  The spectral problem for an infinite periodic medium perturbed by a compact
defect is considered. For a high contrast small $\ve$-size periodicity and a
finite size defect we consider the critical $\ve^2$-scaling for the contrast.
We employ (high contrast) two-scale 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 "$ \ve$
square root" error bounds. 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
error bounds for the eigenfunctions are discussed, by combining our results
with those of M. Cherdantsev ({\it Mathematika. 2009;55:29--57}) based on the
technique of strong two-scale resolvent convergence and associated two-scale
compactness properties.
