Abstract

It is shown that the time needed for light to pass through the optical barrier associated with an antiresonant quarter-wave-stack dielectric mirror, as measured by the group-delay, or phase time, asymptotically reaches a limit that is independent of the barrier thickness and hence of the number of layers. This limit, which scales as the inverse of the refractive-index difference between successive layers, is equal to the mean value of the asymptotic group delays needed for light to reflect off each side of the barrier. This superluminal transmission does not violate causality, as the transmitted intensity is always lower than the intensity that would have been transmitted in vacuum in the absence of the barrier.

© 1999 Optical Society of America

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  1. L. A. Mac Coll, Phys. Rev. 40, 621 (1932).
    [CrossRef]
  2. T. E. Hartman, J. Appl. Phys. 33, 3427 (1962).
    [CrossRef]
  3. A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. Lett. 71, 708 (1993).
    [CrossRef] [PubMed]
  4. Ch. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, Phys. Rev. Lett. 73, 2308 (1994).
    [CrossRef] [PubMed]
  5. M. Büttiker and R. Landauer, Phys. Rev. Lett. 49, 1739 (1982).
    [CrossRef]
  6. M. Büttiker, Phys. Rev. B 27, 6178 (1983).
    [CrossRef]
  7. R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, Physica B 175, 257 (1991).
    [CrossRef]
  8. B. Lee and W. Lee, J. Opt. Soc. Am. B 14, 777 (1997).
    [CrossRef]
  9. A. Enders and G. Nimtz, J. Phys. I (France) 3, 1089 (1993).
    [CrossRef]
  10. P. Tournois, IEEE J. Quantum Electron. 33, 519 (1997).
    [CrossRef]
  11. Wang Yun-ping and Zhang Dian-lin, Phys. Rev. A 52, 2597 (1995).
    [CrossRef] [PubMed]
  12. F. Abelès, Ann. Phys. (Paris) 5, 596 and 706 (1950).
  13. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).
  14. A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 49, 3283 (1994).
    [CrossRef] [PubMed]
  15. A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 51, 3525 (1995).
    [CrossRef] [PubMed]
  16. E. H. Hauge and J. A. Støvneng, Rev. Mod. Phys. 61, 917 (1989).
    [CrossRef]
  17. R. Landauer and Th. Martin, Rev. Mod. Phys. 66, 217 (1994).
    [CrossRef]
  18. R. Y. Chiao and A. M. Steinberg, in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1997), Vol. XXXVII, p. 347.
  19. G. Nimtz and W. Heitmann, Prog. Quantum Electron. 21, 81 (1997).
    [CrossRef]

1997 (3)

B. Lee and W. Lee, J. Opt. Soc. Am. B 14, 777 (1997).
[CrossRef]

P. Tournois, IEEE J. Quantum Electron. 33, 519 (1997).
[CrossRef]

G. Nimtz and W. Heitmann, Prog. Quantum Electron. 21, 81 (1997).
[CrossRef]

1995 (2)

A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 51, 3525 (1995).
[CrossRef] [PubMed]

Wang Yun-ping and Zhang Dian-lin, Phys. Rev. A 52, 2597 (1995).
[CrossRef] [PubMed]

1994 (3)

A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 49, 3283 (1994).
[CrossRef] [PubMed]

R. Landauer and Th. Martin, Rev. Mod. Phys. 66, 217 (1994).
[CrossRef]

Ch. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, Phys. Rev. Lett. 73, 2308 (1994).
[CrossRef] [PubMed]

1993 (2)

A. Enders and G. Nimtz, J. Phys. I (France) 3, 1089 (1993).
[CrossRef]

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. Lett. 71, 708 (1993).
[CrossRef] [PubMed]

1991 (1)

R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, Physica B 175, 257 (1991).
[CrossRef]

1989 (1)

E. H. Hauge and J. A. Støvneng, Rev. Mod. Phys. 61, 917 (1989).
[CrossRef]

1983 (1)

M. Büttiker, Phys. Rev. B 27, 6178 (1983).
[CrossRef]

1982 (1)

M. Büttiker and R. Landauer, Phys. Rev. Lett. 49, 1739 (1982).
[CrossRef]

1962 (1)

T. E. Hartman, J. Appl. Phys. 33, 3427 (1962).
[CrossRef]

1950 (1)

F. Abelès, Ann. Phys. (Paris) 5, 596 and 706 (1950).

1932 (1)

L. A. Mac Coll, Phys. Rev. 40, 621 (1932).
[CrossRef]

Abelès, F.

F. Abelès, Ann. Phys. (Paris) 5, 596 and 706 (1950).

Büttiker, M.

M. Büttiker, Phys. Rev. B 27, 6178 (1983).
[CrossRef]

M. Büttiker and R. Landauer, Phys. Rev. Lett. 49, 1739 (1982).
[CrossRef]

Chiao, R. Y.

A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 51, 3525 (1995).
[CrossRef] [PubMed]

A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 49, 3283 (1994).
[CrossRef] [PubMed]

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. Lett. 71, 708 (1993).
[CrossRef] [PubMed]

R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, Physica B 175, 257 (1991).
[CrossRef]

Dian-lin, Zhang

Wang Yun-ping and Zhang Dian-lin, Phys. Rev. A 52, 2597 (1995).
[CrossRef] [PubMed]

Enders, A.

A. Enders and G. Nimtz, J. Phys. I (France) 3, 1089 (1993).
[CrossRef]

Hartman, T. E.

T. E. Hartman, J. Appl. Phys. 33, 3427 (1962).
[CrossRef]

Hauge, E. H.

E. H. Hauge and J. A. Støvneng, Rev. Mod. Phys. 61, 917 (1989).
[CrossRef]

Heitmann, W.

G. Nimtz and W. Heitmann, Prog. Quantum Electron. 21, 81 (1997).
[CrossRef]

Krausz, F.

Ch. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, Phys. Rev. Lett. 73, 2308 (1994).
[CrossRef] [PubMed]

Kwiat, P. G.

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. Lett. 71, 708 (1993).
[CrossRef] [PubMed]

R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, Physica B 175, 257 (1991).
[CrossRef]

Landauer, R.

R. Landauer and Th. Martin, Rev. Mod. Phys. 66, 217 (1994).
[CrossRef]

M. Büttiker and R. Landauer, Phys. Rev. Lett. 49, 1739 (1982).
[CrossRef]

Lee, B.

Lee, W.

Mac Coll, L. A.

L. A. Mac Coll, Phys. Rev. 40, 621 (1932).
[CrossRef]

Martin, Th.

R. Landauer and Th. Martin, Rev. Mod. Phys. 66, 217 (1994).
[CrossRef]

Nimtz, G.

G. Nimtz and W. Heitmann, Prog. Quantum Electron. 21, 81 (1997).
[CrossRef]

A. Enders and G. Nimtz, J. Phys. I (France) 3, 1089 (1993).
[CrossRef]

Spielmann, Ch.

Ch. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, Phys. Rev. Lett. 73, 2308 (1994).
[CrossRef] [PubMed]

Steinberg, A. M.

A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 51, 3525 (1995).
[CrossRef] [PubMed]

A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 49, 3283 (1994).
[CrossRef] [PubMed]

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. Lett. 71, 708 (1993).
[CrossRef] [PubMed]

R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, Physica B 175, 257 (1991).
[CrossRef]

Stingl, A.

Ch. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, Phys. Rev. Lett. 73, 2308 (1994).
[CrossRef] [PubMed]

Støvneng, J. A.

E. H. Hauge and J. A. Støvneng, Rev. Mod. Phys. 61, 917 (1989).
[CrossRef]

Szipöcs, R.

Ch. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, Phys. Rev. Lett. 73, 2308 (1994).
[CrossRef] [PubMed]

Tournois, P.

P. Tournois, IEEE J. Quantum Electron. 33, 519 (1997).
[CrossRef]

Yun-ping, Wang

Wang Yun-ping and Zhang Dian-lin, Phys. Rev. A 52, 2597 (1995).
[CrossRef] [PubMed]

Ann. Phys. (Paris) (1)

F. Abelès, Ann. Phys. (Paris) 5, 596 and 706 (1950).

IEEE J. Quantum Electron. (1)

P. Tournois, IEEE J. Quantum Electron. 33, 519 (1997).
[CrossRef]

J. Appl. Phys. (1)

T. E. Hartman, J. Appl. Phys. 33, 3427 (1962).
[CrossRef]

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

J. Phys. I (France) (1)

A. Enders and G. Nimtz, J. Phys. I (France) 3, 1089 (1993).
[CrossRef]

Phys. Rev. (1)

L. A. Mac Coll, Phys. Rev. 40, 621 (1932).
[CrossRef]

Phys. Rev. A (3)

Wang Yun-ping and Zhang Dian-lin, Phys. Rev. A 52, 2597 (1995).
[CrossRef] [PubMed]

A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 49, 3283 (1994).
[CrossRef] [PubMed]

A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 51, 3525 (1995).
[CrossRef] [PubMed]

Phys. Rev. B (1)

M. Büttiker, Phys. Rev. B 27, 6178 (1983).
[CrossRef]

Phys. Rev. Lett. (3)

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. Lett. 71, 708 (1993).
[CrossRef] [PubMed]

Ch. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, Phys. Rev. Lett. 73, 2308 (1994).
[CrossRef] [PubMed]

M. Büttiker and R. Landauer, Phys. Rev. Lett. 49, 1739 (1982).
[CrossRef]

Physica B (1)

R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, Physica B 175, 257 (1991).
[CrossRef]

Prog. Quantum Electron. (1)

G. Nimtz and W. Heitmann, Prog. Quantum Electron. 21, 81 (1997).
[CrossRef]

Rev. Mod. Phys. (2)

E. H. Hauge and J. A. Støvneng, Rev. Mod. Phys. 61, 917 (1989).
[CrossRef]

R. Landauer and Th. Martin, Rev. Mod. Phys. 66, 217 (1994).
[CrossRef]

Other (2)

R. Y. Chiao and A. M. Steinberg, in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1997), Vol. XXXVII, p. 347.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

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Figures (6)

Fig. 1
Fig. 1

Stack of alternate layers with refractive indices n1 and n2 and with respective thicknesses d1 and d2.

Fig. 2
Fig. 2

Equivalent reflection planes describing the reflection from and transmission through the quarter-wave-stack.

Fig. 3
Fig. 3

Group delays as a function of the number of layers for (a) (2p) and (b) (2p+1) quarter-wave layers, assuming normal incidence, λ0=800 nm, and q=0. Horizontal lines, the asymptotic limits of Eqs. (10)–(15). Transmission is superluminal as soon as tDT is smaller than t0T.

Fig. 4
Fig. 4

Dependence of the asymptotic tunneling time on the angle of incidence for the quarter-wave stack of Fig. 3(a), λ0=800 nm, and q=0.

Fig. 5
Fig. 5

Measured and calculated (*) delay difference Δτ (20° incidence, p polarization, λ0=800 nm and q=0) in the experiment of Spielmann et al.4 Calculated (†) Δτ with the present theory.

Fig. 6
Fig. 6

Numerical simulation of the transmission of a 28-THz-bandwidth Gaussian pulse through barrier (HL)11 of Fig. 5 (20° incidence, p polarization, λ0=800 nm, and q=0).

Equations (21)

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E0++E0-η0(E0+-E0-)=MEsηs Es,
η=n cos θ,spolarizationn/cos θ,ppolarization.
M=AiBiCD,
t=EsE0+=2η0(η0 A+ηs D)+i(η0ηs B+C),
r=E0-E0+=(η0 A-ηs D)+i(η0ηsB-C)(η0 A+ηs D)+i(η0ηs B+C).
M=cos βiηsin βiη sin βcos β,
β=ω(d/c)n cos θ=ωτ,
M=(-1)q-Δωτiηiη-Δωτ.
M=M1M2=-η2/η1i1η1+1η2Δωτi(η1+η2)Δωτη1/η2.
(tDT)even=η0ηs+η1η2ηs(η1-η2)τ,
(tDT)odd=η0+ηsη1-η2τ,
(tDR)even=(tDR)odd=2η0η1-η2τ.
(tDT)even=η0ηs+η1η2η0(η2-η1)τ,
(tDT)odd=(η0+ηs)η1η2η0ηs(η2-η1)τ,
(tDR)even=(tDR)odd=2η1η2η0(η2-η1)τ.
tan ΦT=η0ηs B+Cη0 A+ηs D,
tan(ΦR-ΦT)=-η0ηs B-Cη0 A-ηsD.
M=(M1M2)p=(-1)p(η2/η1)piapη1-η2Δωτiη1η2apη1-η2Δωτ(η1/η2)p,
ap=(η1/η2)p-(η2/η1)p,
M=(M1M2)pM1=(-1)p+q+1bpΔωτ-iη1(η2/η1)p-iη1(η1/η2)pbp Δωτ,
bp=(η1/η2)p+1/2-(η2/η1)p-1/2(η1/η2)1/2-(η2/η1)-1/2.

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