Transparent boundary conditions for locally perturbed infinite hexagonal periodic media

In this paper, we propose a strategy to determine the Dirichlet-to-Neumann (DtN) operator for infinite, lossy, and locally perturbed hexagonal periodic media. We obtain a factorization of this operator involving two nonlocal operators. The first one is a DtN type operator and corresponds to a half-space problem. The second one is a Dirichlet-to-Dirichlet (DtD) type operator related to the symmetry properties of the problem. The half-space DtN operator is characterized via the Floquet-Bloch transform, a family of elementary strip problems, and a family of stationary Riccati equations. The DtD operator is the solution of an affine operator valued equation which can be reformulated as a nonstandard integral equation.