Abstract

We have derived, for oblique propagation, an equation relating the averaged energy flux density to energy fluxes arising in the process of scattering by a lossless finite photonic structure. The latter fluxes include those associated with the dispersion relation of the structure, reflection, and interference between the incident and reflected waves. We have also derived an explicit relation between the energy flux density and the group velocity, which provides a simple and systematical procedure for studying theoretically and experimentally the properties of the energy transport through a wide variety of finite photonic structures. Such a relation may be regarded as a generalization of the corresponding one for infinite periodic systems to finite photonic structures. A finite, N-period, photonic crystal was used to illustrate the usefulness of our results.

© 2014 Optical Society of America

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    [CrossRef] [PubMed]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [CrossRef] [PubMed]
  3. K. Sakoda, Optical Properties of Photonic Crystals (Springer, Berlin, 2001).
    [CrossRef]
  4. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 2008).
  5. L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).
  6. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, 1984).
  7. S. Foteinopoulou, C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72,165112 (2005).
    [CrossRef]
  8. G. Torrese, J. Taylor, H. P. Schriemer, M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A: Pure Appl. Opt. 8, 973–980 (2006).
    [CrossRef]
  9. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
    [CrossRef]
  10. P. Y. Chen, R. C. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82,053825 (2010).
    [CrossRef]
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    [CrossRef]
  13. W. Frias, A. Smolyakov, A. Hirose, “Non-local energy transport in tunneling and plasmonic structures,” Opt. Express 19, 15281–15296 (2011).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  16. M. de Dios-Leyva, J. C. Drake-Pérez, “Group velocity and nonlocal energy transport velocity in finite photonic structures,” J. Opt. Soc. Am. B 29, 2275–2281 (2012).
    [CrossRef]
  17. H. G. Winful, “Group delay, stored energy, and the tunneling of evanescent electromagnetic waves,” Phys. Rev. E 68,016615 (2003).
    [CrossRef]
  18. R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1992).
  19. M. de Dios-Leyva, 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]
  20. J. M. Bendickson, J. P. Dowling, 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]
  21. M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, 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]
  22. E. P. Wigner, “Lower Limit for the Energy Derivative of the Scattering Phase Shift,” Phys. Rev. 98, 145–147 (1955).
    [CrossRef]
  23. M. de Dios-Leyva, J. C. Drake-Pérez, “Properties of the dispersion relation in finite one-dimensional photonic crystals,” J. Appl. Phys. 109,103526 (2011).
    [CrossRef]
  24. H. Daninthe, S. Foteinopoulou, C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photonics and Nanostructures-Fundamentals and Applications 4, 123–131 (2006).
    [CrossRef]
  25. A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, Y. S. Kivshar, Beam oscillations and curling in chirped periodic structures with metamaterials,” Phys. Rev. A 79,013820 (2009).
    [CrossRef]
  26. G Torrese, J. Taylor, T. J. Hall, P. Mégret, ”Effective-medium theory for energy velocity in one-dimensional finite lossless photonic crystals,” Phys. Rev. E 73,066616 (2006).
    [CrossRef]

2012

2011

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

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

2010

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

N. Le Thomas, 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]

2009

A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, Y. S. Kivshar, Beam oscillations and curling in chirped periodic structures with metamaterials,” Phys. Rev. A 79,013820 (2009).
[CrossRef]

2008

M. de Dios-Leyva, 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

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

H. Daninthe, S. Foteinopoulou, C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photonics and Nanostructures-Fundamentals and Applications 4, 123–131 (2006).
[CrossRef]

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

2005

S. Foteinopoulou, C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72,165112 (2005).
[CrossRef]

2003

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

2002

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
[CrossRef]

2001

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

1999

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, 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]

1996

J. M. Bendickson, J. P. Dowling, 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]

1987

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]

1979

1970

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
[CrossRef]

1955

E. P. Wigner, “Lower Limit for the Energy Derivative of the Scattering Phase Shift,” Phys. Rev. 98, 145–147 (1955).
[CrossRef]

Asatryan, A. A.

P. Y. Chen, R. C. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, 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, 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, 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, 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, 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, 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. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, 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, 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, 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 Press, 1960).

Cada, M.

G. Torrese, J. Taylor, H. P. Schriemer, M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A: Pure Appl. Opt. 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, 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, 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. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, 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, 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, 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]

Daninthe, H.

H. Daninthe, S. Foteinopoulou, C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photonics and Nanostructures-Fundamentals and Applications 4, 123–131 (2006).
[CrossRef]

Davoyan, A. R.

A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, Y. S. Kivshar, Beam oscillations and curling in chirped periodic structures with metamaterials,” Phys. Rev. A 79,013820 (2009).
[CrossRef]

de Dios-Leyva, M.

M. de Dios-Leyva, J. C. Drake-Pérez, “Group velocity and nonlocal energy transport velocity in finite photonic structures,” J. Opt. Soc. Am. B 29, 2275–2281 (2012).
[CrossRef]

M. de Dios-Leyva, 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, 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. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, 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, 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, J. C. Drake-Pérez, “Group velocity and nonlocal energy transport velocity in finite photonic structures,” J. Opt. Soc. Am. B 29, 2275–2281 (2012).
[CrossRef]

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

Foteinopoulou, S.

H. Daninthe, S. Foteinopoulou, C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photonics and Nanostructures-Fundamentals and Applications 4, 123–131 (2006).
[CrossRef]

S. Foteinopoulou, C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72,165112 (2005).
[CrossRef]

Frias, W.

González-Vasquez, O. E.

M. de Dios-Leyva, 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, 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, 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, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 2008).

John, S.

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

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 2008).

Kivshar, Y. S.

A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, Y. S. Kivshar, Beam oscillations and curling in chirped periodic structures with metamaterials,” Phys. Rev. A 79,013820 (2009).
[CrossRef]

Le Thomas, N.

Loudon, R.

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
[CrossRef]

Mc Phedran, R. C.

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

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 2008).

Mégret, P.

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

Nefedov, I.

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, 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. Mc Phedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82,053825 (2010).
[CrossRef]

Ruppin, R.

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
[CrossRef]

Sabilia, C.

M. Centini, C. Sabilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, 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, Berlin, 2001).
[CrossRef]

Scalora, M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. M. Haus, 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, 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, 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, M. Cada, “Energy transport through structures with finite electromagnetic stop gaps,” J. Opt. A: Pure Appl. Opt. 8, 973–980 (2006).
[CrossRef]

Shadrivov, I. V.

A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, Y. S. Kivshar, Beam oscillations and curling in chirped periodic structures with metamaterials,” Phys. Rev. A 79,013820 (2009).
[CrossRef]

Sibilia, C.

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

Smolyakov, A.

Soukoulis, C. M.

H. Daninthe, S. Foteinopoulou, C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photonics and Nanostructures-Fundamentals and Applications 4, 123–131 (2006).
[CrossRef]

S. Foteinopoulou, C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72,165112 (2005).
[CrossRef]

Steel, M. J.

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

Sukhorukov, A. A.

A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, Y. S. Kivshar, Beam oscillations and curling in chirped periodic structures with metamaterials,” Phys. Rev. A 79,013820 (2009).
[CrossRef]

Taylor, J.

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

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

Torrese, G

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

Torrese, G.

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

Wigner, E. P.

E. P. Wigner, “Lower Limit for the Energy Derivative of the Scattering Phase Shift,” Phys. Rev. 98, 145–147 (1955).
[CrossRef]

Winful, H. G.

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

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 2008).

Yablonovitch, E.

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

Yariv, A.

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

Yeh, P.

J. Appl. Phys.

M. de Dios-Leyva, 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: Pure Appl. Opt.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

J. Phys. A

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
[CrossRef]

Opt. Express

Photonics and Nanostructures-Fundamentals and Applications

H. Daninthe, S. Foteinopoulou, C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photonics and Nanostructures-Fundamentals and Applications 4, 123–131 (2006).
[CrossRef]

Phys. Lett. A

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299, 309–312 (2002).
[CrossRef]

Phys. Rev.

E. P. Wigner, “Lower Limit for the Energy Derivative of the Scattering Phase Shift,” Phys. Rev. 98, 145–147 (1955).
[CrossRef]

Phys. Rev. A

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Phys. Rev. B

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Figures (4)

Fig. 1
Fig. 1

Schematic of the process of scattering by a finite photonic structure, localized between the z = 0 and z = L planes and sandwiched between two semi-infinite layers made of the same optical materials. Arrows indicate the incident, reflected and transmitted waves.

Fig. 2
Fig. 2

Normal energy flux density normalized to S 0 = ( c / 8 π ) ε 1 as a function of the frequency ω in units of ω0, for a finite, N-period, quarter-wave-stack, with n 1 = ε 1 = 1, n 2 = ε 2 = 1.41, N = 5 (left-hand panel), 10 (right-hand panel) and various values of the angle of incidence θi.

Fig. 3
Fig. 3

Lateral energy flux density normalized to S 0 = ( c / 8 π ) ε 1 (black lines) and the ratio Vfin = vgx/vgz (red lines) of the finite, N-period, quarter-wave-stack as functions of ω/ω0, for the same parameters as in Fig. 2, except for θi = 0.

Fig. 4
Fig. 4

Ratio of velocities vgx/vgz for the finite (red lines) and infinite (black lines) λ0/4 photonic structure as a function of ω/ω0, for the same parameters as in Fig. 3.

Equations (62)

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v E = T T + T 0 v g = T v g ( ω )
v g ( ω ) = v g T + T 0
E ( r ) = y E ( z ) exp ( i x K x ) = u ( r ) exp [ i ϕ ( x , z ) ] ,
H ( r ) = v ( r ) exp [ i ϕ ( x , z ) ] ,
u ( r ) = y | E ( z ) | ,
v ( r ) = ic g ( z , ω ) [ | E ( z ) | z + i | E ( z ) | φ ( z ) z ] x + c K x g ( z , ω ) | E ( z ) | z .
ϕ ( x , z ) = φ ( z ) + x K x
× u ( r ) + i k × u ( r ) = i c g ( z , ω ) v ( r ) ,
× v ( r ) + i k × v ( r ) = i c f ( z , ω ) u ( r ) ,
k = ϕ ( x , z ) .
F K α + i 32 π c k K α S = i 32 π c v g α U ,
F K α = 2 i c | E ( z ) | 2 K α [ β ( z ) | E ( z ) | ] z ,
β ( z ) = 1 g ( z , ω ) | E ( z ) | z ,
S = c 8 π Re [ E × H * ]
U = 1 16 π ( f ( z , ω ) ω E E * + g ( z , ω ) ω H H * )
S = c 2 8 π g ( z , ω ) { | E ( z ) | 2 K x x + | E ( z ) | 2 φ ( z ) z z }
k K x = x + 2 φ ( z ) K x z z ,
k K z = 2 φ ( z ) K z z z .
c 2 16 π L G z ( K ) + 1 L [ φ ( L ) φ ( 0 ) ] K z z S L = v g z U L ,
c 2 16 π L G x ( K ) + S x L + 1 L [ φ ( L ) φ ( 0 ) ] K x z S L = v g x U L ,
G α ( K ) = G α ( L , K ) G α ( 0 , K ) ,
G α ( z , K ) = | E ( z ) | 2 K α [ β ( z ) | E ( z ) | ] .
E ( z ) = exp ( i z Q L ) + r exp ( i z Q L ) ,
E ( z ) = t exp [ i Q L ( z L ) ] = | t | exp i [ Φ + Q L ( z L ) ] ,
φ ( L ) φ ( 0 ) = Φ θ LK z θ ,
tan θ = r 2 1 + r 1 .
S z L = z S L = c 2 8 π Q L g L T ,
c 2 Q L 8 π g L 1 L { 1 2 g L Q L G α ( K ) θ K α T } + S α L = v g α U L ,
G α ( K ) = G α ( 0 , K ) = i | 1 + r | 2 K α [ Q L g L ( r r * ) | 1 + r | 2 ]
S α L = v g α U L c 2 8 π r 2 L K α ( Q L g L ) c 2 8 π Q L g L R L Φ R K α ,
K z = v g z ω ,
K x = v g x ω + ( K x ) ω ,
v E z = T T + T 0 v g z = T v g ( ω ) ,
v E x = [ T + ( R / τ d ) Φ R / ω ] T + T 0 v g x T x T + T 0 ,
T 0 = 1 τ d { r 2 g L Q L ω ( Q L g L ) + R Φ R ω } ,
T x = 1 τ d { r 2 g L Q L [ K x ( Q L g L ) ] ω + R Φ R K x } .
T 0 = R τ i τ d
T x = τ i l τ d ,
τ i = r 2 Q L { Q L g L g L ω Q L ω } ,
τ i l = K x r 2 Q L 2
v E z = T 1 τ i / τ d v g z = T v g ( ω ) ,
v E x = 1 1 τ i / τ d v g x + τ i l / τ d 1 τ i / τ d .
τ d = τ D + τ i
S x L S z L = v E x v E z = [ T + ( R / τ d ) Φ R / ω ] T v g x v g z τ d T x L T ,
( t 0 ) = T ^ ( 1 r ) .
1 ) t = 1 T 22 , 2 ) r = T 21 T 22 = T 21 t ,
tan Φ = tan LK z = Y X = F ( ω , K x ) ,
τ d = Φ ω = L v g z = X 2 T ( F ω ) K x ,
v g x v g z = X 2 L T ( F K x ) ω ,
T 22 = cos N β i sin N β sin β g ,
T 21 = i 2 sin N β sin β ( η 1 η ) sin b Q 2 exp ( i a Q 1 ) ,
g = sin a Q 1 cos b Q 2 + 1 2 ( η + 1 η ) cos a Q 1 sin b Q 2 ,
cos β = cos a Q 1 cos b Q 2 1 2 ( η + 1 η ) sin a Q 1 sin b Q 2 = f ( ω , K x )
t = 1 T 22 = T { cos N β + i sin N β sin β g } ,
tan Φ = tan LK z = g tan N β sin β = F ( ω , K x ) ,
T = 1 | T 22 | 2 = 1 cos 2 N β + ( sin 2 N β / sin 2 β ) g 2
X = T cos N β
Φ R = ± π / 2 + Φ a Q 1 = ± π / 2 + K z L a Q 1
S x L S z L = 1 T { ( 1 R τ a τ d ) v g x v g z + K x Q 1 1 L Q 1 ( r 2 a Q 1 R ) }
v g x v g z = T L { [ ( 1 f 2 ) g + f g f ] sin 2 N β 2 ( 1 f 2 ) 3 / 2 N g f 1 f 2 }
S z L S 0 = T cos θ i ,
S x L S 0 = cos θ i { ( 1 R τ a τ d ) v g x v g z + tan θ i L Q 1 ( r 2 a Q 1 R ) } ,

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