Techniques like coherent diffraction, speckle spectroscopy, and ptychography require a **coherent illumination** of the sample. However, the wave fronts emitted by current (third generation) storage rings are still far from showing full lateral coherence. Stated differently, the photon beam **emittance**, that is, the product of the linear source size (rms) and the beam divergence (rms), is still several orders of magnitude above the **diffraction limit**, which is roughly 8 pm rad for 10 keV radiation. Hence, for the applications mentioned above only a coherent portion of the beam can be used while the rest has to be *peeled off*. In other words, one has to increase the lateral coherence length at the sample position by reducing the angular size of the (secondary) x-ray source, that is, the ratio of the lateral extent and the distance of the source. Furthermore, it is important to employ **high quality optical components** that preserve the coherence.

Let us consider an example involving an undulator source at 12.4 keV with a width (FWHM) d_{h} = 120 μm and a height (FWHM) d_{v} = 20 μm. A CRL consisting of 11 rotationally parabolic (2D) beryllium lenses with R = 50 μm at a distance L_{1} = 80 m from the source yields a secondary source at a distance L_{2} = 1042 mm (see Figure 1). A sample is illuminated coherently by the full wave front of the secondary source provided that the effective aperture D_{eff} = 399 μm of the CRL is not larger than the lateral coherence length of the wavefront at L_{1} = 80 m (strictly speaking, this is only true for ideal lenses). The lateral coherence length is λ L_{1} / d_{h} = 67 μm horizontally and λ L_{1} / d_{v} = 400 μm vertically. Hence, while the secondary source is **diffraction limited** in the vertical direction this is **not** the case **in the horizontal direction**, despite the large source-to-sample distance.

A **remedy for the horizontal direction** can be obtained by placing **cylinder parabolic** (1D) beryllium lenses half way between the source and the 2D CRL in order to reduce the angular source size in the horizontal direction (see Figure 2). A CRL consisting of 24 cylinder parabolic (1D) lenses with R = 300 μm at a distance L_{3} = 40 m from the source yields a secondary source at a distance L_{4} = 3042 mm. The horizontal extent (FWHM) of this source is s_{h} = 9.13 μm with a horizontal coherence length λ L_{5} / s_{h} = 404 μm at the rotationally parabolic (2D) CRL. The lateral extent of the secondary source is 286 nm horizontally and 284 nm vertically. Hence, this way we obtain a **circular, Gaussian, diffraction limited secondary source**.

Note that it is impossible to achieve this with the aid of slits. For a circular, coherent secondary source one would have to place a slit with a width of roughly 20 μm right at the physical source (which is impossible) or a slit with a width of less than a micrometer (which doesn't exist) at the secondary source. Furthermore, slits tend to destroy the Gaussian profile and introduce ringing.

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