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The propagation of narrow packets of electromagnetic waves (EMWs) in frequency dispersive medium with the consideration of the complex refractive index is studied. It is shown that counting in the dispersion of the complex refractive index within the context of the conventional expression of the group velocity of narrow wave packets of EMWs propagating in a dispersive medium results in the appearance of additional constraints on the group velocity, which dictates that the physically acceptable group velocity can only be realized in the case of a negligible imaginary part of the group index. In this paper, the conditions that allow one to realize the physically acceptable group velocity are formulated and analyzed numerically for the relevant model of the refractive index of a system of two-level atoms in the optical frequency range. It is shown that in the frequency band where superluminal light propagation is expected, there is a strong dispersion of the refractive index that is accompanied with strong absorption, resulting in a strongly attenuated superluminal light.

The studies of superluminal, backward, and slow narrow packets of light waves propagating in different media in the frequency range of strong dispersion of the refractive index have a long history.^{[1]} Currently, the question on the existence, physical meanings, and potential consequences of such waves has become a hot research topic and sometimes controversial. In this respect, there are two groups of researchers: those who support the existence of superluminal electromagnetic waves^{[2]} and those who reject such reports on grounds that it contradicts the special theory of relativity and consequently the so-called superluminal light has no physical meaning.^{[3]} In 1970 it was reported in Ref. [4] that Gaussian packets of electromagnetic waves can travel in dispersive media with group velocity exceeding the velocity of light in a vacuum with no restrictions whatsoever. Recently, the topic has been discussed theoretically in many papers again and even some have reported on the experimental realization of such waves in strongly dispersive media.^{[2–12]}

On the other hand, others have the view that the appearance of superluminal and negative group velocities of light cannot be attributed to a physical flow in the so-called superluminal region where there is strong dispersion of the refractive index that is accompanied by strong absorption.^{[13–15]} H. Tanka, *et al.* obtained negative group velocity as well as pulse velocity (group velocity) that can exceed the speed of light in a vacuum in Rb vapor, however they showed that at the frequency band where this velocity is attained the intensity of the transmitted pulse is reduced by strong dispersion and consequently concluded that the observed velocity is not the velocity of energy flow, i.e., group velocity.^{[14]} Similarly, it is reported that an electromagnetic wave (EMW) packet may travel at a superluminal velocity as well as negative group velocity, however the information carried by them is not superluminally transmitted and the information velocity carried by the wave front is still positive.^{[16]}

The interesting topic that needs further discussion concerns the superluminal light with group velocity *V*_{g} exceeding the speed of light in a vacuum. Such a value of *V*_{g} is commonly obtained with the help of the conventional formula that does not take into account the dispersion of the imaginary part of the refractive index. In this paper, we consider the evolutions of narrow packets of electromagnetic waves with the consideration of the dispersion of the complex refractive index *n*(*ω*). It is shown that this dispersion significantly distorts the shape of the Gaussian wave packet, which is demonstrated by introducing the imaginary part of the group index *n*_{g2} along with the conventional real group index *n*_{g1}.^{[5]} The latter completely controls the group velocity with the help of the real part of *n*(*ω*) and its derivative with respect to the frequency *ω*. Our theoretical and numerical analyses show that the physically consistent group velocity *V*_{g} for a narrow packet of EMW can only be realized in the frequency bands where *n*_{g2} ≪ *n*_{g1}. By considering the relevant model of the complex refractive index of an assembly of two-level atoms, it is shown that in the frequency bands where the conventional formula of *V*_{g} gives the superluminal group velocity, the condition *n*_{g2} ≪ *n*_{g1} is violated, confirming the fact that |*V*_{g}| > *c* has no physical meaning.

The rest of this paper is organized as follows. Section 2 is devoted to the study of the propagation of narrow packets of electromagnetic waves with the consideration of the dispersion of the imaginary part of the refractive index and the conditions on which the physically consistent group velocity can be realized are introduced. In Section 3, we consider the group indices and group velocity of light in an assembly of two-level atoms and determine the frequency bands where the physically acceptable group velocity is attained. The conclusions that are withdrawn from the results obtained in the paper are summarized in Section 4.

The profile of a Gaussian wave packet as a function of position *x* and time *t* propagating in a medium with dispersion can be obtained using the following equation:

*ω*

_{0}is the central frequency of the wave packet,

*E*(

*ω*) is its amplitude,

*σ*is the width at frequency

*ω*, and

*c*is the speed of light in a vacuum. The refractive index of the medium

*n*(

*ω*) is a complex function of

*ω*, given by

*n*

_{1}(

*ω*) and

*n*

_{2}(

*ω*) are the real and imaginary parts of the refractive index, respectively.

For the realistic expressions of *n*(*ω*) and arbitrary *σ*, the integration in Eq. (*ω*_{0}

*n*′(

*ω*

_{0}) is the first derivative of

*n*(

*ω*) evaluated at

*ω*=

*ω*

_{0}. For narrow wave packets the higher order terms in the expansion (

*I*denotes the following integral

*q*=

*ω*−

*ω*

_{0}and the parameters

*a*and

*b*are defined by

*n*

_{g1}(

*ω*) is known as the group index.

^{[5]}Below, we introduce the imaginary part of the complex group index

*n*

_{g2}(

*ω*) along with the real one

*n*

_{g1}(

*ω*).

Equation (

*a*is positive. It requires that the following condition must hold true

Basically, equation (*x*, when equation (*a* = 1/(2*σ*^{2}). In view of this, equation (

*Ω*′ =

*ω*

_{0}+

*σ*

^{2}

*n*

_{g2}

*x*/

*c*. The quantities

*n*

_{1},

*n*

_{2},

*n*

_{g1}, and

*n*

_{g2}are to be evaluated at the central frequency

*ω*

_{0}.

Let us compare Eq. (^{[12,17]} which follows from Eq. (*n*_{g2} = 0 that

*V*

_{g}according to the well known relation

^{[5]}

*ω*

_{0}[

*n*

_{1}(

*ω*

_{0})

*x*/

*c*−

*t*]} moving with the group velocity

*V*

_{g}. The maximum of the envelope decreases with

*x*, since

*n*

_{2}(

*ω*

_{0}) > 0 (the first exponent in Eq. (

However, the consideration of the dispersion of the imaginary part of the refractive index *n*_{2}(*ω*) considerably changes this picture. Let us rewrite the last exponent in Eq. (

*V*

_{g}, defined by Eq. (

Equation (*σ* ≪ *ω*_{0} and small traveling distance *x* ∼ *c*/[*ω*_{0}*n*_{2}(*ω*_{0})]. Inequality (*V*_{g}. The superluminal group velocity *V*_{g} > *c* is obtained from Eq. (

In the next sections we focus on the consistency of Eqs. (

Let us consider a system consisting of weakly interacting two-level atoms with the bottom and upper levels represented by *a* and *b*, respectively. The refractive index of such weakly interacting two-level atoms is given by

*χ*(

*ω*) for a system in equilibrium is expressed as

^{[17]}

*N*is the density number of atoms,

*ω*

_{ba}> 0 is the transition frequency between the energy levels

*b*and

*a*,

*ρ*

_{aa}and

*ρ*

_{bb}are the diagonal components of the density matrix describing the difference in population between the levels

*a*and

*b*at thermal equilibrium,

*μ*

_{ba}is the atom dipole matrix element,

*Δ*is the detuning factor,

*T*

_{1}is the lifetime of the upper level,

*T*

_{2}is the characteristic time of dephasing dipole moment resulting in the transition line width 1/

*T*

_{2},

*Ω*is the on-resonance Rabi frequency, and

*E*is the amplitude of monochromatic electric field of the incident electromagnetic wave of frequency

*ω*.

Below, we consider an equilibrium case when *ρ*_{bb} = 0 and *ρ*_{aa} = 1, that is, when the upper level is not populated. In the case of |*χ*| ≪ 1, which is consistent with the model, using Eqs. (^{[17]}

*α*= 2

*πN*|

*μ*

_{ba}|

^{2}

*T*

_{2}/

*ħ*and

*A*

^{2}= 1 +

*T*

_{1}

*T*

_{2}

*Ω*

^{2}. Equation (

*α*. Taking typical values, |

*μ*

_{ba}| = 5.5 × 10

^{−18}esu (for s → 3p transition of atomic sodium) and

*T*

_{2}= 32 ns,

^{[17]}we obtain

*α*= 6 × 10

^{−15}N, where

*N*is expressed in cm

^{−3}. It is clear that for

*N*≤ 10

^{13}cm

^{−3}and

*A*≥ 1, inequality (

The real and imaginary parts of the group index *n*_{g} are obtained with the help of Eqs. (

*z*=

*T*

_{2}

*Δ*. It is seen that far from the resonance frequency

*ω*=

*ω*

_{ba}, where |

*z*| ≫ 1, the approximation

*n*

_{g1}∼ 1 and inequality |

*n*

_{g2}| ≪ 1 can be satisfied by choosing the appropriate value of

*α*, such that

*α*≪ 1. Therefore, inequality (

*n*

_{g1}∼ 1 and |

*n*

_{g2}| ≪

*n*

_{g1}holds true. At these frequencies the group velocity (

The frequencies in the vicinity of the resonance *z* = 0 are more interesting. To analyze the group indices in Eq. (*T*_{2}*ω*_{ba} ≈ 10^{7} (we set *T*_{2} ≈ 10^{−8} s and *ω*_{ba} ≈ 10^{15} rad/s). Keeping only the leading terms in Eq. (

*β*=

*αT*

_{2}

*ω*

_{ba}can be rather large though

*α*is small.

Figures *n*_{g1} and imaginary *n*_{g2} parts of the group index versus *z* for *β* = 100 and different values of *A*, calculated using Eq. (*A* = 1. It corresponds to the case where the intensity of the incident wave |*E*|^{2} or the on-resonance Rabi frequency is small compared with the unit, i.e., *T*_{1}*T*_{2}*Ω*^{2} ≪ 1. The corresponding *χ*(*ω*) defined by Eq. (^{[17]} In this case, the real part of the group index *n*_{g1} is of the known shape.^{[17]} For frequencies |*z*| ≫ 4, *n*_{g1} ≫ *n*_{g2}, consequently the corresponding group velocity *V*_{g} given by Eq. (*z*| ≤ 4, the situation completely changes. Here (with the exception of a narrow frequency band in the vicinity of *z* = 0) |*n*_{g2}| > |*n*_{g1}| so that the wave is strongly attenuated and inequality (*V*_{g}| > *c* has no physical meaning. However, in the frequency range where |*n*_{g1}| < 1, equation (*z* = 0, *n*_{g2} ≈ 0 and inequality (*V*_{g} ≈ −*c*/100 that describes the negative slow light.

Figure *n*_{g1} and *n*_{g2} versus *z* for the same parameters as those in Fig. *A*^{2} = 3, which corresponds to a rather intense optical field. In this case, the peaks of *n*_{g1} and *n*_{g2} decrease and become broad compared with Fig. *n*_{g2} compared with *n*_{g1} for frequency |*z*| ≤ 6. In the vicinity of |*z*| ≈ 1.6, the conventional formula (*n*_{g2} ∼ 10, near |*z*| = 1.6 resulting in a strongly attenuated wave. On the other hand, in the vicinity of the resonance *z* = 0, for frequency bands where |*n*_{g2}| ≪ |*n*_{g1}|, equation (*V*_{g} ≈ −*c*/30, corresponding to negative slow light.

Figure *n*_{g1} and *n*_{g2} versus *z* for a much higher intensity of the incident optical wave with *A*^{2} = 5. One can observe that the peaks of the group indices become smaller and wider. This effect can be explained by the theory that the widening of the spectral line results in the presence of an intense radiation field.^{[17]} Moreover, the negative peak of *n*_{g2} becomes broader than those in Figs. *n*_{g1} < *n*_{g2} and consequently the group velocity loses its physical meaning. For frequency far from the resonance, i.e., for *z* ≫ 6, the imaginary group index fast approaches to zero, being negative and equation (*V*_{g}| ≤ *c*.

It is necessary to note that again in the vicinity of the resonance *z* = 0, equation (

*n*

_{g1}versus

*z*depicted in Figs.

The profile of a narrow wave packet propagating in a dispersive medium is analyzed with the help of the standard procedure but with the consideration of the dispersion of the imaginary part of the refractive index. In addition to the conventional group index *n*_{g1}, it contains a new expression *n*_{g2}, which corresponds to the imaginary part of the complex group index. This quantity significantly changes the profile of the wave packet compared with the conventional case, with *n*_{g2} = 0.

The main result of the paper is that the group velocity calculated using the conventional formula *V*_{g} = *c*/*n*_{g1} has the physically acceptable value only for frequency *ω*, where |*n*_{g2}(*ω*)| ≪ |*n*_{g1}(*ω*)|. We verify the validity of this inequality with the help of the refractive index of a typical two-level atom model at optical frequency for the equilibrium case and show that it is violated for the frequency range where the conventional formula of *V*_{g} gives superluminal group velocity |*V*_{g}| > *c*. That is, in the frequency domain where superluminal light is expected, the presence of strong absorption attenuates the wave and makes it impossible to realize the physical superluminal light.

On the other hand, in the vicinity of the resonant frequency where the imaginary part of the refractive index of an equilibrium system is large and results in strong absorption, the imaginary group index *n*_{g2} ≈ 0, which allows one to realize the physically consistent group velocity corresponding to the slow negative light with negative real group index |*n*_{g1}| ≫ 1.

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