Abstract

We study the propagation of light pulses through a transparent anomalous dispersion medium where the group velocity of the pulse exceeds c and can even become negative. Because the medium is transparent, we can apply the Kelvin’s method of stationary phase to obtain the general properties of the pulse propagation process for interesting conditions when the group velocity: U<c,U=±∞, and even becomes negative: U<0. A numerical simulation illustrating pulse propagation at a negative group velocity is also presented. We show how rephasing can produce these unusual pulse propagation phenomena.

© 2001 Optical Society of America

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References

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  1. L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000) [and references therein].
    [Crossref] [PubMed]
  2. A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal pulse propagation at a negative group velocity,” Phys. Rev. A. (to be published).
  3. R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34–37 (1993).
    [Crossref] [PubMed]
  4. E. L. Bolda, R. Y. Chiao, and J. C. Garrison, “Two theorems for the group velocity in dispersive media,” Phys. Rev. A 48, 3890–3894 (1993).
    [Crossref] [PubMed]
  5. A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071–2075 (1994).
    [Crossref] [PubMed]
  6. E. 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]
  7. R. Y. Chiao, “Population inversion and superluminality,” in Amazing Light, a Volume Dedicated to C. H. Townes on His Eightieth Birthday, (Springer, New York, 1996).
  8. C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
    [Crossref]
  9. S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
    [Crossref]
  10. B L. Brillouin, Wave Propagation and Group Velocity (Academic Press, New York, 1960), Chapter 1 has a detailed description of Kelvin’s method of stationary phase..
  11. Segev, P. W. Milonni, J. F. Babb, and R. Y. Chiao, “Quantum noise and superluminal propagation” Phys. Rev. A 62, 022114 (2000).
    [Crossref]
  12. A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light propagation,” Phys. Rev. Lett. (to be published) http:/xxx. lanl.gov/abs/physics/0101068.
  13. K. MacDonald, “Negative Group Velocity,” Am. J. Phys. (to be published), http:/xxx.lanl.gov/abs/physics/0008013.
  14. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1974, 2nd Edition), p 316–317.
  15. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), §84.

2000 (2)

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000) [and references therein].
[Crossref] [PubMed]

Segev, P. W. Milonni, J. F. Babb, and R. Y. Chiao, “Quantum noise and superluminal propagation” Phys. Rev. A 62, 022114 (2000).
[Crossref]

1994 (2)

A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071–2075 (1994).
[Crossref] [PubMed]

E. 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 (2)

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34–37 (1993).
[Crossref] [PubMed]

E. L. Bolda, R. Y. Chiao, and J. C. Garrison, “Two theorems for the group velocity in dispersive media,” Phys. Rev. A 48, 3890–3894 (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]

1970 (1)

C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
[Crossref]

Babb, J. F.

Segev, P. W. Milonni, J. F. Babb, and R. Y. Chiao, “Quantum noise and superluminal propagation” Phys. Rev. A 62, 022114 (2000).
[Crossref]

Bolda, E.

E. 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]

Bolda, E. L.

E. L. Bolda, R. Y. Chiao, and J. C. Garrison, “Two theorems for the group velocity in dispersive media,” Phys. Rev. A 48, 3890–3894 (1993).
[Crossref] [PubMed]

Brillouin, B L.

B L. Brillouin, Wave Propagation and Group Velocity (Academic Press, New York, 1960), Chapter 1 has a detailed description of Kelvin’s method of stationary phase..

Chiao, R. Y.

Segev, P. W. Milonni, J. F. Babb, and R. Y. Chiao, “Quantum noise and superluminal propagation” Phys. Rev. A 62, 022114 (2000).
[Crossref]

A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071–2075 (1994).
[Crossref] [PubMed]

E. 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, R34–37 (1993).
[Crossref] [PubMed]

E. L. Bolda, R. Y. Chiao, and J. C. Garrison, “Two theorems for the group velocity in dispersive media,” Phys. Rev. A 48, 3890–3894 (1993).
[Crossref] [PubMed]

R. Y. Chiao, “Population inversion and superluminality,” in Amazing Light, a Volume Dedicated to C. H. Townes on His Eightieth Birthday, (Springer, New York, 1996).

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light propagation,” Phys. Rev. Lett. (to be published) http:/xxx. lanl.gov/abs/physics/0101068.

Chu, S.

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

Dogariu, A.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000) [and references therein].
[Crossref] [PubMed]

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal pulse propagation at a negative group velocity,” Phys. Rev. A. (to be published).

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light propagation,” Phys. Rev. Lett. (to be published) http:/xxx. lanl.gov/abs/physics/0101068.

Garrett, C. G. B.

C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
[Crossref]

Garrison, J. C.

E. 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]

E. L. Bolda, R. Y. Chiao, and J. C. Garrison, “Two theorems for the group velocity in dispersive media,” Phys. Rev. A 48, 3890–3894 (1993).
[Crossref] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1974, 2nd Edition), p 316–317.

Kuzmich, A.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000) [and references therein].
[Crossref] [PubMed]

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal pulse propagation at a negative group velocity,” Phys. Rev. A. (to be published).

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light propagation,” Phys. Rev. Lett. (to be published) http:/xxx. lanl.gov/abs/physics/0101068.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), §84.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), §84.

MacDonald, K.

K. MacDonald, “Negative Group Velocity,” Am. J. Phys. (to be published), http:/xxx.lanl.gov/abs/physics/0008013.

McCumber, D. E.

C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
[Crossref]

Milonni, P. W.

Segev, P. W. Milonni, J. F. Babb, and R. Y. Chiao, “Quantum noise and superluminal propagation” Phys. Rev. A 62, 022114 (2000).
[Crossref]

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light propagation,” Phys. Rev. Lett. (to be published) http:/xxx. lanl.gov/abs/physics/0101068.

Segev,

Segev, P. W. Milonni, J. F. Babb, and R. Y. Chiao, “Quantum noise and superluminal propagation” Phys. Rev. A 62, 022114 (2000).
[Crossref]

Steinberg, A. M.

A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071–2075 (1994).
[Crossref] [PubMed]

Wang, L. J.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000) [and references therein].
[Crossref] [PubMed]

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal pulse propagation at a negative group velocity,” Phys. Rev. A. (to be published).

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light propagation,” Phys. Rev. Lett. (to be published) http:/xxx. lanl.gov/abs/physics/0101068.

Wong, S.

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

Nature (1)

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000) [and references therein].
[Crossref] [PubMed]

Phys. Rev. A (6)

C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
[Crossref]

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34–37 (1993).
[Crossref] [PubMed]

E. L. Bolda, R. Y. Chiao, and J. C. Garrison, “Two theorems for the group velocity in dispersive media,” Phys. Rev. A 48, 3890–3894 (1993).
[Crossref] [PubMed]

A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071–2075 (1994).
[Crossref] [PubMed]

E. 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]

Segev, P. W. Milonni, J. F. Babb, and R. Y. Chiao, “Quantum noise and superluminal propagation” Phys. Rev. A 62, 022114 (2000).
[Crossref]

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 (7)

B L. Brillouin, Wave Propagation and Group Velocity (Academic Press, New York, 1960), Chapter 1 has a detailed description of Kelvin’s method of stationary phase..

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal pulse propagation at a negative group velocity,” Phys. Rev. A. (to be published).

R. Y. Chiao, “Population inversion and superluminality,” in Amazing Light, a Volume Dedicated to C. H. Townes on His Eightieth Birthday, (Springer, New York, 1996).

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light propagation,” Phys. Rev. Lett. (to be published) http:/xxx. lanl.gov/abs/physics/0101068.

K. MacDonald, “Negative Group Velocity,” Am. J. Phys. (to be published), http:/xxx.lanl.gov/abs/physics/0008013.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1974, 2nd Edition), p 316–317.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), §84.

Supplementary Material (2)

» Media 1: MOV (323 KB)     
» Media 2: AVI (2116 KB)     

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Figures (4)

Figure 1.
Figure 1.

Pulse propagation through a transparent anomalous dispersion medium of a length L. Pulse propagation through the same length in vacuum is also shown for comparison.

Figure 2.
Figure 2.

Pulse envelope evolution during propagation through a transparent anomalous dispersion medium of a length L. The medium is assumed to have a group velocity index ng =0 resulting in a zero transit time at a group velocity U=∞. Red curve displays intensity of the pulse propagating through vacuum while blue curve shows the intensity propagating through the medium. The pulse propagation is obtained by solving Maxwell’s equations.

Figure 3.
Figure 3.

Pulse envelope evolution during propagation through a transparent anomalous dispersion medium of a length L at a negative group velocity. The medium is assumed to have a group velocity index ng =-3 resulting in a group velocity U=-c/3. Inside the medium, rephasing produces a pulse peak propagating backward at c/3. Red curve displays intensity of the pulse propagating through vacuum while blue curve shows the intensity propagating through the medium. The pulse propagation is obtained by solving Maxwell’s equations. (Animation: QuckTime 0.3MB).

Figure 4.
Figure 4.

Pulse propagation through a transparent anomalous dispersion medium at a negative group velocity. The medium is assumed to have a group velocity index ng =-3 resulting in a group velocity U=-c/3. Top graph: pulse propagation through vacuum and through the medium where a rephasing process produces two pulse peaks, one inside the medium and one on the far side outside. Lower graph: the evolution of three wave components of the pulse. A blue ray outside the medium becomes a red ray (with longer wavelength) inside the medium, and vice versa. The medium is taken to satisfy Eq.(10), closely resembling the experimental conditions of Ref. [1,2]. (Animation: QuickTime 2.0MB)

Equations (10)

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E ( + ) ( z , t ) = 0 E ( ω ) e i [ ω t k ( ω ) z ] d ω .
I ( z , t ) = 2 ε o c E ( + ) ( z , t ) 2 .
= c · d ϕ d ω = c · d [ ω t k ( ω ) z ] d ω = c ( t z U ) = 0 .
U = ( dk d ω ) ω o 1 = c n + ω dn d ω
k ( ω ) = k ( ω o ) + 1 U ( ω ω o ) + 1 2 ( d 2 k d ω 2 ) ( ω ω o ) 2 + .
E ( + ) ( z + L , t ) = g · e i [ ω o t k ( ω o ) L ] E ( + ) ( z , t L U ) .
I ( z + L , t ) = I ( z , t L U ) .
d ( λ n ) d λ = c n 2 ( dk d ω ) = n g n 2 0 .
d ( λ n ) d λ = 0 .
d ( λ n ) d λ < 0 .

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