Nanometer focusing properties of Fresnel zone plates described by dynamical diffraction theory

The x-ray focusing properties of linear Fresnel zone plates have been derived by solving the Helmholtz equation for the field propagating through the zones. We consider the imaging of a point object into the first diffraction order of a volume zone plate having its zones parallel to the optical axis. For plane wave illumination, the focal spot size is limited by the same material-dependent but wavelength-independent value that affects waveguide focusing. In marked contrast, for the one-to-one imaging condition, corresponding to specular reflection of the x rays from the zone boundaries, the image is found to have a minimal spot size approximately equal to the outermost zone width. Unlike x-ray waveguides, zone plates therefore do not appear to possess a fundamental limit to the smallest spot size to which they can focus.