Abstract

By employing both the ray model and the electromagnetic theory in a slab optical waveguide, we show that the Goos-Hänchen time, which has been recently argued in the literature, really exists and the time associated with total reflection of a plane wave upon nonabsorbing plasma mirror is exactly the sum of the group delay time and the Goos-Hänchen time. Based on this concept, it is also indicated that the causality is preserved not only for the frustrated Gires-Tournois interferometer case but also for the case of total reflection of a plane TM wave on a nonabsorbing plasma mirror.

© 2006 Optical Society of America

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References

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  1. P. Tournois, "Negative group delay times in frustrated Gires-Tournois and Fabry-Perot interferometers," IEEE J. Quantum Electron. 33, 519-526 (1997).
    [CrossRef]
  2. K. J. Resch, J. S. Lundeen, and A. M. Steinberg, "Total reflection cannot occur with a negative delay time," IEEE J. Quantum Electron. 37, 794-799 (2001).
    [CrossRef]
  3. P. Tournois, "Apparent causality paradox in frustrated Gires-Tournois interferometers," Opt. Lett. 30, 815-817 (2005).
    [CrossRef] [PubMed]
  4. H. Kogelnik and H. P. Weber, "Rays, stored energy, and power flow in dielectric waveguides," J. Opt. Soc. Am. 64, 174-185 (1974).
    [CrossRef]
  5. P. K. Tien, "Light waves in thin films and integrated optics," Appl. Opt. 10, 2395-2413 (1971).
    [CrossRef] [PubMed]
  6. H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
    [CrossRef]

2005 (1)

2001 (1)

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, "Total reflection cannot occur with a negative delay time," IEEE J. Quantum Electron. 37, 794-799 (2001).
[CrossRef]

2000 (1)

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

1997 (1)

P. Tournois, "Negative group delay times in frustrated Gires-Tournois and Fabry-Perot interferometers," IEEE J. Quantum Electron. 33, 519-526 (1997).
[CrossRef]

1974 (1)

1971 (1)

Kogelnik, H.

Kwok, C. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Lai, H. M.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Loo, Y. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Lundeen, J. S.

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, "Total reflection cannot occur with a negative delay time," IEEE J. Quantum Electron. 37, 794-799 (2001).
[CrossRef]

Resch, K. J.

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, "Total reflection cannot occur with a negative delay time," IEEE J. Quantum Electron. 37, 794-799 (2001).
[CrossRef]

Steinberg, A. M.

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, "Total reflection cannot occur with a negative delay time," IEEE J. Quantum Electron. 37, 794-799 (2001).
[CrossRef]

Tien, P. K.

Tournois, P.

P. Tournois, "Apparent causality paradox in frustrated Gires-Tournois interferometers," Opt. Lett. 30, 815-817 (2005).
[CrossRef] [PubMed]

P. Tournois, "Negative group delay times in frustrated Gires-Tournois and Fabry-Perot interferometers," IEEE J. Quantum Electron. 33, 519-526 (1997).
[CrossRef]

Weber, H. P.

Xu, B. Y.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

P. Tournois, "Negative group delay times in frustrated Gires-Tournois and Fabry-Perot interferometers," IEEE J. Quantum Electron. 33, 519-526 (1997).
[CrossRef]

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, "Total reflection cannot occur with a negative delay time," IEEE J. Quantum Electron. 37, 794-799 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Phys. Rev. E (1)

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

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Figures (1)

Fig. 1.
Fig. 1.

Symmetric slab waveguide with both cladding layers constituted by ideal nonabsorbing plasma mirrors and zigzag-propagation of the rays: (a) TE guided mode; (b) TM guided mode.

Equations (30)

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tan ( φ 2 ) = { ( 1 u 2 cos 2 θ ) 1 2 u cos θ ( TE wave ) u ( 1 u 2 cos 2 θ ) 1 2 ( 1 u 2 ) cos θ ( TE wave ) .
t g = { 2 cos θ ω p ( 1 u 2 cos 2 θ ) 1 2 ( TE wave ) 2 cos θ ω p ( 1 u 2 cos 2 θ ) 1 2 ( 1 u 2 cos 2 θ ) ( cos 2 θ u 2 cos 2 θ ) ( TE wave ) .
t GH = tan θ ω φ θ
t GH = { 2 tan θ sin θ ω p ( 1 u 2 cos 2 θ ) 1 2 ( TE wave ) 2 tan θ sin θ ω p ( 1 u 2 cos 2 θ ) 1 2 ( u 2 1 ) ( cos 2 θ u 2 cos 2 θ ) ( TE wave ) .
x = { 2 tan θ k 0 ( sin 2 θ n p 2 ) 1 2 ( TE wave ) 2 tan θ k 0 ( sin 2 θ n p 2 ) 1 2 n p 2 ( 1 n p 2 ) n p 4 cos 2 θ + ( sin 2 θ n p 2 ) ( TE wave ) .
l = 2 h tan θ + 2 Δ x = 2 h eff tan θ
h eff = { h + 2 k 0 ( N 2 n p 2 ) 1 2 ( TE mode ) h + 2 k 0 ( N 2 n p 2 ) 1 2 n p 2 ( 1 n p 2 ) n p 4 ( 1 N 2 ) + ( N 2 n p 2 ) ( TE mode ) .
2 κh + 2 φ = 2 , ( m = 0 , 1 , 2 , )
tan ( φ 2 ) = { ( N 2 n p 2 1 N 2 ) 1 2 ( TE mode ) 1 n p 2 ( N 2 n p 2 1 N 2 ) 1 2 ( TE mode ) .
1 v g = β ω = N c + k 0 N ω
κ ω = ( 1 N 2 ) 1 2 c k 0 N ( 1 N 2 ) 1 2 N ω .
φ ω = 2 N ( 1 N 2 ) 1 2 ( N 2 n p 2 ) 1 2 N ω + ( 1 N 2 ) 1 2 ( 1 n p 2 ) ( N 2 n p 2 ) 1 2 n p 2 ω
N ω = 1 N 2 k 0 c N .
v g = c N .
τ total = 2 h eff tan θ v g
= 2 h c cos θ + 4 cos θ ω p ( 1 u 2 cos 2 θ ) 1 2 + 4 tan θ sin θ ω p ( 1 u 2 cos 2 θ ) 1 2 .
= 2 h c cos θ + 2 t g + 2 t GH
φ ω = n p 2 ( 1 n p 2 ) n p 4 ( 1 N 2 ) + ( N 2 n p 2 ) 2 N ( 1 N 2 ) 1 2 ( N 2 n p 2 ) 1 2 N ω + ( 1 N 2 ) 1 2 n p 4 ( 1 N 2 ) + ( N 2 n p 2 ) 2 N 2 n p 2 ( N 2 n p 2 ) 1 2 n p 2 ω
N ω = 1 N 2 k 0 c N h eff [ h + 2 1 n P 2 n p 4 ( 1 N 2 ) + ( N 2 n p 2 ) 2 N 2 n p 2 k 0 ( N 2 n p 2 ) 1 2 ] .
1 v g = N c + 1 N 2 c N h eff [ h + 2 1 n p 2 n p 4 ( 1 N 2 ) + ( N 2 n p 2 ) 2 N 2 n p 2 k 0 ( N 2 n p 2 ) 1 2 ] .
τ total = 2 h eff tan θ v g
= 2 tan θ c N [ h + 2 n p 2 ( 2 N 2 1 ) + 2 N 2 ( 1 N 2 ) n p 4 ( 1 N 2 ) + ( N 2 n p 2 ) 1 n P 2 k 0 ( N 2 n p 2 ) 1 2 ] .
τ total = 2 h c cos θ + 4 tan θ sin θ ω p ( 1 u 2 cos 2 θ ) 1 2 ( u 2 1 ) ( cos 2 θ u 2 cos 2 θ )
+ 4 cos θ ω p ( 1 u 2 cos 2 θ ) 1 2 + ( 1 u 2 cos 2 θ ) ( cos 2 θ u 2 cos 2 θ ) .
= 2 h c cos θ + 2 t GH + 2 t g
τ total = 2 h eff c cos θ + 4 cos θ ω p ( 1 u 2 cos 2 θ ) 1 2 ( u 2 1 ) ( cos 2 θ u 2 cos 2 θ )
+ 4 cos θ ω p ( 1 u 2 cos 2 θ ) 1 2 + ( 1 u 2 cos 2 θ ) ( cos 2 θ u 2 cos 2 θ ) .
t GH = 2 cos θ ω p ( 1 u 2 cos 2 θ ) 1 2 ( u 2 1 ) ( cos 2 θ u 2 cos 2 θ ) ,
t GH = n 1 O 1 p O c + n 1 O ' O 1 p c + t GH ,
= n 1 Δ x ( sin θ 1 sin θ ) c

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