Abstract

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

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

Ann. Phys.

J. M. Deutch and F. E. Low, �??Barrier penetration and superluminal velocity,�?? Ann. Phys. (NY) 210, 184-202 (1993).
[CrossRef]

IEEE Trans. Circuit Theory

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).

J. Appl. Phys.

T. E. Hartman, �??Tunneling of a wave packet,�?? J. Appl. Phys. 33, 3427-3433 (1962).
[CrossRef]

Nature

R. Landauer, �??Light faster than light?�?? Nature 365, 692-693 (1993).
[CrossRef]

Opt. Express

Phys. Lett. A

G. Diener, �??Energy transport in dispersive media and superluminal group velocities,�?? Phys. Lett. A 235, 118-124 (1997).
[CrossRef]

Phys. Rev.

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

F. T. Smith, �??Lifetime matrix in collision theory,�?? Phys. Rev. 118, 349-356 (1960).
[CrossRef]

Phys. Rev. A

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]

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]

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]

A. D. Jackson, A. Lande, and B. Lautrup, �??Apparent superluminal behavior in wave propagation,�?? Phys. Rev. A 64, 044101, 1-4 (2001).
[CrossRef]

Phys. Rev. E

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]

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]

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]

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

Phys. Rev. Lett.

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. Szipocs, A. Stingl, and F. Krausz, �??Tunneling of optical pulses through photonic band gaps,�?? Phys. Rev. Lett. 73, 2308-2311 (1994).
[CrossRef] [PubMed]

H. G. Winful, �??The nature of �??superluminal�?? barrier tunneling,�?? Phys. Rev. Lett., to be published.
[PubMed]

Physica B

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]

Proc. IEEE

H. J. Carlin, �??Network theory without circuit elements,�?? Proc. IEEE 55, 482-497 (1967).
[CrossRef]

Progress in Optics

R. Y. Chiao and A. M. Steinberg, �??Tunneling times and superluminality,�?? Progress in Optics 37, 347-406 (E. Wolf, ed., Elsevier, Amsterdam, 1997).
[CrossRef]

Other

C. G. Montgomery, R. H. Dicke, and E. M. Purcell, Principles of Microwave Circuits (McGraw-Hill, New York, 1948).

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Figures (1)

Fig. 1
Fig. 1

Schematic of a photonic bandgap structure (PBG).

Equations (22)

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n = n 0 + n 1 cos ( 2 β 0 z ) ,
E z t = Re [ E F z t e i ( β 0 z ω 0 t ) + E B z t e i ( β 0 z + ω 0 t ) ]
E F z + 1 v E F t = E B
E B z 1 v E B t = E F ,
2 E F z 2 1 v 2 2 E F t 2 = κ 2 E F .
K 2 = ( Ω 2 Ω c 2 ) v 2 ,
exp ( γz ) exp ( i Ω t )
d E F dz = E B + i ( Ω v ) E F ,
d E B dz = E F i ( Ω v ) E B .
E F ( z ) = E 0 [ γ cosh γ ( z L ) + i ( Ω v ) sinh γ ( z l ) ] g ,
E B ( z ) = i [ E 0 κ sinh γ ( z L ) ] g ,
ϕ t = tan 1 [ ( Ω γv ) tanh γL ] .
ϕ r = ϕ t + π 2 .
U = 1 2 ε vol [ E F 2 + E B 2 ] dv ,
U = ( 1 2 ε E 0 2 A ) [ κ 2 γ 2 tanh γL γ L ( Ω γv ) 2 sech 2 γL ] cos 2 ϕ t ,
τ g = d ϕ t d Ω .
τ g = 1 v [ κ 2 γ 2 tanh γL γ L ( Ω γv ) 2 sech 2 γL ] cos 2 ϕ t .
U = ( 1 2 ε E 0 2 Av ) τ g .
U = P in τ g .
τ D = U P in
τ D = ( R 2 d ϕ r d Ω + T 2 d ϕ t d Ω ) ,
τ g = τ D = d ϕ d Ω ,

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