Development and study of refractive phase retrieval and X-ray multibeam ptychography

The advent of phase retrieval techniques has increased the resolution in X-ray microscopy by nearly an order of magnitude. With conventional imaging, X-ray microscopes cannot reach their full potential due to the lack of strong focusing options for X-rays. While the wavelength of X-rays is a thousand times smaller than visible light, the resolution of X-ray microscopes is only ten times smaller than the resolution of light microscopes. Phase retrieval algorithms are not limited by the numerical aperture of the focusing optics and can therefore resolve much smaller features.Conventional phase retrieval algorithms reconstruct the complex transmission function of the sample. Most microscopic samples are nearly transparent for X-rays and are only visible through their phase contrast. The phase shift however is 2π periodic, which means a specific phase value cannot be distinguished from another one that is larger or smaller by 2π. This phenomenon is called phase wrapping and is problematic for samples that have a maximum phase shift of more than 2π. Phase unwrapping algorithms aim to solve these ambiguities, but are not always successful, because phase wraps can lead to artefacts and phase singularities during the phase retrieval. Phase wrapping is especially challenging for tomography, because the wrapped phase is not a simple projection and also not proportional to the sample density.In this thesis, new algorithms are developed that reconstruct the projected refractive index of the sample instead of the complex transmission. The projected refractive index is in a trivial way directly proportional to the refractive index of the sample. In contrast to the phase of the transmission function, the projected refractive index requires no phase unwrapping, as it is not limited to a specific range. This thesis presents three refractive algorithms for three widely used phase retrieval techniques: ptychography, holography and coherent diffractive imaging. All three refractive algorithms are demonstrated and evaluated on simulated data sets, in the case of refractive ptychography also on two experimental data sets. Refractive ptychography and refractive holography surpass conventional algorithms for samples with a phase shift of more than 2π. In the case of refractive coherent diffractive imaging, the new algorithm surpasses the conventional one even for samples with no phase wrapping.In the second part of this thesis, a new method to scan large samples is presented. Due to the long scan times, large samples are problematic to measure with scanning techniques such as ptychography. To scan large samples in a shorter time, Bevis et al. have developed multibeam ptychography, a technique that uses multiple simultaneous beams [Ultramicroscopy, 184, 164 (2018)]. This thesis presents a set-up for X-ray multibeam ptychography that allows arbitrary numbers and geometries of multibeams. This is made possible through the use of a lens array manufactured with two-photon lithography. The lens array is evaluated for a two and a six beam geometry using experimental data. Compared to a single beam, the multibeam set-up scans the same area up to three times faster