Optimization Based Evaluation of Grating Interferometric Phase Stepping Series and Analysis of Mechanical Setup Instabilities

The diffraction contrast modalities accessible by X-ray grating interferometers are not imaged directly but have to be inferred from sine-like signal variations occurring in a series of images acquired at varying relative positions of the interferometer's gratings. The absolute spatial translations involved in the acquisition of these phase stepping series usually lie in the range of only a few hundred nanometers, wherefore positioning errors as small as 10 nm will already translate into signal uncertainties of 1-10% in the final images if not accounted for. Classically, the relative grating positions in the phase stepping series are considered input parameters to the analysis and are, for the Fast Fourier Transform that is typically employed, required to be equidistantly distributed over multiples of the gratings' period. In the following, a fast converging optimization scheme is presented simultaneously determining the phase stepping curves' parameters as well as the actually performed motions of the stepped grating, including also erroneous rotational motions which are commonly neglected. While the correction of solely the translational errors along the stepping direction is found to be sufficient with regard to the reduction of image artifacts, the possibility to also detect minute rotations about all axes proves to be a valuable tool for system calibration and monitoring. The simplicity of the provided algorithm, in particular when only considering translational errors, makes it well suitable as a standard evaluation procedure also for large image series.