In the early 1990s, Church and Takacs pointed out that the specification of surface figure and finish of\nx-ray mirrors must be based on their performance in the beamline optical system. We demonstrate the limitations\nof specification, characterization, and performance evaluation based on conventional statistical approaches,\nincluding root-mean-square roughness and residual slope variation, evaluated over spatial frequency bandwidths\nthat are system specific, and a more refined description of the surface morphology based on the\npower spectral density distribution. We show that these limitations are fatal, especially in the case of highly\ncollimated coherent x-ray beams, like beams from x-ray free electron lasers (XFELs). The limitations arise\ndue to the deterministic character of the surface profile data for a definite mirror, while the specific correlation\nproperties of the surface are essential for the performance of the entire x-ray optical system. As a possible\nway to overcome the problem, we treat a method, suggested by Yashchuk and Yashchuk in 2012, based on\nan autoregressive moving average modeling of the slope measurements with a limited number of parameters.\nThe effectiveness of the approach is demonstrated with an example specific to the x-ray optical systems under\ndesign at the European XFEL.
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