Aberrations in saw-tooth refractive lenses in short focal length x-ray focusing

Saw-tooth refractive lenses (SRL) provide a comparatively attractive option for X-ray focusing for various reasons, including their simple, continuous tunability in energy and focal length. Optimal focusing of a conventional SRL at short focal lengths is limited by the SRL’s length in relation to the focal length. Three approaches to overcome this limitation are described. Analytical solutions verified with ray-tracing are presented. These are bending, variation of the saw-tooth tip angles, and variation of the period.