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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 16 — Aug. 1, 2011
  • pp: 15281–15296

Non-local energy transport in tunneling and plasmonic structures

Winston Frias, Andrei Smolyakov, and Akira Hirose  »View Author Affiliations

Optics Express, Vol. 19, Issue 16, pp. 15281-15296 (2011)

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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

OCIS Codes
(240.0310) Optics at surfaces : Thin films
(240.6690) Optics at surfaces : Surface waves
(240.7040) Optics at surfaces : Tunneling

ToC Category:
Optics at Surfaces

Original Manuscript: March 4, 2011
Revised Manuscript: June 22, 2011
Manuscript Accepted: July 5, 2011
Published: July 26, 2011

Winston Frias, Andrei Smolyakov, and Akira Hirose, "Non-local energy transport in tunneling and plasmonic structures," Opt. Express 19, 15281-15296 (2011)

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  1. R. Chiao and A. Steinberg, Progress in Optics , E. Wolf, ed. (Elsevier, 1997).
  2. A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994). [CrossRef] [PubMed]
  3. 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]
  4. L. Ragni, “Group delay of evanescent signals in a waveguide with barrier,” Phys. Rev. E 79, 046609 (2009). [CrossRef]
  5. M. T. Reiten, D. Grischkowsky, and R. A. Cheville, “Optical tunneling of single-cycle terahertz bandwidth pulses,” Phys. Rev. E 64, 036604 (2001). [CrossRef]
  6. A. Enders and G. Nimtz, “Evanescent-mode propagation and quantum tunneling,” Phys. Rev. E 48, 632–634 (1993). [CrossRef]
  7. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–463 (2008). [CrossRef]
  8. J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon. 2, 287–318 (2010). [CrossRef]
  9. R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009). [CrossRef] [PubMed]
  10. 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]
  11. 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]
  12. 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]
  13. C. Ling, M. Zheng, and K. Yu, “Slowing light in diatomic nanoshelled chains,” Opt. Commun. 283, 1945–1949 (2010). [CrossRef]
  14. C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. 12, 104003 (2010). [CrossRef]
  15. L. Brillouin and A. Sommerfeld, Wave Propagation and Group Velocity (Academic Press, 1960).
  16. E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955). [CrossRef]
  17. P. Yeh and A. Yariv, Optical Waves in Crystals (Wiley-Interscience, 1984).
  18. 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]
  19. 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]
  20. 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]
  21. T. Sauter and F. Paschke, “Can Bessel beams carry superluminal signals?” Phys. Lett. A 285, 1–6 (2001). [CrossRef]
  22. A. M. Steinberg, “How much time does a tunneling particle spend in the barrier region?” Phys. Rev. Lett. 74, 2405–2409 (1995). [CrossRef] [PubMed]
  23. 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]
  24. 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]
  25. 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]
  26. D. Mugnai, A. Ranfagni, and R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000). [CrossRef] [PubMed]
  27. L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000). [CrossRef] [PubMed]
  28. H. G. Winful, “Apparent superluminality and the generalized hartman effect in double-barrier tunneling,” Phys. Rev. E 72, 046608 (2005). [CrossRef]
  29. H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. 436, 1–69 (2006). [CrossRef]
  30. 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]
  31. 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]
  32. 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]
  33. T. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. Rep. 33, 3427–3433 (1962). [CrossRef]
  34. H. G. Winful, “Delay time and the Hartman effect in quantum tunneling,” Phys. Rev. Lett. 91, 260401 (2003). [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). [CrossRef]
  40. M. Razavy, Quantum Theory of Tunneling (WorldScientific, 2003). [CrossRef]
  41. Y. Chen and S. Blair, “Nonlinearity enhancement in finite coupled-resonator slow-light waveguides,” Opt. Express 12, 3353–3366 (2004). [CrossRef] [PubMed]

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