Effective Maxwell’s Equations for Perfectly Conducting Split Ring Resonators

Lipton, Robert; Schweizer, Ben

doi:10.1007/s00205-018-1237-1

Cited by 11 publications

(20 citation statements)

References 30 publications

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“…The notion of a geometric average was first introduced by Bouchitté, Bourel, and Felbacq in [5]. The notion turned out to be very useful, it was extended in [17] to more general geometries. Although we focus on simple Helmholtz domains in the main part of this work, we define the geometric average for general geometries.…”

Section: The Geometric Averagementioning

confidence: 99%

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Effective Maxwell’s equations in general periodic microstructures

Schweizer

Urban

2017

Applicable Analysis

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Abstract:We study the time harmonic Maxwell equations in a meta-material consisting of perfect conductors and void space. The meta-material is assumed to be periodic with period η > 0; we study the behaviour of solutions (E η , H η ) in the limit η → 0 and derive an effective system. In geometries with a non-trivial topology, the limit system implies that certain components of the effective fields vanish. We identify the corresponding effective system and can predict, from topological properties of the meta-material, whether or not it permits the propagation of waves.

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Section: The Geometric Averagementioning

confidence: 99%

“…Proof. Thanks to the preparations of the last section, we can essentially follow [17] to derive (4.5a)-(4.5c). The remaining relations (4.5d) and (4.5e) follow from the characterisation of the solution spaces of the cell problems.…”

Section: Derivation Of the Effective Equationsmentioning

confidence: 99%

Effective Maxwell’s equations in general periodic microstructures

Schweizer

Urban

2017

Applicable Analysis

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“…In order to analyze split ring geometries for perfectly conducting inclusions, the concept of a geometric average was extended in [18] to cover the case of compactly contained inclusions without the assumption of simple connectedness. The concept of a geometric average was extended further in [22] in order to cover the case that the inclusions touch the boundary of the periodicity cell (0, 1) 3 .…”

Section: Introductionmentioning

confidence: 99%

The geometric average of curl-free fields in periodic geometries

2021

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In periodic homogenization problems, one considers a sequence ( u η ) η {(u^{\eta})_{\eta}} of solutions to periodic problems and derives a homogenized equation for an effective quantity u ^ {\hat{u}} . In many applications, u ^ {\hat{u}} is the weak limit of ( u η ) η {(u^{\eta})_{\eta}} , but in some applications u ^ {\hat{u}} must be defined differently. In the homogenization of Maxwell’s equations in periodic media, the effective magnetic field is given by the geometric average of the two-scale limit. The notion of a geometric average has been introduced in [G. Bouchitté, C. Bourel and D. Felbacq, Homogenization of the 3D Maxwell system near resonances and artificial magnetism, C. R. Math. Acad. Sci. Paris 347 2009, 9–10, 571–576]; it associates to a curl-free field Y ∖ Σ ¯ → ℝ 3 {Y\setminus\overline{\Sigma}\to\mathbb{R}^{3}} , where Y is the periodicity cell and Σ an inclusion, a vector in ℝ 3 {\mathbb{R}^{3}} . In this article, we extend previous definitions to more general inclusions, in particular inclusions that are not compactly supported in the periodicity cell. The physical relevance of the geometric average is demonstrated by various results, e.g., a continuity property of limits of tangential traces.

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“…We treat here properties of the three-dimensional solutions. We emphasize that the transmission properties of a high-contrast medium cannot be captured in the framework of perfect conductors, since the latter excludes resonances on the scale of the periodicity (except if three different length-scales are considered as in [28]).…”

Section: Introductionmentioning

confidence: 99%

Mathematical analysis of transmission properties of electromagnetic meta-materials

Ohlberger¹,

Schweizer²,

Urban³

et al. 2020

Networks &Amp; Heterogeneous Media

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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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Effective Maxwell’s Equations for Perfectly Conducting Split Ring Resonators

Cited by 11 publications

References 30 publications

Effective Maxwell’s equations in general periodic microstructures

Effective Maxwell’s equations in general periodic microstructures

The geometric average of curl-free fields in periodic geometries

Mathematical analysis of transmission properties of electromagnetic meta-materials

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