<p>As research in electromagnetics has expanded, it has given rise to the examination of metamaterials, which possess nontrivial electromagnetic material properties such as engineered permittivity and permeability. Aside from their application in the microwave industry, metamaterials have been associated with novel phenomena since their invention, including sub-wavelength focusing in negative refractive index slabs, evanescent wave amplification in negative index media, and invisibility cloaking and its demonstration at microwave frequency with controlled material properties in space.</p><p>Effective medium theory plays a key role in the development and application of metamaterials, simplifying the electromagnetic analysis of complex engineered metamaterial composites. Any metamaterial composite can be treated as a homogeneous or inhomogeneous medium, while every unit structure in the composite is represented by its permittivity and permeability tensor. Hence, studying an electromagnetic wave's interaction with complex composites is equivalent to studying the interaction between the wave and an artificial material.</p><p>This dissertation first examines the application of a magnetic metamaterial lens in wireless power transfer (WPT) technology, which is proposed to enhance the mutual coupling between two magnetic dipoles in the system. I examine and investigate the boundary effect in the finite sized magnetic metamaterial lens using a numerical simulator. I propose to implement an anisotropic and indefinite lens in a WPT system to simplify the lens design and relax the lens dimension requirements. The numerical results agree with the analytical model proposed by Smith et al. in 2011, where lenses are assumed to be infinitely large.</p><p>By manipulating the microwave properties of a magnetic metamaterial, the nonlinear properties come into the scope of this research. I chose split-ring resonators (SRR) loaded with varactors to develop nonlinear metamaterials. Analogous to linear metamaterials, I developed a nonlinear effective medium model to characterize nonlinear processes in microwave nonlinear metamaterials. I proposed both experimental and numerical methods here for the first time to quantify nonlinear metamaterials' effective properties. I experimentally studied three nonlinear processes: power-dependent frequency tuning, second harmonic generation, and three-wave mixing. Analytical results based on the effective medium model agree with the experimental results under the low power excitation assumption and non-depleted pump approximation. To overcome the low power assumption in the effective medium model for nonlinear metamaterials, I introduced general circuit oscillation models for varactor/diode-loaded microwave metamaterial structures, which provides a qualitative prediction of microwave nonlinear metamaterials' responses at relatively high power levels when the effective medium model no longer fits.</p><p>In addition to 1D nonlinear processes, this dissertation also introduces the first 2D microwave nonlinear field mapping apparatus, which is capable of simultaneously capturing both the magnitude and phase of generated harmonic signals from nonlinear metamaterial mediums. I designed a C-band varactor loaded SRR that is matched to the frequency and space limitation of the 2D mapper. The nonlinear field generation and scattering properties from both a single nonlinear element and a nonlinear metamaterial medium composite are experimentally captured in this 2D mapper, and the results qualitatively agree with numerical results based on the effective medium model.</p> / Dissertation

The present research is concerned with developing analysis methods for analyzing and exploring finite-amplitude elastic wave propagation through periodic media. Periodic arrangements of materials with high acoustic impedance contrasts can be employed to control wave propagation. These systems are often termed phononic crystals or metamaterials, depending on the specific design and purpose. Design of these systems usually relies on computation and analysis of dispersion band structures which contain information about wave propagation speed and direction. The location and influence of complete (and partial) band gaps is a particularly interesting characteristic. Wave propagation is prohibited for frequencies that correspond to band gaps; thus, periodic systems behave as filters, wave guides, and lenses at certain frequencies. Controlling these behaviors has typically been limited to the manufacturing stage or the application of external stimuli to distort material configurations. The inclusion of nonlinear elements in periodic unit cells offers an option for passive tuning of the dispersion band structure through amplitude-dependence. Hence, dispersion analysis methods which may be utilized in the design of nonlinear phononic crystals and metamaterials are required. The approach taken herein utilizes Bloch wave-based perturbation analysis methods for obtaining closed-form expressions for dispersion amplitude-dependence. The influence of material and geometric nonlinearities on the dispersion relationship is investigated. It is shown that dispersion shifts result from both self-action (monochromatic excitation) and wave-interaction (multi-frequency excitation), the latter enabling dynamic anisotropy in periodic media. A particularly novel aspect of this work is the ease with which band structures of discretized systems may be analyzed. This connection enables topology optimization of unit cells with nonlinear elements. Several important periodic systems are considered including monoatomic lattices, multilayer materials, and plane stress matrix-inclusion configurations. The analysis methods are further developed into a procedure which can be implemented numerically with existing finite-element analysis software for analyzing geometrically-complex materials.