Beyond local effective material properties for metamaterials

K. Mnasri, A. Khrabustovskyi, C. Stohrer, M. Plum, and C. Rockstuhl

Phys. Rev. B 97, 075439 – Published 26 February 2018

Abstract

To discuss the properties of metamaterials on physical grounds and to consider them in applications, effective material parameters are usually introduced and assigned to a given metamaterial. In most cases, only weak spatial dispersion is considered. It allows to assign local material properties, e.g., a permittivity and a permeability. However, this turned out to be insufficient. To solve this problem, we study here the effective properties of metamaterials with constitutive relations beyond a local response and take strong spatial dispersion into account. This research requires two contributions. First, bulk properties in terms of eigenmodes need to be studied. We particularly investigate the isofrequency surfaces of their dispersion relation are investigated and compared to those of an actual metamaterial. The significant improvement to effectively describe it provides evidence for the necessity to use nonlocal material laws in the effective description of metamaterials. Second, to be able to capitalize on such constitutive relations, also interface conditions need to be known. They are derived in this contribution for our form of the nonlocality using a generalized (weak) formulation of Maxwell's equations. Based on such interface conditions, Fresnel expressions are obtained that predict the amplitude of the reflected and transmitted plane wave upon illuminating a slab of such a nonlocal metamaterial. This all together offers the necessary means for the in-depth analysis of metamaterials characterized by strong spatial dispersion. The general formulation we choose here renders our approach applicable to a wide class of metamaterials.

Received 1 June 2017
Revised 12 February 2018

DOI:https://doi.org/10.1103/PhysRevB.97.075439

Physics Subject Headings (PhySH)

Metamaterials Nanophotonics Plasmonics

Classical electromagnetism

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

K. Mnasri^1,2,*, A. Khrabustovskyi^3,†, C. Stohrer², M. Plum³, and C. Rockstuhl^1,4

¹Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany
²Institute of Applied and Numerical Mathematics, Karlsruhe Institute of Technology, Englerstrasse 2, 76131, Karlsruhe, Germany
³Institute for Analysis, Department of Mathematics, Karlsruhe Institute of Technology, Englerstrasse 2, 76131 Karlsruhe, Germany
⁴Institute of Nanotechnology, Karlsruhe Institute of Technology, 76344 Eggenstein-Leopoldshafen, Germany

^*Corresponding author: karim.mnasri@kit.edu
^†Present address: Institute of Applied Mathematics, Graz University of Technology, Steyrergasse 30/III, 8010 Graz, Austria.

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Issue

Vol. 97, Iss. 7 — 15 February 2018

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Images

Figure 1
Examples of isofrequency contours decomposed into real (red) and imaginary (blue) parts for the inner $+$ solutions (solid red and dotted blue lines) as well as for the inner $-$ solutions (dashed red and dash-dot blue lines) of Eqs. (15) and (17). The left figure shows the isofrequency contours of the TE mode for $p_{0}^{TE} = (1 + 0.5 i) μ m^{- 2}$ , $p_{1}^{TE} = (- 1.6 - 1.5 i) μ m^{- 2}$ , and $q_{1}^{TE} = (2 + 0.6 i) μ m^{- 4}$ . The center and the right figures show the isofrequency contours of the TM modes for $p_{0}^{TM} = (- 1 - 0.5 i) μ m^{- 2}$ , $p_{1}^{TM} = (- 3.1 + 0.1 i) μ m^{- 4}$ , $q_{0}^{TM} = - 3.1 + 0.5 i$ and $q_{1}^{TM} = 3 + 0.5 i$ , and $p_{0}^{TM} = (3.5 - 0.5 i) μ m^{- 2}$ , $p_{1}^{TM} = (- 11.1 + 0.1 i) μ m^{- 4}$ , $q_{0}^{TM} = - 2.1 + 0.1 i$ and $q_{1}^{TM} = 3 + 0.5 i$ , respectively.
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Figure 2
Examples of isofrequency contours decomposed into real (red) and imaginary (blue) parts for the inner $+$ solutions (solid red and dotted blue lines) as well as for the inner $-$ solutions (dashed red and dash-dot blue lines) of Eq. (30). The upper figure shows the isofrequency contours for $p_{0}^{TM} = (- 1 + 0.5 i) μ m^{- 2}$ , $p_{1}^{TM} = 2.1 + 0.1 i$ , $q_{0}^{TM} = (2.1 + 2.5 i) μ m^{- 2}$ , and $q_{1}^{TM} = (- 2 + 0.5 i) μ m^{- 2}$ , while the bottom one for $p_{0}^{TM} = (1 + 1.11 i) μ m^{- 2}$ , $p_{1}^{TM} = - 1.1 + i$ , $q_{0}^{TM} = (1.1 - 0.3 i) μ m^{- 2}$ , and $q_{1}^{TM} = (- 0.2 + 2.5 i) μ m^{- 2}$ .
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Figure 3
Fishnet metamaterial consisting of a biperiodic structure with periods $Λ_{x} = Λ_{y} = 600 nm$ and $Λ_{z} = 200 nm$ with rectangular holes with the width $w_{y} = 100 nm$ and $w_{x} = 316 nm$ . It compromises a stack of layers made of two $45 − nm$ Ag layers separated by a thin dielectric spacer, $30 nm$ of ${MgF}_{2}$ . The remaining space is filled with air.
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Figure 4
(Left) Dispersion relation of the fundamental mode for $k_{x} = k_{y} = 0$ calculated by a plane-wave expansion Ansatz using the Fourier modal method (FMM) algorithm. It shows a resonance around $k_{0} = 4.3 μ m^{- 1}$ in which $ℜ (k_{z}) < 0$ . (Right) Isofrequency contour in the $x z$ plane at the resonance wave number.
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Figure 5
Isofrequency contours $k_{z} = k_{z} (k_{y})$ at the resonance frequency of the fishnet corresponding to a wave number of $k_{0} = 4.3 μ m^{- 1}$ . Solid (dashed) curve represents the real (imaginary) part. The blue (crosses) curves are obtained from fitting Eq. (30) to the reference curve (black). It shows a good agreement up to $k_{y} = 0.3 k_{0}$ . The red (bullets) curves are obtained from fitting Eq. (17) to the reference curve and show a good agreement up to $k_{y} = 0.4 k_{0}$ . Meanwhile, the green (diamonds) curves, which are obtained from WSD, are showing only an agreement within the paraxial regime, i.e., for $k_{y} < 0.1 k_{0}$ .
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Figure 6
Deviations from reference (FMM) in logarithmic scale. Over all frequencies, modeling metamaterials with nonlocal material laws makes more sense for a realistic homogenization.
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