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) dh = 120 μm and a height (FWHM) dv = 20 μm. A CRL consisting of 11 rotationally parabolic (2D) beryllium lenses with R = 50 μm at a distance L1 = 80 m from the source yields a secondary source at a distance L2 = 1042 mm (see Figure 1). A sample is illuminated coherently by the full wave front of the secondary source provided that the effective aperture Deff = 399 μm of the CRL is not larger than the lateral coherence length of the wavefront at L1 = 80 m (strictly speaking, this is only true for ideal lenses). The lateral coherence length is λ L1 / dh = 67 μm horizontally and λ L1 / dv = 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.

Secondary source
Fig. 1. Secondary source

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 L3 = 40 m from the source yields a secondary source at a distance L4 = 3042 mm. The horizontal extent (FWHM) of this source is sh = 9.13 μm with a horizontal coherence length λ L5 / sh = 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.

Diffraction limited secondary source
Fig. 2. 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.