The background scattering of plane electromagnetic waves by arbitrarily periodic shaped surfaces is investigated. The scattered waves will be assumed to propagate in discrete Floquet modes. Electromagnetic fields are solved for first using the method of separation of variables and then expressed in a very compact form by introducing the modified spherical vector wave functions. The scattered waves are obtained using an exact method or a simplified model based on Rayleigh's hypothesis. For the general treatment, we express the field of the scattered waves in terms of the unknown value of the field and its normal derivative at the boundary, by making use of the Green's functions. The proposed formula allows mathematically exact calculation of the near-field, in the context of scalar wave theory.
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