We consider the diffraction of a p -polarized electromagnetic wave incident from the vacuum side onto a one-dimensional grating ruled on the planar surface of a dielectric medium characterized by an isotropic, frequency-dependent dielectric constant ∊(ω). The plane of incidence is assumed to be perpendicular to the grooves of the grating. On the basis of the Rayleigh hypothesis, integral equations are derived for the amplitudes of the reflected and transmitted beams. Formal solutions of these equations are obtained in the form of series of powers of the grating profile function. Recursion relations for determining the successive terms in these expansions are established. It is then pointed out that, if the series for the amplitudes of the reflected and transmitted waves are truncated after any finite number of terms, they have poles at the frequency of a surface polariton on a flat surface and not at the frequency of a surface polariton on the grating surface, in disagreement with the requirements of formal scattering theory. It is shown how the direct iterative series solutions for these coefficients can be rearranged into quotients of two series in powers of the surface-profile function in which the series in the denominator vanishes at the frequency of a surface polariton on the grating surface as required. Recursion relations are derived for the terms in both of these series. The solutions presented here are convenient in not requiring the inversion of large matrices and are well suited for machine computation in addition to possessing the required analytic properties.
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