Diffraction theory applied to X-ray imaging with clessidra prism array lenses

Clessidra (hourglass) lenses, i.e. two large prisms each composed of smaller identical prisms or prism-like objects, can focus X-rays. As these lenses have a periodic structure perpendicular to the incident radiation, they will diffract the beam like a diffraction grating. Refraction in the prisms is responsible for blazing, i.e. for the concentration of the diffracted intensity into only a few diffraction peaks. It is found that the diffraction of coherent radiation in clessidra lenses needs to be treated in the Fresnel, or near-field, regime. Here, diffraction theory is applied appropriately to the clessidra structure in order to show that blazing in a perfect structure with partly curved prisms can indeed concentrate the diffracted intensity into only one peak. When the lens is entirely composed of identical perfect prisms, small secondary peaks are found. Nevertheless, the loss in intensity in the central peak will not lead to any significant widening of this peak. Clessidras with perfect prisms illuminated by full coherent X-ray radiation can then provide spatial resolutions, which are consistent with the increased aperture, and which are far below the height of the single small prisms.