Transmission-line analysis of $\epsilon$-near-zero–filled narrow channels

Following our recent interest in metamaterial-based devices supporting resonant tunneling, energy squeezing, and supercoupling through narrow waveguide channels and bends, here we analyze the fundamental physical mechanisms behind this phenomenon using a transmission-line model. These theoretical findings extend our theory, allowing us to take fully into account frequency dispersion and losses and revealing the substantial differences between this unique tunneling phenomenon and higher-frequency Fabry-Perot resonances. Moreover, they represent the foundations for other possibilities to realize tunneling through arbitrary waveguide bends, both in $E$ and $H$ planes of polarization, waveguide connections, and sharp abruptions and to obtain analogous effects with geometries arguably simpler to realize.

Figure 1

(Color online) (a) Geometry of the problem: a narrow channel of height $h_{\mathit{ch}}$ and length $l$ connects two waveguide sections of height $h⪢h_{\mathit{ch}}$. (b) Corresponding transmission-line model, in analogy with [13]. The structure is uniform along the $x$ direction.

Figure 2

(Color online) Power transmission through a narrow channel in a rectangular waveguide as in Fig. 1, with $l_{ab}=h_{\mathit{ch}}=0.8\mathrm{mm}$ and $w=2h=10.16\mathrm{cm}$. The outer sections are filled with a material with $2{\epsilon }_0$, whereas the transition region has permittivity ${\epsilon }_0$. $l$ is varied as indicated in the legend. Solid lines refer to full-wave simulations performed with [25], dashed lines use the TL model with $C_{ab}=0$, and dotted lines use $C_{ab}=1.6\mathrm{pF}$.

Figure 3

(Color online) (a), (b) Phase of the magnetic field and (c), (d) electric field distribution (snapshot in time) for the geometry of Fig. 2 with $l=12.7\mathrm{cm}$ at the two tunneling frequencies $f_0\simeq 1.46\mathrm{GHz}$ [(a), (c): supercoupling effect] and $f\simeq 1.82\mathrm{GHz}$ [(b), (d): Fabry-Perot resonance]. Brighter colors refer to larger field amplitudes.

Figure 4

(Color online) Power transmission evaluated using [25] through a single abruption in a rectangular waveguide as in Fig. 2, with $h_{\mathit{ch}}=0.8\mathrm{mm}$ and $w=2h=10.16\mathrm{cm}$. The outer section (orange, darker) is filled with a material with $2{\epsilon }_0$, whereas the ultranarrow channel (blue, lighter) has permittivity ${\epsilon }_0$. $l_{ab}$ is varied as indicated in the legend.

Figure 5

(Color online) (a) Power transmission and (b) phase of the transmission coefficient evaluated with [25] through the channel of Fig. 3, comparing the cases of a parallel-plate waveguide filled by an ideal Drude metamaterial and a rectangular waveguide as in Fig. 3, for which the material dispersion is provided by the waveguide width $w$. In the case transition channels are used, their thickness is $l_{ab}=h_{\mathit{ch}}$. The thin gray line in panel (a) indicates the variation of ${\epsilon }_{\mathit{ch}}∕{\epsilon }_0$ with frequency.

Figure 6

(Color online) Power transmission, evaluated using [25], through a similar geometry as in Fig. 2, but varying the width $w$ in the different waveguide sections instead of varying the material filling the waveguide, as depicted in the inset.

Figure 7

(Color online) Geometry and electric field distribution (snapshot in time, normal component) through a supercoupler with a 90° bend in the $H$ plane. The average rotation radius in this case is $R=17.8\mathrm{cm}$. The corresponding time-domain animation has been reported in [27].

Figure 8

(Color online) (a) Power transmission for the geometry of Fig. 7. (b) Real part of Poynting vector distribution on the $H$ plane with $R=12.7\mathrm{cm}$ at the cutoff frequency $f_0$.

Figure 9

(Color online) (a) Geometry and electric field distribution (snapshot in time, normal component) and (b) amplitude and phase of the transmission through a supercoupler, consistent with the geometry of Fig. 7, but with a 90° bend in the $H$ plane following a 90° bend in the $E$ plane.