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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 8, Iss. 6 — Mar. 12, 2001
  • pp: 344–350
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Superluminal light pulse propagation via rephasing in a transparent anomalously dispersive medium

A. Dogariu, A. Kuzmich, H. Cao, and L. J. Wang  »View Author Affiliations


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

Recently, it was demonstrated that the group velocity of a light pulse can become superluminal (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]

,2

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).

] following earlier theoretical predictions [3

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]

7

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).

]. In the experiment, the pulse propagated through an atomic Cesium (Cs) vapor cell with little absorption or pulse distortion. The experiment utilized a dual Raman amplifier arrangement closely resembled that first analyzed by Chiao and co-workers in a series of papers earlier [3

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]

7

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).

]. The group delay, however, has been measured to be a markedly large negative value of -62 ns, resulting in pulse advance. This pulse advance is to be compared with the vacuum transit time of 0.2 ns over the 6-cm long vapor cell, yielding a negative group velocity of -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]

,2

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).

]. A similar effect has been studied theoretically [8

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]

] for a Gaussian pulse and demonstrated experimentally [9

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

] for the case of an absorbing medium earlier. Such an effect was known and discussed as early as in the 1910’s [10

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..

].

Of course, as first noted by Sommerfeld and Brillouin [10

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..

], such a superluminal group velocity is not at odds with causality. Causality only requires that the 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]

,2

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).

] due to various quantum mechanical noises associated with a gain medium [11

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

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.

].

The concept of a superluminal or even negative group velocity appears to have a need to be reckoned with even from a pedagogical point of view [13

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

,14

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

]. In particular, we note that the phenomenon appears to be rather counterintuitive and this is partly due to the counterintuitive wave nature of light. This is further due to the fact that anomalous dispersion ordinarily occurs only inside heavy absorption lines where the medium is opaque [10

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..

]. The unusual behavior of an anomalous dispersion medium is hence lesser known.

Hence, it appears necessary to emphasize that pulse propagation effect is essentially a wave interference phenomenon where various wave components of a light pulse interfere with each other. Simply put conceptually, when most of the components of various wavelengths have their phase aligned with one another, a peak appears for the pulse’s envelope. Conversely, when these components are out of phase, a minimum appears. This intuitive picture was obtained as early as by Lord Kelvin [10

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..

].

In this article, we will attempt to present an intuitive understanding of the wave interference and pulse propagation inside a transparent anomalous dispersion medium.

2 Pulse propagation in dispersive media

We start by considering the pulse propagation through a dispersive medium illustrated in 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.

If we write the Fourier decomposition of the propagating pulse’s electric field, we obtain for its “positive frequency” part:

E(+)(z,t)=0E(ω)ei[ωtk(ω)z]dω.
(1)

The total electric field will simply become E (+)(z,t)+c.c., and we further have for the intensity (energy flux) of the pulse at the exit surface of the medium:

I(z,t)=2εocE(+)(z,t)2.
(2)

From Eq. (1), we may find the positions and times at which a majority of the components have the same phase and reinforce one another [10

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..

]. Hence, at these times and positions, the pulse will exhibit its strongest part. Conversely, at positions and times other than these conditions, the wave components exhibit destructive interference and a weaker intensity shows. Such a condition given by the Kelvin’s method of stationary phase [10

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..

] would read

=c·dϕdω=c·d[ωtk(ω)z]dω=c(tzU)=0.
(3)

Here,

U=(dkdω)ωo1=cn+ωdndω
(4)

is the group velocity and is the rephasing length. As noted by Brillouin in his summary work on wave propagation [10

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..

], 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).

On a more rigorous approach, let us consider the Taylor expansion of the wavevector:

k(ω)=k(ωo)+1U(ωωo)+12(d2kdω2)(ωωo)2+.
(5)

If the dispersion is linear, i.e., higher order terms are negligible, then we have for the electric field after propagating a distance L:

E(+)(z+L,t)=g·ei[ωotk(ωo)L]E(+)(z,tLU).
(6)

Here, 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

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

]:

I(z+L,t)=I(z,tLU).
(7)

This counterintuitive situation of zero transit time can be understood by closely examining the rephasing length =c·/. 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<): 1=c·t-z. In region-II (L>z>0), we have 2=c·(t-z/U)=c·t-ng ·z, where ng =n+ωdn/ is the group velocity index. And furthermore in region-III (z>L), we obtain 3=c·t + (1-ngL-z.

In the case of zero transit time U=∞, we have for the group velocity index ng =0. Then inside the medium, the rephasing length =c·/ 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=∞.

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.

An even more interesting situation emerges when the anomalous dispersion becomes so strong that ng =n+ωdn/<0. Hence, we have a negative group velocity U=c/ng <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 2=c·t-ng ·z can become zero at a position zo =c·t/ng . Note here since we have both ng < 0 and t<0, we have for this rephasing position: 0<zo <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>ngL/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 zo=c·t′/ng decreases: zo<zo . Hence, the peak inside the medium moves at a negative velocity: c/ng <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 ng <0.

Furthermore, in region-III where z>L, another region of rephasing appears under the condition: 3=c·t + (1-ngL-z =0. This gives the position z =L+c·t-ng ·L. It is clear from the rephasing condition: t>ngL/c that we have z >L (note t, ng <0).

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).

Finally, we note here another interesting aspect of the condition ng =n+ωdn/≤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:

d(λn)dλ=cn2(dkdω)=ngn20.
(8)

Hence, in an anomalous dispersion medium under the relatively strong condition ng =0, we have

d(λn)dλ=0.
(9)

This implies that for all incident rays, inside the medium, their wavelengths are all the same despite their difference in vacuum. Hence, the phase differences between these components are maintained throughout the medium.

Furthermore, under the stronger condition: ng <0, we have

d(λn)dλ<0.
(10)

Under this condition, an incident ray with a shorter vacuum wavelength (bluer) becomes a longer wavelength (redder) ray inside the medium, and vice versa. This unusual condition manifests into reproducing a peak inside the medium even before the incident pulse’s peak reaches the medium’s incident surface. It is further evident from Figure 3 that the peak of the transmitted pulse will have also left the exit surface before the incoming pulse’s peak arrives.

In Figure 4, we show schematically the propagation of a light pulse through such a medium. We also show the evolution of three wave components of the light pulse. It is evident that the wavelengths of the three wave components vary according to Eq.(10) and two rephased pulses are produced both inside and outside (on the far side) of the medium.

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)

Finally, it is worth stressing again that these unusual effects are due to the wave nature of light.

3 Conclusions

In this article, we analyzed the propagation of light pulses inside a transparent anomalous dispersion medium. We stress that the pulse propagation phenomenon is largely a wave interference effect. Because the medium is transparent, we can apply the 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.

We further note that a superluminal group velocity is not at odds with causality or special relativity [10

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..

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.

]. Simply put, causality only requires that the signal velocity of light be limited by 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

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..

]. 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.

LJW wish to thank R. A. Linke and J. A. Giordmaine for helpful discussion. H. Cao is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544.

References and Links

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.

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

  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.

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