High resolution while maintaining high peak reflectivities can be achieved for Lamellar Multilayer Amplitude Gratings (LMAG) in the soft-x-ray (SXR) region. Using the coupled waves approach (CWA), it is derived that for small lamellar widths only the zeroth diffraction order needs to be considered for LMAG performance calculations, referred to as the single-order regime. In this regime, LMAG performance can be calculated by assuming a conventional multilayer mirror with decreased density, which significantly simplifies the calculations. Novel analytic criteria for the design of LMAGs are derived from the CWA and it is shown, for the first time, that the resolution of an LMAG operating in the single-order regime is not limited by absorption as in conventional multilayer mirrors. It is also shown that the peak reflectivity of an LMAG can then still be as high as that of a conventional multilayer mirror (MM). The performance of LMAGs operating in the single-order regime are thus only limited by technological factors.

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An analytic theory describing soft x-ray diffraction by Lamellar Multilayer Gratings (LMG) has been developed. The theory is derived from a coupled waves approach for LMGs operating in the single-order regime, where an incident plane wave can only excite a single diffraction order. The results from calculations based on these very simple analytic expressions are demonstrated to be in excellent agreement with those obtained using the rigorous coupled-waves approach. The conditions for maximum reflectivity and diffraction efficiency are deduced and discussed. A brief investigation into p-polarized radiation diffraction is also performed.

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The optical properties of a-periodic, depth-graded multilayer mirrors operating at 13.5 nm wavelength are investigated using different compositions and designs to provide a constant reflectivity over an essentially wider angular range than periodic multilayers. A reflectivity of up to about 60% is achieved in these calculation in the [0, 18 degrees] range of the angle of incidence for the structures without roughness. The effects of different physical and technological factors (interfacial roughness, natural interlayers, number of bi-layers, minimum layer thickness, inaccuracy of optical constants, and thickness errors) are discussed. The results from an experiment on the fabrication of a depth-graded Mo/Si multilayer mirror with a wide angular bandpass in the [0, 16 degrees] range are presented and analyzed.

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