Abstract

We examine the energy exchanged between an electromagnetic pulse and a linear dielectric medium in which it propagates. While group velocity indicates the presence of field energy (the locus of which can move with arbitrary speed), the velocity of energy transport maintains strict luminality. This indicates that the medium treats the leading and trailing portions of the pulse differently. The principle of causality requires the medium to respond to the instantaneous spectrum, the spectrum of the pulse truncated at each new instant as a given locale in the medium experiences the pulse.

© 2001 Optical Society of America

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References

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  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]
  2. S. Chu and S. Wong, “Linear Pulse Propagation in an Absorbing Medium,” Phys. Rev. Lett. 48, 738–741 (1982).
    [Crossref]
  3. R. Y. Chiao, “Superluminal (but Causal) Propagation of Wave Packets in Transparent Media with Inverted Atomic Populations,” Phys. Rev. A. 48, R34–R37 (1993).
    [Crossref] [PubMed]
  4. E. L. Bolda, J. C. Garrison, and R. Y. Chiao, “Optical Pulse Propagation at Negative Group Velocities due to a Nearby Gain Line,” Phys. Rev. A 49, 2938–2947 (1994).
    [Crossref] [PubMed]
  5. R. Y. Chiao and A. M. Steinberg, “Tunneling Times and Superluminality,” Progress in Optics37, pp. 347–406 (Emil Wolf ed., Elsevier, Amsterdam, 1997).
    [Crossref]
  6. L. J. Wang, A. Kuzmmich, and A. Dogariu, “Gain-Assisted Superluminal Light Propagation,” Nature 406, 277–279 (2000).
    [Crossref] [PubMed]
  7. M. Ware, S. A. Glasgow, and J. Peatross “The Role of Group Velocity in Tracking Field Energy in Linear Dielectrics,” Opt. Express (Submitted).
  8. S. A. Glasgow, M. Ware, and J. Peatross, “Poynting’s Theorem and Luminal Energy Transport Velocity in Causal Dielectrics,” Phys. Rev. E, (to be published 2001).
  9. J. Peatross, M. Ware, and S. A. Glasgow, “The Role of the Instantaneous Spectrum in Pulse Propagation in Causal Linear Dielectrics,” J. Opt. Soc. Am. A 18, 1719–1725 (2001).
    [Crossref]
  10. L. V. Hau, S. E. Haris, Z. Dutton, and C. H. Behroozi, “Light Speed Reduction to 17 Metres per Second in an Ultracold Atomic Gas,” Nature 397, 594–598 (1999).
    [Crossref]
  11. R. Loudon, “The Propagation of Electromagnetic Energy through an Absorbing Dielectric,” J. Phys. A 3, 233–245 (1970).
    [Crossref]
  12. C. H. Page, “Instantaneous Power Spectra,” J. Appl. Phys. 23, 103–106 (1952).
    [Crossref]
  13. M. B. Priestley, “Power Spectral Analysis of Non-Stationary Random Processes,” J. Sound Vib. 6, 86–97 (1967).
    [Crossref]
  14. J. H. Eberly and K. Wodkiewicz, “The Time-Dependent Physical Spectrum of Light,” J. Opt. Soc. Am. 67, 1252–1260 (1977).
    [Crossref]
  15. L. Brillouin, Wave Propagation and Group Velocity (Academic Press, New York, 1960).
  16. K. R. Bownstein, “Some Time Evolution Properties of an Electromagnetic Wave,” Am. J. Phys. 65, 510–515 (1997).
    [Crossref]
  17. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed., (Pergamon, Oxford, 1982), p. 274.
  18. J. D. Jackson, Classical Electrodynamics, 3rd Ed. (Wiley, New York, 1998), pp. 323, 330–335.
  19. M. Born and E. Wolf, Principles of Optics, 7th Ed. (Cambridge, 1999), pp. 19–24.
  20. M. D. Crisp, “Concept of Group Velocity in Resonant Pulse Propagation,” Phys. Rev. A4, (1971).
    [Crossref]
  21. G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A223, (1996).
    [Crossref]

2001 (1)

2000 (1)

L. J. Wang, A. Kuzmmich, and A. Dogariu, “Gain-Assisted Superluminal Light Propagation,” Nature 406, 277–279 (2000).
[Crossref] [PubMed]

1999 (1)

L. V. Hau, S. E. Haris, Z. Dutton, and C. H. Behroozi, “Light Speed Reduction to 17 Metres per Second in an Ultracold Atomic Gas,” Nature 397, 594–598 (1999).
[Crossref]

1997 (1)

K. R. Bownstein, “Some Time Evolution Properties of an Electromagnetic Wave,” Am. J. Phys. 65, 510–515 (1997).
[Crossref]

1994 (1)

E. L. Bolda, J. C. Garrison, and R. Y. Chiao, “Optical Pulse Propagation at Negative Group Velocities due to a Nearby Gain Line,” Phys. Rev. A 49, 2938–2947 (1994).
[Crossref] [PubMed]

1993 (1)

R. Y. Chiao, “Superluminal (but Causal) Propagation of Wave Packets in Transparent Media with Inverted Atomic Populations,” Phys. Rev. A. 48, R34–R37 (1993).
[Crossref] [PubMed]

1982 (1)

S. Chu and S. Wong, “Linear Pulse Propagation in an Absorbing Medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[Crossref]

1977 (1)

1970 (2)

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]

R. Loudon, “The Propagation of Electromagnetic Energy through an Absorbing Dielectric,” J. Phys. A 3, 233–245 (1970).
[Crossref]

1967 (1)

M. B. Priestley, “Power Spectral Analysis of Non-Stationary Random Processes,” J. Sound Vib. 6, 86–97 (1967).
[Crossref]

1952 (1)

C. H. Page, “Instantaneous Power Spectra,” J. Appl. Phys. 23, 103–106 (1952).
[Crossref]

Behroozi, C. H.

L. V. Hau, S. E. Haris, Z. Dutton, and C. H. Behroozi, “Light Speed Reduction to 17 Metres per Second in an Ultracold Atomic Gas,” Nature 397, 594–598 (1999).
[Crossref]

Bolda, E. L.

E. L. Bolda, J. C. Garrison, and R. Y. Chiao, “Optical Pulse Propagation at Negative Group Velocities due to a Nearby Gain Line,” Phys. Rev. A 49, 2938–2947 (1994).
[Crossref] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th Ed. (Cambridge, 1999), pp. 19–24.

Bownstein, K. R.

K. R. Bownstein, “Some Time Evolution Properties of an Electromagnetic Wave,” Am. J. Phys. 65, 510–515 (1997).
[Crossref]

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity (Academic Press, New York, 1960).

Chiao, R. Y.

E. L. Bolda, J. C. Garrison, and R. Y. Chiao, “Optical Pulse Propagation at Negative Group Velocities due to a Nearby Gain Line,” Phys. Rev. A 49, 2938–2947 (1994).
[Crossref] [PubMed]

R. Y. Chiao, “Superluminal (but Causal) Propagation of Wave Packets in Transparent Media with Inverted Atomic Populations,” Phys. Rev. A. 48, R34–R37 (1993).
[Crossref] [PubMed]

R. Y. Chiao and A. M. Steinberg, “Tunneling Times and Superluminality,” Progress in Optics37, pp. 347–406 (Emil Wolf ed., Elsevier, Amsterdam, 1997).
[Crossref]

Chu, S.

S. Chu and S. Wong, “Linear Pulse Propagation in an Absorbing Medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[Crossref]

Crisp, M. D.

M. D. Crisp, “Concept of Group Velocity in Resonant Pulse Propagation,” Phys. Rev. A4, (1971).
[Crossref]

Diener, G.

G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A223, (1996).
[Crossref]

Dogariu, A.

L. J. Wang, A. Kuzmmich, and A. Dogariu, “Gain-Assisted Superluminal Light Propagation,” Nature 406, 277–279 (2000).
[Crossref] [PubMed]

Dutton, Z.

L. V. Hau, S. E. Haris, Z. Dutton, and C. H. Behroozi, “Light Speed Reduction to 17 Metres per Second in an Ultracold Atomic Gas,” Nature 397, 594–598 (1999).
[Crossref]

Eberly, J. H.

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]

Garrison, J. C.

E. L. Bolda, J. C. Garrison, and R. Y. Chiao, “Optical Pulse Propagation at Negative Group Velocities due to a Nearby Gain Line,” Phys. Rev. A 49, 2938–2947 (1994).
[Crossref] [PubMed]

Glasgow, S. A.

J. Peatross, M. Ware, and S. A. Glasgow, “The Role of the Instantaneous Spectrum in Pulse Propagation in Causal Linear Dielectrics,” J. Opt. Soc. Am. A 18, 1719–1725 (2001).
[Crossref]

M. Ware, S. A. Glasgow, and J. Peatross “The Role of Group Velocity in Tracking Field Energy in Linear Dielectrics,” Opt. Express (Submitted).

S. A. Glasgow, M. Ware, and J. Peatross, “Poynting’s Theorem and Luminal Energy Transport Velocity in Causal Dielectrics,” Phys. Rev. E, (to be published 2001).

Haris, S. E.

L. V. Hau, S. E. Haris, Z. Dutton, and C. H. Behroozi, “Light Speed Reduction to 17 Metres per Second in an Ultracold Atomic Gas,” Nature 397, 594–598 (1999).
[Crossref]

Hau, L. V.

L. V. Hau, S. E. Haris, Z. Dutton, and C. H. Behroozi, “Light Speed Reduction to 17 Metres per Second in an Ultracold Atomic Gas,” Nature 397, 594–598 (1999).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd Ed. (Wiley, New York, 1998), pp. 323, 330–335.

Kuzmmich, A.

L. J. Wang, A. Kuzmmich, and A. Dogariu, “Gain-Assisted Superluminal Light Propagation,” Nature 406, 277–279 (2000).
[Crossref] [PubMed]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed., (Pergamon, Oxford, 1982), p. 274.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed., (Pergamon, Oxford, 1982), p. 274.

Loudon, R.

R. Loudon, “The Propagation of Electromagnetic Energy through an Absorbing Dielectric,” J. Phys. A 3, 233–245 (1970).
[Crossref]

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]

Page, C. H.

C. H. Page, “Instantaneous Power Spectra,” J. Appl. Phys. 23, 103–106 (1952).
[Crossref]

Peatross, J.

J. Peatross, M. Ware, and S. A. Glasgow, “The Role of the Instantaneous Spectrum in Pulse Propagation in Causal Linear Dielectrics,” J. Opt. Soc. Am. A 18, 1719–1725 (2001).
[Crossref]

S. A. Glasgow, M. Ware, and J. Peatross, “Poynting’s Theorem and Luminal Energy Transport Velocity in Causal Dielectrics,” Phys. Rev. E, (to be published 2001).

M. Ware, S. A. Glasgow, and J. Peatross “The Role of Group Velocity in Tracking Field Energy in Linear Dielectrics,” Opt. Express (Submitted).

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed., (Pergamon, Oxford, 1982), p. 274.

Priestley, M. B.

M. B. Priestley, “Power Spectral Analysis of Non-Stationary Random Processes,” J. Sound Vib. 6, 86–97 (1967).
[Crossref]

Steinberg, A. M.

R. Y. Chiao and A. M. Steinberg, “Tunneling Times and Superluminality,” Progress in Optics37, pp. 347–406 (Emil Wolf ed., Elsevier, Amsterdam, 1997).
[Crossref]

Wang, L. J.

L. J. Wang, A. Kuzmmich, and A. Dogariu, “Gain-Assisted Superluminal Light Propagation,” Nature 406, 277–279 (2000).
[Crossref] [PubMed]

Ware, M.

J. Peatross, M. Ware, and S. A. Glasgow, “The Role of the Instantaneous Spectrum in Pulse Propagation in Causal Linear Dielectrics,” J. Opt. Soc. Am. A 18, 1719–1725 (2001).
[Crossref]

M. Ware, S. A. Glasgow, and J. Peatross “The Role of Group Velocity in Tracking Field Energy in Linear Dielectrics,” Opt. Express (Submitted).

S. A. Glasgow, M. Ware, and J. Peatross, “Poynting’s Theorem and Luminal Energy Transport Velocity in Causal Dielectrics,” Phys. Rev. E, (to be published 2001).

Wodkiewicz, K.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th Ed. (Cambridge, 1999), pp. 19–24.

Wong, S.

S. Chu and S. Wong, “Linear Pulse Propagation in an Absorbing Medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[Crossref]

Am. J. Phys. (1)

K. R. Bownstein, “Some Time Evolution Properties of an Electromagnetic Wave,” Am. J. Phys. 65, 510–515 (1997).
[Crossref]

J. Appl. Phys. (1)

C. H. Page, “Instantaneous Power Spectra,” J. Appl. Phys. 23, 103–106 (1952).
[Crossref]

J. Opt. Soc. Am. (1)

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

J. Phys. A (1)

R. Loudon, “The Propagation of Electromagnetic Energy through an Absorbing Dielectric,” J. Phys. A 3, 233–245 (1970).
[Crossref]

J. Sound Vib. (1)

M. B. Priestley, “Power Spectral Analysis of Non-Stationary Random Processes,” J. Sound Vib. 6, 86–97 (1967).
[Crossref]

Nature (2)

L. V. Hau, S. E. Haris, Z. Dutton, and C. H. Behroozi, “Light Speed Reduction to 17 Metres per Second in an Ultracold Atomic Gas,” Nature 397, 594–598 (1999).
[Crossref]

L. J. Wang, A. Kuzmmich, and A. Dogariu, “Gain-Assisted Superluminal Light Propagation,” Nature 406, 277–279 (2000).
[Crossref] [PubMed]

Phys. Rev. A (2)

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, J. C. Garrison, and R. Y. Chiao, “Optical Pulse Propagation at Negative Group Velocities due to a Nearby Gain Line,” Phys. Rev. A 49, 2938–2947 (1994).
[Crossref] [PubMed]

Phys. Rev. A. (1)

R. Y. Chiao, “Superluminal (but Causal) Propagation of Wave Packets in Transparent Media with Inverted Atomic Populations,” Phys. Rev. A. 48, R34–R37 (1993).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

S. Chu and S. Wong, “Linear Pulse Propagation in an Absorbing Medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[Crossref]

Other (9)

R. Y. Chiao and A. M. Steinberg, “Tunneling Times and Superluminality,” Progress in Optics37, pp. 347–406 (Emil Wolf ed., Elsevier, Amsterdam, 1997).
[Crossref]

M. Ware, S. A. Glasgow, and J. Peatross “The Role of Group Velocity in Tracking Field Energy in Linear Dielectrics,” Opt. Express (Submitted).

S. A. Glasgow, M. Ware, and J. Peatross, “Poynting’s Theorem and Luminal Energy Transport Velocity in Causal Dielectrics,” Phys. Rev. E, (to be published 2001).

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed., (Pergamon, Oxford, 1982), p. 274.

J. D. Jackson, Classical Electrodynamics, 3rd Ed. (Wiley, New York, 1998), pp. 323, 330–335.

M. Born and E. Wolf, Principles of Optics, 7th Ed. (Cambridge, 1999), pp. 19–24.

M. D. Crisp, “Concept of Group Velocity in Resonant Pulse Propagation,” Phys. Rev. A4, (1971).
[Crossref]

G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A223, (1996).
[Crossref]

L. Brillouin, Wave Propagation and Group Velocity (Academic Press, New York, 1960).

Supplementary Material (3)

» Media 1: MOV (1429 KB)     
» Media 2: MOV (1453 KB)     
» Media 3: MOV (2032 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

(a) The imaginary part of χ (ω) (b) Spectrum of the initial pulse in units of (E 0/γ)2

Fig. 2.
Fig. 2.

(a) Animation of energy densities for the Gaussian pulse traversing the medium (distances are in units of c/γ and energy densities are in units of E02/ 0) (b) Instantaneous spectrum of the pulse at the point where it enters the medium (1.5 MB)

Fig. 3.
Fig. 3.

(a) Animation of energy densities for the Gaussian pulse traversing the medium (distances are in units of c/γ and energy densities are in units of E02/ 0) (b) Instantaneous spectrum of the pulse at the point where it enters the medium (1.5 MB)

Fig. 4.
Fig. 4.

(a) Animation of a Gaussian pulse traversing an amplifying medium. (b) Instantaneous spectrum of the pulse as it enters the medium (1.5 MB)

Fig. 5.
Fig. 5.

Animation of a truncated Gaussian pulse traversing an amplifying medium, linear scale in the upper frame and logarithmic in the lower frame (2.0 MB)

Equations (44)

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· S ( r , t ) + u ( r , t ) t = 0 ,
S ( r , t ) E ( r , t ) × B ( r , t ) μ 0
u ( r , t ) = u field ( r , t ) + u exchange ( r , t ) + u ( r , ) .
u field ( r , t ) B 2 ( r , t ) 2 μ 0 + 0 E 2 ( r , t ) 2 .
u exchange ( r , t ) t E ( r , t ) · P ( r , t ) t ' d t .
A S · da = t V u d 3 r ,
v E S / u .
v E v E u d 3 r u d 3 r = S d 3 r u d 3 r
v E = r · S d 3 r u d 3 r = r u t d 3 r u d 3 r
v E = r t ,
r = r u d 3 r / u d 3 r .
S u field t r u field d 3 r u field d 3 r .
r field t r u field ( r , t ) d 3 r / u field ( r , t ) d 3 r .
Δ r r field t 0 + Δ t r field t 0 .
Δ r = Δ r G + Δ r R .
Δ r G Δ t [ k R e ω ( k ) ] ρ ( k , t ) d 3 k ,
ρ ( k , t ) u field ( k , t ) / u field ( k , t ) d 3 k .
E ( r , ω ) = 1 2 π e iωt E ( r , t ) d t ,
E ( r , t ) = 1 2 π e iωt E ( r , ω ) d ω .
P ( r , ω ) = 0 χ ( r , ω ) E ( r , ω ) .
E ( r , ω ) = E * ( r , ω )
P ( r , ω ) = P * ( r , ω )
χ ( r , ω ) = χ * ( r , ω ) .
u exchange ( r , t ) = t [ 1 2 π E ( r , ω ) e t d ω ]
· [ i 0 2 π ω χ ( r , ω ) E ( r , ω ) e iωt d ω ] d t .
u exchange ( r , t ) = i 0 d ω ω χ ( r , ω ) E ( r , ω ) · d ω E ( r , ω ) 1 2 π t e i ( ω + ω ) t d t .
u exchange ( r , + ) = i 0 ω χ ( r , ω ) E ( r , ω ) · E ( r , ω ) d ω .
u exchange ( r , + ) = 0 ω Im χ ( r , ω ) E ( r , ω ) 2 d ω .
u exchange ( r , t ) = 0 ω Im χ ( r , ω ) E t ( r , ω ) 2 d ω ,
E t ( r , ω ) 1 2 π t E ( r , t ) e iωt d t .
E ( k , t ) = 1 ( 2 π ) 3 / 2 e i k · r E ( r , t ) d 3 r ,
E ( r , t ) = 1 ( 2 π ) 3 / 2 e i k · r E ( k , t ) d 3 k .
E ( k , t ) = E * ( k , t )
E ( k , t 0 + Δ t ) = m E m ( k , t 0 ) e m ( k ) Δ t .
B ( k , t ) = m k × E m ( k , t ) / ω m ( k ) ,
ω m 2 c 2 [ 1 + χ ( ω m ) ] = k 2 .
E ( k , t 0 + Δ t ) = E 0 ( k , t 0 ) e ( k ) Δ t .
r field t r u field ( r , t ) d 3 r / u field ( r , t ) d 3 r .
r field t = R [ E ( k , t ) ] ,
R [ E ( k , t ) ] i d 3 k j = x , y , z [ 0 2 E j * ( k , t ) · k E j ( k , t ) + 1 2 μ 0 B j * ( k , t ) · k B j ( k , t ) ] u field ( r , t ) d 3 r .
u field ( k , t ) = 0 E ( k , t ) · E * ( k , t ) 2 + B ( k , t ) · B * ( k , t ) 2 μ 0 .
Δ r r field t 0 + Δ t r field t 0 = R [ E ( k , t 0 + Δ t ) ] R [ E ( k , t 0 ) ]
Δ r = Δ r G ( t ) + Δ r R ( t 0 ) .
Δ r R ( t 0 ) R [ e Im ω ( k ) Δ t E ( k , t 0 ) ] R [ E ( k , t 0 ) ] .

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