Periodic arrays are structures consisting of geometrically identical subdomains, usually named periodic cells. In this paper, by taking the Helmholtz equation as a model, we consider the definition and evaluation of scattering operators for general semi-infinite periodic arrays. The well-posedness of the Helmholtz equation is established via the limiting absorption principle. A method based on the doubling procedure and extrapolation technique is first proposed to compute the scattering operators of Sommerfeld-to-Sommerfeld type. The advantages of this method are the robustness and simplicity of implementation. However, it suffers from the heavy computational cost and the resonance wavenumbers. To overcome these shortcomings, we propose another more efficient method based on a conjecture about the asymptotic behavior of limiting absorption principle solutions. Numerical evidences suggest that this method presents the same results as the first one.
"Evaluation of scattering operators for semi-infinite periodic arrays." Commun. Math. Sci. 7 (2) 347 - 364, June 2009.