Numerical Analysis of Two-Dimensional Photonic Crystal Waveguide Devices Using Periodic Boundary Conditions

Introduction Photonic crystal is a periodic structure consisting of high contrast dielectrics, in which the electromagnetic wave cannot transmit in a specific wavelength range. It is therefore known that, if localized defects are introduced in the photonic crystal, the electromagnetic fields are strongly confined around the defects. For example, point defects in the photonic crystal work as resonance cavities and line defects work as waveguides. Also, appropriate arrangements of the defects function as photonic crystal waveguide devices, such as branching filter, resonator filter. Since any energy cannot escape through the surrounding photonic crystal, the leakage loss is suppressed. This feature may contribute to miniaturize the integrated circuit in millimeter wave, sub-millimeter wave, and optical regions. For the straight waveguides, the structure maintains the periodicity in the propagation direction, and the Floquet theorem asserts that the electromagnetic fields in the structure can be expressed by superposition of the Floquet-modes[1]. General structures of the photonic crystal waveguide devices are considered as cascade connections of the straight waveguides. Yasumoto and Watanabe[2,3] presented a numerical method to analyze discontinuities in photonic crystal waveguides. This method is based on the Fourier series expansion method (FSEM), and the fields are expressed in the Fourier series expansions by introducing artificial boundaries with periodic condition. The amplitudes of the Floquet-modes are related by a scattering-matrix (S-matrix) for each waveguide section, and S-matrix for cascade connection of waveguides is derived by a recursive calculation for each waveguide section. On FSEM, the Floquet-modes of photonic crystal waveguides are obtained by an eigenvalue analysis of the transfer matrix for the periodicity cell in the propagation direction.