Truncation Effects in a Semi-infinite Periodic Array of Thin Strips: A discrete Wiener-Hopf Formulation

rigorous solution for the current induced on a semi-infinite array of narrow metallic strips is obtained using the Wiener-Hopf factorization method in the Z-transformed domain. The method can be applied to arrays with fixed current shape on each element (e.g, single mode elements), and shows rigorously the physics of waves associated to truncated periodic structures. The solution is obtained via a rigorous factorization, that is improved by using a closed form result based on an approximated factorization. The current on the truncated array is rigorously represented as the sum of the current pertaining to the infinite array plus a contribution induced by the truncation of the array. Asymptotics shows that the truncation-induced current contribution has a diffractive behavior decaying algebraically with the element number, away from the truncation. Uniform asymptotics shows that this diffractive current is effectively represented in terms of Fresnel functions, permitting also a closed form representation in proximity of and at transverse inward resonance, i.e., when a grazing grating lobe points toward the array. Illustrative examples and comparisons with a method of moment solution show the accuracy of our results.