Near-perfect tunneling and amplification of evanescent electromagnetic waves in a waveguide filled by a metamaterial: Theory and experiments

Amplification of evanescent electromagnetic waves by a metamaterial slab is still a controversial issue. Metallic waveguides operating in cutoff provide a unique physical environment to produce and study these waves because isolated evanescent modes can be excited in such environment. In this paper, the tunneling of electromagnetic waves through a rectangular waveguide partially filled by a metamaterial is analyzed. Enhanced tunneling due to the amplification of evanescent waves in the metamaterial filled waveguide section is demonstrated. Since evanescent modes in rectangular waveguides are the superposition of two plane waves, the aforementioned results also show the amplification and tunneling of evanescent plane waves by a metamaterial slab in free space.

Figure 1

A metamaterial filled waveguide sandwiched between two empty waveguides operating in cutoff, and two input/output waveguides above cutoff. The length of each waveguide sections is $l_i$.

Figure 2

Field amplitude distribution inside the structure of Fig. 1, when section 3 is filled by an isotropic metamaterial with $ϵ=-{ϵ}_0$, $\mu =-{\mu }_0$. (a) Perfect tunneling when $l_3=2l_2=2l_4$. (b) Field amplitude when the last transition is eliminated $(l_4\to \infty )$. Dashed lines show the amplitude when section 3 is empty. The used geometrical dimensions are $a=24\mathrm{mm}$, $l_2=l_4=15\mathrm{mm}$, $l_3=30\mathrm{mm}$, and the operating frequency is $5\mathrm{GHz}$. The dielectric constant of sections 1 and 5 is $ϵ=2{ϵ}_0$.

Figure 3

(a) Effect of metamaterial losses on the field amplitude distribution inside the waveguide of Fig. 1. For simplicity, it has been imposed that magnetic and electric loss tangents are equal, so that $ϵ={ϵ}_0(-1+i\delta )$ and $\mu ={\mu }_0(-1+i\delta )$. In (b), the last transition is eliminated $(l_4\to \infty )$.

Figure 4

Effect of the $l_3∕{\lambda }_0$ ratio (numbers close to the lines) on the transmittance of the structure of Fig. 1 for different values of the loss tangent (${\delta }_m={\delta }_e=\delta$ for simplicity). Waveguide parameters are $2a∕{\lambda }_0=0.8$ and $l_2+l_4=l_3$.

Figure 5

Transmitted power through the device of Fig. 1 versus the ratio $(l_2+l_4)∕l_3$ for several values of $l_3∕{\lambda }_0$ (numbers close to the lines). Waveguide width is $a=0.8{\lambda }_0∕2$.

Figure 6

Experimental setup, vertical cross section (a), detail of the BC-SRR (b), and horizontal cross section (c). The number of BC-SRRs in each arrow can vary between $m=0$ (empty waveguide) and $m=10$ (fully filled waveguide). The figure shows the specific case with $m=3$, where amplification and near-perfect tunneling are produced.

Figure 7

Experimental values for the modulus of the transmission coefficient through the experimental setup of Fig. 6 for $m=0,10$ (dashed lines) and for $m=3$ (solid line).

Figure 8

Experimental (solid line) and calculated (dashed line) values for the modulus of the transmission coefficient through the experimental setup of Fig. 6 for $m=10$.

Figure 9

Experimental (solid line) and calculated (dashed line) values for the modulus of the transmission coefficient through the experimental setup of Fig. 6 for $m=5,4,3$.

Figure 10

“Double peak region” for the analyzed case, $m=3$ (zoom of Fig. 9), and some near cases. The numbers close to the curves are the length of the metamaterial section, $l_3$, measured in unit cells, $l_3=3d,3.2d,\dots$, $d$ being the distance between two nearest resonators $(d=6\mathrm{mm})$. At the frequency $5.534\mathrm{GHz}$, the impedances $Z_2$, $Z_3$, and $Z_4$ are matched.

Figure 11

Theoretical field amplitude inside the setup of Fig. 6 $(m=3)$ at the two perfect tunneling frequencies (see Fig. 10).

Figure 12

Theoretical field amplitude inside a structure similar to the one shown in Fig. 6 for $m=3$, but now the last transition is eliminated $(l_4\to \infty )$.

Figure 13

Electromagnetic simulation results for the modulus of the transmission coefficient through the device of Fig. 6 for $m=3$ and without losses (solid line). The predictions of the proposed metamaterial effective medium theory are also shown (dashed lines).

Figure 14

(Color online) Intensity of the simulated magnetic field at the central plane of the waveguides, at $f=5.455\mathrm{GHz}$. At this frequency the input waveguide is above cutoff, and the remaining waveguide sections operate in cutoff. The graphic shows the $z$-averaged intensity of the magnetic field.