Scattering by a double-periodic layer in $\mathbb{R}^3$

We study the Helmholtz equation in a closed waveguide of the form $\Omega = \mathbb{R}^2 \times (0,h) \subset \mathbb{R}^3$ where the refractive index $n = n(x_1,x_2,x_3)$ is periodic with respect to $x_1$ and $x_2$. A functional analytic approach is used to prove the limiting absorption principle under an assumption on the set of propagative wave vectors. This assumption is formulated without the use of the band functions. Finally, we use the stationary phase method to obtain the asymptotic behavior of the limiting absorption solution as $|x|$ tends to infinity.