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Two-dimensional power spectral density measurements of x-ray optics with the Micromap interferometric microscope

A procedure and software have been developed to transform the area distribution of the residual surface heights available from the measurement with the Micromap interferometric microscope into a two-dimensional (2D) power spectral density (PSD) distribution of the surface height. The procedure incorporates correction of one of the spectral distortions of the PSD measurement. The distortion appears as a shape difference between the tangential and sagittal PSD spectra deduced from the 2D PSD distribution for an isotropic surface. A detailed investigation of the origin of the anisotropy was performed, and a mathematical model was developed and used to correct the distortion. The correction employs a modulation transfer function (MTF) of the detector deduced analytically based on an experimentally confirmed assumption about the origin of the anisotropy due to the asymmetry of the read-out process of the instrument's CCD camera. The correction function has only one free parameter, the effective width of the gate-shaped apparatus function which is the same for both directions. The value of the parameter, equal to 1.35 pixels, was found while measuring the 2D PSD distribution of the instrument self-noise, independent of spatial frequency. The effectiveness of the developed procedure is demonstrated with a number of PSD measurements with different X-ray optics including mirrors and a grating.