Time harmonic Maxwell’s equations in an open periodic waveguide: a limiting absorption principle and the radiation condition

We study Maxwell’s equations in an open waveguide in $\mathbb{R}^3$. The electromagnetic parameters (permittivity, dielectricity, and conductivity) are periodic with respect to $x_3$ in an infinite cylinder along the $x_3$−axis and constant outside of the cylinder. The Floquet-Bloch transform is used to reduce the problem to a family of quasi-periodic problems. The quasi-periodic Calderon operator for the exterior of the cylinder reduces the problems to a bounded region. A general functional analytic framework is applied to prove convergence of the solutions when the conductivity tends to zero and allows the formulation of a proper radiation condition. A particular difficuly is introduced because of the singular dependence of the quasi-periodic Calderon operator on the Floquet-Bloch parameter at cut-off values.