  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.
