Notes

1Here we restrict ourselves to stationary radiation fields, which are generated at third-generation synchrotron sources.

2In this equation and below we omit the frequency dependence [omega] for brevity. The GSM will be applied to narrow-bandwidth radiation, where [omega] is the average frequency.

3It is noteworthy that the CSD of a GSM source can be factorized into two transverse components. We will present calculations for one transverse direction and will drop the subscript for brevity.

4In principle, the transmission function can depend on the frequency of radiation [omega]. However, for CRLs in the X-ray range for a bandwidth lower than 10-3 the frequency dependence can be neglected (Kohn, 2012BB13).

5These expressions hold even without the strong lens approximation used earlier in the paper and can be obtained from equations (27)[link] and (28)[link] by a straightforward calculation (Singer, 2012BB23).

6We should note that the geometrical lens size limits the aperture [\Omega_{\rm A}]. According to our estimates it corresponds to an r.m.s. width of about 100 µm (see Appendix A[link]).

7For this set of lenses, aperture sizes of more than 100 µm are larger than the geometrical size of the lens and are shown to illustrate the asymptotic behaviour.