## 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

## 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 [1,2] following earlier theoretical predictions [3–7]. 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 [3–7]. 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 [1,2]. A similar effect has been studied theoretically [8] for a Gaussian pulse and demonstrated experimentally [9] for the case of an absorbing medium earlier. Such an effect was known and discussed as early as in the 1910’s [10].

Of course, as first noted by Sommerfeld and Brillouin [10], 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 [1,2] due to various quantum mechanical noises associated with a gain medium [11,12].

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 [13,14]. 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]. 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].

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.

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

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:

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]. 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] would read

Here,

is the group velocity and *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).

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

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

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]:

Ordinarily, the group velocity is subluminal: *U*<*c*, because for all transparent media that is passive (absorbing at all frequencies) the dispersion is normal [2,15]: *dn*/*dω*>0. This implies that the pulse intensity at position *z*+*L* is related to the pulse intensity at position *z* by a *normal* time delay of *L*/*U*>*L*/*c*. *L*/*c* is the vacuum transit time.

However, in an unusual situation [1,2], a transparent anomalously dispersive medium can be created [1–7]. In this case, we have for the refractive index: *dn*/*dω*<0. Under a slightly stronger condition: *n* +*ω*·*dn*/*dω*<1, we have for the group velocity *U*>*c*. In this case, Eq.(7) implies that at the position *z*+*L*, the pulse’s intensity is related to that of the pulse’s intensity at position *z* by a time delay that is *shorter* than the vacuum transit time *L*/*c*. This results in light pulse propagation at a superluminal group velocity. One extreme of this case would be *U*=∞ under the condition of *n*_{g}
=*n*+*ωdn*/*dω*=0 which results in an unusual situation: *I*(*z*+*L*, *t*)=I(z, t). This corresponds to the situation of *zero transit time*.

This counterintuitive situation of *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*.

In the case of *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*=∞.

An even more interesting situation emerges when the anomalous dispersion becomes so strong that *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 ${z}_{o}^{\prime}$=*c*·*t*′/*n*_{g}
decreases: ${z}_{o}^{\prime}$<*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.

Furthermore, in region-III where *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).

Finally, we note here another interesting aspect of the condition *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:

Hence, in an anomalous dispersion medium under the relatively strong condition *n*_{g}
=0, we have

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: *n*_{g}
<0, we have

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.

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 [10–12]. 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]. 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.

More recently, we studied this problem and found that in order to be consistent with common laboratory practices, one must define the signal operationally [11,12]. In the case that pertains to the experimental situation [1,2] where the medium is transparent, we must consider the excess noise generated by the gain lines that are necessarily associated with the anomalously dispersive medium. We found that, in these cases, the excess spontaneous emission noise from the amplifying gain resonances delays the unset of the signal by retarding the time at which a prescribed signal-to-noise ratio (SNR) is reached [11,12]. Specifically, in the experiment the pulse advance is linear to the Raman gain coefficient. When one attempts to increase the pulse advancement by increasing the Raman gain, the associated spontaneous emission noise also increases. Hence, if we were to define that an optical signal has arrived when the light pulse reaches certain intensity level, this signal level set point must be increased in the presence of spontaneous emission noise in order to preserve the signal-to-noise ratio. This will necessarily result in a delay for the light pulse to reach the set point (which must be increased in the presence of spontaneous emission noise). This delay cancels the pulse advance. Therefore, it can concluded [11,12] that the quantum noise associated with an amplifier (which is related to the “no-cloning” theorem and the linear nature of quantum mechanics) is associated with the basic requirement of causality that states that no “signal” can be transmitted faster than *c*.

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.

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