Asymptotic expansion of wave scattering in a periodic plane

We give a counter part of Sommerfeld outgoing radiation condition for waves propagating in a two dimensional (2d) periodic medium under generical assumptions and provide a uniqueness theorem for outgoing solutions. Part of this work reproduces the work of A. V. Shanin et al. [Phys. Rev. B 110, 024310 (2024)] which the author just became aware by the time of submission of this present work.