Numerical solution of diffraction problems: a method of variation of boundaries III Doubly periodic gratings

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Abstract
We present a new numerical method for the solution of the problem of diffraction of light by a doubly periodic surface. This method is based on a high-order rigorous perturbative technique, whose application to singly periodic gratings was treated in the first two papers of this series [ J. Opt. Soc. Am. A 10, 1168, 2307 ( 1993)]. We briefly discuss the theoretical basis of our algorithm, namely, the property of analyticity of the diffracted fields with respect to variations of the interfaces. While the algebraic derivation of some basic recursive formulas is somewhat involved, it results in expressions that are easy to implement numerically. We present a variety of numerical examples (for bisinusoidal gratings) in order to demonstrate the accuracy exhibited by our method as well as its limited requirements in terms of computing power. Generalization of our computer code to crossed gratings other than bisinusoidal is in principle immediate, but the full domain of applicability of our algorithm remains to be explored. Comparison with results presented previously for actual experimental configurations shows a substantial improvement in the resolution of our numerics over that given by other methods introduced in the past.