Lattice sums and scattering coefficients for the rectangular planar array

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Abstract
We consider the plane wave mode solution for the three−dimensional problem of multiple scattering by a doubly periodic planar array of bounded obstacles. The mode amplitudes are expressed in terms of the multiple scattering coefficients for one obstacle of the array, and these are specified algebraically by the single scattered coefficients and the spherical wave lattice sums that characterize the array. We derive rapidly convergent forms of the lattice sums by exploiting our earlier results for the periodic line of bounded obstacles and for the grating of parallel cylinders. Then we develop closed form approximations for the multiple scattering coefficients for small scatterers which exhibit the effects of the array in coupling the multipole coefficients of the isolated scatterers.

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