Abstract

We have derived an explicit relation between the group delay velocity vgd(ω), determined by the slope of the dispersion curves, and the nonlocal energy transport velocity vE(ω) in a lossless, finite, one-dimensional photonic crystal. Using this relation, we find a simple link between vgd(ω) and the group velocity vg(ω)(ω) defined in terms of the electromagnetic dwell time. It is shown that vE(ω)vgd(ω) for any frequency and vE(ω)=vgd(ω)=vg(ω)(ω) at the resonance frequencies of the transmission coefficient. It was established that the band structure of a finite periodic crystal is of great importance to describe and understand the properties of these velocities. In particular, it was shown that the occurrence of superluminal group delay velocities in these structures is closely related to the existence of null gaps in their dispersion relation. Calculations performed for a half-wave-quarter-wave stack show that the energy velocity remains always subluminal.

© 2012 Optical Society of America

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References

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  1. K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).
  2. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
    [CrossRef]
  3. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).
  4. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
    [CrossRef]
  5. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).
  6. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 450 (1970).
    [CrossRef]
  7. P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
    [CrossRef]
  8. G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63, 036610 (2001).
    [CrossRef]
  9. N. Le Thomas and R. Houdré, “Group velocity and energy transport velocity near the band edge of a disordered coupled cavity waveguide: an analytical approach,” J. Opt. Soc. Am. B 27, 2095–2101 (2010).
    [CrossRef]
  10. G. Torrese, J. Taylor, T. J. Hall, and P. Mégret, “Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E 73, 066616 (2006).
    [CrossRef]
  11. G. Torrese, J. Taylor, H. P. Schriemer, and M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A 8, 973–980 (2006).
    [CrossRef]
  12. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  13. W. Frias, A. Smolyakov, and A. Hirose, “Non-local energy transport in tunneling and plasmonic structures,” Opt. Express 19, 15281–15296 (2011).
    [CrossRef]
  14. M. de Dios-Leyva and O. E. González-Vasquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 77, 125102 (2008).
    [CrossRef]
  15. M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
    [CrossRef]
  16. R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1992).
  17. H. G. Winful, “Group delay, stored energy, and the tunneling of evanescent electromagnetic waves,” Phys. Rev. E 68, 016615 (2003).
    [CrossRef]
  18. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).
  19. L. D. Landau and E. M. Lifschitz, Quantum Mechanics, Nonrelativistic Theory (Pergamon, 1981).
  20. T. Xu, S. Yang, S. V. Nair, and H. E. Ruda, “Confined modes in finite-size photonic crystals,” Phys. Rev. B 72, 045126 (2005).
    [CrossRef]
  21. M. de Dios-Leyva and J. C. Drake-Pérez, “Properties of the dispersion relation in finite one-dimensional photonic crystals,” J. Appl. Phys. 109, 103526 (2011).
    [CrossRef]
  22. H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. 436, 1–69 (2006).
    [CrossRef]

2011 (2)

W. Frias, A. Smolyakov, and A. Hirose, “Non-local energy transport in tunneling and plasmonic structures,” Opt. Express 19, 15281–15296 (2011).
[CrossRef]

M. de Dios-Leyva and J. C. Drake-Pérez, “Properties of the dispersion relation in finite one-dimensional photonic crystals,” J. Appl. Phys. 109, 103526 (2011).
[CrossRef]

2010 (2)

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[CrossRef]

N. Le Thomas and R. Houdré, “Group velocity and energy transport velocity near the band edge of a disordered coupled cavity waveguide: an analytical approach,” J. Opt. Soc. Am. B 27, 2095–2101 (2010).
[CrossRef]

2008 (1)

M. de Dios-Leyva and O. E. González-Vasquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 77, 125102 (2008).
[CrossRef]

2006 (3)

H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. 436, 1–69 (2006).
[CrossRef]

G. Torrese, J. Taylor, T. J. Hall, and P. Mégret, “Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E 73, 066616 (2006).
[CrossRef]

G. Torrese, J. Taylor, H. P. Schriemer, and M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A 8, 973–980 (2006).
[CrossRef]

2005 (1)

T. Xu, S. Yang, S. V. Nair, and H. E. Ruda, “Confined modes in finite-size photonic crystals,” Phys. Rev. B 72, 045126 (2005).
[CrossRef]

2003 (1)

H. G. Winful, “Group delay, stored energy, and the tunneling of evanescent electromagnetic waves,” Phys. Rev. E 68, 016615 (2003).
[CrossRef]

2001 (1)

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

1999 (1)

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

1997 (1)

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

1996 (1)

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

1979 (1)

1970 (1)

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 450 (1970).
[CrossRef]

Asatryan, A. A.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[CrossRef]

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Bertolotti, M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Bloemer, M. J.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Botten, L. C.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[CrossRef]

Bowden, C. M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

Cada, M.

G. Torrese, J. Taylor, H. P. Schriemer, and M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A 8, 973–980 (2006).
[CrossRef]

Centini, M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Chen, P. Y.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[CrossRef]

Collin, R. E.

R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1992).

D’Aguanno, G.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

de Dios-Leyva, M.

M. de Dios-Leyva and J. C. Drake-Pérez, “Properties of the dispersion relation in finite one-dimensional photonic crystals,” J. Appl. Phys. 109, 103526 (2011).
[CrossRef]

M. de Dios-Leyva and O. E. González-Vasquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 77, 125102 (2008).
[CrossRef]

de Sterke, C. M.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[CrossRef]

Dowling, J. P.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Drake-Pérez, J. C.

M. de Dios-Leyva and J. C. Drake-Pérez, “Properties of the dispersion relation in finite one-dimensional photonic crystals,” J. Appl. Phys. 109, 103526 (2011).
[CrossRef]

Fan, S.

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

Frias, W.

González-Vasquez, O. E.

M. de Dios-Leyva and O. E. González-Vasquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 77, 125102 (2008).
[CrossRef]

Hall, T. J.

G. Torrese, J. Taylor, T. J. Hall, and P. Mégret, “Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E 73, 066616 (2006).
[CrossRef]

Haus, J. M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

Hirose, A.

Houdré, R.

Joannopoulos, J. D.

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

Landau, L. D.

L. D. Landau and E. M. Lifschitz, Quantum Mechanics, Nonrelativistic Theory (Pergamon, 1981).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

Le Thomas, N.

Lifschitz, E. M.

L. D. Landau and E. M. Lifschitz, Quantum Mechanics, Nonrelativistic Theory (Pergamon, 1981).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

Loudon, R.

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 450 (1970).
[CrossRef]

McPhedran, R. C.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[CrossRef]

Mégret, P.

G. Torrese, J. Taylor, T. J. Hall, and P. Mégret, “Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E 73, 066616 (2006).
[CrossRef]

Nair, S. V.

T. Xu, S. Yang, S. V. Nair, and H. E. Ruda, “Confined modes in finite-size photonic crystals,” Phys. Rev. B 72, 045126 (2005).
[CrossRef]

Nefedov, I.

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Poulton, C. G.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[CrossRef]

Ruda, H. E.

T. Xu, S. Yang, S. V. Nair, and H. E. Ruda, “Confined modes in finite-size photonic crystals,” Phys. Rev. B 72, 045126 (2005).
[CrossRef]

Sabilia, C.

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Sakoda, K.

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

Scalora, M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Schriemer, H. P.

G. Torrese, J. Taylor, H. P. Schriemer, and M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A 8, 973–980 (2006).
[CrossRef]

Sibilia, C.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

Smolyakov, A.

Steel, M. J.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[CrossRef]

Taylor, J.

G. Torrese, J. Taylor, T. J. Hall, and P. Mégret, “Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E 73, 066616 (2006).
[CrossRef]

G. Torrese, J. Taylor, H. P. Schriemer, and M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A 8, 973–980 (2006).
[CrossRef]

Torrese, G.

G. Torrese, J. Taylor, H. P. Schriemer, and M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A 8, 973–980 (2006).
[CrossRef]

G. Torrese, J. Taylor, T. J. Hall, and P. Mégret, “Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E 73, 066616 (2006).
[CrossRef]

Villeneuve, P. R.

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

Winful, H. G.

H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. 436, 1–69 (2006).
[CrossRef]

H. G. Winful, “Group delay, stored energy, and the tunneling of evanescent electromagnetic waves,” Phys. Rev. E 68, 016615 (2003).
[CrossRef]

Xu, T.

T. Xu, S. Yang, S. V. Nair, and H. E. Ruda, “Confined modes in finite-size photonic crystals,” Phys. Rev. B 72, 045126 (2005).
[CrossRef]

Yang, S.

T. Xu, S. Yang, S. V. Nair, and H. E. Ruda, “Confined modes in finite-size photonic crystals,” Phys. Rev. B 72, 045126 (2005).
[CrossRef]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Yeh, P.

J. Appl. Phys. (1)

M. de Dios-Leyva and J. C. Drake-Pérez, “Properties of the dispersion relation in finite one-dimensional photonic crystals,” J. Appl. Phys. 109, 103526 (2011).
[CrossRef]

J. Opt. A (1)

G. Torrese, J. Taylor, H. P. Schriemer, and M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A 8, 973–980 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. A (1)

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 450 (1970).
[CrossRef]

Nature (1)

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

Opt. Express (1)

Phys. Rep. (1)

H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. 436, 1–69 (2006).
[CrossRef]

Phys. Rev. A (1)

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[CrossRef]

Phys. Rev. B (2)

M. de Dios-Leyva and O. E. González-Vasquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 77, 125102 (2008).
[CrossRef]

T. Xu, S. Yang, S. V. Nair, and H. E. Ruda, “Confined modes in finite-size photonic crystals,” Phys. Rev. B 72, 045126 (2005).
[CrossRef]

Phys. Rev. E (5)

H. G. Winful, “Group delay, stored energy, and the tunneling of evanescent electromagnetic waves,” Phys. Rev. E 68, 016615 (2003).
[CrossRef]

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

G. Torrese, J. Taylor, T. J. Hall, and P. Mégret, “Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E 73, 066616 (2006).
[CrossRef]

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Other (6)

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1992).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

L. D. Landau and E. M. Lifschitz, Quantum Mechanics, Nonrelativistic Theory (Pergamon, 1981).

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

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Figures (5)

Fig. 1.
Fig. 1.

Dispersion curves for a half-wave-quarter-wave photonic crystal with indices of refraction n1=1 and n2=1.42857, for different values of the number N of unit cells. The frequency ω, normalized to a reference frequency ω0, is presented as a function of the reduced wavevector Kdπ for the two lowest bands. Panel (a) corresponds to the infinite crystal, whereas panels (b) and (c) refer to the finite crystal for N=5 and 20, respectively.

Fig. 2.
Fig. 2.

Same as Fig. 1, but for N=5, 10, and 20, and in a frequency range containing the bandgap of zero width localized at Kdπ=1.

Fig. 3.
Fig. 3.

Normalized group delay velocity vgd (solid lines) associated with the dispersion curves of Figs. 1 and 2 as functions of the normalized frequency ω/ω0, for different values of N. The normalized group velocity v (dotted lines) of the infinite crystal is also depicted.

Fig. 4.
Fig. 4.

Group delay velocity vgd (solid lines) and energy transport velocity vE (dotted lines) normalized to c as functions of ω/ω0, for N=5 and 10.

Fig. 5.
Fig. 5.

Normalized energy transport velocity vE (solid lines) associated with the dispersion curves of Figs. 1 and 2 as functions of ω/ω0, for increasing values of N. The normalized group velocity v (dotted lines) of the infinite crystal is also depicted.

Equations (46)

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v⃗g=kω(k⃗),
v⃗E=S⃗U,
E⃗(r⃗)=y^E(z)=u⃗(r⃗)exp(iφ(x,z)),
H⃗(r⃗)=υ⃗(r⃗)exp(iφ(x,z)),
u⃗(r⃗)=y^|E(z)|,
υ⃗(r⃗)=icg(z,ω)[d|E(z)|dz+i|E(z)|dφ(z)dz]x^.
g(z,ω)=ωμ(z),
.[E⃗K×H⃗*+E⃗*×H⃗K]=ic[f(z,ω)KE⃗·E⃗*+g(z,ω)KH⃗·H⃗*],
·F⃗K+i32πck⃗K·S⃗=i32πcvgd(ω)U,
S⃗=c8πRe[E⃗×H⃗*],
U=116π[f(z,ω)ωE⃗·E⃗*+g(z,ω)ωH⃗·H⃗*]
k⃗K=φ(x,z)K=K(dφ(z)dz)z^,
F⃗K=(υ⃗υ⃗*)×u⃗K(υ⃗υ⃗*)K×u⃗=2icvgdG(z,ω)z^,
G(z,ω)=|E(z)|2ω[α(z)|E(z)|],
α(z)=1g(z,ω)|E(z)|z.
2cLvgdG(ω)+32πc1L[φ(L)φ(0)]Kz^·S⃗=32πcvgdU=32πcvgdz^·S⃗vE,
G(ω)=G(L,ω)G(0,ω).
E(z)=exp(izQL)+rexp(izQL),
E(z)=texp[iQL(zL)]=|t|expi[Φ(ω)+QL(zL)],
φ(L)φ(0)=Φ(ω)θ(ω)=LKθ(ω),
tanφ(0)=tanθ(ω)=r21+r1,
G(L,ω)=0,G(ω)=G(0,ω),
G(ω)=i|1+r|2ω[QLgLrr*|1+r|2]=2r2ω(QLgL)+2QLgL|1+r|2ω[r2|1+r|2].
z^·S⃗=c2QL8πωμLT(ω)
vE=T(ω)(1θ(ω)/τd)T(ω)+τ(ω)/2τdvgd=T(ω)T(ω)+T0(ω)vgd
T0(ω)=R(ω)τdΦR(ω)ω+r2τd1QL(QLωQLgLgLω),
vE=T(ω)vg(ω),
vg(ω)=vgdT(ω)+T0(ω)
(t0)=T̂(1r),
T̂=sinNβsinβS01WS0sin(N1)βsinβI.
S0=(11iQLμLiQLμL)
cosβ=12Tr(W).
t(ω)=1T22,r(ω)=T21T22=T21t(ω),
W=(W11W12W21W22)=MA(a)MB(b),
MA(a)=(cosaQ1μ1Q1sinaQ1Q1μ1sinaQ1cosaQ1),
MB(b)=(cosbQ2μ2Q2sinbQ2Q2μ2sinbQ2cosbQ2),
cosβ=cosaQ1cosbQ212(μ2Q1μ1Q2+μ1Q2μ2Q1)sinaQ1sinbQ2=f(ω),
T22=cosNβisinNβsinβg(ω)=1t=exp[iΦ(ω)]|t|,
T21=12sinNβsinβ(μ2Q1μ1Q2μ1Q2μ2Q1)sinbQ2(sinaQ1+icosaQ1),
g(ω)=sinaQ1cosbQ2+12(μ2Q1μ1Q2+μ1Q2μ2Q1)cosaQ1sinbQ2.
qK=Kdπ=1Nπarctan[g(ω)tanNβsinβ]±nN,
τd=Lvgd=[(1f2)(dg/dω)+fg(df/dω)]tanNβ1f2[1+g2tan2Nβf2]N(1+tan2Nβ)g(df/dω)[1+g2tan2Nβf2]=dΦ(ω)dω,
n=int[NRe(β)π+12],
v(ω)=Re(d1f2df(ω)dω),
T0(ω)=R(ω)τdω[Φ(ω)aQ1]=R(ω)τd(τdτa),
vg(ω)=vgd1R(ω)τa/τd,

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