Complex photonic band diagrams for finite-size photonic crystals with arbitrary defects

Based on an effective propagation constant and a frequency-dependent dielectric constant, the plane wave method is extended to calculate the complex photonic band diagram and the density of states for electromagnetic waves propagating in a two-dimensional finite structure with nonoverlapping dielectric cylinders of arbitrary radii, which may be randomly embedded in a dielectric matrix. The effective propagation constant is obtained by using the Twersky formula [J. Math. Phys. 3, 700 (1962)] and the scattering matrix method. A case study shows that an embedded defect with optimal radius in a finite photonic crystal can increase the first photonic band gap as compared to the similar structure without the defect. This work is expected to provide a useful tool in determining the photonic properties of a disorder dielectric medium with arbitrary embedded nanoparticles and nanowires.