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.

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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-R37 (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.A48, 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.A1, 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, 2 nd Edition), p 316-317.
  15. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), §84.

Other

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).

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]

E. L. Bolda, R. Y. Chiao, and J. C. Garrison, “Two theorems for the group velocity in dispersive media,” Phys.Rev.A48, 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]

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).

C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian pulse through an anomalous dispersion medium,”Phys.Rev.A1, 305-313 (1970).
[CrossRef]

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

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.

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.

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, 2 nd 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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