Characterization of metamaterials made of stacked layers of dogbone conductor pairs

The metamaterials composed of planar arrays of tightly coupled dogbone-shaped conductors have recently attracted considerable interest owing to their potential of supporting backward wave propagation [1]-[3]. The detailed study of the plane wave interaction with a layer of dogbone pairs has been performed in [3], [4] along with the eigenwave analysis of an infinite stack of array layers. In particular, it has been found that such metamaterial may exhibit negative effective refractive index near the transmission resonances.


Introduction
The metamaterials composed of planar arrays of tightly coupled dogbone-shaped conductors have recently attracted considerable interest owing to their potential of supporting backward wave propagation [1]- [3]. The detailed study of the plane wave interaction with a layer of dogbone pairs has been performed in [3], [4] along with the eigenwave analysis of an infinite stack of array layers. In particular, it has been found that such metamaterial may exhibit negative effective refractive index near the transmission resonances.
In this paper we present the results of the theoretical and experimental studies of the stacked arrays of the dogbone shaped conductor pairs. The obtained results demonstrate for the first time that in the stacks of finite thickness, strong coupling between the closely spaced arrays causes the additional transmittance resonances. The comprehensive interpretation of the observed transmission resonances and underlying physical mechanisms is proposed on the basis of the simulated current distributions in the dogbone conductors.

Metamaterial Structures and Measurement Setup
The metamaterial structures studied in this paper are composed of stacked doubly periodic planar arrays of dogbone-shaped conductor pairs (Fig. 1a). The detailed parametric study in [3], [4] has revealed that such an artificial medium may exhibit negative effective refractive index (NRI) and the resonance transmittance. To examine the properties of these metamaterials experimentally, the specimens of the periodic planar arrays have been designed using the commercial simulator CST Microwave Studio. The simulations have been performed using the constitutive unit cell (Fig. 1b) of a square shape with A = B = 5.64 mm. The dogbone shaped conductors have the dimensions (in mm): A1 = 0.75, A2 = 5.55, B2 = 0.6, H = 0.381. The array substrates have been made of two laminates with slightly dissimilar parameters: (1) Taconic TLY-5 with permittivity ε r = 2.2 and loss tangent 0.0009, and (2) TLX-9 with permittivity ε r = 2.5 and loss tangent 0.0019. The use of the two laminates with significantly different loss tangents has enabled us to assess the effect of dielectric loss on the metamaterial performance.
978-1-4244-4968-2/10/$25.00 ©2010 IEEE To make the frequency of the lowest magnetic resonance mode the same for both laminate materials, the length of the dogbone lateral arms has been tailored for each substrate: B1 = 2.75 mm for TLY-5, and B1 = 2.5 mm for TLX-9. The size C, used as a variable parameter, represents in Fig. 1 either the distance between the stacked layers or the periodicity in the z-direction for an infinite medium. In multilayer arrangements the stacked individual layers are interleaved by the 1 mm thick sheets (C = 1.42 mm) of rigid foam Rohacelle TM with relative permittivity 1.05 and loss tangent 0.0008. Finally after aligning the unit cells in the stacked arrays, the assembly has been sandwiched between two Rohacelle TM slabs of thickness 12 mm each and fitted in a wooden holder as shown in Fig. 2.
The measurement setup comprises a pair of horn antennas spaced apart for about 1.6 m and the fixture with the test specimen in the middle. The holder frame size and the sample distance from antennas have been chosen such that the specimens be located in the far-field region of the horns and the square shaped arrays (67 unit cells along each side in the x0y plane) cover the entire far-zone spot illuminated by the full-angle 3 dB beamwidth of the horn antenna. The boresights of the horns have been aligned in both vertical and horizontal planes and the beam axis has pointed at the centre of the sample holder.
Since the array of dogbone pairs is a strongly anisotropic structure, it exhibits the resonance response in the specified frequency band only if the incident wave contains the x-component of the electric field, see Fig. 1. At the orthogonal polarization of the incident electric field oriented across the dogbone pairs, the array is nearly perfectly transparent. This has been confirmed by the measurements. Because in the measurement setup an incident electric field is vertically polarized, the samples have been mounted in the holder so that the xaxis of the dogbone pairs be always vertical, while the frame rotation axis be either vertical for TE-polarized or horizontal for TM-polarized incident waves.

Experimental Results and Discussion
Transmittance measurements have been carried out for a single layer array of dogbone pairs, and the stacked two and four layer arrays at different wave polarizations and incidence angles. To gain insight in the transmittance properties of the stacked arrays of finite thickness and identify the modes of the observed resonances, the experimental results have been analyzed in comparison with the simulations of the respective structures and, particularly, the current distributions on the dogbone conductor pairs in the layered arrangements.
The features of the stacked arrays of the dogbone shaped conductor pairs are illustrated by the characteristics of a four layer structure. Fig. 2 displays the transmission coefficient of the assembly comprised of the two arrays on TLY-5 substrates and the other two -on TLX-9 laminates, interleaved by 1-mm-thick Rohacell foam sheets (the layers are equally spaced and the distance between their centers is C = 1.42 mm). The measurement and simulation results show good qualitative agreement. The peaks at f 1 and f 2 correspond to the perturbed transmission resonances of the single and two layer arrays. The identity of these modes is corroborated by the respective current distributions on the dogbone pairs in Fig. 3. The peak at f 3 = 9.9 GHz represents an additional resonance occurring due to the coupling between the interlayer magnetic dipoles. Although the current patterns at f 2 and f 3 look similar in individual interlayer loops, their mutual orientations differ, viz. the current circulation has the same direction in both loops at f 3 but it is opposite in the two loops at f 2 . When the layer separation increases, the transmission resonance at f 3 is suppressed, whereas the resonance at f 2 remains practically unchanged and at f 1 the transmittance magnitude slightly decreases.
The transmission peak at f 4 = 10.9 GHz significantly differs. The current distributions in Fig. 3 show that at f 4 the adjacent individual dogbone conductors (not dogbone pairs) of different layers compose the interlayer magnetic dipoles strongly coupled with the magnetic dipoles created inside the dogbone pairs.