The radiation condition for Helmholtz equations above locally perturbed periodic surfaces

The radiation condition is the key question in the mathematical modelling for scattering problems in unbounded domains. Mathematically, it plays the role as the "boundary condition" at the infinity, which guarantees the well-posedness of the mathematical problem; physically, it describes the behaviour of the physical waves. In this paper, we focus on the radiation conditions for scattering problems with periodic media embedded in two dimensional half-spaces. According to Hu et al. (2021), the radiating solution satisfies the Sommerfeld radiation condition: $$\frac{\partial u}{\partial r}-i k u=o(r^{-1/2}).$$ Although there are literature which have studied this problem, there is no specific method for dealing with periodic structures. Due to this reason, the important properties for the periodic structures are always ignored. Moreover, the existing method is not extendable to bi-periodic structures in three dimensional spaces. In this paper, we study the radiation condition for the scattering problem with periodic medium, which is modelled by the Helmholtz equation. We introduce a novel method based on the Floquet-Bloch transform, which, to the best of the author's knowledge, is the first method that works particularly for periodic media. With this method, we improve the Sommerfeld radiation condition for the scattered field from periodic surface to: $$\frac{\partial u}{\partial r}-i k u=O(r^{-3/2}).$$ Moreover, the prospect of extending this method to 3D cases is optimistic.