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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 5 — Feb. 27, 2012
  • pp: 5586–5600
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Pulse propagation through a slab with time-periodic dielectric function ε(t)

Jorge R. Zurita-Sánchez, J. H. Abundis-Patiño, and P. Halevi  »View Author Affiliations


Optics Express, Vol. 20, Issue 5, pp. 5586-5600 (2012)
http://dx.doi.org/10.1364/OE.20.005586


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Abstract

We describe pulse propagation through a slab with periodic dielectric function ε(t), thus extending our previous investigation for monochromatic incidence [Phys. Rev. A 79, 053821 (2009)]. Based on the concepts of phase and group delays, we prove that, for an incident quasi-monochromatic pulse, the transmitted pulse can be expressed as a superposition of partial pulses that are exact replicas of the incident pulse and that exit the slab with a time delay. These partial pulses have harmonic carrier frequencies ωcnΩ (n is an integer, ωc is the carrier frequency of the incident pulse, and Ω = 2π/T is the slab modulation frequency). We find numerically that these partial pulses can be fast (peak velocity vn > c or vn < 0) or slow (vnc). Further, we investigate the peak velocity vp of the outcoming pulse for several cases. We find that this peak velocity vp and the partial peak velocities vn do not diverge —as occurs to the group velocity vg of the bulk dynamic-periodic medium when ωc = Ω/2. We expect that these results could be verified in the microwave regime [see Halevi et al., Proc. SPIE 8095, 80950I (2011)].

© 2012 OSA

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 (vgc) and fast light (either vg > c or vg < 0) [c is the vacuum light velocity and vg 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 vg > c, vg = ∞, or vg < 0 [1

1. C. G. B. Garrett and D. E. McCumber, “Propagation of a gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970). [CrossRef]

]. Later, this prediction was confirmed experimentally by Chu and Wong [2

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

]. 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

3. S. P. Tewari and G. S. Agarwal, “Control of phase matching and nonlinear generation in dense media by resonant fields,” Phys. Rev. Lett. 56, 1811–1814 (1986). [CrossRef] [PubMed]

5

5. M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, “Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms,” Phys. Rev. Lett. 74, 666–669 (1995). [CrossRef] [PubMed]

]. This possibility was the milestone for achieving great pulse velocity control. For example, in an ultracold gas of atoms a very low group velocity [vgc/(1.76 × 107)] was measured [6

6. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999). [CrossRef]

], while in a hot gas of caesium atoms, fast light was observed vg ≈ −c/310 [7

7. L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000). [CrossRef] [PubMed]

] where this superluminal velocity in a medium with anomalous dispersion is simply explained by light interference [8

8. A. Dogariu, A. Kuzmich, H. Cao, and L. J. Wang, “Superluminal light pulse propagation via rephasing in a transparent anomalously dispersive medium,” Opt. Express 8, 344–350 (2001). [CrossRef] [PubMed]

]. 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

9. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003). [CrossRef] [PubMed]

] and in optical fibers [10

10. K. Y. Song, M. G. Herráez, and L. Thévenaz, “Observation of pulse delaying and advancement in optical fibers using stimulated brillouin scattering,” Opt. Express 13, 82–88 (2005). [CrossRef] [PubMed]

]. 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

11. R. W. Boyd and D. J. Gauthier, “Slow and Fast Light,” in Progress in Optics, Vol. 43, E. Wolf, ed. (Elsevier, Amsterdam, 2002), pp. 497–530. [CrossRef]

].

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

12. J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009). [CrossRef]

, 13

13. J. R. Zurita-Sánchez and P. Halevi, “Resonances in the optical response of a slab with time-periodic dielectric function ε(t),” Phys. Rev. A 81, 053834 (2010). [CrossRef]

]. 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

14. F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’Reilly, “Dynamics of light propagation in spatiotemporal dielectric structures,” Phys. Rev. E 75, 046607 (2007). [CrossRef]

]. Similar effects are reported for pulses propagating in a linear medium with time-varying refractive index [15

15. Y. Xiao, G. P. Agrawal, and D. N. Maywar, “Spectral and temporal changes of optical pulses propagating through time-varying linear media,” Opt. Lett. 36, 505–507 (2011). [CrossRef] [PubMed]

]. Spectral shifts of pulses, resulting from the passage through dynamic cavities and waveguides, were observed [16

16. S. F. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photonics 1, 293–296 (2007). [CrossRef]

19

19. T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010). [CrossRef]

]. 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

20. W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, “Dynamical electric and magnetic metamaterial response at terahertz frequencies,” Phys. Rev. Lett. 96, 107401 (2006). [CrossRef] [PubMed]

, 21

21. V. J. Logeeswaran, A. N. Stameroff, M. S. Islam, W. Wu, A. M. Bratkovsky, P. J. Kuekes, S. Y. Wang, and R. S. Williams, “Switching between positive and negative permeability by photoconductive coupling for modulation of electromagnetic radiation,” Appl. Phys. A 87, 209–216 (2007). [CrossRef]

]. 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

22. E. Poutrina, S. Larouche, and D. R. Smith, “Parametric oscillator based on a single-layer resonant metamaterial,” Opt. Comm. 283, 1640–1646 (2010). [CrossRef]

]. Phase conjugation and negative refraction effects can occur in a metamaterial composed of split-ring resonators with nonlinear varactors [23

23. A. R. Katko, S. Gu, J. P. Barret, B.-I. Popa, G. Shvets, and S. A. Cummer, “Phase conjugation and negative refraction using nonlinear active metamaterials,” Phys. Rev. Lett. 105, 123905 (2010). [CrossRef] [PubMed]

].

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

12. J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009). [CrossRef]

]. Due to the periodic modulation, harmonics ωnΩ 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

13. J. R. Zurita-Sánchez and P. Halevi, “Resonances in the optical response of a slab with time-periodic dielectric function ε(t),” Phys. Rev. A 81, 053834 (2010). [CrossRef]

]. 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 εef(t) with a large modulation strength, thus overcoming the aforementioned limitation for optical frequencies [24

24. P. Halevi, U. Algredo-Badillo, and J. R. Zurita-Sánchez, “Optical response of a slab with time-periodic dielectric function ε(t): towards a dynamic metamaterial,” in Active Photonic Materials IV, G. S. Subramania and S. Foteinopoulou, eds., Proc. SPIE8095, 80950I (2011).

]. 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 εef(t) = C(t)/(εoa) (a is the lattice constant and εo 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.

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

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
Er(0,t)=n=rn(ω)Eo(ω)exp[i(ωnΩ)t]dω,
(6)
Et(D,t)=n=tn(ω)Eo(ω)exp[i(ωnΩ)t]dω,
(7)
where rn(ω)Enr(ω)/Eo(ω) and tn(ω)Ent(ω)/Eo(ω) are the reflection and transmission coefficients for the harmonic ωnΩ, respectively.

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 ωc (carrier frequency), that is,
Ei(y=0,t)=𝒠(t)exp(iωct).
(8)
By performing the variable transformation ω = ωc + α, Eq. (2) for Eo(ω) yields
Eo(α+ωc)=𝒠˜(α),
(9)
where 𝒠̃(α) is the Fourier transform of 𝒠(t), namely,
𝒠˜(α)=12π𝒠(t)exp(iαt)dt.
(10)
By also carrying out the transformation ω = α + ωc to Eq. (7) and using Eq. (9), the transmitted pulse becomes
Et(y=D,t)=n=exp[i(ωcnΩ)t]tn(ωc+α)𝒠˜(α)exp[iαt]dα.
(11)
We express tn(ω) as
tn(ω)=An(ω)exp[iϕn(ω)],
(12)
where An(ω) ≡ |tn(ω)| and ϕn(ω) ≡ arg[tn(ω)] (arg[...] denotes argument). Since 𝒠(t) is a slowly time-varying signal, 𝒠̃(α) is sharply localized around α = 0, namely most spectral components of the pulse should satisfy the condition that
|α|=|ωωc|ωc.
(13)
As a consequence, we can expand tn(ω) in a power series around ωc. The first order approximation renders
tn(ωc+α)An(ωc)exp[iϕn(ωc)]+[An(ωc)+iϕn(ωc)An(ωc)]exp[iϕn(ωc)]α.
(14)
Here, the superscript ”′” denotes derivative. We further consider that the term containing An(ωc) can be neglected. This implies that
|An(ωc)An(ωc)||ϕn(ωc)|.
(15)
With this assumption and Eq. (12), the approximation (14) is reduced to
tn(ωc+α)An(ωc)exp[iϕ(ωc)][1+iϕn(ωc)α]tn(ωc)exp[iϕn(ωc)α].
(16)
Hence, the transmitted pulse at the end of the slab becomes
Et(y=D,t)=n=An(ωc)exp{i[(ωcnΩ)tϕn(ωc)]}×𝒠˜(α)exp{iα[tϕn(ωc)]}dα,=n=tn(ωc)exp[i(ωcnΩ)t]𝒠[tϕn(ωc)].
(17)
Then, the transmitted field can be described as a superposition of pulses that correspond to each of the harmonics created by the dynamic slab. The amplitude of n-th pulse is An(ωc). Moreover, the partial carrier signals oscillate with frequencies ωcnΩ and acquire phases ϕn(ωc) (phase delays [25

25. A. Papoulis, Signal Analysis (McGraw-Hill, Tokio, 1977).

]), whereas the pulse envelope shape remains unchanged, but suffers temporal shifts ϕn(ωc) (group delays [25

25. A. Papoulis, Signal Analysis (McGraw-Hill, Tokio, 1977).

]). These outcoming pulses are characterized by the transmission coefficient tn(ωc).

3. Quasi-monochromatic Gaussian pulses

We apply the theory to the case of an incident Gaussian pulse having the form
Ei(y,t)=exp[(yct)2/(c2τ2)]cos[ωc(y/ct)],
(19)
where τ is the temporal width of the pulse and an unitary amplitude is assumed. The peak hits the left surface of the slab (y = 0) at the time t = 0. Equation (19) corresponds to the real part of Eq. (8), that is, 𝒠(t) = exp(−t2/τ2). The spectral amplitude of the Gaussian pulse yields [Eq. (2)]
Eo(ω)=τ4π{exp[τ2(ωωc)2/4]+exp[τ2(ω+ωc)2/4]},
(20)
where the spectral width is Δω = 2/τ. By Eq. (13), we assume that ωcτ ≫ 1.

Fig. 1 Incident and transmitted pulses as a function of the normalized time for a slab with normalized modulation frequency Ω̃ = 2. (a) The incident electric pulse Ei with carrier frequency ωc/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 ωcτ = 10 and ts Δω = 0.2.
Fig. 2 Incident and transmitted pulses as a function of the normalized time for a slab with normalized modulation frequency Ω̃ = 8 and modulation strength M = 0.648. (a) The incident electric pulse Ei with carrier frequency ωc/Ω = 0.67 and spectral bandwidth Δω/Ω = 0.1. (b) The transmitted pulse. Here ωcτ = 13.4 and ts Δω = 0.8.
Fig. 3 Incident and transmitted pulses as a function of the normalized time for a slab with normalized modulation frequency Ω̃ = 8 and modulation strength M = 0.648. (a) The incident electric pulse Ei with carrier frequency ωc/Ω = 0.67 and spectral bandwidth Δω/Ω = 0.01. (b) The transmitted pulse (inset shows the whole pulse). Here ωcτ = 134 and ts Δω = 0.08.

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 ωcnΩ (n = 0,±1,±2,...) and suffers a phase delay ϕn(ωc), the envelope experiences a time delay ϕn(ωc), and the amplitude is An(ωc). In terms of the normalized parameters and Ω̃, the partial Gaussian pulse at the slab exit (y = D) for the harmonic n can be expressed as
Ent(D,t˜)=An(ω^c)exp[ts2(t˜Δτ˜n)2/τ2]cos[(ω^cn)Ω˜t˜ϕn(ω^c)],
(24)
where ω̂cωc/Ω is the normalized carrier frequency and Δτ̃n = (1/Ω̃)n/dω̂c is the normalized temporal shift of the partial pulse. Tables 1 and 2 show the characterizing parameters of the partial pulses for the lowest orders in which the transmitted pulses of Figs. 1(c) and 3(b) are decomposed, respectively. From Tables 1 and 2, we notice that Δτ̃n that can be either positive or negative.

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 ωc/Ω = 0.5 and spectral width Δω/Ω = 0.1; slab with normalized modulation frequency Ω̃ = 2. Here ωcτ = 10 and ts 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 ωc/Ω = 0.67 and spectral width Δω/Ω = 0.01; slab with normalized modulation frequency Ω̃ = 8. Here ωcτ = 134 and ts 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 Δτ̃n < 0 or 0<Δτ˜n<1/ε¯=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 vn > c or vn < 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 Δτ˜n1/ε¯, 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 ωc = Ωm/2 with integer m. This frequency condition is satisfied in Fig. 1 and Table 1 with ωc = Ω/2. However, as can be seen in the Table 1, the peak velocities do not diverge, that is, the time delay Δτ̃n 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

1. C. G. B. Garrett and D. E. McCumber, “Propagation of a gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970). [CrossRef]

, 2

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

]). 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 ωcnΩ 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 ωc/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 ≈ 1), two small pulses emerge. For the moderate modulation and both normalized modulation frequencies, these pulses are peaked at ≈ 3 and ≈ 5. From Eq. (23), we recall that ≈ 1 is just the normalized time for light propagating at the speed vs=c/ε¯ 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/vs (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/vs, 3D/vs, 5D/vs,... (normalized times = 1,3,5,...). Thus, it is not all that surprising that, for τD/vs and moderate modulation, several of these transmission peaks are found. (On the contrary, if τD/vs, as occurs for the quasi-monochromatic case, then the emerging wave forms strongly overlap and a single pulse is observed, see Figs. 13. 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 = 2.98 and 5.05 for Ω̃ = 2 and = 2.99 and 4.99 for Ω̃ = 8 (for the moderate modulation) are so near to = 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 ≃ 3 are considerably diminished in comparison to the primary peak at ≃ 1, and the peaks at ≃ 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 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 ωc/Ω = 5, and the normalized time is taken with respect to Ω̃ = 2. (a) Ω̃ = 2 and ts Δω = 10. (b) Ω̃ = 8 and ts Δω = 40 (the inset is a closeup of the second pulse). Here ωcτ = 2.

5. Peak velocity

According to our result Eq. (17), every harmonic component of frequency ωnΩ 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 Et(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, ωcτ ≫ 1, and Figs. 13 confirm that, indeed, for ωcτ ≃ 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 vp of a Gaussian pulse with small spectral width (Δω/Ω = 0.1 and 0.01) and carrier frequency ωc/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 to that it takes for the peak of the Gaussian pulse to traverse the slab of width D. The velocity is obtained from vp = D/to. In Fig. 5 we plot the peak velocity vp normalized to the velocity vs as a function of the normalized thickness ( Ωε¯/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

26. P. Halevi, “Transit velocity of a pulse through a transparent plate,” Opt. Lett. 11, 759–760 (1986). [CrossRef] [PubMed]

], which deals with the peak velocity for a wide Gaussian pulse (τD/vs in addition to ωcτ ≫ 1), interacting with a static plate in vacuum. In Ref. [26

26. P. Halevi, “Transit velocity of a pulse through a transparent plate,” Opt. Lett. 11, 759–760 (1986). [CrossRef] [PubMed]

], the following peak velocity has been derived:
vp=vs[2ε¯1+ε¯cos2(ωcε¯D/c)+1+ε¯2ε¯sin2(ωcε¯D/c)].
(25)
Equation (25) is plotted for comparison with the dynamic plate. For Δω/Ω = 0.1 and (Ωε¯/c)D1, the peak velocity vp in absence of modulation is seen to converge to that obtained for the weak modulation strength. Indeed, Fig. 5 confirms that, in the limit D → 0, all four curves converge to vp/vs=2ε¯/(1+ε¯)=0.733, obtained from Eq. (25).

Fig. 5 Peak velocity of a Gaussian pulse with carrier frequency ωc/Ω = 0.5, as a function of the normalized thickness ( Ωε¯/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 τD/vs. The inset is the bulk dispersion relation [normalized frequency ω/Ω vs. normalized wavevector kc/(Ωε¯)] for the strong modulation as in Ref. [12].

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 ωnΩ, and the infinite number of plane waves with wave vectors kp(ω) for each of these frequencies. However, even in the absence of modulation, for a pulse with τD/vs [see Fig. 5 and Eq. (25)], the peak velocity is a periodic function of the slab thickness D, with period λ/2, where λ=2πc/(ωcε¯) is the wavelength in the plate. A generalization of Eq. (25) to allow for absorption (see Ref. [27

27. P. Halevi and L. D. Valenzuela, “Propagation of a broad light pulse through a plate,” J. Opt. Soc. Am. B 8, 1512–1515 (1991). [CrossRef]

]) 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 ( Ωε¯/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 ( Ωε¯/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 vp 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

12. J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009). [CrossRef]

]), thus affecting the speeds of the frequency components of the pulse. Also, from Fig. 5, the peak velocity vp 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 vp 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 vp as a function of the modulation strength M (with ( Ωε¯/c)D and Δω/Ω as parameters) and as function of the normalized bandwidth Δω/Ω (with ( Ωε¯/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

12. J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009). [CrossRef]

] at integer multiples of half the modulation frequency (ω = Ωn/2 with n an integer) have no repercussions for the peak velocity vp 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 vs=c/ε¯, the pulse peak velocity vp is never superluminal (contrary to the case of Table 1 for ωc = Ω/2 where the partial peak velocity for n = −2 is v−2 = 1.7c), at least for our closely studied carrier frequency ωc = 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

12. J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009). [CrossRef]

, 13

13. J. R. Zurita-Sánchez and P. Halevi, “Resonances in the optical response of a slab with time-periodic dielectric function ε(t),” Phys. Rev. A 81, 053834 (2010). [CrossRef]

]. Herein we extend that study to an incident pulse. We define the reflectance 𝒭 (transmittance 𝒯) as
(𝒯)=|Ar(t)(ω)|2dω/|Eo(ω)|2dω.
(26)
Here Ar(ω) [At(ω)] is the Fourier transform of the reflected [transmitted] field Er(0,t) [Et(D,t)] at the entrance [exit] interface. By applying the variable transformation ωω + nΩ to Eqs. (6,7), it follows that
Ar(ω)=n=rn(ω+nΩ)Eo(ω+nΩ),At(ω)=n=tn(ω+nΩ)Eo(ω+nΩ).
(27)
In Table 3, we have computed the sum of the reflectance and transmittance (𝒭 + 𝒯), using Eqs. (26) and (27), under a variety of conditions. Table 3 indicates that the sum of the reflected and transmitted energies exceeds the energy of the incident pulse (𝒭 + 𝒯 > 1) with the exception of the case ωc/Ω = 1 and M = 0.016 in which the gained energy is practically zero. Obviously, this energy excess is supplied by the external agent. Also, we notice from Table 3 that as the modulation is increased, more energy is provided by the external modulating agent.

Table 3. Sum of the reflectance and transmittance (𝒭 + 𝒯) for incident Gaussian pulses with carrier frequency ωc/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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7. Conclusions

We also investigated the peak velocity vp of the transmitted pulse when τD/vs and ωcτ ≫ 1, that is, when a single well defined pulse exits the slab. We found that the peak velocity vp follows an oscillatory behavior as a function of the normalized thickness ( Ωε¯/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/ωc. The peak velocity vp turned out be faster or slower than vs, but never faster than c for the analyzed cases. In these cases, the carrier frequency was ωc = 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 vp. 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.

Acknowledgments

J. H. A.-P. thanks CONACyT for support through the scholarship 212472. P. H. is grateful to Tomás López-Ríos for his hospitality at CNRS Grenoble during a sabbatical year. He also acknowledges support through the CONACyT projects 79180 and 103644-F.

References and links

1.

C. G. B. Garrett and D. E. McCumber, “Propagation of a gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970). [CrossRef]

2.

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

3.

S. P. Tewari and G. S. Agarwal, “Control of phase matching and nonlinear generation in dense media by resonant fields,” Phys. Rev. Lett. 56, 1811–1814 (1986). [CrossRef] [PubMed]

4.

S. E. Harris, J. E. Field, and A. Imamoğlu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990). [CrossRef] [PubMed]

5.

M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, “Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms,” Phys. Rev. Lett. 74, 666–669 (1995). [CrossRef] [PubMed]

6.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999). [CrossRef]

7.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000). [CrossRef] [PubMed]

8.

A. Dogariu, A. Kuzmich, H. Cao, and L. J. Wang, “Superluminal light pulse propagation via rephasing in a transparent anomalously dispersive medium,” Opt. Express 8, 344–350 (2001). [CrossRef] [PubMed]

9.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003). [CrossRef] [PubMed]

10.

K. Y. Song, M. G. Herráez, and L. Thévenaz, “Observation of pulse delaying and advancement in optical fibers using stimulated brillouin scattering,” Opt. Express 13, 82–88 (2005). [CrossRef] [PubMed]

11.

R. W. Boyd and D. J. Gauthier, “Slow and Fast Light,” in Progress in Optics, Vol. 43, E. Wolf, ed. (Elsevier, Amsterdam, 2002), pp. 497–530. [CrossRef]

12.

J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009). [CrossRef]

13.

J. R. Zurita-Sánchez and P. Halevi, “Resonances in the optical response of a slab with time-periodic dielectric function ε(t),” Phys. Rev. A 81, 053834 (2010). [CrossRef]

14.

F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’Reilly, “Dynamics of light propagation in spatiotemporal dielectric structures,” Phys. Rev. E 75, 046607 (2007). [CrossRef]

15.

Y. Xiao, G. P. Agrawal, and D. N. Maywar, “Spectral and temporal changes of optical pulses propagating through time-varying linear media,” Opt. Lett. 36, 505–507 (2011). [CrossRef] [PubMed]

16.

S. F. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photonics 1, 293–296 (2007). [CrossRef]

17.

M. W. McCutcheon, A. G. Pattantyus-Abraham, G. W. Rieger, and J. F. Young, “Emission spectrum of electromagnetic energy stored in a dynamically perturbed optical microcavity,” Opt. Express 15, 11472–11480 (2007). [CrossRef] [PubMed]

18.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009). [CrossRef] [PubMed]

19.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010). [CrossRef]

20.

W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, “Dynamical electric and magnetic metamaterial response at terahertz frequencies,” Phys. Rev. Lett. 96, 107401 (2006). [CrossRef] [PubMed]

21.

V. J. Logeeswaran, A. N. Stameroff, M. S. Islam, W. Wu, A. M. Bratkovsky, P. J. Kuekes, S. Y. Wang, and R. S. Williams, “Switching between positive and negative permeability by photoconductive coupling for modulation of electromagnetic radiation,” Appl. Phys. A 87, 209–216 (2007). [CrossRef]

22.

E. Poutrina, S. Larouche, and D. R. Smith, “Parametric oscillator based on a single-layer resonant metamaterial,” Opt. Comm. 283, 1640–1646 (2010). [CrossRef]

23.

A. R. Katko, S. Gu, J. P. Barret, B.-I. Popa, G. Shvets, and S. A. Cummer, “Phase conjugation and negative refraction using nonlinear active metamaterials,” Phys. Rev. Lett. 105, 123905 (2010). [CrossRef] [PubMed]

24.

P. Halevi, U. Algredo-Badillo, and J. R. Zurita-Sánchez, “Optical response of a slab with time-periodic dielectric function ε(t): towards a dynamic metamaterial,” in Active Photonic Materials IV, G. S. Subramania and S. Foteinopoulou, eds., Proc. SPIE8095, 80950I (2011).

25.

A. Papoulis, Signal Analysis (McGraw-Hill, Tokio, 1977).

26.

P. Halevi, “Transit velocity of a pulse through a transparent plate,” Opt. Lett. 11, 759–760 (1986). [CrossRef] [PubMed]

27.

P. Halevi and L. D. Valenzuela, “Propagation of a broad light pulse through a plate,” J. Opt. Soc. Am. B 8, 1512–1515 (1991). [CrossRef]

OCIS Codes
(350.5500) Other areas of optics : Propagation
(160.3918) Materials : Metamaterials
(130.7405) Integrated optics : Wavelength conversion devices

ToC Category:
Physical Optics

History
Original Manuscript: December 14, 2011
Revised Manuscript: January 19, 2012
Manuscript Accepted: January 26, 2012
Published: February 22, 2012

Citation
Jorge R. Zurita-Sánchez, J. H. Abundis-Patiño, and P. Halevi, "Pulse propagation through a slab with time-periodic dielectric function ε(t)," Opt. Express 20, 5586-5600 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-5-5586


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References

  1. C. G. B. Garrett and D. E. McCumber, “Propagation of a gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A1, 305–313 (1970). [CrossRef]
  2. S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett.48, 738–741 (1982). [CrossRef]
  3. S. P. Tewari and G. S. Agarwal, “Control of phase matching and nonlinear generation in dense media by resonant fields,” Phys. Rev. Lett.56, 1811–1814 (1986). [CrossRef] [PubMed]
  4. S. E. Harris, J. E. Field, and A. Imamoğlu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett.64, 1107–1110 (1990). [CrossRef] [PubMed]
  5. M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, “Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms,” Phys. Rev. Lett.74, 666–669 (1995). [CrossRef] [PubMed]
  6. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature397, 594–598 (1999). [CrossRef]
  7. L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature406, 277–279 (2000). [CrossRef] [PubMed]
  8. A. Dogariu, A. Kuzmich, H. Cao, and L. J. Wang, “Superluminal light pulse propagation via rephasing in a transparent anomalously dispersive medium,” Opt. Express8, 344–350 (2001). [CrossRef] [PubMed]
  9. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett.90, 113903 (2003). [CrossRef] [PubMed]
  10. K. Y. Song, M. G. Herráez, and L. Thévenaz, “Observation of pulse delaying and advancement in optical fibers using stimulated brillouin scattering,” Opt. Express13, 82–88 (2005). [CrossRef] [PubMed]
  11. R. W. Boyd and D. J. Gauthier, “Slow and Fast Light,” in Progress in Optics, Vol. 43, E. Wolf, ed. (Elsevier, Amsterdam, 2002), pp. 497–530. [CrossRef]
  12. J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A79, 053821 (2009). [CrossRef]
  13. J. R. Zurita-Sánchez and P. Halevi, “Resonances in the optical response of a slab with time-periodic dielectric function ε(t),” Phys. Rev. A81, 053834 (2010). [CrossRef]
  14. F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’Reilly, “Dynamics of light propagation in spatiotemporal dielectric structures,” Phys. Rev. E75, 046607 (2007). [CrossRef]
  15. Y. Xiao, G. P. Agrawal, and D. N. Maywar, “Spectral and temporal changes of optical pulses propagating through time-varying linear media,” Opt. Lett.36, 505–507 (2011). [CrossRef] [PubMed]
  16. S. F. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photonics1, 293–296 (2007). [CrossRef]
  17. M. W. McCutcheon, A. G. Pattantyus-Abraham, G. W. Rieger, and J. F. Young, “Emission spectrum of electromagnetic energy stored in a dynamically perturbed optical microcavity,” Opt. Express15, 11472–11480 (2007). [CrossRef] [PubMed]
  18. T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett.102, 043907 (2009). [CrossRef] [PubMed]
  19. T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A81, 043837 (2010). [CrossRef]
  20. W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, “Dynamical electric and magnetic metamaterial response at terahertz frequencies,” Phys. Rev. Lett.96, 107401 (2006). [CrossRef] [PubMed]
  21. V. J. Logeeswaran, A. N. Stameroff, M. S. Islam, W. Wu, A. M. Bratkovsky, P. J. Kuekes, S. Y. Wang, and R. S. Williams, “Switching between positive and negative permeability by photoconductive coupling for modulation of electromagnetic radiation,” Appl. Phys. A87, 209–216 (2007). [CrossRef]
  22. E. Poutrina, S. Larouche, and D. R. Smith, “Parametric oscillator based on a single-layer resonant metamaterial,” Opt. Comm.283, 1640–1646 (2010). [CrossRef]
  23. A. R. Katko, S. Gu, J. P. Barret, B.-I. Popa, G. Shvets, and S. A. Cummer, “Phase conjugation and negative refraction using nonlinear active metamaterials,” Phys. Rev. Lett.105, 123905 (2010). [CrossRef] [PubMed]
  24. P. Halevi, U. Algredo-Badillo, and J. R. Zurita-Sánchez, “Optical response of a slab with time-periodic dielectric function ε(t): towards a dynamic metamaterial,” in Active Photonic Materials IV, G. S. Subramania and S. Foteinopoulou, eds., Proc. SPIE8095, 80950I (2011).
  25. A. Papoulis, Signal Analysis (McGraw-Hill, Tokio, 1977).
  26. P. Halevi, “Transit velocity of a pulse through a transparent plate,” Opt. Lett.11, 759–760 (1986). [CrossRef] [PubMed]
  27. P. Halevi and L. D. Valenzuela, “Propagation of a broad light pulse through a plate,” J. Opt. Soc. Am. B8, 1512–1515 (1991). [CrossRef]

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