High-blaze-angle multilayer coated echelette gratings used in very high (i.e., from 10th to 60th) orders and operating with not too small (≈ 5°) glancing angles of incidence turn out to be necessary for achieving ultrahigh resolution and collecting important flux in x-ray fluorescence spectroscopy. Predicting their diffracting behavior appears to be a challenge to grating theoreticians, because such gratings are associated with the three main difficulties encountered in grating-efficiency numerical computation, namely, low wavelength-to-groove-spacing ratio, large modulation, and multiple interpenetrating profiles. We propose a method to resolve the problem that combines the differential theory of gratings, the R-matrix propagation algorithm, and an asymmetric truncation of the Fourier series of the field. The numerical results are checked against many criteria. For a particular mounting they are compared with those given by a phenomenological formula that was developed for bare echelette gratings and turns out still to apply for multilayer-coated gratings. The theory is developed not only for periodic stacks but also for stacks made of layers with varying thicknesses, such as those used in supermirrors.
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