During the past few decades applications of photonic crystals have been rapidly expanding. Photonic crystals are periodic dielectric or metallic structures that are designed to control and manipulate the propagation of light. As in normal crystal lattice where the periodic arrangement of atoms lead to energy band-gap, the periodic distribution of dielectric or metallic inclusions creates frequency ranges in which propagation of electromagnetic waves is rohibited. It means the wave whose frequency falls within the band gap, regardless of its wave number, exponentially decays in photonic crystal. The properties which control the behavior of light become photonic crystal be widely used in future optical circuits and devices such as modulators, optical couplers, semi-conductor lasers, and filters and so on. The aim of this thesis is to specify the properties and band diagram of photonic waveguides. At first the characteristics of two dimensional (2D) photonic crystals in the polarization of TE and TM are obtained. It is denoted that the band structure depends on the specific geometry and composition of the photonic crystal such as the lattice size, the shape and size of lattice elements, and the contrast in refractive index. Various analytical or computational techniques have been developed to formulate the electromagnetic scattering, guiding, and coupling problems in periodic structures. We apply Plane-Wave Expansion (PWE) method to modeling the properties of photonic crystals and related optical devices. In PWE method, Maxwell's equations are transformed into the eigenvalue problem from which dispersion characteristics of the periodic structure are extracted. Calculating photonic band structures of periodic systems containing dispersive and highly absorptive material which has frequency dependent dielectric function with or without imaginary part, led to the generalize eigenvalue problem. To calculate the photonic band structure of these systems we employ a standard linearization technique which solves the general nonlinear eigenvalue problem by diagonalization of an equivalent, enlarged matrix. The use of complex dielectric function results in complex frequencies that the imaginary part determines the lifetime of the wave. Even though 3D photonic crystals can have complete band gaps and can confine light in all three dimensions but the hardship of their fabrication led to use of simple struct Keywords: Photonic crystal, Photonic Band Gap, Plane-Wave Expansion (PWE), Supercell.
