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

We consider the solution of the Helmholtz equation with absorption −Δu(x)−n2(x)(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.