## Superluminal light pulse propagation via rephasing in a transparent anomalously dispersive medium

Optics Express, Vol. 8, Issue 6, pp. 344-350 (2001)

http://dx.doi.org/10.1364/OE.8.000344

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### 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.

© Optical Society of America

## 1 Introduction

*U*>

*c*) or even negative in a transparent material [1

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

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]

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]

*c*/310 [1

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

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]

*signal velocity*be limited by

*c*, the vacuum speed of light. Signal velocity is still bound by

*c*in the present experiment [1

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

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.

## 2 Pulse propagation in dispersive media

*E*

^{(+)}(

*z*,

*t*)+

*c*.

*c*., and we further have for the intensity (energy flux) of the pulse at the exit surface of the medium:

*Kelvin’s method of stationary phase*[10] would read

*ℓ*is the rephasing length. As noted by Brillouin in his summary work on wave propagation [10], Lord Kelvin had repeatedly used this method in many problems. However, it should be noticed here that the method can only apply under the condition when the medium is transparent (no absorption).

*L*:

*g*is an extra factor denoting gain. For the rest of the article, we will assume

*g*=1 for simplicity. In the experiment, this can be achieved by placing a broadband absorber (or amplifier for the case of loss) to compensate for this extraneous factor associated with dispersion. Following Eq.(6), it is now possible to write for the pulse intensity after propagating through a length

*L*in the medium [14]:

*zero transit time*can be understood by closely examining the rephasing length

*ℓ*=

*c*·

*dϕ*/

*dω*. We consider the situation in Figure 1 and suppose that the peak of the incident pulse arrives at the entrance interface

*z*=0 at a time

*t*=0. Then we have for region-I (

*z*<):

*ℓ*

_{1}=

*c*·

*t*-

*z*. In region-II (

*L*>

*z*>0), we have

*ℓ*

_{2}=

*c*·(

*t*-

*z*/

*U*)=

*c*·

*t*-

*n*

_{g}·

*z*, where

*n*

_{g}=

*n*+

*ωdn*/

*dω*is the group velocity index. And furthermore in region-III (

*z*>

*L*), we obtain

*ℓ*

_{3}=

*c*·

*t*+ (1-

*n*

_{g})·

*L*-

*z*.

*zero transit time U*=∞, we have for the group velocity index

*n*

_{g}=0. Then inside the medium, the rephasing length

*ℓ*=

*c*·

*dϕ*/

*dω*becomes independent of position

*z*. In other words, the relative phase differences between various frequency component will remain the same throughout the length of the medium. Hence, the envelope intensity of the pulse throughout the medium’s length is equal to that at the entrance at any given moment. Beyond the exit surface

*z*=

*L*, an analytical continuation of the pulse’s leading edge is reproduces because in region-III, phase differences between various component of the pulse are reproduced again. Fig.2 shows the calculated pulse envelope intensity for the case of

*U*=∞.

*n*

_{g}=

*n*+

*ωdn*/

*dω*<0. Hence, we have a negative group velocity

*U*=

*c*/

*n*

_{g}<0. In this case, we have a

*negative transit time*. Namely, at and beyond a certain time

*t*<0 before the peak of the pulse enters the medium, the rephasing length inside the medium

*ℓ*

_{2}=

*c*·

*t*-

*n*

_{g}·

*z*can become zero at a position

*z*

_{o}=

*c*·

*t*/

*n*

_{g}. Note here since we have both

*n*

_{g}< 0 and

*t*<0, we have for this rephasing position: 0<

*z*

_{o}<

*L*. In other words, at this position, the relative phase differences between different frequency components vanish and a peak is reproduced due to constructive interference. The rephasing condition hence requires that the peak of the incident pulse be sufficiently near the entrance surface of the medium such that: 0>

*t*>

*n*

_{g}

*L*/

*c*. As the incident pulse approaches the medium, at the later time

*t*’ such that 0>

*t*

^{’}>

*t*, the coordinate of the peak inside the medium

*c*·

*t*′/

*n*

_{g}decreases:

*z*

_{o}. Hence, the peak inside the medium moves at a negative velocity:

*c*/

*n*

_{g}<0. Finally at the time

*t*=0 when the peak of the incident pulse reaches the input surface, the peak of the back-propagating pulse inside the medium also reaches the surface and due to destructive interference, they cancel one another. Figure 3 illustrates the pulse propagation in a medium with a negative group velocity index

*n*

_{g}<0.

*z*>

*L*, another region of rephasing appears under the condition:

*ℓ*

_{3}=

*c*·

*t*+ (1-

*n*

_{g})·

*L*-

*z*

^{″}=0. This gives the position

*z*

^{″}=

*L*+

*c*·

*t*-

*n*

_{g}·

*L*. It is clear from the rephasing condition:

*t*>

*n*

_{g}

*L*/

*c*that we have

*z*

^{″}>

*L*(note

*t*,

*n*

_{g}<0).

*n*

_{g}=

*n*+

*ωdn*/

*dω*≤0. Namely, inside a medium with a refractive index

*n*, the wavelength of a light ray becomes

*λ*/

*n*where

*λ*is the light ray’s vacuum wavelength. It is easy to derive the relation:

*n*

_{g}=0, we have

*n*

_{g}<0, we have

## 3 Conclusions

*Kelvin’s method of stationary phase*to gain an intuitive understanding of the behavior of the pulse’s peak. Using this method, we analyzed the conditions of rephasing under various situations of anomalous dispersion especially those corresponding to

*zero transit time*and

*negative transit time*. The rephasing processes under these unusual conditions can sully explain the recently observed superluminal light pulse propagation. These counterintuitive phenomena should be viewed as direct results of wave interference in an anomalous dispersion medium.

*not*at odds with causality or special relativity [10–12

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.

*c*, instead of the group velocity. Although the difference is subtle, the signal velocity is different from the group velocity, as first noted again by Sommerfeld and Brillouin [10]. They pointed out that for a smooth pulse described by an “analytic signal,” signal velocity cannot be defined since the threshold that marks the unset of a signal can extent infinitely back in time on its leading edge. Therefore, in order to send a totally unexpected signal, a step function must be used and they noted that the proper definition of the signal velocity must be that of a “front velocity,” which propagates exactly at

*c*, producing precursors.

## References and Links

1. | L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature |

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 |

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

5. | A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A |

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 |

7. | R. Y. Chiao, “Population inversion and superluminality,” in |

8. | C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian pulse through an anomalous dispersion medium,” Phys. Rev. A |

9. | S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. |

10. | B L. Brillouin, |

11. | Segev, P. W. Milonni, J. F. Babb, and R. Y. Chiao, “Quantum noise and superluminal propagation” Phys. Rev. A |

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, |

15. | L. D. Landau and E. M. Lifshitz, |

**OCIS Codes**

(020.1670) Atomic and molecular physics : Coherent optical effects

(270.5530) Quantum optics : Pulse propagation and temporal solitons

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 25, 2001

Published: March 12, 2001

**Citation**

Arthur Dogariu, Alexander Kuzmich, H. Cao, and L. Wang, "Superluminal light pulse propagation via rephasing in a transparent anomalously dispersive medium," Opt. Express **8**, 344-350 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-6-344

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### References

- 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.

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