Wavelength separation of plus and minus orders of soft-x-ray-EUV multilayer-coated gratings at near-normal incidence

Abstract

The validity of approximating the efficiency of a multilayer grating operating at close to normal incidence in the soft-Xray-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 AFM-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.

High-resolution, high-reflectivity operation of lamellar multilayer amplitude gratings: identification of the single-order regime

“…(8) means less by a factor of 3, at least. The peak reflectivity for calculations only considering the zeroth order then differs by less than 1% compared to calculations considering many (11) orders.…”

“…For the design of LMAGs with enhanced performance, in terms of resolution and reflectivity, adequate theory for the modeling of the diffraction of the incident SXR beam is required. At present, several rigorous approaches such as modal theory or integral method are used to simulate LMAGs [6][7][8], in particular in the soft X-ray region [3][4][5][9][10][11][12][13][14]. However, the modal theory is poorly suited for LMAGs with groove shapes that differ from rectangular or for the case of smooth interfaces between neighboring materials arising due to implantation and interdiffusion of atoms.…”

“…This CWA allows the implementation of arbitrary lamellar shapes, arbitrary depth distributions of the dielectric permittivity in the multilayer structure. In addition, it can be used without limitations on the grating period, lamellar width or number of bi-layers in the multilayer structure imposed by other models [8,10,11,13]. The CWA is well suited for calculations of LMAG performance in the soft X-ray region, because of the very small polarizability of matter in this region.…”

Single-order lamellar multilayer gratings

“…At present, several rigorous approaches such as modal theory or integral method are used to simulate LMGs [54][55][56][57], in particular in the SXR region [11,12,[58][59][60][61][62][63]. However, the modal theory is poorly suited for LMGs with non-rectangular groove shapes or for smooth interfaces between neighboring materials.…”