Propagation of electromagnetic fields in bulk terahertz metamaterials: A combined experimental and theoretical study

The availability of novel technologies that enable the fabrication of bulk metamaterials made it necessary to develop a theoretical language to discuss their properties appropriately. Here we use the fundamental Bloch mode approximation to explore bulk properties of metamaterials that operate at terahertz frequencies. We demonstrate that this approximation is valid in the case of terahertz meta-foil metamaterials. While relying on theoretical and experimental results, we show that the fundamental Bloch mode approximation can explain minuscule details that are unique to the bulk metamaterial. Various applications such as cylindrical lenses, terahertz cloaks, and sensors will benefit from our analysis.

Figure 1

(a) Schematic view of a single layer of a 1SE meta-foil. (b) Unit cell of the 1SE meta-foil. (c) SEM image of a fabricated meta-foil. The geometrical parameters are: $t=5$ $\mu$m, $d=5$ $\mu$m, $h=5$ $\mu$m, $b=15$ $\mu$m, and $a=31$ $\mu$m.

Figure 2

The real and imaginary parts of the transmission matrix elements for the single-layer meta-foil based on numerical simulations.

Figure 3

(a) and (b) show the simulated (solid) and measured (dot) transmission, respectively, from a single layer of the meta-foil for the two principal polarizations (black-$x$, blue-$y$). (c) and (d) show the simulated (solid) and measured (dot) transmission, respectively, for two layers of the meta-foil for both principal polarizations (black-$x$, blue-$y$).

Figure 4

Crossing points of the simulated transmission spectrum for the two different principal polarizations as a function of number of layers of the meta-foil. The black dashed lines correspond to the crossing points observed in the imaginary part of propagation constants for the two different principal polarizations [see Fig. 5a]. The black crosses represent the experimental crossing points for the single layer and two layers of the meta-foil.

Figure 5

(a) The imaginary parts $k_{\mathrm{FM}}$ of the fundamental Bloch mode and of the effective propagation constant $k_{\mathrm{eff}}$ as retrieved by inverting reflection and transmission from a single layer for two different polarizations (black-x, blue-y). The corresponding real part for the $y$-polarized wave is shown in (b) and for the $x$-polarized wave in (c). Please note, multiple branches are indicated in the figures that display the real part. They correspond to a possible solution in the parameter retrieval. Following earlier literature, we choose the $m=0$ branch for the $y$-polarized and the $m=-1$ branch for the $x$-polarized wave, which corresponds to a negative index.

Figure 6

(a)–(d) Real and imaginary parts of the normalized impedance of fundamental Bloch mode $Z_{\mathrm{FM}}$ and of the impedance $Z_{\mathrm{All}}$ calculated from the reflected light at the single interface between uniform half space (air) and bulk metamaterial for the $x$- [(a) and (b)] and $y$-polarized [(c) and (d)] wave, respectively, where all Bloch modes were taken into account.

Figure 7

(a) Real part of the isofrequency surface $k_z=k_z(k_x,k_y)$ at a frequency of $3.2$ THz. (b) Real part of $k_z=k_z(k_x,k_y=0)$. (c) Real part of $k_z=k_z(k_x=0,k_y)$. All quantities are in $\mu$m${}^{-1}$.