A Hermitian eigenvalue equation converted from Maxwell equations is numerically solved by using the plane-wave method to determine the eigenstates of two-dimensional photonic crystal structures with defects. To enhance computational efficiency, fast Fourier transform and preconditioned conjugate gradient method are employed. By considering the convergence properties of the plane-wave method, the eigen-frequency of a defect mode in an appropriate supercell is obtained within an accuracy of 0.2%.
A square lattice of dielectric columns embedded in air is analyzed. Localized modes of point defects, line defects, and surface defects are calculated and discussed. A point defect and a line defect can act as a micro-cavity and a waveguide, respectively. It is also confirmed that more complicated defect structures such as a tight bending waveguide and a ring resonator can be studied by using the plane-wave method with supercells.
