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Like visible light, X-rays can also be focused by refraction in transmission lenses. For visible light this requires convex lenses while for X-rays one needs to use concave lenses instead. Both lens types can be lightened by the material removal strategy introduced by Fresnel, which results in a lens subdivided into zones. Until now, for the focusing of X-rays, stacks of standard lenses and of Fresnel lenses have mostly been produced. The first are dubbed compound refractive lenses, abbreviated as CRL. State-of-the-art systems of this kind now achieve almost theoretical performance for the focus size and the transmission. On the other hand, the latter Fresnel systems, which promise to provide larger apertures, are still in their infancy. This report discusses systematically the properties of two possible schemes for their realisation. It then compares the optimized apertures of these two schemes with those for CRLs. The best Fresnel lenses in this study are found to provide experimentally more than 50% of the expected refraction efficiency at 8.5 keV photon energy. The photon flux in their focus is then almost identical to that of perfect Be CRLs with the same focal length. This report will also interpret experimental data reported previously for other Fresnel lenses.

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