A theoretical investigation of one-dimensional planar photonic crystal waveguides is carried out. These waveguides consist of a dielectric layer sandwiched between two semiinfinite periodic dielectric structures. Using a novel approach, exact analytical solutions for guided modes in such waveguides are presented. The se rigorous solutions allow one to distinguish clearly between the index-guiding regime and guidance due to the photonic crystal effect.
In the first part of this investigation, a rigorous analysis of the reflection of uniform plane waves from a semi- infinite periodic dielectric structure is undertaken. Both parallel and perpendicular polarizations for the incident plane wave are considered. Exact expressions for the reflection coefficients corresponding to two polarization cases are obtained using an impedance approach.
The results for the reflection coefficient are then used to study propagation properties of guided modes in one-dimensional photonic crystal waveguides with semiinfinite periodic cladding regions. Characteristic equations, from which propagation constants of guided modes can be obtained, and solutions for electromagnetic fields of these modes are derived. Solutions for both TE (transverse electric) and TM (transverse magnetic) modes are presented. Numerical results for the propagation constant and field distributions of several lower-order modes are presented. The solutions unique to photonic crystal waveguides are emphasized.
