We recently introduced a method of variation of boundaries for the solution of diffraction problems [ J. Opt. Soc. Am. A 10, 1168 ( 1993)]. This method, which is based on a theorem of analyticity of the electromagnetic field with respect to variations of the interfaces, has been successfully applied in problems of diffraction of light by perfectly conducting gratings. We continue our investigation of diffraction problems. Using our previous results on analytic dependence with respect to the grating groove depth, we present a new numerical algorithm that applies to dielectric and metallic gratings. We also incorporate Padé approximation in our numerics. This addition enlarges the domain of applicability of our methods, and it results in computer codes that can predict more accurately the response of diffraction gratings in the resonance region. In many cases results are obtained that are several orders of magnitude more accurate than those given by other methods available at present, such as the integral or differential formalisms. We present a variety of numerical applications, including examples for several types of grating profile and for wavelengths of light ranging from microwaves to ultraviolet, and we compare our results with experimental data. We also use Padé approximants to gain insight into the analytic structure and the spectrum of singularities of the fields as functions of the groove depth. Finally, we discuss some connections between Padé approximation and another summation mechanism, enhanced convergence, which we introduced in the earlier paper. It is argued that, provided that certain numerical difficulties can be overcome, the performance of our algorithms could be further improved by a combination of these summation methods.
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