THEORETICAL ANALYSIS OF THE EFFECTIVE PARAMETERS OF ROD METAMATERIALS

Abstract

This thesis is dedicated to theoretical characterizing of two component metamaterials as arrays of metal rods/wires periodically immersed in a dielectric isotropic matrix. The rods material is supposed to be non-magnetic. Only a circular cross section case of the rods is considered in the thesis while the rods do not touch each other. At the same time, there is no restriction to the radius of cross section of the rods. The microwave frequency range (from 0 to 5 GHz) has only been considered in this study. The metamaterial media/structures are being considered in the thesis as artificial semiconductors with their own effective complex dielectric and magnetic constants. So, the presented metamaterial media/structures in the thesis are considered as perfect crystals with their own dispersive properties studied in the microwave frequency range. The lattice constant of the crystal is equal to the constant of unit cell of metamaterial under consideration. The characterizations of considered metamaterial structures in the thesis are identified with the study of properties of the effective dielectric and magnetic constants as functions of the frequency (in the GHz frequencies) of incident electromagnetic wave and the volume fraction of metal rods in the unit cell. The above characterizing is the key for defining unusual material properties unavailable in real nature: enhancement of the effective parameters; a possibility to get negative values of the effective parameters; ultra-low values of the refractive index. Throughout the thesis, we consider the initial plane electromagnetic wave that is normally incident to the flat boundaries of the chain. The wave has the magnetic induction vector parallel to the axes of rods while the electric intensity vector is perpendicular to the ones. The above effective complex dielectric and magnetic constants have been obtained for the first time by author of the thesis on base of extension of the case of 2-D infinite metamaterial medium to a slab metamaterial scatterer under consideration through the implementation of the Effective Medium Theory (EMT) in appropriate frequency range. The expressions of the appropriate effective constants for the infinite medium are obtained by other authors. These expressions of the effective constants obtained in this thesis takes in account multipole effects for the case of composite with a very small value of the rods volume fraction while dipole effects are taken in account for the case of large volume fraction values. The accuracy of obtained mathematical models was always benchmarked through a comparison with numerical calculations obtained via the implementations of Finite-Domain Time-Difference (FDTD) method for calculating S-parameters of a metamaterial structure under considerations. S-parameters were used to calculate the effective constants by means of the using Nicolson-Ross approach. All of the numerical experiments presented in the thesis have been carried out with the help of the free Meep FDTD software package while analytical modeling has been done using MatLab software. In this thesis, an improved broadband method for determining complex effective refractive index, dielectric and magnetic constants of an arbitrary passive metamaterial has been proposed. Evaluation of the effective parameters is realized using the reflection-transmission S-parameters obtained by simulation or experimental measurements and analytically evaluated interface reflection coefficient of the slab. In consideration of practical party of this thesis, the obtained qualitative and quantitative results in this thesis have allowed to formulating some properties of two component slab metamaterial structures as arrays of metal rods/wires periodically immersed in a dielectric isotropic matrix: 1. The effective electromagnetic properties of infinite 2-D array of copper cylinders immersed in metal-dielectric matrix in the GHz frequencies shows the existence of the enhancement of effective dielectric constant and low absorption in the microwave frequencies. 2. The obtained analytical models of the composite in the thesis gives a good qualitative but a weak quantitative correlation with results of numerical simulations in the case if cylinders touch each other. 3. The above analytical models of infinite metamaterial medium quantitatively describes well the slabs embedded with the above metamaterial medium if there is some relation between the width of slabs and the dimension of unit cell of the metamaterial medium for appropriate frequency range. 4. The considered artificial material medium can be used to increase the directivity of patch antenna and to obtain ULI structures in the GHz frequency range and to design a new type of waveguides. 5. The obtained mathematical models cannot reveal negative values of the effective dielectric and/or magnetic constants (their real parts) in the GHz frequency range. The main theoretical results of this thesis can be presented by two theoretical methods of characterizing of any 2-D slab metamaterial structures in the microwave frequencies via EMT approach: 1. Non-destructive broadband method for the evaluation of the effective complex dielectric and magnetic parameters of 2-D slab metamaterials. 2. The analytical and numerical optimization method for separating a slab metamaterial into its elementary sub-slabs of the order of the unit cell dimension of the slab. It is important to mention that the above methods have been designed irrespective to the shape of inclusion in the unit cell. Moreover these methods allow to easy evaluating the optical and transport properties of slab metamaterial structures including magnetic ones trough using the relation for the total reflection and transmit ion coefficients and the above constants of single layer. The obtained results in this thesis are in a good quantitative and qualitative agreement with the results of experimental research carried out earlier by one of the supervisors. Moreover these results can be used for creating the course of laboratory works with the using of personal computers for students of Engineer and Sciences directions (Industrial Mathematics, Theoretical Physics, Electrical & Electronic Engineering, Material Science) to study the optical and transport properties of slab metamaterial structures in the microwave frequency range.