Focusing hard X rays to nanometer dimensions by adiabatically focusing lenses

We address the question of what is the smallest spot size that hard x rays can be focused to using refractive optics. A thick refractive x-ray lens is considered, whose aperture is gradually ( adiabatically) adapted to the size of the beam as it converges to the focus. These adiabatically focusing lenses are shown to have a relatively large numerical aperture, focusing hard x rays down to a lateral size of 2 nm (FWHM), well below the theoretical limit for focusing with waveguides [C. Bergemann et al., Phys. Rev. Lett. 91, 204801 ( 2003)].