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Abstract
Analytical and numerical simulations were carried out for both, surface profiles measured on a real ultra precise mirror by use of the BESSY-NOM slope measuring profiler as well as for model local surface distortions. The effect of mirror imperfections could be properly handled in the frame of the wave optics approach. In spite of the large distances, for hard X-rays one still needs to carry out full-scale calculations surpassing the far field approximation. It is shown that the slope errors corresponding to medium spatial frequency components are of a special importance for the properties of coherent beam reflection from ultra smooth mirrors. The typical height errors for this component should not exceed 1-2 nm. Calculations show that reflection on such a mirror surface still imposes substantial wave field distortions at distances of several hundred meters from the mirror relevant for European XFEL beamlines. Requirements and trade-off for high precision mirrors and demands to coherent beams propagations are discussed.