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

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

^{3}It 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.

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

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

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

^{7}For 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.