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Fourier-Bessel functions are used to expand the field and dielectric in the 2D cylindrical Maxwell's equations recasting their solutions into eigenvalue problems.  The eigenvalues obtained are the localized states of the given structure with their density indicating optical band gaps.  The technique is shown for both TE and TM polarizations within a triangular lattice photonic crystal.  The accuracy and efficiency of the results are compared relative to finite difference techniques.

Photonic Crystals, Photonic Quasi-Crystals