| A full vectorial finite element analysis is presented for the design and analysis of photonic crystal structures. Finite element C++ class libraries are developed based on the vector formulation of the two and three dimensional wave equation. Whitney 1-forms, often called edge elements, are used as basis functions to avoid spurious modes in eigenanalyses. The current finite element codes can solve 2-D and 3-D eigenvalue and scattering problems with boundary conditions: the perfect electric conductor (PEC), the perfect magnetic conductor (PMC) and the Bloch boundary condition. Open boundary problems can also be solved by implementing the perfectly matched layers (PML). Eigenanalyses are performed for various types of photonic crystal structures such as unit cells, infinite waveguides and defect cavities. The transmission spectra of the photonic crystal guiding structures, straight waveguides, waveguide bends and waveguide branches, are derived using scattering formulation. Experimental verification is also presented for a single and five missing line photonic crystal waveguides. Based on the calculated transmission spectra, we conducted simulated annealing optimization of branches and bends to increase the transmission. We applied previous results to a Mach-Zehnder type optical interferometer. The design of waveguide arms is modified to increase the sensitivity. Change of lattice constant gives rise to the shift of the waveguide band. Thus, the operating frequency can be moved to the bandedge which exhibits more dispersive characteristics. We also investigated the coupled-resonator optical waveguide (CROW) structures for the same purpose. By inserting the defect air holes in the waveguide channel, the shape and the frequency range of the band can be engineered. The increase in the sensitivity of the CROW is analyzed by varying the radii of the defect air holes in the waveguide channel. Also group velocities and their dispersion characteristics are investigated and its use on delay lines and dispersion compensators are proposed. |
