Commun. Comput. Phys., 11 (2012), pp. 629-646. Asymptotics for Helmholtz and Maxwell Solutions in 3-D Open Waveguides Carlos Jerez-Hanckes 1*, Jean-Claude Nedelec 21 Seminar fur Angewandte Mathematik, ETH Zurich, Ramistrasse 101, 8092 Zurich, Switzerland; and Escuela de Ingenieria, Pontificia Universidad Catolica de Chile, Av. Vicuna Mackenna 4860, Santiago, Chile. 2 Centre de Mathematiques Appliquees, Ecole Polytechnique, 91128 Palaiseau, France. Received 23 December 2009; Accepted (in revised version) 15 September 2010 Available online 24 October 2011 doi:10.4208/cicp.231209.150910s Abstract We extend classic Sommerfeld and Silver-Muller radiation conditions for bounded scatterers to acoustic and electromagnetic fields propagating over three iso-tropic homogeneous layers in three dimensions. If $\Vx=(x_1,x_2,x_3)\in\R^3$, with $x_{3}$ denoting the direction orthogonal to the layers, standard conditions only hold for the outer layers in the region $\abs{x_3}>\norm{\Vx}^\gamma$, for $\gamma\in(1/4,1/2)$ and $\Vx$ large. For $\abs{x_3}<\norm{\Vx}^\gamma$ and inside the slab, asymptotic behavior depends on the presence of surface or guided modes given by the discrete spectrum of the associated operator. AMS subject classifications: 35C15, 35Q60, 78A45 Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Open waveguide, radiation condition, Helmholtz equation. *Corresponding author. Email: cjerez@math.ethz.ch (C. Jerez-Hanckes), nedelec@cmap.polytechnique.fr (J.-C. Nedelec)