## 1. Introduction

## 2. Electromagnetic Wave through ${(\mathit{air}/\mathit{metal}/\mathit{air})}^{\mathit{n}}$ Structures

## 3. Photonic Transmittance through Metallic Superlattices

## 4. Transmittance of EM Waves through Left-Handed Photonic Superlattices

## 5. Conclusions

## Funding

## Conflicts of Interest

## Appendix A. Resonant Transmission and Resonant Dispersion Relation

## Appendix B. The Band Structure of the λ/4 SL and the RDR

**Figure A1.**In this figure, we plot the transmission coefficient for the quarter $\lambda $ relation ${d}_{L}={d}_{2}{n}_{2}/{n}_{1}$ and for other relations, indicated in the graphs. The resonant transmission coefficient (blue curves) are plotted along with the dispersion relations for an infinite periodic structure (black curves) and the RDR of the TFPS (red lines). In (

**a**), the narrow bands in the quarter lambda limit agree with the predictions of the Kramer condition and with those of the RDR. In (

**b**–

**d**), we find again that the RDR predicts the resonances of the transmission coefficients.

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**Figure 1.**(

**a**) Incoming, reflected, and transmitted fields at the interface dielectric-conductor. (

**b**) The incidence angle ${\theta}_{i}$ and the effective refraction angle $\psi $ for a silver slab with the dielectric constant shown here.

**Figure 2.**Transmittance as a function of the frequency through a single slab of silver with thickness ${d}_{c}=30$ nm (

**a**) and ${d}_{c}=\phantom{\rule{3.33333pt}{0ex}}$80 nm (

**b**), for incidence angles ${\theta}_{i}=\pi /6,$$\pi /4$, $\pi /3$, and slightly less than $\pi /2$. For the slab thicknesses considered here, the EM field is highly attenuated below the plasma frequency ${\omega}_{p}=5.75\times {10}^{15}$ rad/s and oscillating for $\omega >{\omega}_{p}$. Near the incidence angle of $\pi /2$, the narrow and isolated resonances correspond to localized surface plasmons. Notice that as the incidence angle grows, the resonances move towards the isolated surface plasmon resonances.

**Figure 3.**Transmittance through a single slab as a function of the incidence angle ${\theta}_{i}$, for three different EM frequencies. For the graphs in the (

**a**), the silver slab thickness is ${d}_{c}=$ 80 nm, and in the (

**b**), ${d}_{c}=$ 1000 nm. Notice that the number of oscillations at high frequencies is of the order of ${d}_{c}/\lambda $.

**Figure 4.**Transmittance as a function of the frequency through the metallic superlattices ${(air/silver/air)}^{n}$. In (

**a**) and (

**b**), the thickness of the silver and air slabs is equal to ${d}_{a}=$ 100 nm and ${d}_{c}=$ 10 nm. In (

**a**), the transmittance is shown for $n=8$ and incidence angles ${\theta}_{i}=\pi /6,$$\pi /4$, $\pi /3$, and a slightly less than $\pi /2$. Below the plasma frequency ${\omega}_{p}=5.75\times {10}^{15}$ rad/s, the transmittance is an oscillating function of $\omega $. Near the incidence angle of $\pi /2$, the transmittance is highly resonant. In (

**b**), we repeat the transmittances for ${\theta}_{i}=\pi /6,$$\pi /4$, but now for $n=16$, and we plot also the Kramer condition (black curves) and the dispersion relation of Equation (23) derived in the theory of finite periodic systems (red lines). It is clear that this recurrence relation predicts the bands and the frequencies of all the resonant plasmons.

**Figure 5.**Transmittance as a function of the frequency through the metallic superlattices ${(air/silver/air)}^{n}$ where the silver and dielectric widths are (

**a**) small and equal and (

**b**) different with the silver slabs’ width being larger. The transmittances are shown for different incidence angles ${\theta}_{i}$ indicated on the graphs. Below the plasma frequency ${\omega}_{p}=5.75\times {10}^{15}$ rad/s, for small widths, the transmittance is independent of the incidence angle, while for larger silver width, the metallic superlattice is almost completely transparent. For frequencies above ${\omega}_{p}$, we have wider bands for small layers’ widths and thinner bands for larger silver widths.

**Figure 6.**Transmittance through ${(air/silver/air)}^{n}$ SLs as a function of the frequency $\omega $ and the conductor-layer width ${d}_{c}$. In (

**a**) and (

**c**), the number of unit cells n is odd, while in (

**b**) and (

**d**), it is even. The difference in the behavior of ${T}_{n}$ below and above ${\omega}_{p}$ can be seen much clearly in the upper panels. The line shapes of resonant plasmons with $\omega >{\omega}_{p}$ are thinner, therefore more localized, than at low frequencies. We assume that this is a result from a complex coupling of interface plasmons. A parity effect is also clear for ${d}_{c}$ larger than ∼50nm. At larger conductor widths, the low frequency transmission either vanishes, for n odd, or tends asymptotically to one, for n even. In the lower-panel graphs, we plot, together with the transmission coefficient, the dispersion relation of Equation (23), and we see that, if n is odd, the number of plasmon resonances is the same for SLs with n and with $n+1$ unit cells.

**Figure 7.**A metamaterial superlattice $air{\left(L{R}_{2}\right)}^{n}air$ with n unit cells, where left- and right-handed media alternate. We assume the normal incidence of the electromagnetic field with parallel polarization.

**Figure 8.**The transmission coefficients for two superlattices $air{\left(RR\right)}^{n}air$ (upper panel) and $air{\left(LR\right)}^{n}air$ (lower panel), with the same parameters except for the signs of the refraction indices ${\u03f5}_{1}$ and ${\mu}_{1}$. For the upper panel, we have n = 6, ${n}_{1}=2.22$, ${n}_{2}=1.41$, ${d}_{1}=79$ nm, and ${d}_{2}=497$ nm. For the lower panel, we have n=6, ${n}_{1}=-2.22$, ${n}_{2}=1.41$, ${d}_{1}=79$ nm, and ${d}_{2}=497$ nm.

**Figure 9.**Transmission coefficients (TCs) for the superlattices $air{\left(RR\right)}^{6}air$ and $air{\left(L{R}_{2}\right)}^{6}air$, as functions of the frequency $\omega $, when ${d}_{L}={d}_{1}={d}_{2}{n}_{2}/{n}_{L}=316.2$ nm and ${d}_{2}=497$ nm. The TCs of the superlattices $air{\left(LR\right)}^{6}air$ become a sequence of isolated and equidistant peaks, while the TCs of the superlattices $air{\left(RR\right)}^{6}air$ become a periodic sequence of resonant bands. This characteristic is independent of the number of unit cells.

**Figure 10.**In Panel (

**a**), the tunneling (phase) time through the superlattice $air{\left(LR\right)}^{7}air$, whose transmission coefficient is shown in the lower panel of Figure 9. The tunneling time through the SL, obtained from $\tau =\partial {t}_{aSa}/\partial \omega $, at the resonant frequencies is $\cong -0.284$ fs. In (

**b**), a Gaussian wave packet with the centroid at $\omega \simeq 5.32\times {10}^{15}$ rad/s, is prepared at $t=0$ at a distance ${z}_{0}=10{l}_{c}$ from the SL.

**Figure 11.**The Gaussian WP prepared, at $t=0$, at a distance ${z}_{0}=L=10{l}_{c}$ from the SL, as explained in Figure 9b (see the blue curve), moves towards D, passing through the SL ${\left(LR\right)}^{7}$, and reaches this point (see the red curves), as predicted, at $t={\tau}_{D}=2{z}_{0}/{v}_{g}+\tau \simeq 54$.02 fs. The WP is partially transmitted and partially reflected. Not all frequencies are transmitted because the transmission coefficients at the center of the packet are larger than in the tails. Thus, a dip is formed in the reflected (red) WP.

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