Disordered photonic crystals understood by a perturbation formalism

Photonic band gaps in disordered two-dimensional photonic crystals are investigated for two typical types of randomness: cylinder site displacements (site randomness) and cylinder radius variations (size randomness). The plane-wave expansion method with a supercell technique is applied to calculate the density-of-states (DOS) for the disordered photonic crystals. In particular, numerical simulations on the DOS for square and triangular lattices of dielectric cylinders in air with the $E$-polarization mode show that photonic band gaps are far more sensitive to disorders with a size randomness than with a site randomness. The first and second band gaps both reduce very little even for a site randomness of a strength as large as half the cylinder radius, yet they reduce more than one-half for a size randomness of a strength about one-third the cylinder radius. This substantial contrast can be understood by the analysis of the electromagnetic fields in disordered crystals. Based on such a field analysis, a perturbation formalism is proposed for disordered crystals and it accords well with the DOS calculations for a site randomness of even a moderate strength. At very weak size randomness, the perturbation method also works well to some extent. Such a simple perturbative analysis should provide a systematic way to understand various disordered photonic crystals qualitatively and even semiquantitatively.