Spectral separation of the efficiencies of the inside and outside orders of soft-x-ray-extreme-ultraviolet gratings at near normal incidence

It is shown from both a phenomenological study and exact modeling that the reason for the experimentally observed substantial (a few angstroms or nanometers) separation in wavelength between the maxima of the inside (negative numbered) and outside (positive numbered) diffraction orders of a multilayer-coated grating, operating at near normal incidence and close to the Bragg condition in the soft-x-ray and extreme-ultraviolet (EUV) regions, is related to the different angles of deviation of the orders. This wavelength separation is also a feature of uncoated diffraction gratings, although not clearly noticeable. The widely used approximate approach for calculating the absolute efficiency, the product of the relative grating efficiency and the reflectance of its multilayer coating, has until recently been considered accurate enough for the analysis of soft-x-ray and EUV near-normal-incidence multilayer-coated gratings. The inapplicability of this approximation for the analysis of the precise positions and shapes of the efficiency curves for the inside and outside orders, despite the small ratios of wavelength and groove depth to period and the small angles of incidence, is demonstrated using gratings with realistic groove profiles and operating in the EUV region. The rigorous modified integral method (MIM), which is a variant of boundary integral equation methods and is designed for the calculation of the efficiency of multilayer gratings with arbitrary layer thicknesses and boundary shapes (including microroughness) and over a wide wavelength range, is proposed in a general operator formalism. An analysis of a derived simple phenomenological expression and the exact numerical study indicates that the spectral separation between the inside and the outside orders grows with increasing either wavelength, angle of incidence, groove frequency, or diffraction order number ∣m∣. The efficiency modeling carried out with the commercial program PCGRATE-SX, based on the MIM, gave not only the exact values of the spectral separation between the inside and outside orders of Mo4Ru6∕Be, Mo∕Si, and Mo∕Y multilayer-coated gratings with various real groove profiles measured using atomic force microscopy (AFM) but also good agreement with synchrotron radiation measurements, including high orders as well. To determine the shapes and positions of efficiency curves in the soft-x-ray-EUV range of close to normal-incidence bulk and multilayer-coated gratings with real groove profil