1. Introduction

Over the past two decades, a high degree of control of pulse velocity in various media has been accomplished experimentally. Diverse experimental schemes lead to both slow light (v_g ≪ c) and fast light (either v_g > c or v_g < 0) [c is the vacuum light velocity and v_g is the group velocity]. Originally, Garrett and McCumber theoretically predicted that, in the vicinity of an absorption line, the peak velocity of a Gaussian pulse is equal to the group velocity even if v_g > c, v_g = ∞, or v_g < 0 [1]. Later, this prediction was confirmed experimentally by Chu and Wong [2]. Subsequent studies found that the interaction of a gas of atoms with a strong field could eliminate the absorption and modify the dispersive refractive index seen by a weak-intensity wave [3–5]. This possibility was the milestone for achieving great pulse velocity control. For example, in an ultracold gas of atoms a very low group velocity [v_g ≈ c/(1.76 × 10⁷)] was measured [6], while in a hot gas of caesium atoms, fast light was observed v_g ≈ −c/310 [7] where this superluminal velocity in a medium with anomalous dispersion is simply explained by light interference [8]. Other nonlinear mechanisms, as the coherent population oscillations and stimulated Brillouin scattering, allow to observe the slow-fast light effects in solid media as ruby [9] and in optical fibers [10]. Since solids are propitious for practical implementation, potential applications in optical communications such as pulse delay control devices have been envisioned. Despite these remarkable achievements, a central concern still is pulse propagation without distortion (that is inexorably caused by the dispersion of the medium) [11].

Recently, there is renewed interest in exploring dynamic media that could lead to new methods for controlling and manipulating light propagation. The readers can find numerous references on dynamic media in our recent works [12, 13]. We mention that, in the context of pulse propagation, it was found theoretically that a pulse passing through layered dielectric structures with arbitrary space-time dependence of the refractive index experiences reshaping (compression and broadening) and its central frequency is shifted [14]. Similar effects are reported for pulses propagating in a linear medium with time-varying refractive index [15]. Spectral shifts of pulses, resulting from the passage through dynamic cavities and waveguides, were observed [16–19]. In other recent studies, the incorporation of dynamic inclusions has been explored. Dynamic metamaterials that work at GHz and THz frequencies can be made by filling the split-ring resonator gaps with a photoconductive semiconductor [20, 21]. These dynamic metamaterials could exhibit an effective dielectric function that can be strongly modulated with a frequency ∼ 1 GHz (limited by the lifetime and the mobility of the charge carriers). The parametric response in a single split-ring resonator with a time-varying capacitive element was studied in [22]. Phase conjugation and negative refraction effects can occur in a metamaterial composed of split-ring resonators with nonlinear varactors [23].

In prior studies, we theoretically analyzed the wave propagation in a medium whose dielectric function ε(t) varies periodically in time, namely, ε(t) = ε(t + T) (T is the period). We found that the dispersion relation is a band structure with forbidden wave vector gaps and we studied the electromagnetic response of a dynamic-periodic slab to monochromatic plane wave excitation [12]. Due to the periodic modulation, harmonics ω – nΩ are generated (ω is the angular frequency of the exciting monochromatic wave, Ω = 2π/T is the modulation frequency, n is an integer). Consequently, the slab becomes a polychromatic source. Later, we found that this dynamic-periodic slab can exhibit resonances when the excitation frequency is ω = mΩ/2 with m = 1,3,5,... and, in addition, the modulating frequency obeys a certain condition [13]. On the other hand, the temporal variation of our dielectric function relies on the interaction of an external agent with the medium. For optical frequencies, the modulation strength of the dielectric function that can be reached by current nonlinear methods is a small fraction of the unperturbed dielectric constant, and the typical temporal response of a material allows modulating frequencies not higher than ∼ 10 GHz. On the other hand, very recently we have shown that, in the long-wavelength limit, a low-pass dynamic transmission line for microwaves can be characterized by a time-dependent effective dielectric function ε_ef(t) with a large modulation strength, thus overcoming the aforementioned limitation for optical frequencies [24]. This transmission line is a linear array of cells that are formed each by an inductance L and a varactor whose capacitance C(t) is modulated periodically. The response of this transmission line can be characterized with an effective dielectric function ε_ef(t) = C(t)/(ε_oa) (a is the lattice constant and ε_o is the vacuum permittivity) with the restriction that the dominant wavevectors k satisfy ka ≪ 1. Therefore, the experimental implementation of a medium exhibiting an effective and strongly modulated dielectric function ε(t) is feasible, at least in the microwave regime.

We focus on the transmission of a pulse through a slab whose dielectric function ε(t) varies periodically, namely, ε(t) = ε(t + T) (T is the period). We present a theoretical framework for obtaining the transmitted and reflected pulses when a pulse is incident on this dynamic-periodic slab. This framework is an extension of our aforementioned theory of bulk dynamic-periodic media and response of a dynamic slab excited by a monochromatic plane wave [12]. Furthermore, we present an alternative description of the transmitted pulses for an incident quasi-monochromatic pulse that is based on the concepts of group and phase delays [25]. From this description, the transmitted pulse can be decomposed as the superposition of partial pulses. Particularly, we analyze the pulse propagation through the dynamic-periodic slab for incident Gaussian pulses with narrow (quasi-monochromatic) and with broad bandwidths for selected values of the carrier frequency, slab thickness, and modulation strength. We investigate the propagation velocities of the aforementioned partial pulses for incoming quasi-monochromatic pulses. Also, we study the peak velocity of the pulse when a Gaussian pulse is incident on the slab. We address the question whether these propagation velocities are related to the group velocity obtained from the bulk dispersion relation. We pay special attention to the case when the carrier frequency of the incoming pulse is ω_c = nΩ/2, since the dispersion relation (a band structure with forbidden wave vector gaps) yields infinite group velocities at these frequencies. Finally, we discuss the energy balance between the incident pulse and the outcoming transmitted and reflected pulses.

This paper is organized as follows. Section 2 presents the general theory of pulse propagation through the dynamic-periodic slab (reflected and transmitted pulses). In addition, we present a simplified description of the transmitted pulses that is based on the concepts of phase and group delays, being valid for quasi-monochromatic incident pulses. In Sec. 3, we analyze the pulse propagation of quasi-monochromatic incident Gaussian pulses for particular carrier frequencies, slab thicknesses, and modulation strengths. Conversely, Sec. 4 deals with the analysis of the pulse transmission of a Gaussian pulse with large spectral bandwidth. Investigation of the peak velocity of a pulse propagating through the dynamic slab is presented in Sec. 5. Section 6 is devoted to the discussion of the energy carried by the reflected and transmitted pulses in comparison with that of the incoming pulse. Section 7 contains the conclusions.

2. Theory

2.1. An electromagnetic pulse interacting with a dynamic slab

In Ref. [12] we studied a dynamic medium whose dielectric function ε(t) varies periodically with the time t, the period T being related to the modulation frequency Ω by T = 2π/Ω. For an excitation frequency ω, the modulation induces an infinite number of harmonics with frequencies ω – nΩ, n = 0,±1,±2,... Moreover, for any given ω and a given propagation direction (say, y), an infinite number of plane waves, differing in their wavelengths, can propagate. These are characterized by wave vectors k_p(ω), p = 1,2,... The index p labels the wavevector bands, and the corresponding band structure is characterized by band gaps Δk between adjacent bands p and p + 1. We also showed in Ref. [12] that a monochromatic wave (frequency ω), normally incident at a slab of thickness D in vacuum, gives rise to reflected and transmitted waves with all the frequencies ω – nΩ (all integers n). Furthermore, inside the slab, in addition to waves with wave vectors k_p(ω) there will also be excited counterpropagating waves with wave vectors −k_p(ω). Here we apply the former treatment to an incident pulse or wave packet whose electric field is given by

Our analysis of the reflected and transmitted pulses will take place at the slab borders y = 0, and D, respectively. Then, from the Eqs. (4) and (5) we obtain

2.2. Quasi-monochromatic pulses

We analyze the response to an incident quasi-monochromatic pulse. This incident pulse is represented as the product of a slow time-varying signal 𝒠(t) peaked at t = 0 and a harmonic signal oscillating with frequency ω_c (carrier frequency), that is,

Now let us consider as a temporal reference the time in which the peak of incident envelope hits the entrance interface. Since the envelope of the n-th outcoming partial pulse is just a delayed replica of the incident one, the peak of this partial pulse must exit the slab at a time ϕ′_n(ω_c) after this reference. Therefore, the group delay of the partial pulses can be related to peak velocities as

3. Quasi-monochromatic Gaussian pulses

We apply the theory to the case of an incident Gaussian pulse having the form

We consider that the dielectric function ε(t) of the slab is modulated sinusoidally as

We compare the numerical simulations for the transmitted pulse obtained from the exact response (5) and for the approximate response (17) for particular cases. First we consider that the normalized modulation frequency is Ω̃ = 2 and the incident pulse has a carrier frequency ω_c = Ω/2 and spectral width Δω/Ω = 0.1. This pulse is shown in Fig. 1(a) as function of t̃. Figures 1(b) and 1(c) show the transmitted pulse arising from the exact (5) and the approximate (17) responses for the cases: M = 0.162 (moderate modulation) and 0.648 (strong modulation). Despite that Δω is not very small in comparison with ω_c (ω_cτ = 10), we notice from Figs. 1(b) and 1(c) that practically perfect matches between the exact and the approximate responses are obtained. We also observe in these figures that the transmitted pulse for M = 0.648 is more distorted than for M = 0.162. This is explained by the fact that the amplitudes of the higher order harmonics generated inside the dynamic slab become larger as the modulation strength increases (see also [12]). Now we consider another particular case where ω_c/Ω = 0.67, Δω/Ω = 0.1, M = 0.648, and Ω̃ = 8, that is, the carrier frequency ω_c and the normalized modulation frequency Ω̃ have increased with respect to previous strong modulation case. The incident pulse is depicted in Fig. 2(a), while the transmitted pulse resulting from the exact and approximate responses is shown in Fig. 2(b). For this case, contrary to the prior cases, there is a difference between the exact and the approximate responses. This difference is more evident at times t̃ ≈ −1.3, −0.3, 0.4, 4.3 and 5.2. We attribute the deviation of the approximate response from the exact one to the fact that the condition τ ≫ D/v_s = t_s (equivalently t_s Δω ≪ 2) for obtaining a single well defined outcoming pulse is not satisfied as well as in Fig. 1(c). Now we reduce the pulse spectral width of the incident pulse one order of magnitude, that is, Δω/Ω = 0.01. This is illustrated in Fig. 3(a). Due to this spectral reduction, the approximate and the exact responses coincide as seen in Fig. 3(b). Therefore, we have shown that the transmission of an incident quasi-monochromatic pulse passing through the dynamic-periodic slab can be well described by Eq. (17).

 

Fig. 1 Incident and transmitted pulses as a function of the normalized time t̃ for a slab with normalized modulation frequency Ω̃ = 2. (a) The incident electric pulse E_i with carrier frequency ω_c/Ω = 0.5 and spectral bandwidth Δω/Ω = 0.1. (b) The transmitted pulse for a modulation strength M = 0.162. (c) The transmitted pulse for a modulation strength M = 0.648. Here ω_cτ = 10 and t_s Δω = 0.2.

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Fig. 2 Incident and transmitted pulses as a function of the normalized time t̃ for a slab with normalized modulation frequency Ω̃ = 8 and modulation strength M = 0.648. (a) The incident electric pulse E_i with carrier frequency ω_c/Ω = 0.67 and spectral bandwidth Δω/Ω = 0.1. (b) The transmitted pulse. Here ω_cτ = 13.4 and t_s Δω = 0.8.

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Fig. 3 Incident and transmitted pulses as a function of the normalized time t̃ for a slab with normalized modulation frequency Ω̃ = 8 and modulation strength M = 0.648. (a) The incident electric pulse E_i with carrier frequency ω_c/Ω = 0.67 and spectral bandwidth Δω/Ω = 0.01. (b) The transmitted pulse (inset shows the whole pulse). Here ω_cτ = 134 and t_s Δω = 0.08.

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Now we discuss the implications resulting from the description based on the concepts of group and phase delays. According to Eq. (17), the transmitted pulse can be decomposed as a superposition of exact replicas of the incident pulse in which the partial carrier signal has a frequency ω_c – nΩ (n = 0,±1,±2,...) and suffers a phase delay ϕ_n(ω_c), the envelope experiences a time delay ϕ′_n(ω_c), and the amplitude is A_n(ω_c). In terms of the normalized parameters t̃ and Ω̃, the partial Gaussian pulse at the slab exit (y = D) for the harmonic n can be expressed as

Table 1. The characteristics of the outcoming partial Gaussian pulses [Eq. (24)] for the harmonic n and modulation strength M = 0.648 (strong). Incident Gaussian pulse with carrier frequency ω_c/Ω = 0.5 and spectral width Δω/Ω = 0.1; slab with normalized modulation frequency Ω̃ = 2. Here ω_cτ = 10 and t_s Δω = 0.2.

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Table 2. The characteristics of the outcoming partial Gaussian pulses [Eq. (24)] for the harmonic n and modulation strength M = 0.648 (strong). Incident Gaussian pulse with carrier frequency ω_c/Ω = 0.67 and spectral width Δω/Ω = 0.01; slab with normalized modulation frequency Ω̃ = 8. Here ω_cτ = 134 and t_s Δω = 0.08.

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The transmission of partial pulses (for a given harmonic n) can be both fast and slow. On one hand, fast pulse propagation occurs when Δτ̃_n < 0 or $0<\mathrm{\Delta }{\tilde{\tau }}_n<1/\sqrt{\overline{\epsilon }}=0.4364$ (corresponding to Δt < D/c), namely, when the transmitted pulse peak has emerged before the incident pulse peak has entered into the slab or when the transmitted pulse peak has traveled at a velocity faster than c, respectively. As seen in Table 1, fast light appears for the partial pulse n = −2 for which the peak velocity is v₋₂/c = 1.7013. Also in Table 2, fast light shows up for the partial pulses with harmonics n = −2,−1. For these partial pulses, the temporal shifts are negative (Δτ̃₋₂ = −5.4397 and Δτ̃₋₁ = −0.4390) which yield negative peak velocities. This effect of the pulse peak propagating with velocity v_n > c or v_n < 0 does not violate the causality principle, but comes from the interference of the generated spectral components inside the slab that causes the reconstruction of the partial pulses at advanced times.

On the other hand, slow pulse transmission happens when $\mathrm{\Delta }{\tilde{\tau }}_n\gg 1/\sqrt{\overline{\epsilon }}$, that is, the transmitted pulse peak has traveled at a velocity much slower than c. In Table 2, we call attention to the partial pulse n = −3 whose peak exits the slab with a long delay Δτ̃₋₃ = 8.0334 which corresponds to a peak velocity v₋₃/c = 0.0543. This peak velocity reduction is due to the same aforementioned interference effect, but the partial pulse reconstruction takes place at delayed times.

We stated above that the dispersion relation of the bulk dynamic medium is a band structure which exhibits forbidden wavevector bandgaps. This dispersion relation yields infinite group velocities at frequencies ω_c = Ωm/2 with integer m. This frequency condition is satisfied in Fig. 1 and Table 1 with ω_c = Ω/2. However, as can be seen in the Table 1, the peak velocities do not diverge, that is, the time delay Δτ̃_n is finite for all the partial pulses n. This is not really surprising, given the fact that the group velocity in the bulk medium is in general different than the peak velocity (velocities) of a pulse (pulses) emerging from a slab of the same medium. In fact, the group and peak velocities could coincide only if the incident pulse does not decompose into several pulses in the medium (which is the case in Refs. [1, 2]). However, clearly, that is not the case at hand, for, inside our dynamic slab there is, in principle, an infinite number of partial pulses centered at ω_c – nΩ and moving to the left and right.

4. Gaussian pulses with large spectral band-widths

Now we discuss the transmission of a Gaussian pulse with a large spectral bandwidth (Δω/Ω > 1) which corresponds to a short pulse. Since the outcoming pulse is the result of the superposition of a large number of harmonics that are created in the slab by the extended range of frequencies of the incident pulse, the analysis of the transmission of these pulses passing through the dynamic slab is complicated. We consider an incident Gaussian pulse with carrier frequency ω_c/Ω = 5 and bandwidth Δω/Ω = 5 as depicted in Fig. 4(a). The transmitted pulses with the slab modulated at strengths M = 0.162 (moderate modulation) and M = 0.648 (strong modulation) and for normalized modulation frequency Ω̃ = 2 and 8 are also illustrated in Fig. 4. Interestingly, for the behavior in time we observe that after the main pulse traverses the slab (with peak at t̃ ≈ 1), two small pulses emerge. For the moderate modulation and both normalized modulation frequencies, these pulses are peaked at t̃ ≈ 3 and t̃ ≈ 5. From Eq. (23), we recall that t̃ ≈ 1 is just the normalized time for light propagating at the speed $v_{\text{s}}=c/\sqrt{\overline{\epsilon }}$ to traverse the slab. Such behavior could be expected for a transparent static plate, provided that the temporal width τ = 2/Δω of the pulse is much smaller than the transit time D/v_s (a condition satisfied in Fig. 4). Under such circumstances, because of Fabry-Perot reflections, a series of pulses should emerge with their peaks exiting the slab at the times D/v_s, 3D/v_s, 5D/v_s,... (normalized times t̃ = 1,3,5,...). Thus, it is not all that surprising that, for τ ≪ D/v_s and moderate modulation, several of these transmission peaks are found. (On the contrary, if τ ≫ D/v_s, as occurs for the quasi-monochromatic case, then the emerging wave forms strongly overlap and a single pulse is observed, see Figs. 1–3. Evidently, the physics at hand is very different). However, in our dynamic slab, an infinite number of plane waves p and harmonics n interfere, so it is rather unexpected that the exit times t̃ = 2.98 and 5.05 for Ω̃ = 2 and t̃ = 2.99 and 4.99 for Ω̃ = 8 (for the moderate modulation) are so near to t̃ = 3 and 5 of the static plate. On the other hand, in the case of strong modulation (M = 0.648), greater departure from these times is experienced [see Fig. 4(b)]. Also, the emergent peaks at t̃ ≃ 3 are considerably diminished in comparison to the primary peak at t̃ ≃ 1, and the peaks at t̃ ≃ 5 can be scarcely seen on the scale of Fig. 4. As a final comment, for strong modulation we can observe considerable distortion in the wings of the transmitted pulse, obviously due to the generation of significant high-order harmonics.

 

Fig. 4 Incident and transmitted pulses vs. the normalized time t̃ for M = 0.162 (moderate modulation) and M = 0.648 (strong modulation). The incident Gaussian pulse has a large bandwidth Δω/Ω = 5 and carrier frequency ω_c/Ω = 5, and the normalized time t̃ is taken with respect to Ω̃ = 2. (a) Ω̃ = 2 and t_s Δω = 10. (b) Ω̃ = 8 and t_s Δω = 40 (the inset is a closeup of the second pulse). Here ω_cτ = 2.

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5. Peak velocity

According to our result Eq. (17), every harmonic component of frequency ω – nΩ exits the slab with the very same envelope form as the quasi-monochromatic incident pulse. Then, for a Gaussian pulse incident at the slab, all the emerging partial pulses are also Gaussian. What can we say about the superposition of these pulses, namely, the total electric field E_t(y = D,t), as given by Eq. (17)? In general one cannot expect that all these partial pulses will coalesce into a single pulse. However, this does occur for pulses that are wide in time, ω_cτ ≫ 1, and Figs. 1–3 confirm that, indeed, for ω_cτ ≃ 10 a single well defined pulse exits the slab. The peak velocity of such a pulse is a measurable quantity, and it is worthwhile examining its dependence on the parameters that define the dynamic-periodic slab and the pulse itself.

In what follows, we analyze the peak velocity v_p of a Gaussian pulse with small spectral width (Δω/Ω = 0.1 and 0.01) and carrier frequency ω_c/Ω = 0.5 for which the group velocity is infinite, as seen in the inset of Fig. 5. This velocity is determined from the transit time of the peak of the pulse, namely, we calculate the transmitted electric field as a function of time at y = D and observe the time t_o that it takes for the peak of the Gaussian pulse to traverse the slab of width D. The velocity is obtained from v_p = D/t_o. In Fig. 5 we plot the peak velocity v_p normalized to the velocity v_s as a function of the normalized thickness ( $\mathrm{\Omega }\sqrt{\overline{\epsilon }}/c$)D. The cases compared are no modulation (M = 0), weak modulation (M = 0.016) for Δω/Ω = 0.1, and strong modulations (M = 0.648) for Δω/Ω = 0.1 and 0.01. The velocity of the peak in the absence of modulation is obtained from Ref. [26], which deals with the peak velocity for a wide Gaussian pulse (τ ≫ D/v_s in addition to ω_cτ ≫ 1), interacting with a static plate in vacuum. In Ref. [26], the following peak velocity has been derived:

 

Fig. 5 Peak velocity of a Gaussian pulse with carrier frequency ω_c/Ω = 0.5, as a function of the normalized thickness ( $\mathrm{\Omega }\sqrt{\overline{\epsilon }}/c$)D for weak (M = 0.016) and strong (M = 0.648) modulations with Δω/Ω = 0.1, strong modulation with reduced spectral width Δω/Ω = 0.1, and a static case. For the static case M = 0 we consider a wide pulse in time with τ ≫ D/v_s. The inset is the bulk dispersion relation [normalized frequency ω/Ω vs. normalized wavevector $kc/\left(\mathrm{\Omega }\sqrt{\overline{\epsilon }}\right)$] for the strong modulation as in Ref. [12].

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It is apparent from Fig. 5 that the slab width is an important factor in the resulting peak velocity. We observe that the transit time is not proportional to the slab width; as the slab thickness increases, the velocity of the pulse may rise or fall. Results in Fig. 5 indicate an oscillatory behavior of the velocity of the peak with the thickness of the slab. This behavior is in part associated with interference, in the dynamic slab, of all the harmonics ω – nΩ, and the infinite number of plane waves with wave vectors k_p(ω) for each of these frequencies. However, even in the absence of modulation, for a pulse with τ ≫ D/v_s [see Fig. 5 and Eq. (25)], the peak velocity is a periodic function of the slab thickness D, with period λ/2, where $\lambda =2\pi c/\left({\omega }_{\text{c}}\sqrt{\overline{\epsilon }}\right)$ is the wavelength in the plate. A generalization of Eq. (25) to allow for absorption (see Ref. [27]) also shows that the peak velocity varies with the thickness although is no longer a periodic function of D. Now, expressing the peak velocity (25) in terms of ( $\mathrm{\Omega }\sqrt{\overline{\epsilon }}/c$)D [using Eq. (22)] results in the period of 2π. It is interesting to observe that, for weak modulation (M = 0.016), an approximate spacing of 2π between minima can be indeed noticed, despite the fact that in this case the pulse is not very wide in time. Nevertheless, the shape and amplitude of the oscillations for the weak modulation and for lack of modulation (with a very wide pulse) differ substantially for most of the ( $\mathrm{\Omega }\sqrt{\overline{\epsilon }}/c$)D range.

For the dynamic cases [see Fig. 5], the behavior is more complicated than it is for static conditions. The interference of all waves generated in the medium results in differing peak velocities for different thicknesses. As the modulation increases, the velocity profile changes and the separation between minima deviates from 2π. For strong modulation it is not possible to extend the analysis to very large slab widths due to the increased deformation of the transmitted pulse.

As it has been seen, the peak velocity v_p is sensitive to the modulation strength M. This was expected since the dispersion relation changes with the modulation strength (see Fig. 3 in Ref. [12]), thus affecting the speeds of the frequency components of the pulse. Also, from Fig. 5, the peak velocity v_p depends on the spectral width of the pulse (see strong modulation for Δω/Ω = 0.1 and 0.01). Despite the bandwidths Δω/Ω here differing by an order of magnitude, the peak velocities v_p for these cases are nearly equal. In fact, if we reduced the spectral width by one more order of magnitude we would observe, on the scale of Fig. 5, that the corresponding peak velocity plot would overlap with that for Δω/Ω = 0.01. Thus, for this case, the velocity is not sensitive to the spectral width of the pulse.

We have also calculated the peak velocity v_p as a function of the modulation strength M (with ( $\mathrm{\Omega }\sqrt{\overline{\epsilon }}/c$)D and Δω/Ω as parameters) and as function of the normalized bandwidth Δω/Ω (with ( $\mathrm{\Omega }\sqrt{\overline{\epsilon }}/c$)D and M as parameters). The most important conclusion from this analysis, as well as from Fig. 5 is that, for an incident Gaussian pulse and sinusoidal variation in time of the dielectric function, the infinite group velocities found in Ref. [12] at integer multiples of half the modulation frequency (ω = Ωn/2 with n an integer) have no repercussions for the peak velocity v_p of the transmitted pulse. This is due to the slab boundaries and the broadening of the transmitted pulse spectral content (generation of harmonics by the dynamic slab) which void the description based on a single pulse that traverses the slab at the group velocity in the bulk medium (see Sec. 3). Moreover, while the peak of the pulse can travel at speeds above $v_{\text{s}}=c/\sqrt{\overline{\epsilon }}$, the pulse peak velocity v_p is never superluminal (contrary to the case of Table 1 for ω_c = Ω/2 where the partial peak velocity for n = −2 is v₋₂ = 1.7c), at least for our closely studied carrier frequency ω_c = Ω/2.

6. Energy considerations

Ordinarily, for a transparent material, it is expected that the incoming energy breaks down exactly into the reflected and transmitted energies. However, since our system involves an unspecified external agent modulating the dielectric function, this is an open system that may result in supplying energy to the medium. Hence, conservation of energy is not necessarily satisfied. This issue was discussed previously for response of a dynamic-periodic slab to monochromatic incidence, even resulting in huge, resonant amplification of an incident plane wave [12, 13]. Herein we extend that study to an incident pulse. We define the reflectance 𝒭 (transmittance 𝒯) as

Table 3. Sum of the reflectance and transmittance (𝒭 + 𝒯) for incident Gaussian pulses with carrier frequency ω_c/Ω = 0.5 and 1 and spectral width Δω/Ω = 0.1; a normalized modulation frequency Ω̃ = 2, and modulation strengths M = 0.016 (weak), 0.162 (moderate), and 0.648 (strong).

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We have shown, as expected from our previous studies [12, 13], that pulses traversing a dynamic-periodic slab can gain energy that is delivered by the external modulating agent.

7. Conclusions

We presented a theory of pulse propagation through a slab whose dielectric function varies periodically with time. This theory follows from our previous work limited to monochromatic excitation [12].

For quasi-monochromatic pulses, we found that the transmitted pulse at the exit surface can be decomposed as a superposition of pulses. This representation is derived from the concepts of phase and group delays and the outcoming pulse is completely characterized by the monochromatic transmission coefficients t_n(ω_c) of the harmonics n generated by the carrier frequency ω_c. The carrier signals of these partial pulses have frequencies ω_c – nΩ and acquire phase shifts ϕ_n(ω_c), and these pulses exit the slab with the same form of their envelope as the incident pulse and with temporal shifts ϕ′_n(ω_c). In this sense then, no distortion occurs in the pulse shape. We analyzed the case of an incident monochromatic Gaussian pulse for several normalized modulation frequencies, carrier frequencies, and modulations strengths. We showed that the peak velocity of the partial pulses can be both fast (v_n > c or v_n < 0) or slow (v_n ≪ c). It turns out that infinite group velocities occurring in the band structure at frequencies ω = Ωm/2 (m is an integer) are not related to any of the partial peak velocities v_n. In the other extreme of very short pulses, we studied the transmitted pulses for an incident Gaussian pulse with a large bandwidth for several modulation frequencies and modulation strengths. Since here τ ≪ D/ν_s, we obtained multiple pulses coming out of the slab at different times t due to the multiple round trips of the pulse between the slab surfaces. As a consequence, the amplitude of each pulse successively decreases upon each round. We found that these pulses exit the slab with delay times t̃ ≈ 1,3,5,... (the equality would give the exact pulse exit times for an unmodulated slab). The departure from these times is larger as the modulation strength increases. Also, in this case the pulses become more distorted.

We also investigated the peak velocity v_p of the transmitted pulse when τ ≫ D/v_s and ω_cτ ≫ 1, that is, when a single well defined pulse exits the slab. We found that the peak velocity v_p follows an oscillatory behavior as a function of the normalized thickness ( $\mathrm{\Omega }\sqrt{\overline{\epsilon }}/c$)D (similar to the unmodulated case). This behavior depends strongly on the modulation strength M and weakly on the incident pulse spectral width Δω/ω_c. The peak velocity v_p turned out be faster or slower than v_s, but never faster than c for the analyzed cases. In these cases, the carrier frequency was ω_c = Ω/2 for which the bulk dispersion relation renders an infinite group velocity. We concluded that this divergence has no repercussions for the transmitted pulse peak velocity v_p. At last, we mention that a strong modulation strength M and a large slab thickness can lead to strong pulse deformation.

Further, we performed numerical calculations of the reflected and transmitted energies for an incident Gaussian pulse with particular parameters. We showed that, in general, the sum of these energies can exceed the incident energy. This gained energy is supplied by the external modulating agent.

Current and future developments could lead to the implementation of metamaterials that can be strongly and rapidly modulated and operated at higher frequencies than microwaves. Then, hopefully, our dynamic-periodic slab could be applied as a controlling device for the propagation velocity of light pulses. Also, such a dynamic slab could be used as a generator of pulses with up-conversion of the carrier frequency.