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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 25 — Dec. 16, 2002
  • pp: 1491–1496

Energy storage in superluminal barrier tunneling: Origin of the “Hartman effect”

Herbert G. Winful  »View Author Affiliations

Optics Express, Vol. 10, Issue 25, pp. 1491-1496 (2002)

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We show that the anomalously short delay times observed in barrier tunneling have their origin in energy storage and its subsequent release. The observed group delay is proportional to the energy stored. This delay is not a propagation delay and should not be linked to a velocity since evanescent waves do not propagate. The “Hartman effect”, in which the group delay becomes independent of thickness for opaque barriers, is shown to be a consequence of the saturation of stored energy with barrier length.

© 2002 Optical Society of America

OCIS Codes
(230.1480) Optical devices : Bragg reflectors
(240.7040) Optics at surfaces : Tunneling
(260.2110) Physical optics : Electromagnetic optics
(260.2160) Physical optics : Energy transfer
(350.2460) Other areas of optics : Filters, interference

ToC Category:
Research Papers

Original Manuscript: December 2, 2002
Revised Manuscript: December 7, 2002
Published: December 16, 2002

Herbert Winful, "Energy storage in superluminal barrier tunneling: Origin of the Hartman effect," Opt. Express 10, 1491-1496 (2002)

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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. Th. Martin and R. Landauer, �??Time delay of evanescent electromagnetic waves and analogy to particle tunneling,�?? Phys. Rev. A 45, 2611-2617 (1992). [CrossRef] [PubMed]
  4. 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]
  5. R. Y. Chiao and A. M. Steinberg, �??Tunneling times and superluminality,�?? Progress in Optics 37, 347-406 (E. Wolf, ed., Elsevier, Amsterdam, 1997). [CrossRef]
  6. 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]
  7. Ch. Spielmann, R. Szipocs, A. Stingl, and F. Krausz, �??Tunneling of optical pulses through photonic band gaps,�?? Phys. Rev. Lett. 73, 2308-2311 (1994). [CrossRef] [PubMed]
  8. S. Longhi, M. Marano, P. Laporta, and M. Belmonte, �??Superluminal optical pulse propagation at 1.5 m in periodic fiber Bragg gratings,�?? Phys. Rev. E 64, 055602(R) 1-4 (2001). [CrossRef]
  9. A. Enders and G. Nimtz, �??Evanescent-mode propagation and quantum tunneling,�?? Phys. Rev. E 48, 632-634 (1993). [CrossRef]
  10. Y. Japha and G. Kurizki, �??Superluminal delays of coherent pulses in nondissipative media: a universal mechanism,�?? Phys. Rev. A 53, 586-590 (1996). [CrossRef] [PubMed]
  11. R. Landauer, �??Light faster than light?�?? Nature 365, 692-693 (1993). [CrossRef]
  12. H. G. Winful, �??The nature of �??superluminal�?? barrier tunneling,�?? Phys. Rev. Lett., to be published. [PubMed]
  13. C. G. Montgomery, R. H. Dicke, and E. M. Purcell, Principles of Microwave Circuits (McGraw-Hill, New York, 1948).
  14. G. Kishi and K. Nakazawa, �??Relations between reactive energy and group delay in lumped-constant networks,�?? IEEE Trans. Circuit Theory, CT-10, 67-71 (1963).
  15. H. J. Carlin, �??Network theory without circuit elements,�?? Proc. IEEE 55, 482-497 (1967). [CrossRef]
  16. G. D�??Aguanno, et al, �??Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures,�?? Phys. Rev. E 63, 036610, 1-3 (2001). [CrossRef]
  17. F. T. Smith, �??Lifetime matrix in collision theory,�?? Phys. Rev. 118, 349-356 (1960). [CrossRef]
  18. C. G. B. Garrett and D. E. McCumber, �??Propagation of a Gaussian light pulse through an anomalous dispersion medium,�?? Phys. Rev. A 1, 305-313 (1970). [CrossRef]
  19. G. Diener, �??Energy transport in dispersive media and superluminal group velocities,�?? Phys. Lett. A 235, 118-124 (1997). [CrossRef]
  20. M. Ware, S. A. Glasgow, and J. Peatross, �??Role of group velocity in tracking field energy in linear dielectrics,�?? Opt. Express 9, 506-518 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-10-506">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-10-506</a> [CrossRef] [PubMed]
  21. T. Emig, "Propagation of an electromagnetic pulse through a waveguide with a barrier: A time domain solution within classical electrodynamics,�?? Phys. Rev. E 54, 5780-5787 (1996). [CrossRef]
  22. J. M. Deutch and F. E. Low, �??Barrier penetration and superluminal velocity,�?? Ann. Phys. (NY) 210, 184-202 (1993). [CrossRef]
  23. A. D. Jackson, A. Lande, and B. Lautrup, �??Apparent superluminal behavior in wave propagation,�?? Phys. Rev. A 64, 044101, 1-4 (2001). [CrossRef]

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