|Open periodic waveguides play an important role in applications, in particular in optics.
We investigate how a purely geometric perturbation along a line of a periodic domain
occupied by a homogeneous medium can create guided waves.
This amounts to solving a self-adjoint eigenvalue problem in an unbounded medium. The
guided modes correspond to eigenvalues in the spectral gaps of the unperturbed operator.
Using asymptotic analysis, we shall show how to derive existence results of such eigenvalues for thin propagation domains that degenerate into perturbed periodic graphs at the small thickness limit. We shall present a numerical method for computing such guided waves and provide various numerical results.