Propagation and scattering of lightwaves in cylindrical and spherical periodic structures

Abstract

In the last two decades, photonic crystals (PhCs) have attracted much attention because of their amazing ability to manipulate and control light. Typical PhCs are periodic structures with unit cells containing circular cylinders, spheres or other simple geometries. To understand the basic physical properties of a PhC and to design PhC devices for various applications, efficient numerical methods are needed. Mathematically, we encounter eigenvalue problems and boundary value problems. Recently, various two-dimensional PhC structures with cylindrical inclusions have been analyzed using efficient numerical methods that rely on cylindrical wave expansions and the Dirichlet-to-Neumann (DtN) or Neumann-to-Dirichlet (NtD) maps of the unit cells. So far, the DtN or NtD map method has only been developed for PhCs composed of isotropic materials. In this thesis, we extend the DtN map method to anisotropic PhCs, based on cylindrical wave expansions for circular cylinders of anisotropic media. For three-dimensional PhC structures with spherical inclusions, there exists a few accurate numerical methods based on spherical wave expansions. In this thesis, we present an improved spherical wave least squares method for calculating transmission and reflection spectra of periodic arrays of spheres. The electromagnetic fields inside and outside the periodic arrays are approximated by vector spherical waves and plane waves, respectively, and they are matched at the interfaces in the least squares sense. Finally, we extend the spherical wave least squares method to infinite and periodic linear chains of dielectric spheres and calculate travelling electromagnetic waves around the chains. Keywords: Photonic crystals, Anisotropic media, Dirichlet-to-Neumann map, Least squares method, Spherical scatterer.