Aspherical lens shapes for focusing synchrotron beams

Aspherical surfaces required for focusing collimated and divergent synchrotron beams using a single refractive element (lens) are reviewed. The Cartesian oval, a lens shape that produces perfect point-to-point focusing for monochromatic radiation, is studied in the context of X-ray beamlines. Optical surfaces that approximate ideal shapes are compared. Results are supported by ray-tracing simulations. Elliptical lenses, rather than parabolic, are preferred for nanofocusing X-rays because of the higher peak and lower tails in the intensity distribution. Cartesian ovals will improve the gain when using high-demagnification lenses of high numerical aperture.