We present a new rigorous integral formalism for the theoretical study of dielectric coated gratings and grating couplers. It applies in the resonance domain, where the wavelength of the incident field and the groove spacing are of the same order of magnitude. The computed program issued from this theory extends the domain of application of the previous differential or integral theories. It can be used to investigate, with a very good accuracy, the properties of bare or dielectric coated gratings, for any groove shape and any polarization, in the entire visible, ultraviolet, and infrared regions. Various classical criteria are used to control the validity of the numerical results and comparisons are made with the numerical results obtained using the previous integral and differential formalisms. Two examples of applications are given. First, we show that the new possibilities of our program lead to a better agreement between theoretical results and experimental data. Second, a theoretical study of a certain type of grating coupler is given.
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