X-ray diffractive optics of curved crystals: focusing properties on a diffraction-limited basis

For a given x-ray focusing geometry with curved crystals, we derive, on a diffraction-limited basis, analytical expressions for the intensity distribution at focus that account for both the Bragg and the Fresnel diffractions in an optical system. Within a monochromatic x-ray wavelength range of 0.709–3.294 Å, a doubly bent quartz crystal with (20-2) reflection is chosen to calculate analytically intensity distributions for point, 5-μm, and 20-μm x-ray sources. It is found that, whereas the fundamental width (for a point source) of the intensity distribution is broader than that deduced from a sinc function analogous to those used in optics, for the case of extended sources the width can be simply expressed as a sum of the fundamental width and the width obtained from demagnification of the source.