[en] This second part of a pair of papers complements the first part (see Kirsch 2018 (35 104004)) but can be read independently. Scattering of time-harmonic waves from periodic structures at some fixed real-valued wave number becomes analytically difficult whenever there arise surface waves: These non-zero solutions to the homogeneous scattering problem physically correspond to modes propagating along the periodic structure and clearly imply non-uniqueness of any solution to the scattering problem. As in the first part we consider a medium described by a refractive index which is periodic along the axis of an infinite cylinder in and constant outside of the cylinder. We formulate a proper radiation condition which allows the existence of traveling modes (and is motivated by the limiting absorption principle proven in the first part) and prove uniqueness and existence. (paper)