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:

For the realistic expressions of n(ω) and arbitrary σ, the integration in Eq. (1) is rather complex even in numerical calculation. However, it can be carried out analytically for narrow wave packets. For such cases, we may use the power series expansion of the complex refractive index (2) around ω₀

Equation (5) can be evaluated with the help of the Poisson integral. The result takes the following form

Basically, equation (4) together with Eqs. (6)–(11) describes the evolution of the narrow wave packet propagating in the medium with a complex refractive index. To make the analysis of Eq. (4) more clear and simple, we consider a narrow wave packet traveling small distance x, when equation (6) reduces to a = 1/(2σ²). In view of this, equation (4) is simplified into

Let us compare Eq. (12) with the conventional result,^[12,17] which follows from Eq. (12) by setting n_g2 = 0 that

However, the consideration of the dispersion of the imaginary part of the refractive index n₂(ω) considerably changes this picture. Let us rewrite the last exponent in Eq. (12) in the form

Equation (17) provides a weak damping and can be satisfied for rather narrow wave packet σ ≪ ω₀ and small traveling distance x ∼ c/[ω₀n₂(ω₀)]. Inequality (16) is new and it imposes an important restriction on the conventional definition of V_g. The superluminal group velocity V_g > c is obtained from Eq. (14) in the frequency band where the following inequality holds true:

In the next sections we focus on the consistency of Eqs. (16) and (18) for the relevant mathematical model of the refractive index of an assembly of two-level atoms.

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

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. (19) and (20), we obtain the following expressions for the real and imaginary parts of the refractive index^[17]

The real and imaginary parts of the group index n_g are obtained with the help of Eqs. (8), (9), and (21). The result becomes

The frequencies in the vicinity of the resonance z = 0 are more interesting. To analyze the group indices in Eq. (23) at these frequencies, one has to take into account that T₂ω_ba ≈ 10⁷ (we set T₂ ≈ 10⁻⁸ s and ω_ba ≈ 10¹⁵ rad/s). Keeping only the leading terms in Eq. (23), we obtain

Figures 1–3 show the curves of the real n_g1 and imaginary n_g2 parts of the group index versus z for β = 100 and different values of A, calculated using Eq. (24). Figure 1 shows the curve for 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., T₁T₂Ω² ≪ 1. The corresponding χ(ω) defined by Eq. (20) coincides with the first order susceptibility.^[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. (14) has the physical meaning. However, for frequencies |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 (16) is not valid. Therefore, the group velocity |V_g| > c has no physical meaning. However, in the frequency range where |n_g1| < 1, equation (14) gives the group velocity corresponding to superluminal positive or negative light. In the vicinity of the exact resonance z = 0, n_g2 ≈ 0 and inequality (16) holds true, giving V_g ≈ −c/100 that describes the negative slow light.

Fig. 1.

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	Fig. 1. Curves of ng1 and ng2 versus z for β = 100 and A2 = 1.

Figure 2 shows the curves of n_g1 and n_g2 versus z for the same parameters as those in Fig. 1, but with A² = 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. 1. Again, one cannot ignore n_g2 compared with n_g1 for frequency |z| ≤ 6. In the vicinity of |z| ≈ 1.6, the conventional formula (14) gives superluminal group velocity. However, these velocities have to be discarded as they have no physical meanings since 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 (16) is valid, giving V_g ≈ −c/30, corresponding to negative slow light.

Fig. 2.

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	Fig. 2. Curves of ng1 and ng2 versus z for β = 100 and A2 = 3.

Figure 3 shows the curves of n_g1 and n_g2 versus z for a much higher intensity of the incident optical wave with A² = 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. 1 and 2. We can claim that the applied strong optical field of the incident wave broadens the frequency band where 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 (14) gives the physically admissible value of the group velocity |V_g| ≤ c.

Fig. 3.

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	Fig. 3. Curves of ng1 and ng2 versus z for β = 100 and A2 = 5.

It is necessary to note that again in the vicinity of the resonance z = 0, equation (14) results in a physically consistent group velocity corresponding to the slow negative light. The corresponding group index is obtained from the first expression of Eq. (24) and the group velocity is given by

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.