X-ray focusing optics using flat or bent crystals with two-dimensionally modulated surfaces

Abstract

Calculations of focus characteristics for x-ray focusing systems using either flat or bent crystals with two-dimensional modulated surfaces are presented. Based on the wave-optics approach, the analytical formulas are derived for elliptical Fresnel zone structures (flat crystals) and linear Bragg zone structures (toroidally bent crystals). It is shown that for a given Si(111) reflection and Ti Kα radiation, a micron-size focal spot can be achieved by using only one flat crystal with a modulated surface. The peak intensity is then nearly 100 times higher than that of the flat crystal without modulations. However, for a toroidally bent crystal with both Fresnel and Bragg diffraction amplitudes, the latter becomes dominant, since, due to the bending, the rapid oscillations of the Fresnel diffraction amplitude vanish. It is also shown that a focal spot of 0.3 μm for a bent crystal with linear Bragg zone structures is approximately four times narrower than that of an unmodulated surface. From the analysis, the modulated surfaces for cylindrically and spherically bent crystals are deduced to be a combination of Bragg and Fresnel zones.