A universal analytical theory of soft x-ray multilayer gratings operating in the single-order regime is developed here. The single-order regime, in which an incident wave excites only the diffracted wave, is characterized by the maximum possible diffraction efficiency tending to the reflectivity of a conventional multilayer mirror. The theory is applied to analysis of different multilayer gratings, including lamella multilayer gratings, sliced multilayer gratings, blazed multilayer gratings, and blazed lamella multilayer gratings. A simple analytical formula that describes the diffraction efficiency of an arbitrary multilayer grating as a function of the wavelength and the incidence angle is deduced. Expressions for the peak value of the diffraction efficiency and the generalized Bragg condition corresponding to the maximum efficiency are obtained. The high accuracy of the deduced analytical expressions is justified by comparison with rigorous numerical calculations. Comparative analysis of different types of gratings is performed. The advantages and disadvantages of different gratings in practical applications are discussed. (C) 2015 Optical Society of America
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