Abstract

We show that the physical clock approach can be applied to the problem of optical pulse transmission in the Fabry–Perot cavity. Our theoretical analysis leads directly to a complex-valued traversal time for the pulse. Real and imaginary parts of the traversal time, referred to as the phase time and the loss time, are associated, respectively, with the rotation angle of polarization and the change in the polarization ellipticity of the outgoing pulse in the presence of a magnetic clock. The physical significance of the phase time and the loss time is discussed in relation to the superluminal group velocity and the spectral shift of the pulse.

© 2000 Optical Society of America

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  1. E. H. Hauge and J. A. Støvneng, “Tunneling times: a critical review,” Rev. Mod. Phys. 61, 917–936 (1989).
    [CrossRef]
  2. R. Landauer and Th. Martin, “Barrier interaction time in tunneling,” Rev. Mod. Phys. 66, 217–228 (1994).
    [CrossRef]
  3. M. Büttiker and R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. 49, 1739–1742 (1982).
    [CrossRef]
  4. A. I. Baz’, “Life-time of intermediate states,” Yad. Fiz. 4, 252–260 (1966).
  5. M. Büttiker, “Larmor precession and the traversal time for tunneling,” Phys. Rev. B 27, 6178–6188 (1983).
    [CrossRef]
  6. V. Gasparian, M. Ortuño, J. Ruiz, and E. Cuevas, “Faraday rotation and complex-valued traversal time for classical light waves,” Phys. Rev. Lett. 75, 2312–2315 (1995).
    [CrossRef] [PubMed]
  7. M. Deutsch and J. E. Golub, “Optical Larmor clock: measurement of the photonic tunneling time,” Phys. Rev. A 53, 434–439 (1996).
    [CrossRef] [PubMed]
  8. Ph. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
    [CrossRef]
  9. E. Pollak and W. H. Miller, “New physical interpretation for time in scattering theory,” Phys. Rev. Lett. 53, 115–118 (1984).
    [CrossRef]
  10. D. Sokolovski and L. M. Baskin, “Traversal time in quantum scattering,” Phys. Rev. A 36, 4604–4611 (1987).
    [CrossRef] [PubMed]
  11. K. L. Jensen and F. A. Buot, “Numerical calculation of particle trajectories and tunneling times for resonant tunneling barrier structures,” Appl. Phys. Lett. 55, 669–671 (1989).
    [CrossRef]
  12. J. G. Muga, S. Brouard, and R. Sala, “Transmission and reflection tunneling times,” Phys. Lett. A 167, 24–28 (1992).
    [CrossRef]
  13. K. L. Jensen and F. A. Buot, “The methodology of simulating particle trajectories through tunneling structures using a Wigner distribution approach,” IEEE Trans. Electron Devices 38, 2337–2347 (1991).
    [CrossRef]
  14. A. Enders and G. Nimtz, “Photonic-tunneling experiments,” Phys. Rev. B 47, 9605–9609 (1993).
    [CrossRef]
  15. R. Pelster, V. Gasparian, and G. Nimtz, “Propagation of plane waves and of waveguide modes in quasiperiodic dielectric heterostructures,” Phys. Rev. E 55, 7645–7655 (1997).
    [CrossRef]
  16. 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]
  17. A. M. Steinberg and R. Y. Chiao, “Subfemtosecond determination of transmission delay times for a dielectric mirror (photonic band gap) as a function of the angle of incidence,” Phys. Rev. A 51, 3525–3528 (1995).
    [CrossRef] [PubMed]
  18. W. M. Robertson, “Transmission-line matrix modeling of superluminal electromagnetic-pulse tunneling through the forbidden gap in two-dimensional photonic band structures,” J. Opt. Soc. Am. B 14, 1066–1073 (1997).
    [CrossRef]
  19. V. Laude and P. Tournois, “Superluminal asymptotic tunneling times through one-dimensional photonic bandgaps in quarter-wave-stack dielectric mirrors,” J. Opt. Soc. Am. B 16, 194–198 (1999).
    [CrossRef]
  20. Th. Martin and R. Landauer, “Time delay of evanescent electromagnetic waves and the analogy to particle tunneling,” Phys. Rev. A 45, 2611–2617 (1992).
    [CrossRef] [PubMed]
  21. A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
    [CrossRef] [PubMed]
  22. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1986).
  23. E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
    [CrossRef]
  24. T. E. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. 33, 3427–3433 (1962).
    [CrossRef]
  25. G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A 223, 327–331 (1996).
    [CrossRef]
  26. 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]
  27. 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]
  28. S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
    [CrossRef]
  29. R. Y. Chiao and J. Boyce, “Superluminality, paraelectricity, and Earnshaw’s theorem in media with inverted populations,” Phys. Rev. Lett. 73, 3383–3386 (1994).
    [CrossRef] [PubMed]
  30. E. L. Bolda, “Theory and simulation of superluminal optical pulses in gain media,” Phys. Rev. A 54, 3514–3518 (1996).
    [CrossRef] [PubMed]
  31. D. Mugnai, A. Ranfagni, and L. S. Schuman, “Delay time measurements in a diffraction experiment: a case of optical tunneling,” Phys. Rev. E 55, 3593–3597 (1977).
    [CrossRef]
  32. R. Y. Chiao, “Atomic coherence effects which produce superluminal (but causal) propagation of wavepackets,” Quantum Opt. 6, 359–369 (1994).
    [CrossRef]

1999 (1)

1997 (3)

W. M. Robertson, “Transmission-line matrix modeling of superluminal electromagnetic-pulse tunneling through the forbidden gap in two-dimensional photonic band structures,” J. Opt. Soc. Am. B 14, 1066–1073 (1997).
[CrossRef]

Ph. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
[CrossRef]

R. Pelster, V. Gasparian, and G. Nimtz, “Propagation of plane waves and of waveguide modes in quasiperiodic dielectric heterostructures,” Phys. Rev. E 55, 7645–7655 (1997).
[CrossRef]

1996 (3)

M. Deutsch and J. E. Golub, “Optical Larmor clock: measurement of the photonic tunneling time,” Phys. Rev. A 53, 434–439 (1996).
[CrossRef] [PubMed]

E. L. Bolda, “Theory and simulation of superluminal optical pulses in gain media,” Phys. Rev. A 54, 3514–3518 (1996).
[CrossRef] [PubMed]

G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A 223, 327–331 (1996).
[CrossRef]

1995 (2)

V. Gasparian, M. Ortuño, J. Ruiz, and E. Cuevas, “Faraday rotation and complex-valued traversal time for classical light waves,” Phys. Rev. Lett. 75, 2312–2315 (1995).
[CrossRef] [PubMed]

A. M. Steinberg and R. Y. Chiao, “Subfemtosecond determination of transmission delay times for a dielectric mirror (photonic band gap) as a function of the angle of incidence,” Phys. Rev. A 51, 3525–3528 (1995).
[CrossRef] [PubMed]

1994 (5)

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]

R. Landauer and Th. Martin, “Barrier interaction time in tunneling,” Rev. Mod. Phys. 66, 217–228 (1994).
[CrossRef]

R. Y. Chiao and J. Boyce, “Superluminality, paraelectricity, and Earnshaw’s theorem in media with inverted populations,” Phys. Rev. Lett. 73, 3383–3386 (1994).
[CrossRef] [PubMed]

A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
[CrossRef] [PubMed]

R. Y. Chiao, “Atomic coherence effects which produce superluminal (but causal) propagation of wavepackets,” Quantum Opt. 6, 359–369 (1994).
[CrossRef]

1993 (2)

A. Enders and G. Nimtz, “Photonic-tunneling experiments,” Phys. Rev. B 47, 9605–9609 (1993).
[CrossRef]

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]

1992 (2)

Th. Martin and R. Landauer, “Time delay of evanescent electromagnetic waves and the analogy to particle tunneling,” Phys. Rev. A 45, 2611–2617 (1992).
[CrossRef] [PubMed]

J. G. Muga, S. Brouard, and R. Sala, “Transmission and reflection tunneling times,” Phys. Lett. A 167, 24–28 (1992).
[CrossRef]

1991 (1)

K. L. Jensen and F. A. Buot, “The methodology of simulating particle trajectories through tunneling structures using a Wigner distribution approach,” IEEE Trans. Electron Devices 38, 2337–2347 (1991).
[CrossRef]

1989 (2)

K. L. Jensen and F. A. Buot, “Numerical calculation of particle trajectories and tunneling times for resonant tunneling barrier structures,” Appl. Phys. Lett. 55, 669–671 (1989).
[CrossRef]

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

1987 (1)

D. Sokolovski and L. M. Baskin, “Traversal time in quantum scattering,” Phys. Rev. A 36, 4604–4611 (1987).
[CrossRef] [PubMed]

1984 (1)

E. Pollak and W. H. Miller, “New physical interpretation for time in scattering theory,” Phys. Rev. Lett. 53, 115–118 (1984).
[CrossRef]

1983 (1)

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

1982 (2)

M. Büttiker and R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. 49, 1739–1742 (1982).
[CrossRef]

S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[CrossRef]

1977 (1)

D. Mugnai, A. Ranfagni, and L. S. Schuman, “Delay time measurements in a diffraction experiment: a case of optical tunneling,” Phys. Rev. E 55, 3593–3597 (1977).
[CrossRef]

1970 (1)

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]

1966 (1)

A. I. Baz’, “Life-time of intermediate states,” Yad. Fiz. 4, 252–260 (1966).

1962 (1)

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

1955 (1)

E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
[CrossRef]

Balcou, Ph.

Ph. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
[CrossRef]

Baskin, L. M.

D. Sokolovski and L. M. Baskin, “Traversal time in quantum scattering,” Phys. Rev. A 36, 4604–4611 (1987).
[CrossRef] [PubMed]

Baz’, A. I.

A. I. Baz’, “Life-time of intermediate states,” Yad. Fiz. 4, 252–260 (1966).

Bolda, E. L.

E. L. Bolda, “Theory and simulation of superluminal optical pulses in gain media,” Phys. Rev. A 54, 3514–3518 (1996).
[CrossRef] [PubMed]

Boyce, J.

R. Y. Chiao and J. Boyce, “Superluminality, paraelectricity, and Earnshaw’s theorem in media with inverted populations,” Phys. Rev. Lett. 73, 3383–3386 (1994).
[CrossRef] [PubMed]

Brouard, S.

J. G. Muga, S. Brouard, and R. Sala, “Transmission and reflection tunneling times,” Phys. Lett. A 167, 24–28 (1992).
[CrossRef]

Buot, F. A.

K. L. Jensen and F. A. Buot, “The methodology of simulating particle trajectories through tunneling structures using a Wigner distribution approach,” IEEE Trans. Electron Devices 38, 2337–2347 (1991).
[CrossRef]

K. L. Jensen and F. A. Buot, “Numerical calculation of particle trajectories and tunneling times for resonant tunneling barrier structures,” Appl. Phys. Lett. 55, 669–671 (1989).
[CrossRef]

Büttiker, M.

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

M. Büttiker and R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. 49, 1739–1742 (1982).
[CrossRef]

Chiao, R. Y.

A. M. Steinberg and R. Y. Chiao, “Subfemtosecond determination of transmission delay times for a dielectric mirror (photonic band gap) as a function of the angle of incidence,” Phys. Rev. A 51, 3525–3528 (1995).
[CrossRef] [PubMed]

R. Y. Chiao, “Atomic coherence effects which produce superluminal (but causal) propagation of wavepackets,” Quantum Opt. 6, 359–369 (1994).
[CrossRef]

A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
[CrossRef] [PubMed]

R. Y. Chiao and J. Boyce, “Superluminality, paraelectricity, and Earnshaw’s theorem in media with inverted populations,” Phys. Rev. Lett. 73, 3383–3386 (1994).
[CrossRef] [PubMed]

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]

Chu, S.

S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[CrossRef]

Cuevas, E.

V. Gasparian, M. Ortuño, J. Ruiz, and E. Cuevas, “Faraday rotation and complex-valued traversal time for classical light waves,” Phys. Rev. Lett. 75, 2312–2315 (1995).
[CrossRef] [PubMed]

Deutsch, M.

M. Deutsch and J. E. Golub, “Optical Larmor clock: measurement of the photonic tunneling time,” Phys. Rev. A 53, 434–439 (1996).
[CrossRef] [PubMed]

Diener, G.

G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A 223, 327–331 (1996).
[CrossRef]

Dutriaux, L.

Ph. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
[CrossRef]

Enders, A.

A. Enders and G. Nimtz, “Photonic-tunneling experiments,” Phys. Rev. B 47, 9605–9609 (1993).
[CrossRef]

Garrett, C. G. B.

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]

Gasparian, V.

R. Pelster, V. Gasparian, and G. Nimtz, “Propagation of plane waves and of waveguide modes in quasiperiodic dielectric heterostructures,” Phys. Rev. E 55, 7645–7655 (1997).
[CrossRef]

V. Gasparian, M. Ortuño, J. Ruiz, and E. Cuevas, “Faraday rotation and complex-valued traversal time for classical light waves,” Phys. Rev. Lett. 75, 2312–2315 (1995).
[CrossRef] [PubMed]

Golub, J. E.

M. Deutsch and J. E. Golub, “Optical Larmor clock: measurement of the photonic tunneling time,” Phys. Rev. A 53, 434–439 (1996).
[CrossRef] [PubMed]

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]

Jensen, K. L.

K. L. Jensen and F. A. Buot, “The methodology of simulating particle trajectories through tunneling structures using a Wigner distribution approach,” IEEE Trans. Electron Devices 38, 2337–2347 (1991).
[CrossRef]

K. L. Jensen and F. A. Buot, “Numerical calculation of particle trajectories and tunneling times for resonant tunneling barrier structures,” Appl. Phys. Lett. 55, 669–671 (1989).
[CrossRef]

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]

Kwiat, P. G.

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]

Landauer, R.

R. Landauer and Th. Martin, “Barrier interaction time in tunneling,” Rev. Mod. Phys. 66, 217–228 (1994).
[CrossRef]

Th. Martin and R. Landauer, “Time delay of evanescent electromagnetic waves and the analogy to particle tunneling,” Phys. Rev. A 45, 2611–2617 (1992).
[CrossRef] [PubMed]

M. Büttiker and R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. 49, 1739–1742 (1982).
[CrossRef]

Laude, V.

Martin, Th.

R. Landauer and Th. Martin, “Barrier interaction time in tunneling,” Rev. Mod. Phys. 66, 217–228 (1994).
[CrossRef]

Th. Martin and R. Landauer, “Time delay of evanescent electromagnetic waves and the analogy to particle tunneling,” Phys. Rev. A 45, 2611–2617 (1992).
[CrossRef] [PubMed]

McCumber, D. E.

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]

Miller, W. H.

E. Pollak and W. H. Miller, “New physical interpretation for time in scattering theory,” Phys. Rev. Lett. 53, 115–118 (1984).
[CrossRef]

Muga, J. G.

J. G. Muga, S. Brouard, and R. Sala, “Transmission and reflection tunneling times,” Phys. Lett. A 167, 24–28 (1992).
[CrossRef]

Mugnai, D.

D. Mugnai, A. Ranfagni, and L. S. Schuman, “Delay time measurements in a diffraction experiment: a case of optical tunneling,” Phys. Rev. E 55, 3593–3597 (1977).
[CrossRef]

Nimtz, G.

R. Pelster, V. Gasparian, and G. Nimtz, “Propagation of plane waves and of waveguide modes in quasiperiodic dielectric heterostructures,” Phys. Rev. E 55, 7645–7655 (1997).
[CrossRef]

A. Enders and G. Nimtz, “Photonic-tunneling experiments,” Phys. Rev. B 47, 9605–9609 (1993).
[CrossRef]

Ortuño, M.

V. Gasparian, M. Ortuño, J. Ruiz, and E. Cuevas, “Faraday rotation and complex-valued traversal time for classical light waves,” Phys. Rev. Lett. 75, 2312–2315 (1995).
[CrossRef] [PubMed]

Pelster, R.

R. Pelster, V. Gasparian, and G. Nimtz, “Propagation of plane waves and of waveguide modes in quasiperiodic dielectric heterostructures,” Phys. Rev. E 55, 7645–7655 (1997).
[CrossRef]

Pollak, E.

E. Pollak and W. H. Miller, “New physical interpretation for time in scattering theory,” Phys. Rev. Lett. 53, 115–118 (1984).
[CrossRef]

Ranfagni, A.

D. Mugnai, A. Ranfagni, and L. S. Schuman, “Delay time measurements in a diffraction experiment: a case of optical tunneling,” Phys. Rev. E 55, 3593–3597 (1977).
[CrossRef]

Robertson, W. M.

Ruiz, J.

V. Gasparian, M. Ortuño, J. Ruiz, and E. Cuevas, “Faraday rotation and complex-valued traversal time for classical light waves,” Phys. Rev. Lett. 75, 2312–2315 (1995).
[CrossRef] [PubMed]

Sala, R.

J. G. Muga, S. Brouard, and R. Sala, “Transmission and reflection tunneling times,” Phys. Lett. A 167, 24–28 (1992).
[CrossRef]

Schuman, L. S.

D. Mugnai, A. Ranfagni, and L. S. Schuman, “Delay time measurements in a diffraction experiment: a case of optical tunneling,” Phys. Rev. E 55, 3593–3597 (1977).
[CrossRef]

Sokolovski, D.

D. Sokolovski and L. M. Baskin, “Traversal time in quantum scattering,” Phys. Rev. A 36, 4604–4611 (1987).
[CrossRef] [PubMed]

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]

Steinberg, A. M.

A. M. Steinberg and R. Y. Chiao, “Subfemtosecond determination of transmission delay times for a dielectric mirror (photonic band gap) as a function of the angle of incidence,” Phys. Rev. A 51, 3525–3528 (1995).
[CrossRef] [PubMed]

A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
[CrossRef] [PubMed]

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]

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]

Tournois, P.

Wigner, E. P.

E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
[CrossRef]

Wong, S.

S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[CrossRef]

Appl. Phys. Lett. (1)

K. L. Jensen and F. A. Buot, “Numerical calculation of particle trajectories and tunneling times for resonant tunneling barrier structures,” Appl. Phys. Lett. 55, 669–671 (1989).
[CrossRef]

IEEE Trans. Electron Devices (1)

K. L. Jensen and F. A. Buot, “The methodology of simulating particle trajectories through tunneling structures using a Wigner distribution approach,” IEEE Trans. Electron Devices 38, 2337–2347 (1991).
[CrossRef]

J. Appl. Phys. (1)

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

J. Opt. Soc. Am. B (2)

Phys. Lett. A (2)

G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A 223, 327–331 (1996).
[CrossRef]

J. G. Muga, S. Brouard, and R. Sala, “Transmission and reflection tunneling times,” Phys. Lett. A 167, 24–28 (1992).
[CrossRef]

Phys. Rev. (1)

E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
[CrossRef]

Phys. Rev. A (7)

D. Sokolovski and L. M. Baskin, “Traversal time in quantum scattering,” Phys. Rev. A 36, 4604–4611 (1987).
[CrossRef] [PubMed]

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]

E. L. Bolda, “Theory and simulation of superluminal optical pulses in gain media,” Phys. Rev. A 54, 3514–3518 (1996).
[CrossRef] [PubMed]

A. M. Steinberg and R. Y. Chiao, “Subfemtosecond determination of transmission delay times for a dielectric mirror (photonic band gap) as a function of the angle of incidence,” Phys. Rev. A 51, 3525–3528 (1995).
[CrossRef] [PubMed]

Th. Martin and R. Landauer, “Time delay of evanescent electromagnetic waves and the analogy to particle tunneling,” Phys. Rev. A 45, 2611–2617 (1992).
[CrossRef] [PubMed]

A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49, 3283–3295 (1994).
[CrossRef] [PubMed]

M. Deutsch and J. E. Golub, “Optical Larmor clock: measurement of the photonic tunneling time,” Phys. Rev. A 53, 434–439 (1996).
[CrossRef] [PubMed]

Phys. Rev. B (2)

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

A. Enders and G. Nimtz, “Photonic-tunneling experiments,” Phys. Rev. B 47, 9605–9609 (1993).
[CrossRef]

Phys. Rev. E (2)

R. Pelster, V. Gasparian, and G. Nimtz, “Propagation of plane waves and of waveguide modes in quasiperiodic dielectric heterostructures,” Phys. Rev. E 55, 7645–7655 (1997).
[CrossRef]

D. Mugnai, A. Ranfagni, and L. S. Schuman, “Delay time measurements in a diffraction experiment: a case of optical tunneling,” Phys. Rev. E 55, 3593–3597 (1977).
[CrossRef]

Phys. Rev. Lett. (8)

S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[CrossRef]

R. Y. Chiao and J. Boyce, “Superluminality, paraelectricity, and Earnshaw’s theorem in media with inverted populations,” Phys. Rev. Lett. 73, 3383–3386 (1994).
[CrossRef] [PubMed]

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

V. Gasparian, M. Ortuño, J. Ruiz, and E. Cuevas, “Faraday rotation and complex-valued traversal time for classical light waves,” Phys. Rev. Lett. 75, 2312–2315 (1995).
[CrossRef] [PubMed]

Ph. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
[CrossRef]

E. Pollak and W. H. Miller, “New physical interpretation for time in scattering theory,” Phys. Rev. Lett. 53, 115–118 (1984).
[CrossRef]

M. Büttiker and R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. 49, 1739–1742 (1982).
[CrossRef]

Quantum Opt. (1)

R. Y. Chiao, “Atomic coherence effects which produce superluminal (but causal) propagation of wavepackets,” Quantum Opt. 6, 359–369 (1994).
[CrossRef]

Rev. Mod. Phys. (2)

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

R. Landauer and Th. Martin, “Barrier interaction time in tunneling,” Rev. Mod. Phys. 66, 217–228 (1994).
[CrossRef]

Yad. Fiz. (1)

A. I. Baz’, “Life-time of intermediate states,” Yad. Fiz. 4, 252–260 (1966).

Other (1)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1986).

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Figures (4)

Fig. 1
Fig. 1

Principle of the magnetic clock defining the traversal time in a Fabry–Perot cavity. The complex traversal time with multiple reflections taken into account is introduced by the resultant complex angle in the presence of mirrors forming a cavity.

Fig. 2
Fig. 2

Complex traversal time of a Fabry–Perot cavity. (a) The real component (phase time) and (b) the imaginary component (loss time) are plotted as a function of the optical frequency detuning from the nearest lower resonance frequency 2πN/tr for three different values of the cavity mirror reflectivity R, 0.50, 0.90, and 0.99.

Fig. 3
Fig. 3

Superluminal optical pulse transmission through a Fabry–Perot cavity with R=0.99 for an incident Gaussian pulse of FWHM tp=6tr/2. (a) Temporal evolution of the transmitted pulse exhibiting the arrival time of only 5.4×10-3 of the single-pass time tr/2 and the narrowed pulse width of 5.75tr/2. (b) Dispersion of the phase time and the loss time over the incident pulse spectrum, producing slight discrepancies in the arrival time and the resultant pulse width.

Fig. 4
Fig. 4

Transmission of an optical wave packet carrying an abrupt disturbance. The result is given for a Fabry–Perot cavity with R=0.99 and the incident Gaussian pulse envelope of FWHM tp=6tr/2. Note that the transmitted profile of the pulse is magnified for comparison with the delayed incident pulse and that a severe distortion of the transmitted pulse takes place owing to the broad spectrum of the fast-varying portion of the incident pulse.

Equations (15)

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K(ω)=A(ω)exp[iϕ(ω)]=m=0T2R2m expiωtrm+12=T2 exp(iωtr/2)1-R2 exp(iωtr),
|K(ω)|2=(1-R)2(1-R)2+4R sin2(12ωtr+φr),
tan θ=EyEx=-iE+-E-E++E-=-iK+-K-K++K-,
θθ1-iθ2=(ϕ+-ϕ-)2-iln(A+/A-)2.
ϕ+-ϕ-=ϕ(n+)-ϕ(n-)[ϕ(n0)/n0]ΔnB,
ln(A+/A-)=ln[A(n+)/A(n-)][ ln A(n0)/n0]ΔnB,
ττ1-iτ2n0ωϕ(n0)n0-i ln A(n0)n0.
τ1=ϕ(ω)ω=1-R2(1-R)2+4R sin2(12ωtr+φr)tr2,
τ2= ln A(ω)ω=-2R sin(ωtr+2φr)(1-R)2+4R sin2(12ωtr+φr)tr2.
τav=m=0(m+12)trT2R2m exp[iωtr(m+12)]m=0T2R2m exp[iωtr(m+12)].
τav=1iK(ω)K(ω)ω=-i ln K(ω)ω=ϕ(ω)ω-i ln A(ω)ω=τ.
τ1=1-R1+Rtr2,
G(t)=m=0T2R2mδ(t-[m+1/2]tr),
|K(ω)|2=|K(ωp)|2+|K(ω)|2ωωp(ω-ωp)+|K(ωp)|2[1+2τ2(ω-ωp)].
δωp=Δωpτ2tp,

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