We consider the Maxwell's equations with perfect electric conductor boundary conditions in three-dimensional unbounded domains which are the union of a bounded resonator and one or several semi-infinite waveguides. We are interested in the existence of electromagnetic trapped modes, that is L2 non zero solutions of the problem without source term. These trapped modes are associated to eigenvalues of the Maxwell's operator, that can be either below the continuous spectrum or embedded in it. First for homogeneous waveguides, we will present different families of geometries for which we can prove the existence of eigenvalues. Then we will show that certain non homogeneous waveguides with local perturbations of the dielectric constants can support trapped modes. Let us mention that certain mechanisms we will describe are very specific to Maxwell's equations and have no equivalent in the classical proofs of existence of trapped modes for the scalar Dirichlet or Neumann Laplacians. This is a joint work with Anne-Sophie Bonnet-Ben Dhia and Sonia Fliss.
