We consider the solution of the Helmholtz equation with absorption − u(x)−n(x)2(ω2 + ıε)u(x) = f (x), x = (x, y), in a 2D periodic medium Ω = R2. We assume that f (x) is supported in a bounded domain Ωi and that n(x) is periodic in the two directions in Ωe = Ω \ Ωi . We show how to obtain exact boundary conditions on the boundary of Ωi ,ΣS that will enable us to find the solution on Ωi . Then the solution can be extended in Ω in a straightforward manner from the values on ΣS . The particular case of medium with symmetries is exposed. The exact boundary conditions are found by solving a family of waveguide problems. © 2008 IMACS.