Elecromagnetic metamaterials
Electromagnetic metamaterials are media which are designed to interact with a propagating electromagnetic field (e.g. light) in a way which is often very different from any naturally occurring materials. For example artificial media can be designed such that the refractive index be negative, or that it depend on the direction of propagation of the light in some complex, but precisely specified way.
Light interacts with matter in two basic ways: through the electric field and through the magnetic field. Matter reacts to these fields by becoming polarized. There is an electric polarization and a magnetic polarization. These polarizations are related to the strength of the respective applied fields, and the relationship is a complicated one involving two functions known as susceptibilities. The electric and the magnetic susceptibilities are functions of position and time which encapsulate the electric and magnetic properties of a given material at each point in space. Given a field distribution and susceptibility functions one can immediately determine the polarizations which will result. The mathematical operation involved is difficult to grasp intuitively because it is not a simple multiplication between field and susceptibility; rather it is an integral known as a convolution.
There are some cases when the convolution integral reduces to a simple multiplication. These are the widely studied cases when the wavelength of the electromagnetic field propagating in the material is much longer than any of the microscopic inhomogeneities of the medium (e.g. atoms, small pores, etc.). When this is the case we speak of a homogeneous model of the material, and the susceptibilities take the form of constant rank 2 tensors.
More exotic situations have been under intense study recently however: photonic crystals, since the work of E. Yablonovitch and S. John in 1987, and composite metamaterials, since the work of J. Pendry in the late 90′s.
The considerable theoretical and experimental work following these developments managed to shed much light on the electrodynamics of inhomogeneous media. However, a gray area remained as to the all important transition region between homogeneous and inhomogeneous behavior. No theoretical tool was available for the systematic exploration of this intermediate region.
In the first chapter of my thesis I attempt to clarify these issues and propose a technique for a detailed study and physical understanding of the behavior in the transition region. It relies on the novel notion of inhomogeneous effective medium model and leads to the technologically promising concept of meta-photonic crystal.
The second chapter presents some results on the study of the super-prism effect in rectangular photonic crystals. The third chapter tests numerically some of the ideas presented in the first chapter on negative index composite metamaterials made of thin metal wires and broad-side coupled resonators. The simulations are full vector three dimensional Finite Element Method calculations done using the CST Microwave Studio© software package.
