We illustrate the errors inherent in the conventional empty beam correction of full field X-ray propagation imaging, i.e. the division of intensities in the detection plane measured with an object in the beam by the intensity pattern measured without the object, i.e. the empty beam intensity pattern. The error of this conventional approximation is controlled by the ratio of the source size to the smallest feature in the object, as is shown by numerical simulation. In a second step, we investigate how to overcome the flawed empty beam division by simultaneous reconstruction of the probing wavefront (probe) and of the object, based on measurements in several detection planes (multi-projection approach). The algorithmic scheme is demonstrated numerically and experimentally, using the defocus wavefront of the hard X-ray nanoprobe setup at the European Synchrotron Radiation Facility (ESRF).

Extended wavefronts are used for coherent full field imaging of objects based on solving the inverse Fresnel diffraction problem. To this end, the conventional data correction step is given by division of the recorded object image by the intensity pattern of the empty beam. This division of intensities in the detection plane is a rather crude approximation for the separation of the complex valued object and probing fields. Here we present a quantitative error estimate, along with its mathematical proof, and confirm the prediction with numerical simulations. Finally the problem is illustrated with experimental results. PACS number(s): 42.40.−i, 42.30.Wb, 87.59.−e u 0 (r ) := u(r ,0) = ι(r ,0)O(r ), r = (x,y) ∈ R 2 .

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