Effects of disorder on wave propagation in two-dimensional photonic crystals

The electromagnetic transmittance of disordered two-dimensional photonic crystals composed of circular cylinders is investigated as a function of wavelength and polarization. At short wavelengths, the transmittance shows a band structure similar to that found in the optical absorption spectrum of amorphous semiconductors, with impurity states increasingly appearing on the long wavelength side of the band gaps as the degree of disorder is increased. In the long-wavelength limit, Anderson localization of waves is found, provided that the wavelength is not so large that the random photonic crystal can be viewed as homogeneous. The localization properties in this regime are studied and an analytic expression for the dependence of the localization length on wavelength is derived. In the limit of extremely long wavelengths, the system homogenizes and can be replaced by an equivalent one with uniform effective refractive index, whose form is derived for both polarizations. Analysis of the crossover between localization and homogenization is also presented.