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We present a systematic numerical approach to compute the eigenmodes and therelated eigenfrequencies of a disordered photonic crystal, characterized bysmall fluctuations of the otherwise periodic dielectric profile. The fieldeigenmodes are expanded on the basis of Bloch modes of the correspondingperiodic structure, and the resulting eigenvalue problem is diagonalized on atruncated basis including a finite number of Bloch bands. The Bloch-modeexpansion is very effective for modeling modes of very extended disorderedstructures in a given frequency range, as only spectrally close bands must beincluded in the basis set. The convergence can be easily verified by repeatingthe diagonalization for an increased band set. As illustrations, we apply themethod 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 circularholes. We compute and characterize the eigenfrequencies, spatial field profilesand radiation loss rates of the localized modes induced by disorder. For the W1waveguide, we demonstrate that radiation losses, at the bottom of thespatially-even guided band, are determined by a small hybridization with Blochmodes of the spatially-odd band, induced by disorder in spite of theirfrequency separation. The Bloch-mode expansion method has a very broad range ofapplications: it can be also used to accurately compute the modes of structureswith systematic modulations of the periodic dielectric constant, as in severaldesigns of high-Q cavities.

Electromagnetic modes of a disordered photonic crystal