A. Krokhin and P. Halevi, "Influence of weak dissipation on the photonic band structure of periodic composites," Phys. Rev. B 53,1205-1214 (1996).
[CrossRef]

Kuznetsova, E.

E. Kuznetsova, O. Kocharovskaya, P. Hemmer, and M. O. Scully, "Atomic interference phenomena in solids with a long-lived spin coherence," Phys. Rev. A 66,063802 (2002).
[CrossRef]

La Rocca, G.

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

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

La Rocca, G. C.

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]

Lee, E. S.

M. Hubner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Optical lattices achieved by excitons in periodic quantum well structures," Phys. Rev. Lett. 83,2841-2844 (1999).
[CrossRef]

Lee, H.

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, "Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium," Phys. Rev. A 66,013805 (2002).
[CrossRef]

Lisyansky, A. A.

E. L. Ivchenko, M. M. Voronov, M. V. Eremetchouk, L. I. Deych, and A. A. Lisyansky, "Multiple-quantum-wellbased photonic crystals with simple and compound elementary supercells," Phys. Rev. B 70,195106 (2004).
[CrossRef]

L. I. Deych, M. V. Erementchouk, and A. A. Lisyansky, "Effects of inhomogeneous broadening on reflection spectra of Bragg multiple quantum well structures with a defect," Phys. Rev. B 69,075308 (2004).
[CrossRef]

Lukin, M.

M. Lukin, "Colloquium: trapping and manipulating photon states in atomic ensembles," Rev. Mod. Phys. 75,457 (2003).
[CrossRef]

Lukin, M. D.

A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Nonlinear optics with stationary pulses of light," Phys. Rev. Lett. 94,063902 (2005).
[CrossRef] [PubMed]

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Stationary pulses of light in an atomic medium," Nature 426,638-641 (2003).
[CrossRef] [PubMed]

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

Manson, N. B.

X. F. He, N. B. Manson, and P. T. H. Fisk, "Paramagnetic resonance of photoexcited N-V defects in diamond. I. Level anticrossing in the 3A ground state," Phys. Rev. B 47,8809 (1993).
[CrossRef]

Martijn de Sterke, C.

Muljarov, E. A.

A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, "Optical properties of polaritonic crystal slab," Phys. Stat. Solidi A 190,413-419 (2002).
[CrossRef]

Musser, J.

Omnes, F.

C. Tavares, F. Omnes, J Pernot, and E. Bustarret, "Electronic properties of boron-doped 111-oriented homoepitaxial diamond layers," Diamond Relat. Mater. 15,582-585 (2006).
[CrossRef]

Pernot, J

C. Tavares, F. Omnes, J Pernot, and E. Bustarret, "Electronic properties of boron-doped 111-oriented homoepitaxial diamond layers," Diamond Relat. Mater. 15,582-585 (2006).
[CrossRef]

Phillips, W. D.

I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, and W. D. Phillips, "Photonic band gaps in optical lattices," Phys. Rev. A 52,1394-1410 (1995).
[CrossRef] [PubMed]

Prineas, J. P.

J. P. Prineas, C. Ell, E. S. L EE, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Exciton-polariton eigenmodes in light-coupled in 0.04Ga0.96As/GaAs semiconductor multiple-quantum-well periodic structures," Phys. Rev. B 61,13863-13872 (2000).
[CrossRef]

M. Hubner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Optical lattices achieved by excitons in periodic quantum well structures," Phys. Rev. Lett. 83,2841-2844 (1999).
[CrossRef]

Rolston, S. L.

I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, and W. D. Phillips, "Photonic band gaps in optical lattices," Phys. Rev. A 52,1394-1410 (1995).
[CrossRef] [PubMed]

Scully, M. O.

E. Kuznetsova, O. Kocharovskaya, P. Hemmer, and M. O. Scully, "Atomic interference phenomena in solids with a long-lived spin coherence," Phys. Rev. A 66,063802 (2002).
[CrossRef]

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, "Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium," Phys. Rev. A 66,013805 (2002).
[CrossRef]

Shahriar, M. S.

P. R. Hemmer, A. V. Turukhin, M. S. Shahriar, and J. Musser, "Raman excited spin coherence in NV-Diamond," Opt. Lett. 26,361-363 (2001).
[CrossRef]

B. S. Ham, P. R. Hemmer, and M. S. Shahriar, "Efficient electromagnetically induced transparency in a rare-earth doped crystal," Opt. Commun. 144,227-230 (1997).
[CrossRef]

Shibata, T.

T. Shibata, "Micromachining of diamond thin film," New Diamond Front. Carbon Technol. 10,161-175 (2000).

Sigalas, M.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, "Photonic band gaps and defects in two dimensions: studies of the transmission coefficient," Phys. Rev. B 48,14121-14126 (1993).
[CrossRef]

Sigalas, M. M.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, "Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials," Phys. Rev. B 49,11080-11087 (1994).
[CrossRef]

Soukoulis, C. M.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, "Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials," Phys. Rev. B 49,11080-11087 (1994).
[CrossRef]

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, "Photonic band gaps and defects in two dimensions: studies of the transmission coefficient," Phys. Rev. B 48,14121-14126 (1993).
[CrossRef]

Spreeuw, R. J. C.

I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, and W. D. Phillips, "Photonic band gaps in optical lattices," Phys. Rev. A 52,1394-1410 (1995).
[CrossRef] [PubMed]

Steel, M. J.

Su, X. M.

X. M. Su and B. S. Ham, "Dynamic control of the photonic band gap using quantum coherence," Phys. Rev. A 71,013821 (2005).
[CrossRef]

Tavares, C.

C. Tavares, F. Omnes, J Pernot, and E. Bustarret, "Electronic properties of boron-doped 111-oriented homoepitaxial diamond layers," Diamond Relat. Mater. 15,582-585 (2006).
[CrossRef]

Tikhodeev, S. G.

A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, "Optical properties of polaritonic crystal slab," Phys. Stat. Solidi A 190,413-419 (2002).
[CrossRef]

Tomljenovic-Hanic, S.

Turukhin, A. V.

Voronov, M. M.

E. L. Ivchenko, M. M. Voronov, M. V. Eremetchouk, L. I. Deych, and A. A. Lisyansky, "Multiple-quantum-wellbased photonic crystals with simple and compound elementary supercells," Phys. Rev. B 70,195106 (2004).
[CrossRef]

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]

Wu, J. H.

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]

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.

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]

Yablonskii, A. L.

A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, "Optical properties of polaritonic crystal slab," Phys. Stat. Solidi A 190,413-419 (2002).
[CrossRef]

Zhu, Y.

H. Kang, G. Hernandez, and Y. Zhu, "Slow-light six-wave mixing at low light intensities," Phys. Rev. Lett. 93,073601 (2004).
[CrossRef] [PubMed]

Zibrov, A. S.

A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Nonlinear optics with stationary pulses of light," Phys. Rev. Lett. 94,063902 (2005).
[CrossRef] [PubMed]

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Stationary pulses of light in an atomic medium," Nature 426,638-641 (2003).
[CrossRef] [PubMed]

Diamond Relat. Mater. (1)

C. Tavares, F. Omnes, J Pernot, and E. Bustarret, "Electronic properties of boron-doped 111-oriented homoepitaxial diamond layers," Diamond Relat. Mater. 15,582-585 (2006).
[CrossRef]

Nature (1)

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Stationary pulses of light in an atomic medium," Nature 426,638-641 (2003).
[CrossRef] [PubMed]

New Diamond Front. Carbon Technol. (1)

T. Shibata, "Micromachining of diamond thin film," New Diamond Front. Carbon Technol. 10,161-175 (2000).

Opt. Commun. (1)

B. S. Ham, P. R. Hemmer, and M. S. Shahriar, "Efficient electromagnetically induced transparency in a rare-earth doped crystal," Opt. Commun. 144,227-230 (1997).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (5)

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]

X. M. Su and B. S. Ham, "Dynamic control of the photonic band gap using quantum coherence," Phys. Rev. A 71,013821 (2005).
[CrossRef]

I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, and W. D. Phillips, "Photonic band gaps in optical lattices," Phys. Rev. A 52,1394-1410 (1995).
[CrossRef] [PubMed]

E. Kuznetsova, O. Kocharovskaya, P. Hemmer, and M. O. Scully, "Atomic interference phenomena in solids with a long-lived spin coherence," Phys. Rev. A 66,063802 (2002).
[CrossRef]

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, "Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium," Phys. Rev. A 66,013805 (2002).
[CrossRef]

Phys. Rev. B (8)

X. F. He, N. B. Manson, and P. T. H. Fisk, "Paramagnetic resonance of photoexcited N-V defects in diamond. I. Level anticrossing in the 3A ground state," Phys. Rev. B 47,8809 (1993).
[CrossRef]

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, "Photonic band gaps and defects in two dimensions: studies of the transmission coefficient," Phys. Rev. B 48,14121-14126 (1993).
[CrossRef]

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, "Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials," Phys. Rev. B 49,11080-11087 (1994).
[CrossRef]

A. A. Krokhin and P. Halevi, "Influence of weak dissipation on the photonic band structure of periodic composites," Phys. Rev. B 53,1205-1214 (1996).
[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]

J. P. Prineas, C. Ell, E. S. L EE, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Exciton-polariton eigenmodes in light-coupled in 0.04Ga0.96As/GaAs semiconductor multiple-quantum-well periodic structures," Phys. Rev. B 61,13863-13872 (2000).
[CrossRef]

L. I. Deych, M. V. Erementchouk, and A. A. Lisyansky, "Effects of inhomogeneous broadening on reflection spectra of Bragg multiple quantum well structures with a defect," Phys. Rev. B 69,075308 (2004).
[CrossRef]

E. L. Ivchenko, M. M. Voronov, M. V. Eremetchouk, L. I. Deych, and A. A. Lisyansky, "Multiple-quantum-wellbased photonic crystals with simple and compound elementary supercells," Phys. Rev. B 70,195106 (2004).
[CrossRef]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (7)

M. Hubner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Optical lattices achieved by excitons in periodic quantum well structures," Phys. Rev. Lett. 83,2841-2844 (1999).
[CrossRef]

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]

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

H. Kang, G. Hernandez, and Y. Zhu, "Slow-light six-wave mixing at low light intensities," Phys. Rev. Lett. 93,073601 (2004).
[CrossRef] [PubMed]

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

A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Nonlinear optics with stationary pulses of light," Phys. Rev. Lett. 94,063902 (2005).
[CrossRef] [PubMed]

Phys. Stat. Solidi A (1)

A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, "Optical properties of polaritonic crystal slab," Phys. Stat. Solidi A 190,413-419 (2002).
[CrossRef]

Phys. Today (1)

S. Harris, "Electromagnetically induced transparency," Phys. Today 50,36-42 (1997).
[CrossRef]

Rev. Mod. Phys. (1)

M. Lukin, "Colloquium: trapping and manipulating photon states in atomic ensembles," Rev. Mod. Phys. 75,457 (2003).
[CrossRef]

Other (9)

According to the experiment done by Ham in Ref. [32], we know that inhomogeneous line broadening can be effectively reduced up to the magnitude of the laser beam jitter using an optical repump scheme, corresponding reduction in the effective atomic density. As for the use of a repumper in Ref. [20], the authors explicitly say that for NV diamond this procedure is not frequency selective, however in Ref. [22, 23] the discrepancy in the transparency value (bigger in experiment than in theory) is attributed to possible effects of the repumper. Maybe that the use of the repumper in NV diamond leads to a minor correction of the broadening compared to the case of Pr:YSO.

M. Artoni, G. La Rocca, and F. Bassani have described the detail about labs and lext for atom stacks in Ref. [18].

The scaling affects both the resonant absorption (κ) and the refractive index (η) as shown in Fig. 2(c) for NV diamond. α → 1 corresponds to the actual linewidth profile, smaller α yield a linewidth narrowing with a concomitant peak absorption increase.

It could be realized by using a rather large misalignment between the two beams. While in Fig. 2(b) the periodicity a ≃ 318.5 nm is the situation in which the two beams are exactly counter propagating, because it is just equal to the half-wavelength of the resonant transition from the excited state |ai to the ground-state spin sublevel |ci in NV diamond.

The use of Lorentzian line shapes allows us to obtain analytical results for χ as described in detail in Ref. [22].

Here, χ (ω,ωab(cb)_ = Nμ2 abρab/(2¯hΩp) the density equations and all parameters are shown in detail in our earlier work [7, 8].

F. Bassani and G. Pastori Parravicini, Electronic States and Optical Transitions in Solids (Pergamon Press, Oxford, 1975).

In typical experimental configurations a is set by the periodicity of the standing wave and it is just half the wavelength of the two counter-propagating laser beams creating the optical potential as described in following Ref. [28, 29]. Each slab has a thickness d sufficiently smaller than the periodicity.

M. Born and E. Wolf, Principles of Optics, 6th Edition (Cambridge University Press, Cambridge, 1980).

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Figures (6)

Fig. 1.
Fig. 1.

(Color online) (a) Band-gap reflectivity spectrum for NV diamond stack when the dielectric constant is assumed different value: a) 0.995 (blue), b) 0.998 (black), c) 1.002 (magenta), and d) 1.005 (red). (b) Band-gap reflectivity (solid) and absorption (dashed) spectrum when the dielectric constant of NV is given by a) Im(ε)=0, i.e., nonabsorbent case (blue), or b) Eq. (3) (red). All other parameters are: the periodicity a ≃ 318.5 nm, the thickness of single slab d=a/10, and the array of length L ≃ 6.55×103 a.

Fig. 2.
Fig. 2.

(Color online) (a)Band-gap reflectivity (solid) and absorption (dashed) spectrum for NV diamond stack corresponding to different absorption profile shown down where α=0.05(blue), 0.25(green), 0.5(red), and 1(black). (b)Resonance region blow-up, and (c) is the same as (b) but with the periodicity a ≃ 321.0 nm; The bottom curves under (b, d) represent the corresponding photonic band profiles in the gap region. All other parameters are as in Fig. 1.

Fig. 3.
Fig. 3.

(Color online) (a) Photonic bands (imaginary Bloch wavevector) profiles corresponding to different periodicity a ≃ 318.5 nm (black), 318.9 nm (red), and 318.1 nm (blue). (b) Band-gap reflectivity (solid) and absorption (dashed) spectrum when an array of length L ≃ 6.55×103 a is used (d/a=0.1 in all cases).

Fig. 4.
Fig. 4.

Reflectivity (solid) and absorption (dashed) profiles for NV diamond stack corresponding to different structure of single layer: d/a=0.05, 0.2, 0.5, and 1. Here, a ≃ 318.5 nm and L ≃ 6.55×103 a are used for all the cases.

Fig. 5.
Fig. 5.

(Color online) Band-gap reflectivity (solid) and absorption (dashed) spectrum corresponding to different broadening width: Wab ≃ 100 MHz (green and black), and 375 GHz (blue and red) when the array of length L ≃ 6.55×103 a (a,b) or L ≃ 6.55×104 a (c, d) is used. Figures (b) and (d) are corresponding resonance region blow-up.

Fig. 6.
Fig. 6.

(Color online) Photonic bands profiles (a, c) and resonance region blow-up (b, d) for the region shown in Fig. 5. (a ≃ 318.5 nm, d/a=0.1)

Equations (7)

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χ ( ω ) = d ( ω a b ) f ( ω a b ) d ( ω c b ) f ( ω c b ) χ ¯ ( ω , ω a b ( c b ) ) ,
f ( ω a b ( c b ) ) = W a b ( c b ) π ( Δ ω a b ( c b ) ) 2 + ( W a b ( c b ) ) 2 ,
ε ( ω ) = n 2 ( ω ) = 1 + 4 π χ ( ω ) = ε ( ω ) + i ε ( ω ) .
( E + ( x + a ) E ( x + a ) ) = M ( ω ) · ( E + ( x ) E ( x ) ) = ( e i κ a E + ( x ) e i κ a E ( x ) ) ,
κ a = ± cos 1 [ T r ( M ( ω ) ) 2 ] .
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 ) ,

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