Asymptotic behavior of solutions to open periodic waveguides

In this paper we consider the propagation of waves in an open waveguide in the half space $\mathbb{R}^2_+ = \{x\in\mathbb{R}^2:x_2>0\}$ under Dirichlet- or Neumann boundary condition for $x_2 = 0$. The index of refraction $n = n(x)$ is periodic along the axis of the waveguide (which we choose to be the $x_1$−axis) and equal to one for $x_2 > h_0$ for some $h_0 > 0$. Based on a Limiting Absorption Principle, derived in a former paper, we formulate a radiation condition, prove existence and uniqueness of a solution and describe explicitely the asymptotic behavior along the axis of the waveguide. This behavior depends crucially on the existence of non-evanescent (normal to the axis of the waveguide) modes.