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We have formulated an asymptotic model for implementation in the finite-element method to calculate diffraction from a planar multilayered structure having a shallow surface-relief grating. The thin grating layer containing the shallow grating is replaced by a planar interface with transmission conditions that differ from the standard continuity conditions, thereby eliminating the necessity of representing the grating layer by a fine mesh. The parameters defining the shallow surface-relief grating are thereby removed from the geometry to the transmission conditions. Adoption of the asymptotic model will considerably reduce the computational cost of optimizing the grating shape because there is no need to re-mesh at every optimization step.

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In this paper, we devise an asymptotic model for calculating electromagnetic diffraction and absorption in planar multilayered structures with a shallow surface-relief grating. Far from the grating, we assume that the solution can be written as a power series in terms of the grating thickness δ , the coefficients of this expansion being smooth up to the grating. However, the expansion approximates the solution only sufficiently far from the grating (far field approximation). Near the grating, we assume that there exists another expansion in powers of δ (near field approximation). Moreover, there is an overlapping zone where both expansion are valid. The proposed model is based on matching the two expansions on this overlapping domain. Then, by truncating terms of order δ 2 or higher, we obtain explicitly the equations satisfied by the lowest order terms in the power series. Under appropriate assumptions, we prove second order convergence of the error with respect to δ . Finally, an alternative form, more convenient for implementation, is derived and discretized with finite elements to perform some numerical tests.