Abstract

Various definitions of the velocity of propagation of the electromagnetic field have been adopted in experimental and theoretical studies of tunneling and plasmonic systems. Tunneling problems are often analyzed by invoking the group delay (or dwell time) velocities. On the other hand, slow light and plasmonic systems are considered by using the wave packet group velocity. This paper discusses various definitions for the velocity of the electromagnetic wave propagation and compares them in applications to the problems of slow light and superluminality in resonant and tunneling structures. Energy propagation is, in general, a nonlocal quantity and depends on the global properties of the system, rather than being simply a local quantity. The energy propagation velocity takes into account the non-local characteristics of the wave propagation and offers a natural generalization for those situations when the group velocity is ill defined or gives unphysical results. It is shown that the group delay velocity, which may be superluminal away from the resonance, becomes equal to the energy velocity at the resonant point.

© 2011 OSA

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    [CrossRef]
  35. E. H. Hauge and J. A. Støvneng, “Tunneling times: a critical review,” Rev. Mod. Phys. 61, 917–936 (1989).
    [CrossRef]
  36. V. S. Olkhovsky, E. Recami, and A. K. Zaichenko, “Resonant and non-resonant tunneling through a double barrier,” Europhys. Lett. 70, 712–718 (2005).
    [CrossRef]
  37. L. Landau and E. Lifshitz, Electrodynamics of Continuous Media (in Russian) (Nauka, 1982).
  38. E. Schulz-DuBois, “Energy transport velocity of electromagnetic propagation in dispersive media,” Proc. IEEE 57, 1748–1757 (1969).
    [CrossRef]
  39. G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,” Phys. Rev. E 63, 036610 (2001).
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2010 (6)

C. Ling, M. Zheng, and K. Yu, “Slowing light in diatomic nanoshelled chains,” Opt. Commun. 283, 1945–1949 (2010).
[CrossRef]

C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. 12, 104003 (2010).
[CrossRef]

A. I. Smolyakov, E. A. Fourkal, S. I. Krasheninnikov, and N. Sternberg, “Resonant modes and resonant transmission in multi-layer structures,” Prog. Electromagn. Res. 107, 293–314 (2010).
[CrossRef]

N. Borjemscaia, S. V. Polyakov, P. D. Lett, and A. Migdall, “Single-photon propagation through dielectric bandgaps,” Opt. Express 18, 2279–2286 (2010).
[CrossRef] [PubMed]

J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon. 2, 287–318 (2010).
[CrossRef]

J. Wang, Y. Zhang, J. Zhang, Y. Cai, X. Zhang, and P. Yuan, “Simultaneous observation of superluminal and slow light propagation in a nested fiber ring resonator,” Opt. Express 18, 13180–13186 (2010).
[CrossRef] [PubMed]

2009 (4)

C. M. de Sterke, K. B. Dossou, T. P. White, L. C. Botten, and R. C. McPhedran, “Efficient coupling into slow light photonic crystal waveguide without transition region: role of evanescent modes,” Opt. Express 17, 17338–17343 (2009).
[CrossRef]

N. Malkova, G. W. Bryant, S. Polyakov, and A. Migdall, “Effect of surface modes on photon traversal through stop bands of dielectric stacks,” Phys. Rev. B 80, 165127 (2009).
[CrossRef]

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[CrossRef] [PubMed]

L. Ragni, “Group delay of evanescent signals in a waveguide with barrier,” Phys. Rev. E 79, 046609 (2009).
[CrossRef]

2008 (2)

A. B. Shvartsburg, M. Marklund, G. Brodin, and L. Stenflo, “Superluminal tunneling of microwaves in smoothly varying transmission lines,” Phys. Rev. E 78, 016601 (2008).
[CrossRef]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–463 (2008).
[CrossRef]

2007 (2)

E. Fourkal, I. Velchev, C.-M. Ma, and A. Smolyakov, “Resonant transparency of materials with negative permittivity,” Phys. Lett. A 361, 277–282 (2007).
[CrossRef]

J. F. Galisteo-López, M. Galli, A. Balestreri, M. Patrini, L. C. Andreani, and C. López, “Slow to superluminal light waves in thin 3D photonic crystals,” Opt. Express 15, 15342–15350 (2007).
[CrossRef] [PubMed]

2006 (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]

2005 (2)

H. G. Winful, “Apparent superluminality and the generalized hartman effect in double-barrier tunneling,” Phys. Rev. E 72, 046608 (2005).
[CrossRef]

V. S. Olkhovsky, E. Recami, and A. K. Zaichenko, “Resonant and non-resonant tunneling through a double barrier,” Europhys. Lett. 70, 712–718 (2005).
[CrossRef]

2004 (2)

Y. Chen and S. Blair, “Nonlinearity enhancement in finite coupled-resonator slow-light waveguides,” Opt. Express 12, 3353–3366 (2004).
[CrossRef] [PubMed]

N. Brunner, V. Scarani, M. Wegmüller, M. Legré, and N. Gisin, “Direct measurement of superluminal group velocity and signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).
[CrossRef] [PubMed]

2003 (2)

M. Mojahedi, K. Malloy, G. Eleftheriades, J. Woodley, and R. Chiao, “Abnormal wave propagation in passive media,” IEEE J. Sel. Top. Quantum Electron. 9, 30–39 (2003).
[CrossRef]

H. G. Winful, “Delay time and the Hartman effect in quantum tunneling,” Phys. Rev. Lett. 91, 260401 (2003).
[CrossRef]

2001 (4)

A. Dogariu, A. Kuzmich, H. Cao, and L. Wang, “Superluminal light pulse propagation via rephasing in a transparent anomalously dispersive medium,” Opt. Express 8, 344–350 (2001).
[CrossRef] [PubMed]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. 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. T. Reiten, D. Grischkowsky, and R. A. Cheville, “Optical tunneling of single-cycle terahertz bandwidth pulses,” Phys. Rev. E 64, 036604 (2001).
[CrossRef]

T. Sauter and F. Paschke, “Can Bessel beams carry superluminal signals?” Phys. Lett. A 285, 1–6 (2001).
[CrossRef]

2000 (2)

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000).
[CrossRef] [PubMed]

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

1998 (1)

J. C. Garrison, M. W. Mitchell, R. Y. Chiao, and E. L. Bolda, “Superluminal signals: causal loop paradoxes revisited,” Phys. Lett. A 245, 19–25 (1998).
[CrossRef]

1995 (1)

A. M. Steinberg, “How much time does a tunneling particle spend in the barrier region?” Phys. Rev. Lett. 74, 2405–2409 (1995).
[CrossRef] [PubMed]

1994 (1)

A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
[CrossRef] [PubMed]

1993 (2)

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

A. Enders and G. Nimtz, “Evanescent-mode propagation and quantum tunneling,” Phys. Rev. E 48, 632–634 (1993).
[CrossRef]

1989 (1)

E. H. Hauge and J. A. Støvneng, “Tunneling times: a critical review,” Rev. Mod. Phys. 61, 917–936 (1989).
[CrossRef]

1969 (1)

E. Schulz-DuBois, “Energy transport velocity of electromagnetic propagation in dispersive media,” Proc. IEEE 57, 1748–1757 (1969).
[CrossRef]

1962 (1)

T. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. Rep. 33, 3427–3433 (1962).
[CrossRef]

1955 (1)

E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
[CrossRef]

Andreani, L. C.

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–463 (2008).
[CrossRef]

Balestreri, A.

Bertolotti, M.

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

Blair, S.

Bloemer, M. J.

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

Bolda, E. L.

J. C. Garrison, M. W. Mitchell, R. Y. Chiao, and E. L. Bolda, “Superluminal signals: causal loop paradoxes revisited,” Phys. Lett. A 245, 19–25 (1998).
[CrossRef]

Borjemscaia, N.

Botten, L. C.

Bowden, C. M.

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

Boyd, R. W.

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[CrossRef] [PubMed]

Brillouin, L.

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

Brodin, G.

A. B. Shvartsburg, M. Marklund, G. Brodin, and L. Stenflo, “Superluminal tunneling of microwaves in smoothly varying transmission lines,” Phys. Rev. E 78, 016601 (2008).
[CrossRef]

Brunner, N.

N. Brunner, V. Scarani, M. Wegmüller, M. Legré, and N. Gisin, “Direct measurement of superluminal group velocity and signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).
[CrossRef] [PubMed]

Bryant, G. W.

N. Malkova, G. W. Bryant, S. Polyakov, and A. Migdall, “Effect of surface modes on photon traversal through stop bands of dielectric stacks,” Phys. Rev. B 80, 165127 (2009).
[CrossRef]

Cai, Y.

Cao, H.

Centini, M.

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

Chen, Y.

Cheville, R. A.

M. T. Reiten, D. Grischkowsky, and R. A. Cheville, “Optical tunneling of single-cycle terahertz bandwidth pulses,” Phys. Rev. E 64, 036604 (2001).
[CrossRef]

Chiao, R.

M. Mojahedi, K. Malloy, G. Eleftheriades, J. Woodley, and R. Chiao, “Abnormal wave propagation in passive media,” IEEE J. Sel. Top. Quantum Electron. 9, 30–39 (2003).
[CrossRef]

R. Chiao and A. Steinberg, Progress in Optics , E. Wolf, ed. (Elsevier, 1997).

Chiao, R. Y.

J. C. Garrison, M. W. Mitchell, R. Y. Chiao, and E. L. Bolda, “Superluminal signals: causal loop paradoxes revisited,” Phys. Lett. A 245, 19–25 (1998).
[CrossRef]

A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
[CrossRef] [PubMed]

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

D’Aguanno, G.

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

de Sterke, C. M.

de Sterke, M.

C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. 12, 104003 (2010).
[CrossRef]

Dogariu, A.

Dossou, K. B.

Eggleton, B. J.

C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. 12, 104003 (2010).
[CrossRef]

Eleftheriades, G.

M. Mojahedi, K. Malloy, G. Eleftheriades, J. Woodley, and R. Chiao, “Abnormal wave propagation in passive media,” IEEE J. Sel. Top. Quantum Electron. 9, 30–39 (2003).
[CrossRef]

Enders, A.

A. Enders and G. Nimtz, “Evanescent-mode propagation and quantum tunneling,” Phys. Rev. E 48, 632–634 (1993).
[CrossRef]

Fourkal, E.

E. Fourkal, I. Velchev, C.-M. Ma, and A. Smolyakov, “Resonant transparency of materials with negative permittivity,” Phys. Lett. A 361, 277–282 (2007).
[CrossRef]

Fourkal, E. A.

A. I. Smolyakov, E. A. Fourkal, S. I. Krasheninnikov, and N. Sternberg, “Resonant modes and resonant transmission in multi-layer structures,” Prog. Electromagn. Res. 107, 293–314 (2010).
[CrossRef]

Galisteo-López, J. F.

Galli, M.

Garrison, J. C.

J. C. Garrison, M. W. Mitchell, R. Y. Chiao, and E. L. Bolda, “Superluminal signals: causal loop paradoxes revisited,” Phys. Lett. A 245, 19–25 (1998).
[CrossRef]

Gauthier, D. J.

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[CrossRef] [PubMed]

Gisin, N.

N. Brunner, V. Scarani, M. Wegmüller, M. Legré, and N. Gisin, “Direct measurement of superluminal group velocity and signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).
[CrossRef] [PubMed]

Grischkowsky, D.

M. T. Reiten, D. Grischkowsky, and R. A. Cheville, “Optical tunneling of single-cycle terahertz bandwidth pulses,” Phys. Rev. E 64, 036604 (2001).
[CrossRef]

Hartman, T.

T. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. Rep. 33, 3427–3433 (1962).
[CrossRef]

Hauge, E. H.

E. H. Hauge and J. A. Støvneng, “Tunneling times: a critical review,” Rev. Mod. Phys. 61, 917–936 (1989).
[CrossRef]

Haus, J. W.

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

Khurgin, J. B.

Krasheninnikov, S. I.

A. I. Smolyakov, E. A. Fourkal, S. I. Krasheninnikov, and N. Sternberg, “Resonant modes and resonant transmission in multi-layer structures,” Prog. Electromagn. Res. 107, 293–314 (2010).
[CrossRef]

Kuzmich, A.

Kwiat, P. G.

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

Landau, L.

L. Landau and E. Lifshitz, Electrodynamics of Continuous Media (in Russian) (Nauka, 1982).

Legré, M.

N. Brunner, V. Scarani, M. Wegmüller, M. Legré, and N. Gisin, “Direct measurement of superluminal group velocity and signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).
[CrossRef] [PubMed]

Lett, P. D.

Lifshitz, E.

L. Landau and E. Lifshitz, Electrodynamics of Continuous Media (in Russian) (Nauka, 1982).

Ling, C.

C. Ling, M. Zheng, and K. Yu, “Slowing light in diatomic nanoshelled chains,” Opt. Commun. 283, 1945–1949 (2010).
[CrossRef]

López, C.

Ma, C.-M.

E. Fourkal, I. Velchev, C.-M. Ma, and A. Smolyakov, “Resonant transparency of materials with negative permittivity,” Phys. Lett. A 361, 277–282 (2007).
[CrossRef]

Malkova, N.

N. Malkova, G. W. Bryant, S. Polyakov, and A. Migdall, “Effect of surface modes on photon traversal through stop bands of dielectric stacks,” Phys. Rev. B 80, 165127 (2009).
[CrossRef]

Malloy, K.

M. Mojahedi, K. Malloy, G. Eleftheriades, J. Woodley, and R. Chiao, “Abnormal wave propagation in passive media,” IEEE J. Sel. Top. Quantum Electron. 9, 30–39 (2003).
[CrossRef]

Marklund, M.

A. B. Shvartsburg, M. Marklund, G. Brodin, and L. Stenflo, “Superluminal tunneling of microwaves in smoothly varying transmission lines,” Phys. Rev. E 78, 016601 (2008).
[CrossRef]

McPhedran, R. C.

Migdall, A.

N. Borjemscaia, S. V. Polyakov, P. D. Lett, and A. Migdall, “Single-photon propagation through dielectric bandgaps,” Opt. Express 18, 2279–2286 (2010).
[CrossRef] [PubMed]

N. Malkova, G. W. Bryant, S. Polyakov, and A. Migdall, “Effect of surface modes on photon traversal through stop bands of dielectric stacks,” Phys. Rev. B 80, 165127 (2009).
[CrossRef]

Mitchell, M. W.

J. C. Garrison, M. W. Mitchell, R. Y. Chiao, and E. L. Bolda, “Superluminal signals: causal loop paradoxes revisited,” Phys. Lett. A 245, 19–25 (1998).
[CrossRef]

Mojahedi, M.

M. Mojahedi, K. Malloy, G. Eleftheriades, J. Woodley, and R. Chiao, “Abnormal wave propagation in passive media,” IEEE J. Sel. Top. Quantum Electron. 9, 30–39 (2003).
[CrossRef]

Monat, C.

C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. 12, 104003 (2010).
[CrossRef]

Mugnai, D.

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000).
[CrossRef] [PubMed]

Nimtz, G.

A. Enders and G. Nimtz, “Evanescent-mode propagation and quantum tunneling,” Phys. Rev. E 48, 632–634 (1993).
[CrossRef]

Olkhovsky, V. S.

V. S. Olkhovsky, E. Recami, and A. K. Zaichenko, “Resonant and non-resonant tunneling through a double barrier,” Europhys. Lett. 70, 712–718 (2005).
[CrossRef]

Paschke, F.

T. Sauter and F. Paschke, “Can Bessel beams carry superluminal signals?” Phys. Lett. A 285, 1–6 (2001).
[CrossRef]

Patrini, M.

Polyakov, S.

N. Malkova, G. W. Bryant, S. Polyakov, and A. Migdall, “Effect of surface modes on photon traversal through stop bands of dielectric stacks,” Phys. Rev. B 80, 165127 (2009).
[CrossRef]

Polyakov, S. V.

Ragni, L.

L. Ragni, “Group delay of evanescent signals in a waveguide with barrier,” Phys. Rev. E 79, 046609 (2009).
[CrossRef]

Ranfagni, A.

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000).
[CrossRef] [PubMed]

Razavy, M.

M. Razavy, Quantum Theory of Tunneling (WorldScientific, 2003).
[CrossRef]

Recami, E.

V. S. Olkhovsky, E. Recami, and A. K. Zaichenko, “Resonant and non-resonant tunneling through a double barrier,” Europhys. Lett. 70, 712–718 (2005).
[CrossRef]

Reiten, M. T.

M. T. Reiten, D. Grischkowsky, and R. A. Cheville, “Optical tunneling of single-cycle terahertz bandwidth pulses,” Phys. Rev. E 64, 036604 (2001).
[CrossRef]

Ruggeri, R.

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000).
[CrossRef] [PubMed]

Sauter, T.

T. Sauter and F. Paschke, “Can Bessel beams carry superluminal signals?” Phys. Lett. A 285, 1–6 (2001).
[CrossRef]

Scalora, M.

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

Scarani, V.

N. Brunner, V. Scarani, M. Wegmüller, M. Legré, and N. Gisin, “Direct measurement of superluminal group velocity and signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).
[CrossRef] [PubMed]

Schulz-DuBois, E.

E. Schulz-DuBois, “Energy transport velocity of electromagnetic propagation in dispersive media,” Proc. IEEE 57, 1748–1757 (1969).
[CrossRef]

Shvartsburg, A. B.

A. B. Shvartsburg, M. Marklund, G. Brodin, and L. Stenflo, “Superluminal tunneling of microwaves in smoothly varying transmission lines,” Phys. Rev. E 78, 016601 (2008).
[CrossRef]

Sibilia, C.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. 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.

E. Fourkal, I. Velchev, C.-M. Ma, and A. Smolyakov, “Resonant transparency of materials with negative permittivity,” Phys. Lett. A 361, 277–282 (2007).
[CrossRef]

Smolyakov, A. I.

A. I. Smolyakov, E. A. Fourkal, S. I. Krasheninnikov, and N. Sternberg, “Resonant modes and resonant transmission in multi-layer structures,” Prog. Electromagn. Res. 107, 293–314 (2010).
[CrossRef]

Sommerfeld, A.

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

Steinberg, A.

R. Chiao and A. Steinberg, Progress in Optics , E. Wolf, ed. (Elsevier, 1997).

Steinberg, A. M.

A. M. Steinberg, “How much time does a tunneling particle spend in the barrier region?” Phys. Rev. Lett. 74, 2405–2409 (1995).
[CrossRef] [PubMed]

A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
[CrossRef] [PubMed]

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

Stenflo, L.

A. B. Shvartsburg, M. Marklund, G. Brodin, and L. Stenflo, “Superluminal tunneling of microwaves in smoothly varying transmission lines,” Phys. Rev. E 78, 016601 (2008).
[CrossRef]

Sternberg, N.

A. I. Smolyakov, E. A. Fourkal, S. I. Krasheninnikov, and N. Sternberg, “Resonant modes and resonant transmission in multi-layer structures,” Prog. Electromagn. Res. 107, 293–314 (2010).
[CrossRef]

Støvneng, J. A.

E. H. Hauge and J. A. Støvneng, “Tunneling times: a critical review,” Rev. Mod. Phys. 61, 917–936 (1989).
[CrossRef]

Velchev, I.

E. Fourkal, I. Velchev, C.-M. Ma, and A. Smolyakov, “Resonant transparency of materials with negative permittivity,” Phys. Lett. A 361, 277–282 (2007).
[CrossRef]

Wang, J.

Wang, L.

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L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
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Wegmüller, M.

N. Brunner, V. Scarani, M. Wegmüller, M. Legré, and N. Gisin, “Direct measurement of superluminal group velocity and signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).
[CrossRef] [PubMed]

White, T. P.

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

H. G. Winful, “Apparent superluminality and the generalized hartman effect in double-barrier tunneling,” Phys. Rev. E 72, 046608 (2005).
[CrossRef]

H. G. Winful, “Delay time and the Hartman effect in quantum tunneling,” Phys. Rev. Lett. 91, 260401 (2003).
[CrossRef]

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M. Mojahedi, K. Malloy, G. Eleftheriades, J. Woodley, and R. Chiao, “Abnormal wave propagation in passive media,” IEEE J. Sel. Top. Quantum Electron. 9, 30–39 (2003).
[CrossRef]

Yariv, A.

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

Yeh, P.

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

Yu, K.

C. Ling, M. Zheng, and K. Yu, “Slowing light in diatomic nanoshelled chains,” Opt. Commun. 283, 1945–1949 (2010).
[CrossRef]

Yuan, P.

Zaichenko, A. K.

V. S. Olkhovsky, E. Recami, and A. K. Zaichenko, “Resonant and non-resonant tunneling through a double barrier,” Europhys. Lett. 70, 712–718 (2005).
[CrossRef]

Zhang, J.

Zhang, X.

Zhang, Y.

Zheng, M.

C. Ling, M. Zheng, and K. Yu, “Slowing light in diatomic nanoshelled chains,” Opt. Commun. 283, 1945–1949 (2010).
[CrossRef]

Adv. Opt. Photon. (1)

Europhys. Lett. (1)

V. S. Olkhovsky, E. Recami, and A. K. Zaichenko, “Resonant and non-resonant tunneling through a double barrier,” Europhys. Lett. 70, 712–718 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

M. Mojahedi, K. Malloy, G. Eleftheriades, J. Woodley, and R. Chiao, “Abnormal wave propagation in passive media,” IEEE J. Sel. Top. Quantum Electron. 9, 30–39 (2003).
[CrossRef]

J. Appl. Phys. Rep. (1)

T. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. Rep. 33, 3427–3433 (1962).
[CrossRef]

J. Opt. (1)

C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. 12, 104003 (2010).
[CrossRef]

Nat. Photonics (1)

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–463 (2008).
[CrossRef]

Nature (1)

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

Opt. Commun. (1)

C. Ling, M. Zheng, and K. Yu, “Slowing light in diatomic nanoshelled chains,” Opt. Commun. 283, 1945–1949 (2010).
[CrossRef]

Opt. Express (6)

Phys. Lett. A (3)

E. Fourkal, I. Velchev, C.-M. Ma, and A. Smolyakov, “Resonant transparency of materials with negative permittivity,” Phys. Lett. A 361, 277–282 (2007).
[CrossRef]

J. C. Garrison, M. W. Mitchell, R. Y. Chiao, and E. L. Bolda, “Superluminal signals: causal loop paradoxes revisited,” Phys. Lett. A 245, 19–25 (1998).
[CrossRef]

T. Sauter and F. Paschke, “Can Bessel beams carry superluminal signals?” Phys. Lett. A 285, 1–6 (2001).
[CrossRef]

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. (1)

E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
[CrossRef]

Phys. Rev. A (1)

A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
[CrossRef] [PubMed]

Phys. Rev. B (1)

N. Malkova, G. W. Bryant, S. Polyakov, and A. Migdall, “Effect of surface modes on photon traversal through stop bands of dielectric stacks,” Phys. Rev. B 80, 165127 (2009).
[CrossRef]

Phys. Rev. E (6)

H. G. Winful, “Apparent superluminality and the generalized hartman effect in double-barrier tunneling,” Phys. Rev. E 72, 046608 (2005).
[CrossRef]

A. B. Shvartsburg, M. Marklund, G. Brodin, and L. Stenflo, “Superluminal tunneling of microwaves in smoothly varying transmission lines,” Phys. Rev. E 78, 016601 (2008).
[CrossRef]

L. Ragni, “Group delay of evanescent signals in a waveguide with barrier,” Phys. Rev. E 79, 046609 (2009).
[CrossRef]

M. T. Reiten, D. Grischkowsky, and R. A. Cheville, “Optical tunneling of single-cycle terahertz bandwidth pulses,” Phys. Rev. E 64, 036604 (2001).
[CrossRef]

A. Enders and G. Nimtz, “Evanescent-mode propagation and quantum tunneling,” Phys. Rev. E 48, 632–634 (1993).
[CrossRef]

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

Phys. Rev. Lett. (5)

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000).
[CrossRef] [PubMed]

A. M. Steinberg, “How much time does a tunneling particle spend in the barrier region?” Phys. Rev. Lett. 74, 2405–2409 (1995).
[CrossRef] [PubMed]

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

N. Brunner, V. Scarani, M. Wegmüller, M. Legré, and N. Gisin, “Direct measurement of superluminal group velocity and signal velocity in an optical fiber,” Phys. Rev. Lett. 93, 203902 (2004).
[CrossRef] [PubMed]

H. G. Winful, “Delay time and the Hartman effect in quantum tunneling,” Phys. Rev. Lett. 91, 260401 (2003).
[CrossRef]

Proc. IEEE (1)

E. Schulz-DuBois, “Energy transport velocity of electromagnetic propagation in dispersive media,” Proc. IEEE 57, 1748–1757 (1969).
[CrossRef]

Prog. Electromagn. Res. (1)

A. I. Smolyakov, E. A. Fourkal, S. I. Krasheninnikov, and N. Sternberg, “Resonant modes and resonant transmission in multi-layer structures,” Prog. Electromagn. Res. 107, 293–314 (2010).
[CrossRef]

Rev. Mod. Phys. (1)

E. H. Hauge and J. A. Støvneng, “Tunneling times: a critical review,” Rev. Mod. Phys. 61, 917–936 (1989).
[CrossRef]

Science (1)

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[CrossRef] [PubMed]

Other (5)

R. Chiao and A. Steinberg, Progress in Optics , E. Wolf, ed. (Elsevier, 1997).

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

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

L. Landau and E. Lifshitz, Electrodynamics of Continuous Media (in Russian) (Nauka, 1982).

M. Razavy, Quantum Theory of Tunneling (WorldScientific, 2003).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Reflection and transmission responses of the slab layer: ωp =10 GHz, and d=0.02 m.

Fig. 2
Fig. 2

(a) Normalized energy, group delay and group velocities of the slab layer, ωp =10 GHz and d=0.02 m; (b) Close-up of the region where ɛ → 0; (c) the case with dispersion, ɛ is given by Eq. (35), ω 0 = 0.5ωp , d and ωp as in (a).

Fig. 3
Fig. 3

Normalized energy, group delay and group velocities for ɛ = 10. (a) As a function of frequency; a=0.2 m. (b) As function of the slab width; f=1 GHz.

Fig. 4
Fig. 4

(a) The two layer structure, region 1– 0 < ɛ 1 < 1; region 2 – ɛ 2 < 0. Incidence angles larger than the critical are considered, so waves are evanescent in both regions. (b) The double barrier structure; regions 1,3 – ɛ < 0; region 2 – ɛ = 1.

Fig. 5
Fig. 5

Magnetic field distribution for the double layer structure. The width of the first layer L 1 = 0.78m, the width of the second layer L 2 = 0.02m, ɛ 1 = 0.9, ɛ 2 = −35. The resonance occurs at θ ≃ 74°.

Fig. 6
Fig. 6

Transmissivity of the double layer for ωp 1 = 3.77 × 1010 rad s −1 (6 GHz), ωp 2 = 2 × 109 rad s −1 (0.32 GHz), L 1 = 0.78m, L 2 = 0.02m. (a) As a function of the incident angle with f=1 GHz. (b) As a function of frequency at resonant incidence.

Fig. 8
Fig. 8

Magnetic field distribution for the double barrier structure. The width of the barriers is a = 0.021m, the width of the middle layer is L = 0.25m, ɛ = −35. The resonance occurs at θ = 62.3°.

Fig. 9
Fig. 9

Transmissivity of the double barrier structure for ωp = 3.77 × 1010 rad s −1 (6 GHz), L = 0.25m, a = 0.021m. (a) As a function of the incident angle with f=1 GHz. (b) As a function of frequency at resonant incidence.

Fig. 7
Fig. 7

Normalized energy velocity and group delay velocity of the double layer structure at resonant incidence.

Fig. 10
Fig. 10

Normalized energy and group delay velocities of the double barrier structure as function of frequency for resonant incidence.

Equations (65)

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ψ ( x , t ) = ω f ( ω ω 0 ) exp [ i k x i ω t ] d ω .
ψ R ( x , t ) = ω f ( ω ω 0 ) | R | exp [ i φ r ( ω ) + i k x i ω t ] d ω ,
ψ T ( x , t ) = ω f ( ω ω 0 ) | T | exp [ i φ t ( ω ) + i k x i ω t ] d ω .
d d ω ( φ r ω t ) = 0 ,
d d ω ( φ t + k d ω t ) = 0.
τ g t = d d ω ( φ t + k d ) ,
τ g r = d φ r d ω .
τ g = | R | 2 τ g r + | T | 2 τ g t .
v d = d τ g t .
U t + S = Q ,
S | z = 0 = S | z = d = 1 2 Z 0 | H 0 | 2 | T | 2 = 1 2 Z 0 | E 0 | 2 | T | 2 ,
H x ( z ) = C exp ( κ z ) + D exp ( κ z ) ,
E y ( z ) = i 1 ω ɛ H x z ,
E y ( z ) = i κ ω ɛ ( C exp ( κ z ) D exp ( κ z ) ) .
S z = 1 2 ( E y H x * ) ( C D * C * D ) ( C D * ) .
E = [ 0 , E y , 0 ] e i ( k z ω t ) ,
H = [ H x , 0 , 0 ] e i ( k z ω t ) .
H 0 ( e i k v z + R e i k v z ) z < 0 ,
H 0 ( A e i k z + B e i k z ) 0 < z < d ,
H 0 T e i k v z z > d .
k 2 = ω 2 c 2 ɛ ( ω ) ,
ɛ ( ω ) = 1 ω p 2 ω 2 .
T = e i k v d g ,
R = i ( k ɛ 0 / k v ɛ k v ɛ / k ɛ 0 ) sin k d 2 g ,
g = cos k d i ( k ɛ 0 / k v ɛ + k v ɛ 0 / k ɛ ) sin k d / 2 ,
k v = ω c ,
k = ω c ɛ ( ω ) .
φ 0 = φ t + k v d = arctan ( k ɛ 0 / k v ɛ + k v ɛ 0 / k ɛ 2 tan k d ) .
v d = d τ g t = d ( φ 0 ω ) 1 .
v E = 1 2 Z 0 | H 0 | 2 | T | 2 0 d U d z ,
U = 1 4 ɛ 0 ( ω ɛ ) ω | E y | 2 + 1 4 μ 0 | H x | 2 .
v E c = 4 ( 1 + ɛ ) ( 1 + 1 ɛ ( ω ɛ ) ω ) + ( ɛ 1 ) ( 1 1 ɛ ( ω ɛ ) ω ) sin 2 k d 2 k d .
( ω ɛ ) ω = 1 + ω p 2 ω 2 = 2 ɛ ,
v E c = 2 ɛ 1 + ɛ + sin 2 k d 2 k d ( ɛ 1 ) 2 ,
ɛ = 1 + ω p 2 ω 0 2 ω 2 ,
v E c = 2 ɛ + 1 .
k v 2 = k 0 2 k y 2 , k 0 = ω / c k y = k 0 sin θ i ,
κ 1 2 = k y 2 ɛ 1 ( ω ) k 0 2 ,
κ 2 2 = k y 2 ɛ 2 ( ω ) k 0 2 ,
η i = ɛ i / κ i ,
H 0 ( e i k v z + R e i k v z ) z < 0 ,
H 0 ( A e κ 1 z + B e κ 1 z ) 0 < z < L 1 ,
H 0 ( C e κ 2 ( z L 1 ) + D e κ 2 ( z L 1 ) ) L 1 < z < L ,
H 0 T e i k v z z > L .
T = e i k v L g ,
g = cosh κ 2 L 2 cosh κ 1 L 1 + Δ 1 sinh κ 1 L 1 sinh κ 2 L 2 + i [ Δ 2 sinh κ 1 L 1 cosh κ 2 L 2 ] .
φ 0 = arctan [ Δ 2 sinh κ 1 L 1 cosh κ 2 L 2 + Δ 3 cosh κ 1 L 1 sinh κ 2 L 2 cosh κ 2 L 2 cosh κ 1 L 1 + Δ 1 sinh κ 1 L 1 sinh κ 2 L 2 ] .
Δ 1 = 1 2 ( η 1 η 2 + η 2 η 1 ) ,
Δ 2 = 1 2 ( η 1 η 0 η 0 η 1 ) ,
Δ 3 = 1 2 ( η 2 η 0 η 0 η 2 ) .
κ 1 L 1 = κ 2 L 2 ,
η 1 + η 2 = 0 .
H 0 ( e i k v z + R e i k v z ) z < 0 ,
H 0 ( A 0 e κ z + B 0 e κ z ) 0 < z < a ,
H 0 ( A 1 e i k v z + B 1 e i k v z ) a < z < L + a ,
H 0 ( A 2 e κ ( z L a ) + B 2 e κ ( z L a ) ) L + a < z < L + 2 a ,
H 0 T e i k v z z > L + 2 a .
T = e 2 i k v a g ,
g = cosh 2 ( κ a ) + 1 4 sinh 2 ( κ a ) [ σ 2 cos ( k v L ) δ 2 ] + i sinh ( κ a ) [ δ cosh ( κ a ) + 1 4 σ 2 sinh ( κ a ) sin ( 2 k v L ) ] .
σ = η 0 η 1 + η 1 η 0 ,
δ = η 1 η 0 η 0 η 1 ,
σ 2 = δ 2 + 4.
cot ( k v L ) = 1 2 δ tanh ( κ a ) ,
tan ( k v L ) = 2 ξ 1 ξ 2 , ξ = η 0 η = κ ɛ k v ɛ 0 .
φ 0 = k v L arctan [ sinh ( κ a ) [ δ cosh ( κ a ) + ( 1 / 4 ) σ 2 sinh 2 ( κ a ) sin ( 2 k v L ) ] cosh 2 ( κ a ) + ( 1 / 4 ) sinh 2 ( κ a ) [ σ 2 cos ( 2 k v L ) δ 2 ] ] .

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