EM Models for Multisurface Metamaterial Homogenization Giovanni M. Sardi, Enrica Martini, Stefano Maci Department of Information Engineering and Mathematical Sciences University of Siena Siena, Italy macis@ing.unisi.it Abstract— This work presents a quasi-analytical approach to the homogenization of metamaterials realized by the superposition of identical periodic planar surfaces. It is based on the definition of an equivalent admittance matrix for the single layer, combined with the application of the Bloch theory for the analysis of the periodically loaded equivalent transmission line modelling the layered structure. Different theoretical definitions of equivalence are discussed in connection with practical applications of the homogenization concept. It is shown that the uniqueness of the equivalent homogeneous medium depends on the final objective of the homogenization procedure, i.e. on the specific metamaterial behavior to be mimicked. I. INTRODUCTION The continuous interest on metamaterials, started more than a decade ago, has in parallel inspired a lot of work about their modeling, and the most proper and efficient methods of analysis and design [1]-[2]. The development of a reliable and accurate approach for the modeling of the electromagnetic behavior of artificial materials is a key point to fully exploit their potential. Considerable effort has been made to represent these artificial structures as effective homogeneous media, described by a set of equivalent constitutive parameters [3]-[7]. This approach may reveal very useful in the design of metamaterial-based devices. However, the procedure for the retrieval of the equivalent parameters is not univocally defined; in fact, different techniques can be used depending on the metamaterial characteristics and on the goal of the homogenization process. In this framework, different models can be set up relying on the definition of different “homogenization equivalences”. Volumetric metamaterials can be realized by cascading a number of periodic surfaces made of patch- or slot-type elements, as shown in Fig. 1. This kind of surfaces is widely known and is indicated with different denominations depending on the application, as for instance Frequency Selective Surfaces (FSS) [8], Partially Reflecting Surfaces (PRS) and, more recently, with the new exotic denomination of MetaSurfaces [9]. This class of volumetric metamaterials is particularly interesting due to the ease of fabrication, and the wide variety of realizable electromagnetic behaviors. In this work we describe a general approach for the homogenization of these artificial materials, and we show how it can be applied to satisfy different homogenization equivalences. Fig. 1. Geometry for the multilayer structure under analysis. II. DEFINITION OF THE EQUIVALENCE MODELS The definition of “homogenization equivalence” actually depends on the final objective of the homogenization procedure, since this latter determines which characteristics of the artificial media must be also exhibited by the effective homogeneous medium. In most of the cases, the definition of the equivalent parameters is based on a scattering analysis; in some other cases, the effective homogeneous medium is required to match the dispersion properties of the metamaterial. Moreover, the capability of the homogenization model to correctly represent the structure of the supported fields could also be important. Starting from these considerations, we can define three different homogenization equivalences, namely: • External Equivalence: slabs of the artificial multilayer material and of the equivalent homogeneous medium possess the same scattering and transmission matrices as a function of frequency and wavenumber. • Dispersion Equivalence: the artificial multilayer material and the equivalent homogeneous medium admit the same solutions of the dispersion equation for the two dominant eigenmodes. • Modal equivalence: the field structure of the two modes supported by the equivalent homogeneous medium matches the one of the two dominant eigenmodes of the artificial multilayer material. In the following, a general homogenization procedure that can be adapted to match the different types of equivalence is presented.