Scattering by a bounded highly oscillating periodic medium and the effect of boundary correctors

We study the homogenization of a transmission problem arising in the scattering theory for bounded inhomogeneities with periodic coefficient in the lower-order term of the Helmholtz equation. The squared index of refraction is assumed to be a periodic function of the fast variable, specified over the unit cell with characteristic size ϵ. We obtain improved convergence results that assume lower regularity than previous estimates (which also allow for periodicity in the second-order operator), and we describe the asymptotic behavior of boundary correctors for general domains at all orders. In particular we show that, in contrast to Dirichlet problems, the O(ϵ) boundary corrector is nontrivial and can be observed in the far field. We further demonstrate the latter far field effect is larger than that of the \bulk" corrector|the so-called periodic drift, which is found to emerge only at O(ϵ2). We illustrate the analysis by examples in one and two spatial dimensions.