NUMERICAL MODELS FOR THE PHASE SHIFT OF THE PERMITTIVITY AND PERMEABILITY FOR MAGNETO- ELECTRIC LATTICE IN METAMATERIALS H.M. Katiniotis 1 , I.S. Karanasiou 2 , E.A. Karagianni 3 , H.E. Nistazakis 1 , A.D. Tsigopoulos 3 and M.E. Fafalios 3 1: Department of Electronics, Computers, Telecommunications and Control, Faculty of Physics, National and Kapodistrian University of Athens, Athens, 15784, Greece, e-mails: ckatiniotis@yahoo.gr; enistaz@phys.uoa.gr 2: National Technical University of Athens, Institute of Communications and Computer Systems, Iroon Polytechniou, Politechnioupolis, Zographou, Greece, e-mail: ikaran@esd.ece.ntua.gr 3: Department of Battle Systems, Naval Operations, Sea Studies, Navigation, Electronics and Telecommunications, Hellenic Naval Academy, Hadjikyriakou ave, Piraeus 18539, Greece, e-mail:{evka;atsigo;fafalios}@hna.gr ABSTRACT Metamaterials are artificially made electromagnetic materials consisting of periodically arranged metallic elements which are less than the wavelength of the incident wave in size. In this work, the analytical models for the description of metamaterials are studied and simulated. For the one dimensional (1D) homogeneous metamaterial slab, the mathematical terms of transfer and scattering matrices are introduced. We discuss the behavior of the system in presence of spatial dispersion. INDEX TERMS Metamaterials, Dispersion, negative permittivity, permeability, refractive index, transmission line, Thin Metallic Wire, Split Ring Resonator I. INTRODUCTION ave propagation with simultaneously negative values for ε and μ was discussed and analyzed by Veselago [1]. He called those substances Left Handed (LH) because they allow the propagation of electromagnetic waves with vectors   , ,    EMk creating a left-handed triad. According to Veselago some of the fundamental properties of the (LH) medium are the frequency dispersion of constitutive parameters, the reversal of Doppler effect, the reversal of Vavilov Cerenkov radiation, the reversal of Snell’s law and the subsequent negative refraction at the interface between a Right Handed (RH) and a (LH) medium. Smith et al [2] realized practically for the first time a composite medium based on a periodic array of interspaced conducting non magnetic split ring resonators and continuous wires with simultaneously negative values of effective permeability and permittivity. Before Smith, Pedry had studied [3,4] negative –ε / positive –μ structure (TW) and positive -ε/negative –μ structure (SRR) in the microwave range. After the first experimental demonstration of the (LH) structure by Smith, a large number of both theoretical and experimental reports regarding the characterization of (TW-SRR LH) medium confirmed the (LH) nature of these structures [5,6,7]. In this work we introduce and discuss briefly the concepts of transfer and scattering matrices and we derive the expression of dispersion equation for a periodic medium which is consisted of thin polarized slabs. For this medium we analyze and simulate the mathematical equations which describe the characteristic parameters, such as permittivity and permeability in the presence of spatial dispersion. We study different cases which apply to lattices with electric, magnetic, magnetoelectric and strongly magnetoelectric responses. For each case we ensure that the product kd<1 but positive, where d is the distance between slabs (in m) shown in fig.1 and k is the wave vector (in radians per meter). Next we simulate the phase advance, the permittivity and the permeability of the system as a function of frequency, for slab thicknesses of 0,02 and 0,01cm respectively and a resonant frequency of 10GHz. The distance d between two neighboring slabs is chosen to be between 0,4 – 0,1 cm. II. TRANSFER AND SCATTERING MATRIX MODEL In this approach, we consider the metamaterial as a periodic system of electrically and magnetically polarized slabs [8]. We assume that the electromagnetic wave propagates in the direction along the normal to the sheets, with the fields polarized in the plane of the sheets. The geometry of this model is presented in Fig.1. The transfer matrix T relates the fields on one side of the slab to the other. Since we consider the one- dimensional (1D) case, its dimensions are 2x2. W