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The validity of approximating the efficiency of a multilayer grating operating at close to normal incidence in the soft-X-ray-EUV range with a product of the relative grating efficiency by the reflectance of its multilayer coating has been studied by the rigorous integral method. The widely used approximated approach has until recently been considered accurate enough for analysis of short-wavelength normal-incidence multilayer-coated gratings. Real gratings employed in the soft-X-ray-EUV range are used to demonstrate the inapplicability of this approximation to an analysis of precise positions of efficiency maxima for the external (n > 0) and internal (n < 0) diffraction orders, despite the small ratios of wavelength and groove depth to period. The present authors have performed an analysis of the accuracy inherent in a derived simple expression for spectral separation of the same plus and minus orders with respect to the wavelength, order's number, incident angle, period, and groove depth. The reason for the observed substantial (a few Angstrom or even nm) wavelength separation between the maxima of positive and negative orders is related to oblique, close-tonormal incidence of radiation on a grating operating in the short-wavelength spectral region and different angles of deviation of respective orders. The modeling carried out with the commercial code PCGrate-S(X) v.6.1 permitted not only prediction of the separation between positive and negative orders for a multilayer Mo/Si 4200-gr/mm grating with FM-measured trapezoidal groove profile, which is designed for operation in the EIS spectrometer on the Solar-B spacecraft, but obtaining a good agreement with synchrotron radiation measurements, including high orders as well. A conclusion is drawn that high-precision calculations of the efficiency of multilayer normal-incidence soft-X-ray-EUV range gratings have in some cases to be performed, although this may require increasing the computation time by several times compared to the commonly used approximate approach.
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