We present in this chapter a review of some recent research work about a new
approach to the numerical simulation of time harmonic wave propagation in infinite periodic
media including a local perturbation. The main difficulty lies in the reduction of the
effective numerical computations to a bounded region enclosing the perturbation. Our objective
is to extend the approach by Dirichlet-to-Neumann (DtN) operators, well known in
the case of homogeneous media (as non local transparent boundary conditions). The new
difficulty is that this DtN operator can no longer be determined explicitly and has to be
computed numerically. We consider successively the case of a periodic waveguide and the
more complicated case of the whole space. We show that the DtN operator can be characterized
through the solution of local PDE cell problems, the use of the Floquet-Bloch
transform and the solution of operator-valued quadratic or linear equations. In our text, we
shall outline the main ideas without going into the rigorous mathematical details. The non
standard aspects of this procedure will be emphasized and numerical results demonstrating
the efficiency of the method will be presented.
