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A new formalism based on perfectly matched layers (PMLs) is proposed to derive a fast converging series expansion for the two-dimensional periodic Green's function of layered media. The series combines a modal expansion for the waveguide formed by the layered medium terminated by PMLs with a truncated periodic Green's function series in the spatial domain. The efficiency of the new approach is illustrated by studying the scattering by a grid of metallic wires, both in free space and embedded in a dielectric slab. It is shown that the new technique results in a significant speed up compared to existing approaches.