Electromagnetic wave propagation in two-dimensional photonic crystals

In this dissertation, we have undertaken the challenge to understand the unusual propagation properties of the photonic crystal (PC)---a medium with periodically modulated dielectric function. PCs have frequency regions were propagation is prohibited (gaps) and regions where propagation is allowed (bands). We first study a two-dimensional (2D) photonic crystal system in the gap region. A line defect introduces allowed states in an otherwise prohibited frequency spectrum. The dependence of the defect resonance state on the system\u27s parameters, such as the lateral width of the structure and the profile of the source, is investigated in detail. Subsequently, we examine the band properties of periodic 2D PCs, and study their unusual refractive behavior. In some cases, of anomalous refraction in PCs, the beam refracts on the wrong side of the surface normal, a phenomenon known as negative refraction. This phenomenon occurs in materials with the wave vector, the electric field, and the magnetic field forming a left-handed set of vectors---called left-handed materials (LHM) or negative index materials (NIM). We examine carefully the conditions to obtain left-handed behavior in PCs. We find with a wedge type simulation experiment, that in the PC system negative refraction is neither a prerequisite nor guarantees left-handed behavior. We identify a frequency region where the PC shows left-handed behavior and acts in some respects like a homogeneous medium with a negative refractive index. Using this realistic PC system we show how negative refraction occurs. Our findings indicate that the formation of the negatively refracted beam is not instantaneous and involves a transient time. This way, we address previous controversial issues about negative refraction concerning causality and the speed of light limit. Lastly, we systematically study anomalous refractive phenomena in PCs. We classify these different effects according to their Bragg order and type of propagation (left-handed or not). Moreover, we discuss the validity of our findings in the low index modulation PCs