Nanofocusing compound refractive lenses (CRLs) have short focal lengths and hence require many refracting surfaces to be lined up along the optical axis. The usual spherical or parabolic refracting surfaces introduce aberrations that, in a long CRL, will pile up and lead to unacceptable focal broadening. It has long been known in optics, though not widely in the X-ray synchrotron community, that the ideal lens surface for focusing a point source into a point image lies on a quartic polynomial curve called a Cartesian oval [1]. This is here shown to apply even to the refracting surfaces on the downstream end of a CRL, which accept rays that are already converging toward a focus and bend them toward a new, closer focal point [2]. The following treatment summarizes results recently published [3]. Basic properties of Cartesian ovals will be covered and analytical methods of calculating them will be provided. An "X-ray" approximation of the Cartesian oval will be given for the case of small change in refractive index across the surface, since in this case the general analytical solution becomes numerically unstable. Finally, approximate conic sections will be derived for the paraxial limit. The development of nanofocusing CRLs with large aperture may be guided by these calculations once the technology for fabricating refracting surfaces advances sufficiently.
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