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 (ω² + i{dotless} ε) 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 Ω^e = Ω {set minus} Ωⁱ. We show how to obtain exact boundary conditions on the boundary of Ωⁱ, Σ_S 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 Σ_S. The particular case of medium with symmetries is exposed. The exact boundary conditions are found by solving a family of waveguide problems.