Exact boundary conditions for time-harmonic wave propagation in locally perturbed periodic media

We consider the solution of the Helmholtz equation with absorption -Delta u(x) - n(x)(2)(omega(2) + l epsilon)u(x) = f(x), x = (x, y), in a 2D periodic medium Omega = R(2). We assume that f(x) is supported in a bounded domain Omega(i) and that n(x) is periodic in the two directions ill Omega(e) = Omega \ Omega(i). We show how to obtain exact boundary conditions on the boundary of Omega(i), Sigma(S) that will enable us to find the solution on Omega(i). Then the solution can be extended in Omega in a straightforward manner from the values on Sigma(S). The particular case of medium with symmetries is exposed. The exact boundary conditions are found by solving a family of waveguide problems. (C) 2008 IMACS. Published by Elsevier B.V. All rights reserved.