Mathematical analysis of transmission properties of electromagnetic meta-materials

Mario Ohlberger; Ben Schweizer; Maik Urban; Barbara Verfürth; Mario Ohlberger; Ben Schweizer; Maik Urban; Barbara Verfürth

doi:10.3934/nhm.2020002

Networks and Heterogeneous Media

2020, Volume 15, Issue 1: 29-56. doi: 10.3934/nhm.2020002

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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

Received: 01 September 2018 Revised: 01 October 2019
35B27, 35Q61, 65N30, 78M40

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.
- homogenization,
- Maxwell's equations,
- multiscale method,
- meta-material
Citation: Mario Ohlberger, Ben Schweizer, Maik Urban, Barbara Verfürth. Mathematical analysis of transmission properties of electromagnetic meta-materials[J]. Networks and Heterogeneous Media, 2020, 15(1): 29-56. doi: 10.3934/nhm.2020002

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Abstract

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.

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