Existence of guided waves due to a lineic perturbation of a 3D periodic medium

In this note, we exhibit a three dimensional structure that permits to guide waves. This structure is obtained by a geometrical perturbation of a 3D periodic domain that consists of a three dimensional grating of equi-spaced thin pipes oriented along three orthogonal directions. Homogeneous Neumann boundary conditions are imposed on the boundary of the domain. The diameter of the section of the pipes, of order $\epsilon$ \textgreater{} 0, is supposed to be small. We prove that, for $\epsilon$ small enough, shrinking the section of one line of the grating by a factor of $\sqrt$ $\mu$ (0 \textless{} $\mu$ \textless{} 1) creates guided modes that propagate along the perturbed line. Our result relies on the asymptotic analysis (with respect to $\epsilon$) of the spectrum of the Laplace-Neumann operator in this structure. Indeed, as $\epsilon$ tends to 0, the domain tends to a periodic graph, and the spectrum of the associated limit operator can be computed explicitly.