# American Institute of Mathematical Sciences

March  2020, 15(1): 29-56. doi: 10.3934/nhm.2020002

## Mathematical analysis of transmission properties of electromagnetic meta-materials

 1 Angewandte Mathematik: Institut für Analysis und Numerik, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, DE-48149 Münster, Germany 2 Fakultät für Mathematik, TU Dortmund, Vogelpothsweg 87, DE-44227 Dortmund, Germany 3 Institut für Mathematik, Universität Augsburg, Universitätsstr. 14, DE-86159 Augsburg, Germany

*Corresponding author: Ben Schweizer

Received  September 2018 Revised  October 2019 Published  December 2019

Fund Project: This work was supported by the Deutsche Forschungsgemeinschaft (DFG) in the project "Wellenausbreitung in periodischen Strukturen und Mechanismen negativer Brechung" (grant OH 98/6-1 and SCHW 639/6-1)

We study time-harmonic Maxwell's equations in meta-materials that use either perfect conductors or high-contrast materials. Based on known effective equations for perfectly conducting inclusions, we calculate the transmission and reflection coefficients for four different geometries. For high-contrast materials and essentially two-dimensional geometries, we analyze parallel electric and parallel magnetic fields and discuss their potential to exhibit transmission through a sample of meta-material. For a numerical study, one often needs a method that is adapted to heterogeneous media; we consider here a Heterogeneous Multiscale Method for high contrast materials. The qualitative transmission properties, as predicted by the analysis, are confirmed with numerical experiments. The numerical results also underline the applicability of the multiscale method.

Citation: Mario Ohlberger, Ben Schweizer, Maik Urban, Barbara Verfürth. Mathematical analysis of transmission properties of electromagnetic meta-materials. Networks & Heterogeneous Media, 2020, 15 (1) : 29-56. doi: 10.3934/nhm.2020002
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The cube shows the periodicity cell $Y$. The microstructures $\Sigma_1$, $\Sigma_3$, and $\Sigma_4$ are shown in dark grey. (A) The metal cylinder $\Sigma_1$. (B) The metal plate $\Sigma_3$. (C) The metal part $\Sigma_4$ is the complement of a cylinder
Waveguide domain $G$ with periodic scatterer $\Sigma_{ \eta}$ contained in the middle part $Q_M$ and incident wave from the right
Metal cuboid $\tilde \Sigma_1$, the magnitude of $\operatorname{Re}( \hat{H})$ is plotted. Left: The $H$-field is $\operatorname{e}_3$-polarized and the plot shows values in the plane $x_3 = 0.5$. The analysis of both, (PC) and (HC) yields: transmission is possible. Right: The $H$-field is $\operatorname{e}_2$-polarized and the plot shows values in the plane $x_2 = 0.545$. Since the $H$-field is not parallel to $\operatorname{e}_3$, the analysis of (PC) and (HC) predicts that no transmission is possible. Inlet in the middle: Microstructure in the unit cube
Test of numerical schemes for the metal cuboid $\tilde \Sigma_1$. We consider an $\operatorname{e}_3$-polarized incoming $H$-field and plot the solution in the plane $x_3 = 0.5$; the colors indicate the magnitude of the reference solution $\operatorname{Re}(H^\eta)$ (left) and the zeroth order approximation $\operatorname{Re}(H^0_{{\rm HMM}})$ (right). Inlet in the center: Microsctructure in the unit cube with visualization plane in red
Metal cuboid $\tilde \Sigma_2$. We study an $\operatorname{e}_3$-polarized incident $H$-field and plot the magnitude of $\operatorname{Re}( \hat{H})$ (left) and $\operatorname{Re}(H^\eta)$ (right) in the plane $x_2 = 0.545$. The analysis (PC) predicts transmission in this case, the analysis (HC) does not exclude transmission. Middle: Microstructure in the unit cube with visualization plane in red
Metal plate $\Sigma_3$. The colors indicate the magnitude of $\operatorname{Re}( \hat{H})$ in the plane $x_3 = 0.5$. Left: The $H$-field is $\operatorname{e}_3$-polarized. The analysis (PC) predicts transmission, the analysis (HC) cannot exclude transmission. Right: The $H$-field is $\operatorname{e}_2$-polarized. The analysis (PC and HC) predicts that no transmission is possible
Metal block with holes. Left: The structure $\tilde \Sigma_4$, we plot the magnitude of $\operatorname{Re}(H^\eta)$ in the plane $x_3 = 0.545$ for $\operatorname{e}_3$-polarized incoming $H$-field. The analysis (PC) predicts no transmission, the analysis (HC) cannot exclude transmission. Right: A geometry in which the cylinders $\tilde \Sigma_4$ are rotated in $\operatorname{e}_3$-direction. We plot the magnitude of $\operatorname{Re}(H^\eta)$ in the plane $x_3 = 0.5$ for $\operatorname{e}_3$-polarized incoming $H$-field. Small pictures show the microstructures in the unit cube and the visualization planes in red
Index sets $\mathcal{N}_{\Sigma}$, $\mathcal{L}_{\Sigma}$, and $\mathcal{N}_{Y \setminus \overline{\Sigma}}$ for microstructures $\Sigma_1$ to $\Sigma_4$ of (2.6)–(2.9)
 geometry metal cylinder $\Sigma_1$ metal cylinder $\Sigma_2$ metal plate $\Sigma_3$ air cylinder $\Sigma_4$ $\mathcal{N}_{\Sigma}$ $\{1, 2\}$ $\{2, 3\}$ $\{2\}$ $\emptyset$ $\mathcal{L}_{\Sigma}$ $\{3\}$ $\{1\}$ $\{1, 3\}$ $\{1, 2, 3\}$ $\mathcal{N}_{Y \setminus \overline{\Sigma}}$ $\emptyset$ $\emptyset$ $\{2\}$ $\{2, 3\}$
 geometry metal cylinder $\Sigma_1$ metal cylinder $\Sigma_2$ metal plate $\Sigma_3$ air cylinder $\Sigma_4$ $\mathcal{N}_{\Sigma}$ $\{1, 2\}$ $\{2, 3\}$ $\{2\}$ $\emptyset$ $\mathcal{L}_{\Sigma}$ $\{3\}$ $\{1\}$ $\{1, 3\}$ $\{1, 2, 3\}$ $\mathcal{N}_{Y \setminus \overline{\Sigma}}$ $\emptyset$ $\emptyset$ $\{2\}$ $\{2, 3\}$
Overview of the transmission coefficients $T$ when $H$ is parallel to $\operatorname{e}_3$. We see, in particular, that $T$ is vanishing for the structure $\Sigma_4$, but it is nonzero for the other micro-structures. The constant $\gamma \in \mathbb{C}$ depends on the microstructure and on solutions to cell problems, and is defined in the subsequent sections, $\alpha := | Y \setminus \Sigma|$ is the volume fraction of air, $L > 0$ is the width of the meta-material $Q_M$. We use $k_0 = \omega \sqrt{\varepsilon_0 \mu_0}$ and the numbers $p_0 := \operatorname{e}^{i k_0 L}$, $p_1 := p_0 \operatorname{e}^{ i \sqrt{\alpha \gamma} L}$, and $p_2 := p_0 \operatorname{e}^{ i \sqrt{\gamma}L}$
 microstructure $\Sigma$ transmission coefficient $T$ metal cylinder $\Sigma_1$ $T =4 p_1\sqrt{\alpha\gamma} \Big[(\alpha + \gamma)(1-p_1^2) + 2 \sqrt{\alpha \gamma} (1+ p_1^2)\Big]^{-1}$ metal cylinder $\Sigma_2$ $T = 4 p_2 \sqrt{\gamma} \Big[(1+ \gamma)(1- p_2^2) + 2 \sqrt{\gamma}(1+p_2^2) \Big]^{-1}$ metal plate $\Sigma_3$ $T = 4p_0 \alpha \Big[(1+\alpha^2)(1-p_0^2) + 2 \alpha (1+ p_0^2)\Big]^{-1}$ air cylinder $\Sigma_4$ $T =0$
 microstructure $\Sigma$ transmission coefficient $T$ metal cylinder $\Sigma_1$ $T =4 p_1\sqrt{\alpha\gamma} \Big[(\alpha + \gamma)(1-p_1^2) + 2 \sqrt{\alpha \gamma} (1+ p_1^2)\Big]^{-1}$ metal cylinder $\Sigma_2$ $T = 4 p_2 \sqrt{\gamma} \Big[(1+ \gamma)(1- p_2^2) + 2 \sqrt{\gamma}(1+p_2^2) \Big]^{-1}$ metal plate $\Sigma_3$ $T = 4p_0 \alpha \Big[(1+\alpha^2)(1-p_0^2) + 2 \alpha (1+ p_0^2)\Big]^{-1}$ air cylinder $\Sigma_4$ $T =0$
Summary of analytical predictions of the transmission properties and references to numerical results. The first row provides the geometry. The second row indicates possible transmission polarizations (of $H$) according to the theory of perfect conductors of Section 3.1. The third row indicates the possibility of transmission based on Section 3.2: We mention cases in which we cannot derive weak convergence to $0$. An entry "-" indicates that no analytical result can be applied. The last row provides the reference to the visualization of the numerical calculation for high-contrast media
 geometry metal cylinder $\Sigma_1$ metal cylinder $\Sigma_2$ metal plate $\Sigma_3$ air cyl. $\Sigma_4$ transmission (PC) $\mathbf{e}_3$-polarized $\mathbf{e}_2$ and $\mathbf{e}_3$-polarized $\mathbf{e}_3$-polarized no nontriv. limit (HC) $\mathbf{e}_3$-polarized - $\mathbf{e}_3$-polarized - numerical example Fig. 5.1 Fig. 5.3 Fig. 5.4 Fig. 5.5
 geometry metal cylinder $\Sigma_1$ metal cylinder $\Sigma_2$ metal plate $\Sigma_3$ air cyl. $\Sigma_4$ transmission (PC) $\mathbf{e}_3$-polarized $\mathbf{e}_2$ and $\mathbf{e}_3$-polarized $\mathbf{e}_3$-polarized no nontriv. limit (HC) $\mathbf{e}_3$-polarized - $\mathbf{e}_3$-polarized - numerical example Fig. 5.1 Fig. 5.3 Fig. 5.4 Fig. 5.5
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