Enrica Martini, Giacomo Cigni, Giovanni M. Sardi, Francesco Caminita, Stefano Maci. Department of Information Engineering, University of Siena. Via Roma, 56 ...
A Systematic Approach to Determining the Constitutive Parameters of Metamaterials in the Context of Transformation Electromagnetics Enrica Martini, Giacomo Cigni, Giovanni M. Sardi, Francesco Caminita, Stefano Maci Department of Information Engineering, University of Siena Via Roma, 56, 53100 Siena, Italy {martini, sardi, caminita, macis}@dii.unisi.it Abstract — A novel systematic method for the retrieval of the constitutive electromagnetic parameters of metamaterials is presented. The analysis is focused on the important class of metamaterials consisting of a stack of planar periodic structures. A complete characterization of the effective medium is obtained by retrieving all the components of the permeability and permittivity tensors and accounting for both spatial and time dispersion. The capability of characterizing the anisotropy is especially useful in the implementation via metamaterials of the results of transformation electromagnetics, which requires a full control of all the components of the constitutive dyadic parameters. Numerical results are presented to demonstrate how the retrieved parameters can be used to obtain an accurate prediction of the scattering from a finite metamaterial slab.
I. INTRODUCTION Metamaterials exhibit macroscopic electromagnetic or optical properties that do not exist in nature and find many applications in optics and electromagnetic engineering. Recently, the concept of “transformation optics” (TO) has been introduced [1] that establishes criteria for achieving control of optical ray-paths within optically anisotropic metamaterials via coordinate trans-formations and relevant interpretations in terms of the constitutive electric and magnetic tensors of a metamaterial. This methodology has been applied, for example, to finding possible designs of “metamaterial cloaks”: shells of anisotropic inhomogeneous metamaterials capable of rendering any object within their interior cavities invisible to detection from outside. The TO concept is nowadays evolving in our community into the more general concept of “Transformation Electromagnetics” (TREM). This is associated with the general objective of finding ways of transforming an incident electromagnetic wave in a different field configuration with desired properties by using an effective medium with appropriate constitutive parameters (i.e., dielectric permittivity and magnetic permeability tensors), and finally synthesizing this effective medium through the use of metamaterials. In fact, a periodic distribution of sub-wavelength intrusions in a host homogenous medium can be macroscopically described as an equivalent homogenous structure and characterized in terms of conventional material parameters. One of the key aspects to pursue the TREM strategy is to find a systematic approach to the homogenization of a metamaterial which thoroughly accounts for anisotropy and
spatial dispersion. Indeed, a description based on simplifying assumptions like assigned polarization and wavevector direction leads to macroscopic descriptions that are adequate in some applications, but are not sufficient in the framework of the TREM strategy, where anisotropy and spatial dispersion are fundamental ingredients to address the ray-fields. This paper has the objective to fill this gap by proposing a novel systematic and non-ambiguous process of metamaterial homogenization. The effective constitutive parameters, that are retrieved starting from the solution of an accurate full wave solver, or from scattering measurements, are in general anisotropic and spatially and time dispersive. The analysis here is focused on the important class of metamaterials consisting of a stack of planar periodic structures. This class includes many metamaterials of practical interest, since artificial materials are usually fabricated using planar technology and can be therefore regarded as arrays of periodic screens. II. METHOD OUTLINE The proposed approach starts from the analysis of the plane wave scattering from a single planar periodic structure for fixed frequency and incidence direction. In the hypothesis that the interactions among the planar sheets are substantially restricted to the dominant mode, the single sheet is represented as a shunt load in the transmission line analogy and modelled through an equivalent admittance matrix (Fig. 1). Accordingly, the three-dimensional periodic structure is modelled as a periodically loaded transmission line, which is analysed through the Bloch theory to identify the propagation constant and field structure of the supported modes.
Fig. 1 Equivalent transmission line representation of a single planar periodic structure with coupling admittance matrix Y.
After imposing that the same modes are supported by an effective homogeneous medium with unknown permittivity and permeability dyads, a linear system is obtained whose solution provides the effective constitutive parameters. It is noted that this way the 3D metamaterial is completely characterized trough the analysis of a much simpler 2D structure. In principle, the analysis must be carried out for any incidence direction in order to completely characterize the periodic structure. However, a great computational saving is achieved by using the pole-zero matching technique introduced in [2] to model the dependence of the equivalent admittance matrix of the single periodic sheet on the incidence direction. III. COMPARISON WITH OTHER RETRIEVAL TECHNIQUES Different homogenization approaches have been proposed in the literature. Some of them use quasi-static approximations [4] or analytical models [3], which give insight into the relationship between the physical properties and the geometrical characteristics of the metamaterials, but are not straightforward to use in metamaterials with complicated structures. Other approaches are based on appropriate averages of local fields obtained from a full-wave electromagnetic simulation or analytic calculation within the unit cell [5]. These approaches provide accurate results when the dimension of the periodicity cell is very small compared to the wavelength, however, they do not take into account spatial dispersion, and may fail in correctly predicting the scattering from a metamaterial slab. Another commonly adopted approach is based on the analysis of the reflection from a slab of metamaterial illuminated by a plane wave impinging from a fixed direction, which is normally aligned with one periodicity axis [6]. Neither this method is appropriate for TREM applications, because it leaves an ambiguity in certain components of the constitutive tensors and does not provide information on the parameters dependence on the wavevector direction. The homogenization procedure proposed in this work can be regarded as a generalization of this latter approach, in that the starting point is the analysis of the reflection from a metamaterial slab. However, by correctly modeling space dispersion and anisotropy, it provides a complete characterization of the effective homogeneous medium and its interaction with an electromagnetic field. IV. NUMERICAL RESULTS The proposed homogenization procedure has been applied to various metamaterials consisting of a stack of planar periodic structures. The numerical results reported in this section are relevant to two different structures: in the first one the unit cell is a conducting convoluted square loop [7], while the second one is a conducting split-ring resonator (SRR). The geometry for the convoluted square loop is reported in Fig. 2; the distance between adjacent planar sheets is equal to the period in the transverse direction, L. First, the wavenumber and the characteristic impedance for the Bloch
modes supported by the three-dimensional periodic medium have been derived for different values of the transverse wavevector; then, these parameters have been used for the extraction of the metamaterial constitutive dyadics. Figures 3 and 4 show the characteristic impedance and the z-component of the wavevector of the Bloch modes for kx=kcos(θ) with θ=85°, ky=0.
Fig. 2 Geometry for the unit cell of the first periodic structure. Dimensions are as follows: w=0.1mm, S1=0.84mm, S2=0.63mm, L=6mm.
Fig. 3 Characteristic impedance of the Bloch modes for the convoluted square structure.
Fig. 4 Longitudinal propagation constant of the Bloch modes modes for the convoluted square structure.
Fig. 7 Geometry for the unit cell of the second periodic structure. Dimensions are as follows: R1=1.5mm, R2=2.5mm, s=0.2mm, c=0.4mm, L=8mm. Fig. 5 Reflection coefficient from a metamaterial slab consisting of 5 convoluted square layers (TE polarization). Comparison between full-wave analysis and effective homogeneous slab. The geometry for the problem is shown in the inset.
Fig. 6 Reflection coefficient from a metamaterial slab consisting of 5 convoluted square layers (TM polarization). Comparison between full-wave analysis and effective homogeneous slab.
In order to verify the validity of the final results, the reflection coefficient from a structure consisting of a finite number of layers (obtained through a full-wave analysis) has been compared with the reflection coefficient from an homogeneous slab having an equivalent thickness and characterized by the retrieved effective parameters. Figs. 5 and 6 show the results for the magnitude of the TE and TM reflection coefficient of a 5-layer structure illuminated by a plane wave impinging from the direction φ=0°, θ=85°. As it is apparent, a very good agreement is obtained between the two curves. A similar agreement has been found for the phases. The unit cell of the second structure is reported in Fig. 7 and Figs 8 and 9 show the characteristic impedance and the zcomponent of the wavevector of the Bloch modes for kx=kcos(θ) with θ=60°, ky=0. Finally, Figs. 10 and 11 report the results for TE and TM reflection coefficient of a 5-layer structure illuminated by a plane wave impinging from the direction φ=0°, θ=60°. Also in this case, a very good agreement is obtained between the two curves.
Fig. 8 Characteristic impedance of the Bloch modes for the split ring resonator structure.
Fig. 9 Longitudinal propagation constant of the Bloch modes for the split ring resonator structure.
ACKNOWLEDGMENT This work is co-financed by U.S. Army Research Laboratory (ARL) through USAITC-A. We thank Steve Weiss for suggesting the research and Neil Vallestero for providing the funding vehicle. REFERENCES [1] [2]
[3]
[4] Fig. 10 Reflection coefficient from a metamaterial slab consisting of 5 split ring resonator layers (TE polarization). Comparison between full-wave analysis and effective homogeneous slab. The geometry for the problem is shown in the inset.
[5] [6]
[7]
Fig. 11 Reflection coefficient from a metamaterial slab consisting of 5 split ring resonator (TM polarization). Comparison between full-wave analysis and effective homogeneous slab.
V. CONCLUSIONS A novel homogenization approach is proposed for the retrieval of the effective parameters of metamaterials consisting of a stack of planar periodic structures. The proposed procedure is systematic and computationally efficient, since it is based on the numerical analysis of a single periodic sheet. Furthermore, by correctly modeling space dispersion and anisotropy, it is able to accurately predict the reflection from a finite metamaterial slab for arbitrary incidence direction and polarization. The method can be generalized to an arbitrary threedimensional periodic structure at the expense of an increased computational complexity. In the general case, the modal structures supported by the effective homogeneous medium can be derived through the full-wave analysis of the periodic structure after applying a proper averaging to extract the dominant Floquet mode contributions.
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