# Multilayer metamaterial absorbers inspired by perfectly matched layers

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## Abstract

We derive periodic multilayer absorbers with effective uniaxial properties similar to perfectly matched layers (PML). This approximate representation of PML is based on the effective medium theory and we call it an effective medium PML. We compare the spatial reflection spectrum of the layered absorbers to that of a PML material and demonstrate that after neglecting gain and magnetic properties, the absorber remains functional. This opens a route to create electromagnetic absorbers for real and not only numerical applications and as an example we introduce a layered absorber for the wavelength of 8 \(\upmu \hbox {m}\) made of \(\hbox {SiO}_2\) and NaCl. We also show that similar cylindrical core-shell nanostructures derived from flat multilayers also exhibit very good absorptive and reflective properties despite the different geometry.

## Keywords

Electromagnetic absorber Metamaterial Electromagnetic modeling Perfectly matched layer UPML## 1 Introduction

Perfectly matched layer (PML) (Berenger 2007) absorbers are now widely used to terminate electromagnetic simulations with an open domain. PMLs suppress reflection and ensure absorption of incident electromagnetic radiation at any angle and any polarization. A variety of PML formulations exist, starting from the early split-field PML (Berenger 1994), and the coordinate stretching approach (Chew and Weedon 1994) up to the convolutional PML (CPML) (Roden and Gedney 2000), and the near PML (NPML) (Cummer 2003). In this paper we refer to the Maxwellian formulation of PML, represented by an artificial material with uniaxial permittivity and permeability tensors, usually termed as uniaxial PML (UPML) (Sacks et al. 1995; Gedney 1996). A PML can be used with both time-domain and frequency domain methods, as well as with finite difference or finite element discretization schemes. It can assume various dispersion models, see e.g. the time-derivative Lorentz material that is capable of absorbing oblique, pulsed electromagnetic radiation having narrow and broad waists (Ziolkowski 1997). A PML can not be applied in some rare cases and for instance it fails to absorb a backward propagating wave for which an adiabatic absorber should be used instead (Zhang et al. 2008; Loh et al. 2009).

Electromagnetic absorbers have a much longer history than any kind of numerical modeling. Their possible applications range from modification of radar echo, through applications related to electromagnetic compatibility, up to photovoltaics. Early real-world absorbers were based on resistive sheets separated from a ground plate by quarter wave distances. With several sheets and multiple resonances it was possible to achieve broadband operation. The idea evolved into the theory of frequency selective surfaces (Munk 2000). Furthermore, it is possible to obtain a tailored impedance at a surface transition region using homogenized periodic one-dimensional or two-dimensional corrugated surfaces (Kristensson 2005). A static periodic magnetization obtained with ferromagnetic or ferrimagnetic materials is another route to obtaining broadband absorbers (Ramprecht and Norgren 2008). A recent overview paper (Watts et al. 2012) can serve as a tutorial on absorbers with the focus on novel metamaterial absorbers based on split-ring and electric-ring resonators.

In this paper we introduce the effective medium PML absorbers (EM-PML), which are metamaterial absorbers with a layered structure that exhibit effective permittivity and permeability tensors similar to a PML material. We calculate the reflection coefficient achieved with these layered absorbers. We look towards their possible physical realizations.

## 2 Approximate representation of UPML

The performance of a multilayer absorber obtained for \(f=0.6\) and \(s=1+0.5i\), and \(s=1+5i\) is illustrated in Fig. 3. Either of the two values of \(s\) enables to construct an efficient broadband absorber which is at the same time subwavelength in size. Layers have both magnetic and electric properties, including gain, and a complex permeability. In the limit of \(a/\lambda \rightarrow 0\) the multilayer approaches the properties of a true UPML (but at the same time, its thickness approaches \(N\cdot a\rightarrow 0\)). When \(s=1+5i\), an absorber consisting of \(N=5\) periods, with a total thickness of \(L=5 a\approx \lambda /20\) reflects \(-\)30 dB for a broad range of incidence angles, and the reflection decreases rapidly with total thickness \(L/\lambda \). However, the evanescent waves are amplified in this situation. If the absorbing power is smaller, e.g. \(s=1+0.5i\), the thickness \(L/\lambda \) has to be larger, but the reflection is less sensitive to the magnetic permeability and gain.

Finally, the multilayer considered here has an elliptical effective dispersion relation, while similar absorbers made of hyperbolic metamaterials have been also recently proposed (Guclu et al. 2012), although with no relation to PML.

## 3 Layered slab and core-shell metamaterial absorbers

Based on the theoretical considerations and material parameters used to calculate the spatial reflection spectrum in Fig. 5, a simple rule of thumb for the range of required permittivities can be drawn up. This simple rule requires one permittivity to have its real part between \(0\) and \(1\), while the other premittivity would have its real part larger than one. The calculations show, that losses should be provided by the second material (with \(\mathrm{Re}(\epsilon )>1\)), while in the first material we merely neglect gain. Materials in general have \(\mathrm{Re}(\epsilon )>1\), with the exception of localized transitions and broader frequency ranges in metals up to the plasma frequency.

Finally, we demonstrate the operation of a layered absorber consisting or real materials. Here, we make use of materials with localized electronic transitions. In the mid-infrared \(\hbox {SiO}_2\) is such a material, which features a strong transition at approximately 9 \(\upmu \hbox {m}\) and in a range between 7.2 and \(8\,\upmu \hbox {m}\) its real part of permittivity is between \(1\) and \(0\). The complementary material of choice is NaCl, which in this range has \(\mathrm{Re}(\epsilon )\approx 1.5\), however, it also is weakly dispersive in this range and due to Kramers-Kroning relations between the real and imaginary parts of permittivity it has a very small imaginary part. Thus, \(\hbox {SiO}_2\) needs to provide dissipation to extinguish the incident beam. Deposition of alternating \(\hbox {SiO}_2\) and NaCl or LiF layers of required thickness may be accomplished using such techniques as chemical vapor deposition or thermal evaporation (Fornarini et al. 1999; Kim and King 2007), although with the required total thickness of the layers apporaching a few wavelengths, a fast method is preferable, at least for absorbers intended for infrared.

In general, a desired refractive index of one of the layers (in the range \(0<\mathfrak {R}(n)<1\)) can be manufactured using the electromagnetic mixing rules. Recently, absorbers made of hyperbolic metamaterials have been proposed (Guclu et al. 2012) and similar to that material, our multilayer has an elliptical dispersion with a large eccentricity.

## 4 Conclusions

We have introduced an approximate representation of the uniaxial perfectly matched layer reflection-free absorber. The representation consists of a one-dimensional stack of uniform and isotropic metamaterial layers. A further simplification to non-magnetic materials with no gain can be assumed for some combinations of filling fraction and absorbing power. We have also shown that similar cylindrical core-shell nanostructures derived from flat multilayers also exhibit very good absorptive and reflective properties. A probable route to implement the absorber experimentally is by using a lossy dielectric as one material and for the other to take a metamaterial, or to make use of electromagnetic mixing rules, or to use some material near its resonance frequency. As an example we have demonstrated a layered absorber for the wavelength of \(8\,\upmu \hbox {m}\) made of \(\hbox {SiO}_2\) and NaCl.

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