Near- and far-field solutions are derived for the diffraction of a Guassian beam by a layer whose permittivity is periodically modulated along the longitudinal direction. We examine a first-order coupled-mode expression of the plane-wave solution by Chu and Tamir and compare it with a highly accurate second-order formulation developed by Kong, who verified its validity and accuracy by means of a rigorous method. Our results show that, while the first-order approach is accurate for Gaussian beams scattered by layers with small periodic modulation, the second-order formulation must be used for strongly modulated media. The latter formulation can easily handle also asymmetric configurations involving a periodic layer which is bounded on its two sides by media having different dielectric constants. Computed results agree qualitatively with experiments carried out by Forshaw, and they account for the observed ripples in the diffracted beam profiles. We also confirm that, in contrast to previous theoretical studies which assumed an incident plane wave of infinite extent, complete conversion of energy into the Bragg field cannot generally occur in the case of bounded beams.
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