Resolvent method for computations of localized defect modes of H-polarization in two-dimensional photonic crystals

We have developed and tested a version of the resolvent (or Green's function) method, based on the shift-inverse of the Maxwellian operator, that ensures stable convergence of iterative computations of localized defect modes of H-polarization in two-dimensional (2D) photonic crystals. The defect states are obtained by solving the eigenvalue problem for an associated compact operator with the expansion in Bloch eigenfunctions of the unperturbed Maxwellian operator. This method can be extended to 3D photonic crystals. We apply the method to a 2D square lattice of square dielectric rods in a dielectric background and compute (with controlled precision of approximately 1%) the defect modes induced by the replacement of one rod (the defect). We investigate the rise and variation of the defect frequencies in a photonic band gap, caused by the increase of the dielectric strength of the defect, for four branches of localized modes of various symmetries.