Abstract

There have been many recent theoretical and experimental reports on the propagation of light pulses at speeds exceeding the speed of light in vacuum c within media with anomalous dispersion, either opaque or with gain. Superluminal propagation has also been reported within vacuum, in the case of inhomogeneous pulses. In this paper we show that the observations of superluminal and non-causal propagation of evanescent pulses under the conditions of frustrated internal reflection are only apparent, and that they can be simply explained employing an explicitly (sub)luminal causal theory. However, the usual one-dimensional approach to the analysis of pulse propagation has to be abandoned and the spatial extent of the incoming pulse along the directions normal to the propagation direction has to be accounted for to correctly interpret the propagation speed of these evanescent waves. We illustrate our theory with animations of the time development of a pulse built upon the Huygen’s construction.

© Optical Society of America

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References

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  1. A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, "Measurement of the single-photon tunneling time," Phys. Rev. Lett. 71, 708-711 (1993) http://link.aps.org/abstract/PRL/v71/i5/p708.
    [CrossRef] [PubMed]
  2. Spielmann, R. Szipocs, A. Stingl, and F. Krausz, "Tunneling of optical pulses through photonic band gaps," Phys. Rev. Lett. 73, 2308 (1994) http://link.aps.org/abstract/PRL/v73/p2308.
    [CrossRef] [PubMed]
  3. T. H. Hartman, "Tunneling of a wave packet," J. Appl. Phys. 33, 3427 (1962).
    [CrossRef]
  4. C. G. B. Garret and D. E. McCumber, "Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium," Phys. Rev. A 1, 305 (1970) http://link.aps.org/abstract/PRA/v1/i2/p305.
    [CrossRef]
  5. S. Chu and S. Wong, "Linear Pulse Propagation in an Absorbing Medium," Phys. Rev. Lett. 48, 738 (1982) http://link.aps.org/abstract/PRL/v48/i11/p738.
    [CrossRef]
  6. A. Katz and R. R. Alfano, "Pulse Propagation in an Absorbing Medium," Phys. Rev. Lett. 49, 1292 (1982) http://link.aps.org/abstract/PRL/v49/p1292.
    [CrossRef]
  7. Rolf Landauer, "Light faster than light?" Nature 365, 692 (1993).
    [CrossRef]
  8. G. Diener, "Superluminal group velocities and information transfer," Phys. Lett. A 223, 327-331 (1996).
    [CrossRef]
  9. Leon Brillouin, "Wave Propagation and Group Velocity" (Academic, New york, 1960) Ch. II, III, and IV.
  10. G. Diener, "Energy Transport in Dispersive Media and Superluminal Group Velocities," Phys. Lett. A 235, 118 (1997).
    [CrossRef]
  11. V. Romero-Rochin, R. P. Duarte-Zamorano, S. Nielsen-Hofseth, and R. G. Barrera, "Superluminal transmission of light pulses through optically opaque barriers," Phys. Rev. E 63, 027601 (2001) http://link.aps.org/abstract/PRE/v63/e027601.
    [CrossRef]
  12. Richard W. Ziolkowski, "Superluminal Transmission of Information through an Electromagnetic Metamaterial," Phys. Rev. E 63, 046604 (2001) http://link.aps.org/abstract/PRE/v63/e046604.
    [CrossRef]
  13. A. N. Oraevsky, "Superluminal Waves in Amplifying Media," Physycs-Uspekhi 41, 1199 (1998).
    [CrossRef]
  14. L. J. Wang and A. Kuzmich abd A. Dogariu, "Gain-Assisted Superluminal Light Propagation," Nature 406, 277 (2000).
    [CrossRef] [PubMed]
  15. A. Dogariu, A. Kuzmich, and L. J. Wang, "Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity," Phys. Rev. A 63, 053806
    [CrossRef]
  16. A. Dogariu and Ak. Kuzmich and H. Cao and L. J. Wang, "Superluminal Light Pulse Propagation Via Rephasing in a Transparent Anomalously Dispersive Medium," Opt. Express 8, 344 (2001) http://www.opticsexpress.org/oearchive/source/30536.htm.
    [CrossRef] [PubMed]
  17. A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, "Signal Velocity , Causality, and Quantum Noise in Superluminal Light Pulse Propagation," Phys. Rev. Lett. 86, 3925-3929 (2001) http://link.aps.org/abstract/PRL/v86/p3925.
    [CrossRef] [PubMed]
  18. G. Nimtz, "Evanescent Modes are not Necessarily Einstein Causal," Eur. Phys. J. B 7, 523 (1999).
    [CrossRef]
  19. Yakir Aharonov and Benni Reznil and Ady Stern, "Quantum Limitatins on Superluminal Propagation," Phys. Rev. Lett. 81, 2190 (1998) http://link.aps.org/abstract/PRL/v81/p2190.
    [CrossRef]
  20. P. W. Milonni and K. Furuya and R. Y. Chiao, "Quantum Theory of Superluminal Pulse Propagation," Opt. Express 8, 59 (2001) http://www.opticsexpress.org/oearchive/source/27132.htm.
    [CrossRef] [PubMed]
  21. Aephraim M Steinberg, "No thing goes faster than light," Physics World 13, 3 (2000) http://www.physicsweb.org/article/world/13/9/3.
  22. Peter W. Milonni, "Causal Discussion of Superluminal Pulses," Physics Today 54, 81 (2001) http://www.physicstoday.org/pt/vol-54/iss-2/p14b.html.
    [CrossRef]
  23. Jacob Broe and Ole Keller, "Superluminality and spatial confinement in optical tunneling," Opt. Commun. 194, 83 (2001).
    [CrossRef]
  24. A. Ranfagni, D. Mugnai, P. Fabeni, and G. P. Pazzi, "Delay-time measurements in narrowed waveguides as a test of tunneling," Appl. Phys. Lett. 58, 774 (1991).
    [CrossRef]
  25. A. Enders and G. Nimtz, "Photonic-tunneling experiments," Phys. Rev. B 47, 9605 (1993) http://link.aps.org/abstract/PRB/v47/p9605.
    [CrossRef]
  26. A. Pablo L. Barbero, Hugo E. Hernandez-Figueroa, and Erasmo Recami, "Propagation Speed of Evanescent Modes," Phys. Rev. E 62, 8628 (2000) http://link.aps.org/abstract/PRE/v62/p8628.
    [CrossRef]
  27. Klass Wynne, John J. J. Carey, Justyna Zawadzka, and Dino Jaroszynski, "Tunneling of Single-Cycle Terahertz Pulses through Waveguides," Opt. Commun. 176, 429 (2000).
    [CrossRef]
  28. S. Bosanac, "Propagation of Electromagnetic Wave Packets in Nondispersive Dielectric Media," Phys. Rev. A 28, 577 (1983) http://link.aps.org/abstract/PRA/v28/p577.
    [CrossRef]
  29. Ph. Balcou and L. Dutriaux, "Dual Optical Tunneling Times in Frustrated Total Internal Reflection," Phys. Rev. Lett. 78, 851 (1997) http://link.aps.org/abstract/PRL/v78/p851.
    [CrossRef]
  30. John J. Carey, Justyna Zawadzka, Dino A. Jaroszynski, and Klaas Wynne, "Noncausal Time Response in Frustrated Total Internal Reflection?" Phys. Rev. Lett. 84, 1431 (2000) http://link.aps.org/abstract/PRL/v84/p1431.
    [CrossRef] [PubMed]
  31. W. Luis Mochan and Vera L. Brudny, "Comment on Noncausal Time Response in Frustrated Total Internal Reflection?" Phys. Rev. Lett. 87, 119101 (2001) http://link.aps.org/abstract/PRL/v87/e119101.
    [CrossRef]
  32. Vera L. Brudny and W. Luis Mochan, under preparation.
  33. A. Ranfagni, P. Fabeni, G. P. Pazzi, and D. Mugnai "Anomalous pulse delay in microwave propagation: A plausible connection to the tunneling time," Phys. Rev. E 48, 1453 (1993). http://link.aps.org/abstract/PRE/v48/p1453.
    [CrossRef]
  34. D. Mugnai, A. Ranfagni, and R. Ruggeri, Phys. Rev. Lett. 84, 4830 (2000) http://link.aps.org/abstract/PRL/v84/p4830.
    [CrossRef] [PubMed]

Other (34)

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, "Measurement of the single-photon tunneling time," Phys. Rev. Lett. 71, 708-711 (1993) http://link.aps.org/abstract/PRL/v71/i5/p708.
[CrossRef] [PubMed]

Spielmann, R. Szipocs, A. Stingl, and F. Krausz, "Tunneling of optical pulses through photonic band gaps," Phys. Rev. Lett. 73, 2308 (1994) http://link.aps.org/abstract/PRL/v73/p2308.
[CrossRef] [PubMed]

T. H. Hartman, "Tunneling of a wave packet," J. Appl. Phys. 33, 3427 (1962).
[CrossRef]

C. G. B. Garret and D. E. McCumber, "Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium," Phys. Rev. A 1, 305 (1970) http://link.aps.org/abstract/PRA/v1/i2/p305.
[CrossRef]

S. Chu and S. Wong, "Linear Pulse Propagation in an Absorbing Medium," Phys. Rev. Lett. 48, 738 (1982) http://link.aps.org/abstract/PRL/v48/i11/p738.
[CrossRef]

A. Katz and R. R. Alfano, "Pulse Propagation in an Absorbing Medium," Phys. Rev. Lett. 49, 1292 (1982) http://link.aps.org/abstract/PRL/v49/p1292.
[CrossRef]

Rolf Landauer, "Light faster than light?" Nature 365, 692 (1993).
[CrossRef]

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

Leon Brillouin, "Wave Propagation and Group Velocity" (Academic, New york, 1960) Ch. II, III, and IV.

G. Diener, "Energy Transport in Dispersive Media and Superluminal Group Velocities," Phys. Lett. A 235, 118 (1997).
[CrossRef]

V. Romero-Rochin, R. P. Duarte-Zamorano, S. Nielsen-Hofseth, and R. G. Barrera, "Superluminal transmission of light pulses through optically opaque barriers," Phys. Rev. E 63, 027601 (2001) http://link.aps.org/abstract/PRE/v63/e027601.
[CrossRef]

Richard W. Ziolkowski, "Superluminal Transmission of Information through an Electromagnetic Metamaterial," Phys. Rev. E 63, 046604 (2001) http://link.aps.org/abstract/PRE/v63/e046604.
[CrossRef]

A. N. Oraevsky, "Superluminal Waves in Amplifying Media," Physycs-Uspekhi 41, 1199 (1998).
[CrossRef]

L. J. Wang and A. Kuzmich abd A. Dogariu, "Gain-Assisted Superluminal Light Propagation," Nature 406, 277 (2000).
[CrossRef] [PubMed]

A. Dogariu, A. Kuzmich, and L. J. Wang, "Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity," Phys. Rev. A 63, 053806
[CrossRef]

A. Dogariu and Ak. Kuzmich and H. Cao and L. J. Wang, "Superluminal Light Pulse Propagation Via Rephasing in a Transparent Anomalously Dispersive Medium," Opt. Express 8, 344 (2001) http://www.opticsexpress.org/oearchive/source/30536.htm.
[CrossRef] [PubMed]

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, "Signal Velocity , Causality, and Quantum Noise in Superluminal Light Pulse Propagation," Phys. Rev. Lett. 86, 3925-3929 (2001) http://link.aps.org/abstract/PRL/v86/p3925.
[CrossRef] [PubMed]

G. Nimtz, "Evanescent Modes are not Necessarily Einstein Causal," Eur. Phys. J. B 7, 523 (1999).
[CrossRef]

Yakir Aharonov and Benni Reznil and Ady Stern, "Quantum Limitatins on Superluminal Propagation," Phys. Rev. Lett. 81, 2190 (1998) http://link.aps.org/abstract/PRL/v81/p2190.
[CrossRef]

P. W. Milonni and K. Furuya and R. Y. Chiao, "Quantum Theory of Superluminal Pulse Propagation," Opt. Express 8, 59 (2001) http://www.opticsexpress.org/oearchive/source/27132.htm.
[CrossRef] [PubMed]

Aephraim M Steinberg, "No thing goes faster than light," Physics World 13, 3 (2000) http://www.physicsweb.org/article/world/13/9/3.

Peter W. Milonni, "Causal Discussion of Superluminal Pulses," Physics Today 54, 81 (2001) http://www.physicstoday.org/pt/vol-54/iss-2/p14b.html.
[CrossRef]

Jacob Broe and Ole Keller, "Superluminality and spatial confinement in optical tunneling," Opt. Commun. 194, 83 (2001).
[CrossRef]

A. Ranfagni, D. Mugnai, P. Fabeni, and G. P. Pazzi, "Delay-time measurements in narrowed waveguides as a test of tunneling," Appl. Phys. Lett. 58, 774 (1991).
[CrossRef]

A. Enders and G. Nimtz, "Photonic-tunneling experiments," Phys. Rev. B 47, 9605 (1993) http://link.aps.org/abstract/PRB/v47/p9605.
[CrossRef]

A. Pablo L. Barbero, Hugo E. Hernandez-Figueroa, and Erasmo Recami, "Propagation Speed of Evanescent Modes," Phys. Rev. E 62, 8628 (2000) http://link.aps.org/abstract/PRE/v62/p8628.
[CrossRef]

Klass Wynne, John J. J. Carey, Justyna Zawadzka, and Dino Jaroszynski, "Tunneling of Single-Cycle Terahertz Pulses through Waveguides," Opt. Commun. 176, 429 (2000).
[CrossRef]

S. Bosanac, "Propagation of Electromagnetic Wave Packets in Nondispersive Dielectric Media," Phys. Rev. A 28, 577 (1983) http://link.aps.org/abstract/PRA/v28/p577.
[CrossRef]

Ph. Balcou and L. Dutriaux, "Dual Optical Tunneling Times in Frustrated Total Internal Reflection," Phys. Rev. Lett. 78, 851 (1997) http://link.aps.org/abstract/PRL/v78/p851.
[CrossRef]

John J. Carey, Justyna Zawadzka, Dino A. Jaroszynski, and Klaas Wynne, "Noncausal Time Response in Frustrated Total Internal Reflection?" Phys. Rev. Lett. 84, 1431 (2000) http://link.aps.org/abstract/PRL/v84/p1431.
[CrossRef] [PubMed]

W. Luis Mochan and Vera L. Brudny, "Comment on Noncausal Time Response in Frustrated Total Internal Reflection?" Phys. Rev. Lett. 87, 119101 (2001) http://link.aps.org/abstract/PRL/v87/e119101.
[CrossRef]

Vera L. Brudny and W. Luis Mochan, under preparation.

A. Ranfagni, P. Fabeni, G. P. Pazzi, and D. Mugnai "Anomalous pulse delay in microwave propagation: A plausible connection to the tunneling time," Phys. Rev. E 48, 1453 (1993). http://link.aps.org/abstract/PRE/v48/p1453.
[CrossRef]

D. Mugnai, A. Ranfagni, and R. Ruggeri, Phys. Rev. Lett. 84, 4830 (2000) http://link.aps.org/abstract/PRL/v84/p4830.
[CrossRef] [PubMed]

Supplementary Material (2)

» Media 1: MPG (790 KB)     
» Media 2: MPG (720 KB)     

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Figures (2)

Fig. 1.
Fig. 1.

Animation illustrating the non-evanescent transmission of light across an air gap from ∑1 to ∑2 separating two semi-infinite dielectrics (1.9Mb). A single incoming narrow plane wavefront, with angle of incidence θ, excites secondary waves w 1 within the air gap, which grow with speed c around successive points p 1 at the surface ∑1. As they sweep points p 2 on the other side of the air gap ∑2, these waves in turn excite further secondary waves w 2 within the second dielectric growing with speed c/n. S 1 and S 2 are perfectly opaque screens. The nominal propagation directions are indicated by red arrows. The dots indicate the centers of the secondary waves launched at equally spaced times. The figure corresponds to n=2 and θ=27°<θc .

Fig. 2.
Fig. 2.

Animation illustrating the evanescent transmission of light across an air gap separating two semi-infinite dielectrics (2.1Mb), as in Fig. 1, but with θ=34°>θc . The vertical dashed line connects the point at which the incident wavefront intersects the front face of the air gap i with the corresponding point on the back face t. The thin black arrow connects the superluminally excited position t to its true field source s.

Equations (3)

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P ( x , z , t ) = c π z Θ ( ct x 2 + z 2 ) ( ct ) 2 x 2 z 2
ϕ ( x , z , t ) = { 0 t < T ( x , z ; x 1 s ) ϕ 0 F ( x , z , t ; x 1 s ) T ( x , z ; x 1 s ) < t < T ( x , z ; x 2 s ) ϕ 0 ( F ( x , z , t ; x 1 s ) F ( x , z , t ; x 2 s ) ) t > T ( x , z ; x 2 s ) ,
F 0 ( x , z , t ) = γ z π γ 2 ( x v t ) 2 + z 2

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