We develop a homogenization expansion approach to photonic waveguides whose transverse structures are N-fold rotationally symmetric. Examples include microstructured or holey optical fibers with air holes arranged in one or more concentric rings. We carry out a homogenization expansion for large N about the limit. Our multiple scale analysis applies to the scalar approximation of structures in which the microfeatures have arbitrary geometry and large index contrasts and lead to a natural efficient computational algorithm for the waveguide modes and spectral characteristics. In this paper we focus on structures that possess leaky modes. The leading order equations describe the modes of an averaged structure. We derive an expansion in powers of of corrections to the leading order behavior and show that the leading order nontrivial contribution arises at order We numerically calculate this leading order correction to the complex effective indices (scattering resonances) for the leaky modes of various microstructured photonic waveguides whose imaginary parts give the leakage rates. We observe that in many instances a two-term truncation of the homogenization expansion gives good agreement with full simulations, even for fairly small values of N, whereas the leading order (averaged) theory yields a substantial underestimate of the leakage rates.
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