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

We consider the solution of the Helmholtz equation with absorption − Δ u ( x ) − n ( x ) 2 ( ω 2 + ı ε ) u ( x ) = f ( x ) , x = ( x , y ) , in a 2D periodic medium Ω = R 2 . We assume that f ( x ) is supported in a bounded domain Ω i and that n ( x ) is periodic in the two directions in Ω e = Ω ∖ Ω i . We show how to obtain exact boundary conditions on the boundary of Ω i , Σ S that will enable us to find the solution on Ω i . Then the solution can be extended in Ω in a straightforward manner from the values on Σ S . The particular case of medium with symmetries is exposed. The exact boundary conditions are found by solving a family of waveguide problems.