Localised modes due to defects in high contrast periodic media via homogenization

The spectral problem for an infinite periodic medium perturbed by a compact defect is considered. This may be seen for example as a simplified scalar cross-sectional model of the problem for localised modes in photonic crystal fibers. For a high contrast small size periodicity and a finite size defect we consider the critical (the so called double poros-ity type) scaling. We employ (high contrast) homogenization for deriving asymptotically explicit limit equations for the localised modes (exponentially decaying eigenfunctions) and associated eigenvalues. Those are expressed in terms of the eigenvalues and eigenfunctions of a perturbed version of a " two-scale " limit operator introduced by V.V. Zhikov, with an emergent explicit nonlinear dependence on the spectral parameter for the spectral problem at the macroscale. Using the method of asymptotic expansions supplemented by a " high contrast " boundary layer analysis we establish the existence of the actual eigenvalues near the eigenvalues of the limit operator, with tight " ε square root " error bounds (ε is the small parameter). An example for circular or spherical defects in a periodic medium with isotropic homogenized properties is given and displays explicit limit eigenvalues and eigenfunctions. Further results on improved convergence of eigenfunctions via the technique of strong two-scale re-solvent convergence and associated two-scale compactness properties are discussed.