In refractive optics X-rays are guided by refraction at surfaces between different materials. This effect is described by Snell's law. For X-rays the difference of the refractive index to one is called index decrement . As the index decrement for X-rays for all materials is only slightly below one, hundreds of well aligned lens elements with very small radii of curvature are needed to form a refractive X-ray lens with a focal length in the range below a meter (fig. 1). For visible light the index of refraction is clearly larger than one for lens materials and focusing lenses are thicker in their centre than at their edges, resulting in their typically biconvex shape. Due to the fact, that the index of refraction is smaller than one for X-rays, focusing lenses are thinner at the centre than at their edges, giving a biconcave shape. A parabolic geometry of the refracting surfaces is well suited to focus X-rays hitting the lens parallel to the optical axis. Parabolic surfaces avoid spherical aberration (=when rays have a shorter focal length with increasing distance to the optical axis).

Fig. 1: Single lens element with long focal length f_{1} (top) and CRL with short focal length f_{2} (bottom)

For thin lenses the focal length f can be calculated as

with the radius of curvature R and the number of lens elements N. When the lens length L is in the range of the focal length, the formula for a thick lens made out of identical lens elements standing close to each other can be applied:

Some typical parameters of CRLs are listed in table 1.

Parameter |
Symbol |
Typical values |

Photon energy | E_{photon} |
5-100 keV |

Refractive index decrement | δ_{refractiv}(λ)=1-n |
10^{-5} to 10^{-9} |

Entrance aperture | A_{CRL}_{} |
50 µm to 2 mm |

Numerical aperture | NA_{CRL} |
≤10^{-3} |

Entrance and exit focal length | f_{1}, f_{2} |
5 mm to metres |

Lens materials used: |
- |
Be, Si, SU-8 (resist), Al, Ni |

Beam diameter | d_{U, CRL} |
30 µm to 2 mm |

Table 1: Typical parameters of CRLs [VDI 2007b]

[VDI 2007b] | VDI/VDE 5575 Blatt 7:2007-12 Röntgenoptische Systeme, Refraktive Röntgenoptiken (X-ray optical systems; Refractive X-ray optics), Berlin, Beuth Verlag, 2007 |