Abstract

Tunneling times in absorptive and dispersive media are considered, as are the relations between them. Group delay and dwell time are used as the most appropriate tunneling time characterizations. A general expression that relates these two quantities, valid for all linear media (with both positive and negative index of refraction), is derived, but particular attention is given to negative index metamaterials. The example of a nonmagnetic, lossless medium with dispersive surroundings was chosen to illustrate the derivation of self-interference time. Existence of the Hartman effect and negative group delay in a certain range of frequencies, in metamaterials, is numerically verified.

© 2008 Optical Society of America

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References

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  1. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  2. U. K. Chettiar, A. V. Kildishev, H.-K. Yuan, W. Cai, S. Xiao, V. P. Drachev, and V. M. Shalaev, “Dual-band negative index metamaterial: Double negative at 813 nm and single-negative at 772 nm,” Opt. Lett. 32, 1671-1673 (2007).
    [CrossRef] [PubMed]
  3. A. Enders and G. Nimtz, “Evanescent-mode propagation and quantum tunneling,” Phys. Rev. E 48, 632-634 (1993).
    [CrossRef]
  4. Ch. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308-2311 (1994).
    [CrossRef] [PubMed]
  5. J. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E 70, 046603 (2004).
    [CrossRef]
  6. E. H. Hauge and J. A. Støvneng, “Tunneling times: A critical review,” Rev. Mod. Phys. 61, 917-936 (1989).
    [CrossRef]
  7. F. T. Smith, “Lifetime matrix in collision theory,” Phys. Rev. 118, 349-356 (1960).
    [CrossRef]
  8. M. Büttiker, “Larmor precession and the traversal time for tunneling,” Phys. Rev. B 27, 6178-6188 (1983).
    [CrossRef]
  9. H. G. Winful, “Group delay, stored energy and tunneling of evanescent waves,” Phys. Rev. E 68, 016615 (2003).
    [CrossRef]
  10. R. Ruppin, “Electromagnetic energy density in a adispersive and absorptive material,” Phys. Lett. 299, 309-312 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. C.-F. Li, “Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects,” Phys. Rev. Lett. 91, 133903 (2003).
    [CrossRef] [PubMed]
  14. T. E. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. 33, 3427-3433 (1962).
    [CrossRef]
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2008

G. Isić, V. Milanović, J. Radovanović, Z. Ikonić, D. Indjin, and P. Harrison, “Time delay in thin slabs with self-focusing Kerr-type nonlinearity,” Phys. Rev. A 77, 033821 (2008).
[CrossRef]

2007

2004

J. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E 70, 046603 (2004).
[CrossRef]

2003

H. G. Winful, “Group delay, stored energy and tunneling of evanescent waves,” Phys. Rev. E 68, 016615 (2003).
[CrossRef]

C.-F. Li, “Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects,” Phys. Rev. Lett. 91, 133903 (2003).
[CrossRef] [PubMed]

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

2002

R. Ruppin, “Electromagnetic energy density in a adispersive and absorptive material,” Phys. Lett. 299, 309-312 (2002).
[CrossRef]

2001

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[CrossRef] [PubMed]

1994

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

1993

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

1989

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

1983

M. Büttiker, “Larmor precession and the traversal time for tunneling,” Phys. Rev. B 27, 6178-6188 (1983).
[CrossRef]

1965

1962

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

1960

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

Büttiker, M.

M. Büttiker, “Larmor precession and the traversal time for tunneling,” Phys. Rev. B 27, 6178-6188 (1983).
[CrossRef]

Cai, W.

Chettiar, U. K.

Drachev, V. P.

Enders, A.

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

Harrison, P.

G. Isić, V. Milanović, J. Radovanović, Z. Ikonić, D. Indjin, and P. Harrison, “Time delay in thin slabs with self-focusing Kerr-type nonlinearity,” Phys. Rev. A 77, 033821 (2008).
[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]

Ikonic, Z.

G. Isić, V. Milanović, J. Radovanović, Z. Ikonić, D. Indjin, and P. Harrison, “Time delay in thin slabs with self-focusing Kerr-type nonlinearity,” Phys. Rev. A 77, 033821 (2008).
[CrossRef]

Indjin, D.

G. Isić, V. Milanović, J. Radovanović, Z. Ikonić, D. Indjin, and P. Harrison, “Time delay in thin slabs with self-focusing Kerr-type nonlinearity,” Phys. Rev. A 77, 033821 (2008).
[CrossRef]

Isic, G.

G. Isić, V. Milanović, J. Radovanović, Z. Ikonić, D. Indjin, and P. Harrison, “Time delay in thin slabs with self-focusing Kerr-type nonlinearity,” Phys. Rev. A 77, 033821 (2008).
[CrossRef]

Kildishev, A. V.

Krausz, F.

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

Li, C.-F.

C.-F. Li, “Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects,” Phys. Rev. Lett. 91, 133903 (2003).
[CrossRef] [PubMed]

Malitson, I. H.

Milanovic, V.

G. Isić, V. Milanović, J. Radovanović, Z. Ikonić, D. Indjin, and P. Harrison, “Time delay in thin slabs with self-focusing Kerr-type nonlinearity,” Phys. Rev. A 77, 033821 (2008).
[CrossRef]

Mojahedi, M.

J. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E 70, 046603 (2004).
[CrossRef]

Nimtz, G.

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

Radovanovic, J.

G. Isić, V. Milanović, J. Radovanović, Z. Ikonić, D. Indjin, and P. Harrison, “Time delay in thin slabs with self-focusing Kerr-type nonlinearity,” Phys. Rev. A 77, 033821 (2008).
[CrossRef]

Ruppin, R.

R. Ruppin, “Electromagnetic energy density in a adispersive and absorptive material,” Phys. Lett. 299, 309-312 (2002).
[CrossRef]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Shalaev, V. M.

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Smith, F. T.

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

Spielmann, Ch.

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

Stingl, A.

Ch. Spielmann, R. Szipöcs, 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]

Szipöcs, R.

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

Winful, H. G.

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

H. G. Winful, “Group delay, stored energy and tunneling of evanescent waves,” Phys. Rev. E 68, 016615 (2003).
[CrossRef]

Woodley, J.

J. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E 70, 046603 (2004).
[CrossRef]

Xiao, S.

Yuan, H.-K.

J. Appl. Phys.

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

J. Opt. Soc. Am.

Opt. Lett.

Phys. Lett.

R. Ruppin, “Electromagnetic energy density in a adispersive and absorptive material,” Phys. Lett. 299, 309-312 (2002).
[CrossRef]

Phys. Rev.

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

Phys. Rev. A

G. Isić, V. Milanović, J. Radovanović, Z. Ikonić, D. Indjin, and P. Harrison, “Time delay in thin slabs with self-focusing Kerr-type nonlinearity,” Phys. Rev. A 77, 033821 (2008).
[CrossRef]

Phys. Rev. B

M. Büttiker, “Larmor precession and the traversal time for tunneling,” Phys. Rev. B 27, 6178-6188 (1983).
[CrossRef]

Phys. Rev. E

H. G. Winful, “Group delay, stored energy and tunneling of evanescent waves,” Phys. Rev. E 68, 016615 (2003).
[CrossRef]

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

J. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E 70, 046603 (2004).
[CrossRef]

Phys. Rev. Lett.

Ch. Spielmann, R. Szipöcs, 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, “Delay time and the Hartman effect in quantum tunneling,” Phys. Rev. Lett. 91, 260401 (2003).
[CrossRef]

C.-F. Li, “Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects,” Phys. Rev. Lett. 91, 133903 (2003).
[CrossRef] [PubMed]

Rev. Mod. Phys.

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

Science

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

The model.

Fig. 2
Fig. 2

(a) The dwell time and (b) the group delay dependences on incident radiation frequency for three different incident angles ( θ = 0 , π 12 , and π 6 ).

Fig. 3
Fig. 3

(a) Absorption, dwell time, (b) group delay and the real part of the refractive index versus incident light frequency for a left-handed obstacle within the visible frequency range.

Fig. 4
Fig. 4

Self-interference time versus incident light frequency.

Equations (21)

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ε s = ε ( 1 ω p 2 ω 2 ω r 2 + i ω Γ e ) ,
μ s = 1 F ω 0 2 ω 2 ω 0 2 + i ω Γ m .
τ d = W P in ,
ω = ε 0 4 ε eff E 2 + μ 0 4 μ eff H 2 ,
W = ε 0 S 4 ( ε μ ) eff 0 L E y 2 d x S μ eff γ 0 2 ω 2 μ s 2 μ 0 Im ( R ) E 0 2 S μ eff 4 ω 2 μ s 2 μ 0 K 1 ,
( ε μ ) eff = ε eff + μ eff Re ( ε s μ s ) μ s 2 ,
K 1 = Re { ( μ s μ b ) [ E y * E y μ ( x ) ] x = 0 + ( μ b μ s ) [ E y * E y μ ( x ) ] x = L } ,
0 L E y 2 d x = [ 2 γ 0 τ g + 2 d γ 0 d ω Im ( R ) ] 1 k 0 2 d k 0 2 d ω Re ( γ s 2 ) + d Re ( γ s 2 ) d ω E 0 2 2 k 0 2 0 L Im ( ε s μ s ) Im ( d E y * d ω E y ) d x K 2 1 k 0 2 d k 0 2 d ω Re ( γ s 2 ) + d Re ( γ s 2 ) d ω ,
τ g = T 2 d φ 0 d ω + R 2 d φ r d ω ,
K 2 = Re { ( μ s μ b ) E y μ ( x ) d E y * d ω d d ω [ ( μ s μ b ) E y μ ( x ) ] E y * } x = 0 + Re { ( μ b μ s ) E y μ ( x ) d E y * d ω d d ω [ ( μ b μ s ) E y μ ( x ) ] E y * } x = L ,
A = 1 γ 0 E 0 2 0 L [ Im ( γ s 2 ) E y 2 Im ( μ μ E y E y * ) ] d x ,
P in = γ 0 ε 0 c 2 S 2 ω E 0 2 .
τ d = τ g + 1 γ 0 d γ 0 d ω Im ( R ) K 3 μ eff ω μ s 2 Im ( R ) + K 2 + 2 k 0 2 0 L Im ( ε s μ s ) Im ( d E y * d ω E y ) d x 2 γ 0 c 2 E 0 2 ω ( ε μ ) eff ( 1 k 0 2 d k 0 2 d ω Re ( γ s 2 ) + d Re ( γ s 2 ) d ω ) μ eff 2 ω γ 0 μ s 2 E 0 2 K 1 ,
K 3 = 1 ( ε μ ) eff ( 2 K 4 + ω d K 4 d ω ) ,
K 4 = Re ( ε s μ s ε b μ b sin 2 θ ) .
τ g = τ d + τ i
d 2 E y d x 2 1 μ d μ d x d E y d x + ( k 0 2 ε μ β 2 ) E y = 0 .
W m = S μ eff 4 ω 2 μ s 2 μ 0 [ β 2 0 L E y 2 d x + 0 L d E y d x 2 d x ] .
0 L d E y d x 2 d x = ( 2 γ 0 Im ( R ) + i γ 0 A ) E 0 2 0 L E y d 2 E y * d x 2 d x ,
W m = S μ eff 4 ω 2 μ s 2 μ 0 [ 2 γ 0 Im ( R ) E 0 2 + k 0 2 0 L Re ( ε μ ) E y 2 d x 0 L Re ( E y ( μ μ E y ) * ) d x ] ,
W e = ε 0 ε eff S 4 0 L E y 2 d x .

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