Electromagnetic modes of a disordered photonic crystal
We present a systematic numerical approach to compute the eigenmodes and the related eigenfrequencies of a disordered photonic crystal, characterized by small fluctuations of the otherwise periodic dielectric profile. The field eigenmodes are expanded on the basis of Bloch modes of the corresponding periodic structure, and the resulting eigenvalue problem is diagonalized on a truncated basis including a finite number of Bloch bands. The Bloch-mode expansion is very effective for modeling modes of very extended disordered structures in a given frequency range, as only spectrally close bands must be included in the basis set. The convergence can be easily verified by repeating the diagonalization for an increased band set. As illustrations, we apply the method to the W1 line-defect waveguide and to the L3 coupled-cavity waveguide, both based on a photonic crystal slab with a triangular lattice of circular holes. We compute and characterize the eigenfrequencies, spatial field profiles, and radiation loss rates of the localized modes induced by disorder. For the W1 waveguide, we demonstrate that radiation losses, at the bottom of the spatially even guided band, are determined by a small hybridization with Bloch modes of the spatially odd band, induced by disorder in spite of their frequency separation. The Bloch-mode expansion method has a very broad range of applications: It can be also used to accurately compute the modes of structures with systematic modulations of the periodic dielectric constant, as in several designs of high-Q cavities.