Analysis of active and passive metamaterials

Abstract

Metamaterials are composite materials acting as effectively continuous media, which are capable of realizing entirely new phenomena such as negative refraction and transformation optics. This class of new materials may therefore in principle make available concepts such as the perfect lens and invisibility cloaks. These materials are thus promised to achieve electromagnetic properties which most certainly go beyond materials found in nature, or ever encountered before, except from perhaps in science fiction. Such media may in general be built up of both active and passive components. New physical phenomena may require new physical models. A thorough investigation should therefore be made regarding the well established mathematical models from earlier research on electromagnetic properties of continuous, as well as structured media. This thesis contributes in various ways to the development of a solid theoretical framework, which can be used in the analysis of such novel materials. Two lines of inquiry are followed. The first part of the thesis considers isotropic, possibly active media, which are described by given material parameters; a permittivity (ω) and a permeability μ(ω). A mathematical framework based on Fourier-Laplace analysis is introduced for representing the response from such media in terms of (possibly complex) frequency- and transversal wavenumber components. The idealization of monochromatic plane waves may for active systems be dangerous due to the presence of growing waves, and the possibility of approaching this limit is investigated by deforming the integration surface in complex frequency-complex wavenumber space. We also give the most general criterion for absolute instabilities. The general theory is used to analyze example media with weak or strong gain. In particular it is used to show the existence of isotropic media which in principle exhibit simultaneous refraction, meaning they refract positively and negatively at the same time. In the second part, the importance of spatial dispersion in metamaterials consisting of passive components is considered. We discuss the importance of higher order multipole terms in homogenization theories for metamaterials. While it is common to include polarization, magnetic dipole and perhaps electric quadrupole terms, it is shown that certain higher order terms are generally significant when second order spatially dispersive effects (e.g. magnetism) are concerned. Based on this notion it is not necessarily clear how the magnetic permeability should be defined. We therefore state and compare four different definitions of the permeability, and analyze their properties in general. As a further investigation of their physical relevance we compare how well the parameters predict the reflection from semi-infinite periodic metamaterial structures. The predictions are based on the Fresnel equation using these four permeabilities. It is found that the Fresnel equation gives accurate results for 2D metamaterials which mimic natural magnetism, in a frequency range where the magnetic moment dominates the O(k2) part of the total Landau-Lifshitz permittivity. For a 1D layered structure, or for large frequencies, the correspondece is poor.