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In this paper we consider the propagation of waves in a closed full- or half-waveguide where the index of refraction is periodic along the axis of the waveguide. Motivated by the limiting absorption principle, proven in the Appendix by a functional analytic perturbation theorem, we formulate a radiation condition which assures uniqueness of a solution and allows the existence of propagating modes. Our approach is quite different to the known one as, e.g., considered in [6] and allows an extension to open wave guides (see [10]). After application of the Floquet-Bloch transform we consider the Floquet-Bloch variable α as a parameter in the resulting quasi-periodic boundary value problem and study the behaviour of the solution when α tends to an exceptional value by a singular perturbation result which we have found in [4].