  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.
