We develop an electromagnetic analysis and its numerical implementation for gratings of arbitrary shape etched into a plane multilayer. The multilayer consists of a stack of identical bilayers, each bilayer including two materials (dielectrics and/or metals) with different thicknesses. The analysis is based on the differential theory of gratings, which is generalized to the case of many superimposed gratings by computation of the T matrix of each elementary grating linking the field above the modulated region to the field below it. The product of all the T matrices gives the T matrix of the stack, from which the outgoing wave condition permits derivation of the grating efficiencies. The method, developed for TE polarization, can be used throughout the entire spectrum, but it is particularly useful in the soft-x-ray region. Examples of performances of multilayer gratings in this spectral domain are computed.
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