Asymptotic Behavior of Solutions to Open Periodic Waveguide Problems

In this paper we consider the propagation of waves in an open waveguide in the half space ℝ$^2_+$ = {𝑥 ∈ 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 nonevanescent (normal to the axis of the waveguide) modes.