Aberration-free short focal length x-ray lenses.

We treat the problem of defining the ideal x-ray refractive lens design for point focusing of low emittance x-ray beams at third- and fourth-generation synchrotron sources. The task is accomplished by using Fermat's principle to define a lens shape that is completely free from geometrical aberrations. Current microfabrication resolution limits are identified, and a design that tolerates the inherent fabrication imperfections is proposed. The refractive lens design delivers nanometer-sized focused x-ray beams and is compatible with current microfabrication techniques.