Abstract

We calculate the time required for a broad light pulse to traverse a plate bounded by dissimilar media. If the three media are transparent, then the peak velocity is a periodic function of the thickness of the plate, the period being equal to a half-wavelength in the plate. The minimum and maximum speeds depend on the optical constants of the three media; however, their geometric average is the group velocity in the slab medium. In the presence of small absorption the minimum (maximum) speed increases (decreases) linearly with thickness. We also discuss situations of experimental interest, including that of a thin metallic film on a glass substrate, in which case the oscillations of the transit velocity are strongly attenuated.

© 1991 Optical Society of America

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References

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  1. S. C. Bloch, “Eighth velocity of light,” Am. J. Phys. 45, 538 (1977).
    [CrossRef]
  2. See, for example, Ultrashort Light Pulses, S. L. Shapiro, ed. (Springer, Berlin, 1977).
    [CrossRef]
  3. E. P. Ippen and C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses, S. L. Shapiro, ed. (Springer, Berlin, 1977), p. 83.
    [CrossRef]
  4. D. Anderson, J. Askne, and M. Lisak, “Wave packets in an absorptive and strongly dispersive medium,” Phys. Rev. A 12, 1546 (1975).
    [CrossRef]
  5. L. A. Vainshtein, “Propagation of pulses,” Usp. Fiz. Nauk 118, 339 (1976) [Sov. Phys. Usp. 19, 189 (1976)].
    [CrossRef]
  6. L. Casperson and A. Yariv, “Pulse propagation in a high-gain medium,” Phys. Rev. Lett. 26, 293 (1971).
    [CrossRef]
  7. R. G. Ulbrich and G. W. Fehrenbach, “Polariton wave packet propagation in the exciton resonance of a semiconductor,” Phys. Rev. Lett. 43, 963 (1979).
    [CrossRef]
  8. Y. Segawa, Y. Aoyagi, and S. Namba, “Anomalously slow group velocity of upper branch polariton in CuCl,” Solid State Commun. 32, 229 (1979).
    [CrossRef]
  9. S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738 (1982).
    [CrossRef]
  10. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975) (see comment at the end of Sec. 1.3.4).
  11. 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).
    [CrossRef]
  12. D. G. Anderson and J. I. H. Askne, “Wave packets in strongly dispersive media,” Proc. IEEE 62, 1518 (1974).
    [CrossRef]
  13. E. S. Birger and L. A. Vainshtein, “Propagation of high-frequency perturbations in absorbing and active media,” Zh. Tekh. Fiz. 43, 2217 (1973) [Sov. Phys. Tech. Phys. 18, 1405 (1974)].
  14. B. Segard and B. Macke, “Observation of negative velocity pulse propagation,” Phys. Lett. A 109, 213 (1985).
    [CrossRef]
  15. P. Halevi and R. Fuchs, “Pulse propagation in an absorbing film,” Phys. Rev. Lett. 55, 338 (1985).
    [CrossRef] [PubMed]
  16. In addition to the exponential dependence on D, the temporal discrepancy is also proportional to D cos[2 Re(k2)D].
  17. P. Halevi, “Transit velocity of a light pulse through a transparent plate,” Opt. Lett. 11, 759 (1986).
    [CrossRef] [PubMed]
  18. P. Halevi and J. A. Gaspar-Armenta, “Propagation of pulses in solids and plasmas,” in Surface Waves in Plasmas and Solids, S. Vuković, ed. (World Scientific, Singapore, 1986), p. 147.
  19. An equivalent formula for a monochromatic wave is given in Sec. 7.6.1 of Ref. 10. The generalization to a pulse is straightforward.

1986 (1)

1985 (2)

B. Segard and B. Macke, “Observation of negative velocity pulse propagation,” Phys. Lett. A 109, 213 (1985).
[CrossRef]

P. Halevi and R. Fuchs, “Pulse propagation in an absorbing film,” Phys. Rev. Lett. 55, 338 (1985).
[CrossRef] [PubMed]

1982 (1)

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

1979 (2)

R. G. Ulbrich and G. W. Fehrenbach, “Polariton wave packet propagation in the exciton resonance of a semiconductor,” Phys. Rev. Lett. 43, 963 (1979).
[CrossRef]

Y. Segawa, Y. Aoyagi, and S. Namba, “Anomalously slow group velocity of upper branch polariton in CuCl,” Solid State Commun. 32, 229 (1979).
[CrossRef]

1977 (1)

S. C. Bloch, “Eighth velocity of light,” Am. J. Phys. 45, 538 (1977).
[CrossRef]

1976 (1)

L. A. Vainshtein, “Propagation of pulses,” Usp. Fiz. Nauk 118, 339 (1976) [Sov. Phys. Usp. 19, 189 (1976)].
[CrossRef]

1975 (1)

D. Anderson, J. Askne, and M. Lisak, “Wave packets in an absorptive and strongly dispersive medium,” Phys. Rev. A 12, 1546 (1975).
[CrossRef]

1974 (1)

D. G. Anderson and J. I. H. Askne, “Wave packets in strongly dispersive media,” Proc. IEEE 62, 1518 (1974).
[CrossRef]

1973 (1)

E. S. Birger and L. A. Vainshtein, “Propagation of high-frequency perturbations in absorbing and active media,” Zh. Tekh. Fiz. 43, 2217 (1973) [Sov. Phys. Tech. Phys. 18, 1405 (1974)].

1971 (1)

L. Casperson and A. Yariv, “Pulse propagation in a high-gain medium,” Phys. Rev. Lett. 26, 293 (1971).
[CrossRef]

1970 (1)

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).
[CrossRef]

Anderson, D.

D. Anderson, J. Askne, and M. Lisak, “Wave packets in an absorptive and strongly dispersive medium,” Phys. Rev. A 12, 1546 (1975).
[CrossRef]

Anderson, D. G.

D. G. Anderson and J. I. H. Askne, “Wave packets in strongly dispersive media,” Proc. IEEE 62, 1518 (1974).
[CrossRef]

Aoyagi, Y.

Y. Segawa, Y. Aoyagi, and S. Namba, “Anomalously slow group velocity of upper branch polariton in CuCl,” Solid State Commun. 32, 229 (1979).
[CrossRef]

Askne, J.

D. Anderson, J. Askne, and M. Lisak, “Wave packets in an absorptive and strongly dispersive medium,” Phys. Rev. A 12, 1546 (1975).
[CrossRef]

Askne, J. I. H.

D. G. Anderson and J. I. H. Askne, “Wave packets in strongly dispersive media,” Proc. IEEE 62, 1518 (1974).
[CrossRef]

Birger, E. S.

E. S. Birger and L. A. Vainshtein, “Propagation of high-frequency perturbations in absorbing and active media,” Zh. Tekh. Fiz. 43, 2217 (1973) [Sov. Phys. Tech. Phys. 18, 1405 (1974)].

Bloch, S. C.

S. C. Bloch, “Eighth velocity of light,” Am. J. Phys. 45, 538 (1977).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975) (see comment at the end of Sec. 1.3.4).

Casperson, L.

L. Casperson and A. Yariv, “Pulse propagation in a high-gain medium,” Phys. Rev. Lett. 26, 293 (1971).
[CrossRef]

Chu, S.

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

Fehrenbach, G. W.

R. G. Ulbrich and G. W. Fehrenbach, “Polariton wave packet propagation in the exciton resonance of a semiconductor,” Phys. Rev. Lett. 43, 963 (1979).
[CrossRef]

Fuchs, R.

P. Halevi and R. Fuchs, “Pulse propagation in an absorbing film,” Phys. Rev. Lett. 55, 338 (1985).
[CrossRef] [PubMed]

Garret, C. G. B.

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).
[CrossRef]

Gaspar-Armenta, J. A.

P. Halevi and J. A. Gaspar-Armenta, “Propagation of pulses in solids and plasmas,” in Surface Waves in Plasmas and Solids, S. Vuković, ed. (World Scientific, Singapore, 1986), p. 147.

Halevi, P.

P. Halevi, “Transit velocity of a light pulse through a transparent plate,” Opt. Lett. 11, 759 (1986).
[CrossRef] [PubMed]

P. Halevi and R. Fuchs, “Pulse propagation in an absorbing film,” Phys. Rev. Lett. 55, 338 (1985).
[CrossRef] [PubMed]

P. Halevi and J. A. Gaspar-Armenta, “Propagation of pulses in solids and plasmas,” in Surface Waves in Plasmas and Solids, S. Vuković, ed. (World Scientific, Singapore, 1986), p. 147.

Ippen, E. P.

E. P. Ippen and C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses, S. L. Shapiro, ed. (Springer, Berlin, 1977), p. 83.
[CrossRef]

Lisak, M.

D. Anderson, J. Askne, and M. Lisak, “Wave packets in an absorptive and strongly dispersive medium,” Phys. Rev. A 12, 1546 (1975).
[CrossRef]

Macke, B.

B. Segard and B. Macke, “Observation of negative velocity pulse propagation,” Phys. Lett. A 109, 213 (1985).
[CrossRef]

McCumber, D. E.

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).
[CrossRef]

Namba, S.

Y. Segawa, Y. Aoyagi, and S. Namba, “Anomalously slow group velocity of upper branch polariton in CuCl,” Solid State Commun. 32, 229 (1979).
[CrossRef]

Segard, B.

B. Segard and B. Macke, “Observation of negative velocity pulse propagation,” Phys. Lett. A 109, 213 (1985).
[CrossRef]

Segawa, Y.

Y. Segawa, Y. Aoyagi, and S. Namba, “Anomalously slow group velocity of upper branch polariton in CuCl,” Solid State Commun. 32, 229 (1979).
[CrossRef]

Shank, C. V.

E. P. Ippen and C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses, S. L. Shapiro, ed. (Springer, Berlin, 1977), p. 83.
[CrossRef]

Ulbrich, R. G.

R. G. Ulbrich and G. W. Fehrenbach, “Polariton wave packet propagation in the exciton resonance of a semiconductor,” Phys. Rev. Lett. 43, 963 (1979).
[CrossRef]

Vainshtein, L. A.

L. A. Vainshtein, “Propagation of pulses,” Usp. Fiz. Nauk 118, 339 (1976) [Sov. Phys. Usp. 19, 189 (1976)].
[CrossRef]

E. S. Birger and L. A. Vainshtein, “Propagation of high-frequency perturbations in absorbing and active media,” Zh. Tekh. Fiz. 43, 2217 (1973) [Sov. Phys. Tech. Phys. 18, 1405 (1974)].

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975) (see comment at the end of Sec. 1.3.4).

Wong, S.

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

Yariv, A.

L. Casperson and A. Yariv, “Pulse propagation in a high-gain medium,” Phys. Rev. Lett. 26, 293 (1971).
[CrossRef]

Am. J. Phys. (1)

S. C. Bloch, “Eighth velocity of light,” Am. J. Phys. 45, 538 (1977).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (1)

B. Segard and B. Macke, “Observation of negative velocity pulse propagation,” Phys. Lett. A 109, 213 (1985).
[CrossRef]

Phys. Rev. A (2)

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).
[CrossRef]

D. Anderson, J. Askne, and M. Lisak, “Wave packets in an absorptive and strongly dispersive medium,” Phys. Rev. A 12, 1546 (1975).
[CrossRef]

Phys. Rev. Lett. (4)

L. Casperson and A. Yariv, “Pulse propagation in a high-gain medium,” Phys. Rev. Lett. 26, 293 (1971).
[CrossRef]

R. G. Ulbrich and G. W. Fehrenbach, “Polariton wave packet propagation in the exciton resonance of a semiconductor,” Phys. Rev. Lett. 43, 963 (1979).
[CrossRef]

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

P. Halevi and R. Fuchs, “Pulse propagation in an absorbing film,” Phys. Rev. Lett. 55, 338 (1985).
[CrossRef] [PubMed]

Proc. IEEE (1)

D. G. Anderson and J. I. H. Askne, “Wave packets in strongly dispersive media,” Proc. IEEE 62, 1518 (1974).
[CrossRef]

Solid State Commun. (1)

Y. Segawa, Y. Aoyagi, and S. Namba, “Anomalously slow group velocity of upper branch polariton in CuCl,” Solid State Commun. 32, 229 (1979).
[CrossRef]

Usp. Fiz. Nauk (1)

L. A. Vainshtein, “Propagation of pulses,” Usp. Fiz. Nauk 118, 339 (1976) [Sov. Phys. Usp. 19, 189 (1976)].
[CrossRef]

Zh. Tekh. Fiz. (1)

E. S. Birger and L. A. Vainshtein, “Propagation of high-frequency perturbations in absorbing and active media,” Zh. Tekh. Fiz. 43, 2217 (1973) [Sov. Phys. Tech. Phys. 18, 1405 (1974)].

Other (6)

In addition to the exponential dependence on D, the temporal discrepancy is also proportional to D cos[2 Re(k2)D].

P. Halevi and J. A. Gaspar-Armenta, “Propagation of pulses in solids and plasmas,” in Surface Waves in Plasmas and Solids, S. Vuković, ed. (World Scientific, Singapore, 1986), p. 147.

An equivalent formula for a monochromatic wave is given in Sec. 7.6.1 of Ref. 10. The generalization to a pulse is straightforward.

See, for example, Ultrashort Light Pulses, S. L. Shapiro, ed. (Springer, Berlin, 1977).
[CrossRef]

E. P. Ippen and C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses, S. L. Shapiro, ed. (Springer, Berlin, 1977), p. 83.
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975) (see comment at the end of Sec. 1.3.4).

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Figures (2)

Fig. 1
Fig. 1

Peak velocity as a function of thickness for a plate bounded by dissimilar media (schematic). All media are assumed to be transparent, and 1 < 2 < 3; otherwise, vc and vs interchange roles. Here λ2 and vg are the wavelength and the group velocity in the plate, evaluated at the carrier frequency, and vc and vs are defined by Eqs. (14) and (15).

Fig. 2
Fig. 2

Peak velocity as a function of thickness for a thin film of silver on a glass substrate. The metal is strongly absorptive, while the glass is transparent; the optical constants are given in the text.

Equations (21)

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E 1 ( z , t ) = - d ω S 0 ( ω ) exp [ - i ( t - n 1 z c ) ] ,
R ( t ) = d u S ( u ) exp ( - i u t ) ,
E 1 ( z , t ) = R ( t - n 1 z c ) exp [ - i ω ¯ ( t - z n 1 c ) ] .
τ m ( ω ) = τ 12 ( ρ 23 ρ 21 ) m τ 23 ,
τ i j = 2 ( i / μ i ) 1 / 2 ( i / μ i ) 1 / 2 + ( j / μ j ) 1 / 2 ,             ρ i j = ( i / μ i ) 1 / 2 - ( j / μ j ) 1 / 2 ( i / μ i ) 1 / 2 + ( j / μ j ) 1 / 2
E 3 ( D , t ) = m = 0 d ω τ m ( ω ) S 0 ( ω ) × exp { i [ k 2 ( ω ) D ( 1 + 2 m ) - ω t ] } ,
t = O ( D / v g ) τ ,
k 2 ( ω ) k ( ω ¯ ) + d k d ω ¯ ( ω - ω ¯ ) = n ω ¯ c + ω - ω ¯ v g ,
E 3 ( D , t ) m = 0 τ m ( ω ¯ ) R [ t - D ( 1 + 2 m ) v g ] × exp { i ω ¯ [ n D ( 1 + 2 m ) c - t ] } .
R ( t ) R ( 0 ) + 1 2 R ( 0 ) t 2 ,
E 3 ( D , t ) τ 1 ( ω ¯ ) τ 2 ( ω ¯ ) exp [ i ω ¯ ( n D c - t ) ] × m = 0 η 2 m { R ( 0 ) + 1 2 R [ t - D v g ( 1 + 2 m ) ] 2 } , η 2 = ρ 23 ( ω ¯ ) ρ 21 ( ω ¯ ) exp [ 2 i k 2 ( ω ¯ ) D ] .
E 3 ( D , t ) τ 12 ( ω ¯ ) τ 23 ( ω ¯ ) exp { i [ k 2 ( ω ¯ ) D - ω ¯ t ] } × 1 1 - η 2 [ R ( 0 ) - 2 R ( 0 ) η 2 1 - η 2 D v g ( t - D v g ) + 1 2 R ( 0 ) ( t - D v g ) 2 ] .
E 3 ( D , t ) 2 R 2 ( 0 ) - 4 R ( 0 ) R ( 0 ) Re ( η 2 1 - η 2 ) D v g ( t - D v g ) + R ( 0 ) R ( 0 ) ( t - D v g ) 2 .
t 0 = D v g Re ( 1 + η 2 1 - η 2 ) .
v = D t 0 = v g 1 - 2 Re η 2 + η 4 1 - η 4 .
v = v c cos 2 [ k 2 ( ω ¯ ) D ] + v s sin 2 [ k 2 ( ω ¯ ) D ] ,
v c = v g 1 - ρ 23 ( ω ¯ ) ρ 21 ( ω ¯ ) 1 + ρ 23 ( ω ¯ ) ρ 21 ( ω ¯ ) = v g ( 2 / μ 2 ) 1 / 2 [ ( 1 / μ 1 ) 1 / 2 + ( 3 / μ 3 ) 1 / 2 ] ( 2 / μ 2 ) + ( 1 / μ 1 ) 1 / 2 ( 3 / μ 3 ) 1 / 2 ,
v s = v g 1 + ρ 23 ( ω ¯ ) ρ 21 ( ω ¯ ) 1 - ρ 23 ( ω ¯ ) ρ 21 ( ω ¯ ) = v g ( 2 / μ 2 ) + ( 1 / μ 1 ) 1 / 2 ( 3 / μ 3 ) 1 / 2 ( 2 / μ 2 ) 1 / 2 [ ( 1 / μ 1 ) 1 / 2 + ( 3 / μ 3 ) 1 / 2 ] .
( v c v s ) 1 / 2 = v g .
1 ± 4 ρ 23 ( ω ¯ ) ρ 21 ( ω ¯ ) 1 - ρ 23 2 ( ω ¯ ) ρ 21 2 ( ω ¯ ) Im [ k 2 ( ω ¯ ) ] D .
Im 2 ( ω ¯ ) Re 2 ( ω ¯ ) 3 1 / 2 ,

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