Perturbation theory for light diffraction with surface-polariton resonances on a bigrating

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Abstract
The diffraction of light of arbitrary polarization and arbitrary plane of incidence from a bigrating surface is described by an existing theory based on Rayleigh’s method and the vectorial Kirchhoff integral. Its implementation requires solving large matrix equations (e.g., 162×162). This paper presents an approximation of that theory, to first order in the corrugation amplitude. It yields a 4×4 matrix equation—allowing one to determine the reflectance for any diffracted order, as well as the surface-field enhancement, when there are surface-polariton resonant excitations. The theory retains all the nonresonant amplitudes (to renormalize the surface-polariton frequency) and is valid directly at—or away from—the boundaries of the two-dimensional Brillouin zones defined by the surface periodicity. The complex dispersion relation, for a surface polariton which propagates in an arbitrary direction across the bigrating, is given by a simple algebraic expression in this perturbation theory.