Abstract

Tunneling time (or group delay) for an optical pulse to transmit the energy through a photonic bandgap is calculated analytically for a one-dimensional multilayered optical superlattice. The analytical solution shows that the calculated tunneling time converges to a finite value with increasing numbers of layers, and we have derived the formula for the converged value of the tunneling time. This effect is similar to the so-called Hartman effect in a quantum system. Because of the destructive interference of multireflected light in the superlattice, the tunneling time is determined by an exponentially decaying evanescent wave, which is the reason for this effect.

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  1. L. A. MacColl, “Note on the transmission and reflection of wave packets by potential barriers,” Phys. Rev. 40, 621–626 (1932).
    [CrossRef]
  2. T. E. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. 33, 3427–3433 (1962).
    [CrossRef]
  3. E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
    [CrossRef]
  4. V. S. Olkhovsky and E. Recami, “Recent developments in the time analysis of tunneling processes,” Phys. Rep. 214, 339–356 (1992).
    [CrossRef]
  5. H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. 436, 1–69 (2006).
    [CrossRef]
  6. A. Ranfagni, D. Mugnai, P. Fabeni, and G. P. Pazzi, “Delay-time measurements in narrowed waveguides as a test of tunneling,” Appl. Phys. Lett. 58, 774–776 (1991).
    [CrossRef]
  7. L. Ragni, “Group delay of evanescent signals in a waveguide with barrier,” Phys. Rev. E 79, 046609 (2009).
    [CrossRef]
  8. Ph. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
    [CrossRef]
  9. D. Mugnai, A. Ranfagni, and L. Ronchi, “The question of tunneling time duration: a new experimental test at microwave scale,” Phys. Lett. A 247, 281–286 (1998).
    [CrossRef]
  10. J. J. Carey, J. Zawadzka, D. A. Jaroszynski, and K. Wynne, “Noncausal time response in frustrated total internal reflection?” Phys. Rev. Lett. 84, 1431–1434 (2000).
    [CrossRef] [PubMed]
  11. 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]
  12. Ch. Spielmann, R. Szipoc¨s, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311 (1994).
    [CrossRef] [PubMed]
  13. M. Mojahedi, E. Schamiloglu, F. Hegeler, and K. J. Malloy, “Time-domain detection of superluminal group velocity for single microwave pulses,” Phys. Rev. E 62, 5758–5766(2000).
    [CrossRef]
  14. S. Longhi, M. Marano, and P. Laporta, “Superluminal optical pulse propagation at 1.5 μm in periodic fiber Bragg gratings,” Phys. Rev. E 64, 055602 (2001).
    [CrossRef]
  15. A. Hache and L. Poirier, “Long-range superluminal pulse propagation in a coaxial photonic crystal,” Appl. Phys. Lett. 80, 518–520 (2002).
    [CrossRef]
  16. R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, “Analogies between electron and photon tunneling: a proposed experiment to measure photon tunneling times,” Physica B 175, 257–262(1991).
    [CrossRef]
  17. Th. Martin and R. Landauer, “Time delay of evanescent electromagnetic waves and the analogy to particle tunneling,” Phys. Rev. A 45, 2611–2617 (1992).
    [CrossRef] [PubMed]
  18. E. H. Hauge and J. A. Støvneng, “Tunneling times: a critical review,” Rev. Mod. Phys. 61, 917–936 (1989).
    [CrossRef]
  19. V. Laude and P. Tournois, “Superluminal asymptotic tunneling times through one-dimensional photonic bandgaps in quarter-wave-stack dielectric mirrors,” J. Opt. Soc. Am. B 16, 194–198(1999).
    [CrossRef]
  20. P. Pereyra, “Closed formulas for tunneling time in superlattices,” Phys. Rev. Lett. 84, 1772–1775 (2000).
    [CrossRef] [PubMed]
  21. S. Esposito, “Universal photonic tunneling time,” Phys. Rev. E 64, 026609 (2001).
    [CrossRef]
  22. H. G. Winful, “The meaning of group delay in barrier tunnelling: a re-examination of superluminal group velocities,” New J. Phys. 8, 101 (2006).
    [CrossRef]
  23. H. G. Winful, “Energy storage in superluminal barrier tunneling: origin of the Hartman effect,” Opt. Express 10, 1491–1496(2002).
    [PubMed]
  24. H. G. Winful, “Nature of ‘superluminal’ barrier tunneling,” Phys. Rev. Lett. 90, 023901 (2003).
    [CrossRef] [PubMed]
  25. P. Pereyra and H. P. Simanjuntak, “Time evolution of electromagnetic wave packets through superlattices: evidence for superluminal velocities,” Phys. Rev. E 75, 056604 (2007).
    [CrossRef]
  26. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).
    [CrossRef]
  27. 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]

2009 (1)

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

2007 (1)

P. Pereyra and H. P. Simanjuntak, “Time evolution of electromagnetic wave packets through superlattices: evidence for superluminal velocities,” Phys. Rev. E 75, 056604 (2007).
[CrossRef]

2006 (2)

H. G. Winful, “The meaning of group delay in barrier tunnelling: a re-examination of superluminal group velocities,” New J. Phys. 8, 101 (2006).
[CrossRef]

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

2003 (1)

H. G. Winful, “Nature of ‘superluminal’ barrier tunneling,” Phys. Rev. Lett. 90, 023901 (2003).
[CrossRef] [PubMed]

2002 (2)

H. G. Winful, “Energy storage in superluminal barrier tunneling: origin of the Hartman effect,” Opt. Express 10, 1491–1496(2002).
[PubMed]

A. Hache and L. Poirier, “Long-range superluminal pulse propagation in a coaxial photonic crystal,” Appl. Phys. Lett. 80, 518–520 (2002).
[CrossRef]

2001 (2)

S. Longhi, M. Marano, and P. Laporta, “Superluminal optical pulse propagation at 1.5 μm in periodic fiber Bragg gratings,” Phys. Rev. E 64, 055602 (2001).
[CrossRef]

S. Esposito, “Universal photonic tunneling time,” Phys. Rev. E 64, 026609 (2001).
[CrossRef]

2000 (3)

P. Pereyra, “Closed formulas for tunneling time in superlattices,” Phys. Rev. Lett. 84, 1772–1775 (2000).
[CrossRef] [PubMed]

M. Mojahedi, E. Schamiloglu, F. Hegeler, and K. J. Malloy, “Time-domain detection of superluminal group velocity for single microwave pulses,” Phys. Rev. E 62, 5758–5766(2000).
[CrossRef]

J. J. Carey, J. Zawadzka, D. A. Jaroszynski, and K. Wynne, “Noncausal time response in frustrated total internal reflection?” Phys. Rev. Lett. 84, 1431–1434 (2000).
[CrossRef] [PubMed]

1999 (1)

1998 (1)

D. Mugnai, A. Ranfagni, and L. Ronchi, “The question of tunneling time duration: a new experimental test at microwave scale,” Phys. Lett. A 247, 281–286 (1998).
[CrossRef]

1997 (1)

Ph. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (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]

1994 (1)

Ch. Spielmann, R. Szipoc¨s, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311 (1994).
[CrossRef] [PubMed]

1993 (1)

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]

1992 (2)

V. S. Olkhovsky and E. Recami, “Recent developments in the time analysis of tunneling processes,” Phys. Rep. 214, 339–356 (1992).
[CrossRef]

Th. Martin and R. Landauer, “Time delay of evanescent electromagnetic waves and the analogy to particle tunneling,” Phys. Rev. A 45, 2611–2617 (1992).
[CrossRef] [PubMed]

1991 (2)

A. Ranfagni, D. Mugnai, P. Fabeni, and G. P. Pazzi, “Delay-time measurements in narrowed waveguides as a test of tunneling,” Appl. Phys. Lett. 58, 774–776 (1991).
[CrossRef]

R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, “Analogies between electron and photon tunneling: a proposed experiment to measure photon tunneling times,” Physica B 175, 257–262(1991).
[CrossRef]

1989 (1)

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

1962 (1)

T. E. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. 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]

1932 (1)

L. A. MacColl, “Note on the transmission and reflection of wave packets by potential barriers,” Phys. Rev. 40, 621–626 (1932).
[CrossRef]

Balcou, Ph.

Ph. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
[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]

Carey, J. J.

J. J. Carey, J. Zawadzka, D. A. Jaroszynski, and K. Wynne, “Noncausal time response in frustrated total internal reflection?” Phys. Rev. Lett. 84, 1431–1434 (2000).
[CrossRef] [PubMed]

Chiao, R. Y.

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]

R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, “Analogies between electron and photon tunneling: a proposed experiment to measure photon tunneling times,” Physica B 175, 257–262(1991).
[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]

Dutriaux, L.

Ph. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
[CrossRef]

Esposito, S.

S. Esposito, “Universal photonic tunneling time,” Phys. Rev. E 64, 026609 (2001).
[CrossRef]

Fabeni, P.

A. Ranfagni, D. Mugnai, P. Fabeni, and G. P. Pazzi, “Delay-time measurements in narrowed waveguides as a test of tunneling,” Appl. Phys. Lett. 58, 774–776 (1991).
[CrossRef]

Hache, A.

A. Hache and L. Poirier, “Long-range superluminal pulse propagation in a coaxial photonic crystal,” Appl. Phys. Lett. 80, 518–520 (2002).
[CrossRef]

Hartman, T. E.

T. E. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. 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]

Hegeler, F.

M. Mojahedi, E. Schamiloglu, F. Hegeler, and K. J. Malloy, “Time-domain detection of superluminal group velocity for single microwave pulses,” Phys. Rev. E 62, 5758–5766(2000).
[CrossRef]

Jaroszynski, D. A.

J. J. Carey, J. Zawadzka, D. A. Jaroszynski, and K. Wynne, “Noncausal time response in frustrated total internal reflection?” Phys. Rev. Lett. 84, 1431–1434 (2000).
[CrossRef] [PubMed]

Krausz, F.

Ch. Spielmann, R. Szipoc¨s, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311 (1994).
[CrossRef] [PubMed]

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]

R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, “Analogies between electron and photon tunneling: a proposed experiment to measure photon tunneling times,” Physica B 175, 257–262(1991).
[CrossRef]

Landauer, R.

Th. Martin and R. Landauer, “Time delay of evanescent electromagnetic waves and the analogy to particle tunneling,” Phys. Rev. A 45, 2611–2617 (1992).
[CrossRef] [PubMed]

Laporta, P.

S. Longhi, M. Marano, and P. Laporta, “Superluminal optical pulse propagation at 1.5 μm in periodic fiber Bragg gratings,” Phys. Rev. E 64, 055602 (2001).
[CrossRef]

Laude, V.

Longhi, S.

S. Longhi, M. Marano, and P. Laporta, “Superluminal optical pulse propagation at 1.5 μm in periodic fiber Bragg gratings,” Phys. Rev. E 64, 055602 (2001).
[CrossRef]

MacColl, L. A.

L. A. MacColl, “Note on the transmission and reflection of wave packets by potential barriers,” Phys. Rev. 40, 621–626 (1932).
[CrossRef]

Malloy, K. J.

M. Mojahedi, E. Schamiloglu, F. Hegeler, and K. J. Malloy, “Time-domain detection of superluminal group velocity for single microwave pulses,” Phys. Rev. E 62, 5758–5766(2000).
[CrossRef]

Marano, M.

S. Longhi, M. Marano, and P. Laporta, “Superluminal optical pulse propagation at 1.5 μm in periodic fiber Bragg gratings,” Phys. Rev. E 64, 055602 (2001).
[CrossRef]

Martin, Th.

Th. Martin and R. Landauer, “Time delay of evanescent electromagnetic waves and the analogy to particle tunneling,” Phys. Rev. A 45, 2611–2617 (1992).
[CrossRef] [PubMed]

Mojahedi, M.

M. Mojahedi, E. Schamiloglu, F. Hegeler, and K. J. Malloy, “Time-domain detection of superluminal group velocity for single microwave pulses,” Phys. Rev. E 62, 5758–5766(2000).
[CrossRef]

Mugnai, D.

D. Mugnai, A. Ranfagni, and L. Ronchi, “The question of tunneling time duration: a new experimental test at microwave scale,” Phys. Lett. A 247, 281–286 (1998).
[CrossRef]

A. Ranfagni, D. Mugnai, P. Fabeni, and G. P. Pazzi, “Delay-time measurements in narrowed waveguides as a test of tunneling,” Appl. Phys. Lett. 58, 774–776 (1991).
[CrossRef]

Olkhovsky, V. S.

V. S. Olkhovsky and E. Recami, “Recent developments in the time analysis of tunneling processes,” Phys. Rep. 214, 339–356 (1992).
[CrossRef]

Pazzi, G. P.

A. Ranfagni, D. Mugnai, P. Fabeni, and G. P. Pazzi, “Delay-time measurements in narrowed waveguides as a test of tunneling,” Appl. Phys. Lett. 58, 774–776 (1991).
[CrossRef]

Pereyra, P.

P. Pereyra and H. P. Simanjuntak, “Time evolution of electromagnetic wave packets through superlattices: evidence for superluminal velocities,” Phys. Rev. E 75, 056604 (2007).
[CrossRef]

P. Pereyra, “Closed formulas for tunneling time in superlattices,” Phys. Rev. Lett. 84, 1772–1775 (2000).
[CrossRef] [PubMed]

Poirier, L.

A. Hache and L. Poirier, “Long-range superluminal pulse propagation in a coaxial photonic crystal,” Appl. Phys. Lett. 80, 518–520 (2002).
[CrossRef]

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 L. Ronchi, “The question of tunneling time duration: a new experimental test at microwave scale,” Phys. Lett. A 247, 281–286 (1998).
[CrossRef]

A. Ranfagni, D. Mugnai, P. Fabeni, and G. P. Pazzi, “Delay-time measurements in narrowed waveguides as a test of tunneling,” Appl. Phys. Lett. 58, 774–776 (1991).
[CrossRef]

Recami, E.

V. S. Olkhovsky and E. Recami, “Recent developments in the time analysis of tunneling processes,” Phys. Rep. 214, 339–356 (1992).
[CrossRef]

Ronchi, L.

D. Mugnai, A. Ranfagni, and L. Ronchi, “The question of tunneling time duration: a new experimental test at microwave scale,” Phys. Lett. A 247, 281–286 (1998).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).
[CrossRef]

Scalora, 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]

Schamiloglu, E.

M. Mojahedi, E. Schamiloglu, F. Hegeler, and K. J. Malloy, “Time-domain detection of superluminal group velocity for single microwave pulses,” Phys. Rev. E 62, 5758–5766(2000).
[CrossRef]

Simanjuntak, H. P.

P. Pereyra and H. P. Simanjuntak, “Time evolution of electromagnetic wave packets through superlattices: evidence for superluminal velocities,” Phys. Rev. E 75, 056604 (2007).
[CrossRef]

Spielmann, Ch.

Ch. Spielmann, R. Szipoc¨s, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311 (1994).
[CrossRef] [PubMed]

Steinberg, A. M.

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]

R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, “Analogies between electron and photon tunneling: a proposed experiment to measure photon tunneling times,” Physica B 175, 257–262(1991).
[CrossRef]

Stingl, A.

Ch. Spielmann, R. Szipoc¨s, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311 (1994).
[CrossRef] [PubMed]

Støvneng, J. A.

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

Szipoc¨s, R.

Ch. Spielmann, R. Szipoc¨s, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311 (1994).
[CrossRef] [PubMed]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).
[CrossRef]

Tournois, 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, “The meaning of group delay in barrier tunnelling: a re-examination of superluminal group velocities,” New J. Phys. 8, 101 (2006).
[CrossRef]

H. G. Winful, “Nature of ‘superluminal’ barrier tunneling,” Phys. Rev. Lett. 90, 023901 (2003).
[CrossRef] [PubMed]

H. G. Winful, “Energy storage in superluminal barrier tunneling: origin of the Hartman effect,” Opt. Express 10, 1491–1496(2002).
[PubMed]

Wynne, K.

J. J. Carey, J. Zawadzka, D. A. Jaroszynski, and K. Wynne, “Noncausal time response in frustrated total internal reflection?” Phys. Rev. Lett. 84, 1431–1434 (2000).
[CrossRef] [PubMed]

Zawadzka, J.

J. J. Carey, J. Zawadzka, D. A. Jaroszynski, and K. Wynne, “Noncausal time response in frustrated total internal reflection?” Phys. Rev. Lett. 84, 1431–1434 (2000).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

A. Ranfagni, D. Mugnai, P. Fabeni, and G. P. Pazzi, “Delay-time measurements in narrowed waveguides as a test of tunneling,” Appl. Phys. Lett. 58, 774–776 (1991).
[CrossRef]

A. Hache and L. Poirier, “Long-range superluminal pulse propagation in a coaxial photonic crystal,” Appl. Phys. Lett. 80, 518–520 (2002).
[CrossRef]

J. Appl. Phys. (1)

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

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

New J. Phys. (1)

H. G. Winful, “The meaning of group delay in barrier tunnelling: a re-examination of superluminal group velocities,” New J. Phys. 8, 101 (2006).
[CrossRef]

Opt. Express (1)

Phys. Lett. A (1)

D. Mugnai, A. Ranfagni, and L. Ronchi, “The question of tunneling time duration: a new experimental test at microwave scale,” Phys. Lett. A 247, 281–286 (1998).
[CrossRef]

Phys. Rep. (2)

V. S. Olkhovsky and E. Recami, “Recent developments in the time analysis of tunneling processes,” Phys. Rep. 214, 339–356 (1992).
[CrossRef]

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. (2)

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

L. A. MacColl, “Note on the transmission and reflection of wave packets by potential barriers,” Phys. Rev. 40, 621–626 (1932).
[CrossRef]

Phys. Rev. A (1)

Th. Martin and R. Landauer, “Time delay of evanescent electromagnetic waves and the analogy to particle tunneling,” Phys. Rev. A 45, 2611–2617 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (6)

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

M. Mojahedi, E. Schamiloglu, F. Hegeler, and K. J. Malloy, “Time-domain detection of superluminal group velocity for single microwave pulses,” Phys. Rev. E 62, 5758–5766(2000).
[CrossRef]

S. Longhi, M. Marano, and P. Laporta, “Superluminal optical pulse propagation at 1.5 μm in periodic fiber Bragg gratings,” Phys. Rev. E 64, 055602 (2001).
[CrossRef]

P. Pereyra and H. P. Simanjuntak, “Time evolution of electromagnetic wave packets through superlattices: evidence for superluminal velocities,” Phys. Rev. E 75, 056604 (2007).
[CrossRef]

S. Esposito, “Universal photonic tunneling time,” Phys. Rev. E 64, 026609 (2001).
[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]

Phys. Rev. Lett. (6)

P. Pereyra, “Closed formulas for tunneling time in superlattices,” Phys. Rev. Lett. 84, 1772–1775 (2000).
[CrossRef] [PubMed]

H. G. Winful, “Nature of ‘superluminal’ barrier tunneling,” Phys. Rev. Lett. 90, 023901 (2003).
[CrossRef] [PubMed]

Ph. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
[CrossRef]

J. J. Carey, J. Zawadzka, D. A. Jaroszynski, and K. Wynne, “Noncausal time response in frustrated total internal reflection?” Phys. Rev. Lett. 84, 1431–1434 (2000).
[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]

Ch. Spielmann, R. Szipoc¨s, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311 (1994).
[CrossRef] [PubMed]

Physica B (1)

R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, “Analogies between electron and photon tunneling: a proposed experiment to measure photon tunneling times,” Physica B 175, 257–262(1991).
[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]

Other (1)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Tunneling time (or group delay) τ of a pulse (solid curve) is defined by a time at which the transmitted pulse appears at z = L (L, width of superlattice) when the peak of incident pulse passes at z = 0 for t = 0 . The dashed curve denotes the decay of pulse height.

Fig. 2
Fig. 2

Unit cell (bottom) for an N periodic multilayered optical superlattice (top). d A and d C are the lengths of each layer. A i , A r , and A t denote the incident, reflecting, and transmitting amplitudes of the light, respectively. A j ( + ) and A j ( ) denote right and left going waves at the jth interface, respectively.

Fig. 3
Fig. 3

(a) Transmittance T N ( k ) near the center of photonic bandgap k = k B , and (b) transmittance T N ( k B ) as a function of layer number N. Here we adopted parameters n A = 2.0 , n C = 1.0 ( K = 5 / 4 ), λ B = 500 [ nm ] , k B = 2 π / λ B , and δ A = δ C = π / 2 . T N ( k ) was a minimum at k = k B and T N ( k B ) decreases exponentially [ T N 4 N from Eq. (30), solid curve] with increasing numbers of period N.

Fig. 4
Fig. 4

(a) Wavenumber k dependence of tunneling time τ N ( k ) in the case of N = 1 , 2, 3, 4, 5, and 10 by using Eq. (31). This picture is plotted using the same parameters as in Fig. 3. The calculated tunneling time becomes minimum around k = k B ( k B = 2 π / λ B , λ B = 500 [ nm ] ). Tunneling time of k = k B converges on a finite value with increasing numbers of layer N. (b) Layer number N dependence of tunneling time τ (solid squares). τ N ( k B ) converges exponentially to a finite value τ ( k B ) given by Eq. (32) with increasing N (see τ ( k B ) τ N ( k B ) τ ( k B ) , open circles correspond to the right axis). For the case of n A = 2.0 and n C = 1.0 , τ ( k B ) is 1.25 [ fs ] .

Fig. 5
Fig. 5

(a) Time dependence of transmitted pulse amplitude at the right end of the superlattice, | A T ( z = L , t ) | 2 , in the case of n A = 1.0 , 1.2, 1.4, 1.6, 1.8, and 2.0 for N = 10 and n C = 1.0 by Eq. (1) with σ = 10 14 / c , λ B = 500 [ nm ] , and k B = 2 π / λ B . (b) Numerically calculated time that the pulse arrives at z = L is plotted in the case from n A = 1.0 to 2.0 for N = 10 and n C = 1.0 . The solid curve is the analytical value given by Eq. (31).

Equations (34)

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A t ( z , t ) = a 0 d k t ( k ) e i k ( z L ) i ω t e ( k k 0 2 σ ) 2 ,
t ( k ) = T ( k ) e i ϕ ( k ) ,
A t ( z , t ) = a 0 t ( k 0 ) e i k 0 ( z L c t ) e β 2 α 2 d k e α 2 4 [ ( k k 0 ) 2 i β α 2 ] 2 ,
α 2 = 1 σ 2 2 [ ln T ( k 0 ) ] 2 i ϕ ( k 0 ) ,
β = ( z L ) c t + ϕ ( k 0 ) i [ ln T ( k 0 ) ] .
arg [ α 2 e 2 i θ s ] = 0 ,
A t ( z , t ) = A t 0 e β 2 α 2 e i k 0 ( z L c t ) ,
A t 0 = 2 a 0 π t ( k 0 ) e i θ s | α | .
e β 2 α 2 = exp [ α r β r + α i β i α r 2 + α i 2 i α i β r + α r β i α r 2 + α i 2 ] .
| A t ( z , t ) | 2 = | A t 0 | 2 e 2 ( α r β r + α i β i ) α r 2 + α i 2 .
( 2 Φ ( z , t ) z 2 ) z = z p = 1 σ 2 2 [ ln T ( k 0 ) ] > 0.
( Φ ( z , t ) z ) z = z p = 0 ,
z p ( t ) = L + c t ϕ ( k 0 ) 2 ϕ ( k 0 ) [ ln T ( k 0 ) ] 1 σ 2 2 [ ln T ( k 0 ) ] .
τ = 1 c ( ϕ ( k 0 ) + 2 ϕ ( k 0 ) [ ln T ( k 0 ) ] 1 σ 2 2 [ ln T ( k 0 ) ] ) .
τ = ϕ ( k 0 ) / c .
( A i + 1 ( + ) A i + 1 ( ) ) = M ( A i ( + ) A i ( ) ) ,
M = ( ( cos δ A + i K sin δ A ) e i δ C i K ¯ sin δ A i K ¯ sin δ A ( cos δ A i K sin δ A ) e i δ C ) ,
K = 1 2 ( n A n C + n C n A ) , K ¯ = 1 2 ( n A n C n C n A ) .
r ( k ) = M 12 / M 22 , t ( k ) = det [ M ] / M 22 ,
M 2 = 1 2 ( Tr M ) M det ( M ) I ,
M N = Ψ N M Ψ N 1 I ,
Ψ N ( k ) sin N Φ ( k ) sin Φ ( k ) ,
cos Φ ( k ) = cos δ A ( k ) cos δ C ( k ) K sin δ A ( k ) sin δ C ( k ) .
t N ( k ) = [ Ψ N ( k ) ( cos δ A ( k ) i K sin δ A ( k ) e i δ C ( k ) ) Ψ N 1 ( k ) ] 1 .
T N ( k ) = [ Ψ N ( k ) 2 ( cos 2 δ A ( k ) + K 2 sin 2 δ A ( k ) ) 2 Ψ N ( k ) Ψ N 1 ( k ) cos Φ ( k ) + Ψ N 1 ( k ) 2 ] 1 2 ,
e i ϕ N ( k ) = T N ( k ) [ Ψ N ( k ) { cos δ A ( k ) i K sin δ A ( k ) } e i δ C ( k ) Ψ N 1 ( k ) ] .
T N ( k B ) = 1 | K Ψ N ( k B ) + Ψ N 1 ( k B ) | 2 ,
Ψ N ( k B ) Ψ N 1 ( k B ) = 2 K 1 Ψ N 1 ( k B ) Ψ N 2 ( k B ) .
lim N Ψ N ( k B ) Ψ N 1 ( k B ) = K K 2 1 ,
T N ( k B ) T N 1 ( k B ) = | Ψ N 1 ( k B ) Ψ N ( k B ) K + Ψ N 2 ( k B ) Ψ N 1 ( k B ) K + Ψ N 1 ( k B ) Ψ N ( k B ) | 2 ,
lim N T N ( k B ) T N 1 ( k B ) = 1 ( K + K 2 1 ) 2 .
T N ( k B ) ( K + K 2 1 ) 2 N .
τ N ( k B ) = ϕ ( k B ) c = λ B 4 c 1 + K K + Ψ N 1 ( k B ) Ψ N ( k B ) .
τ ( k B ) = λ B ( 1 + K ) 4 c K + K 2 1 K 2 + K K 2 1 1 .

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