Model-based reconstruction of magnetisation distributions in nanostructures from electron optical phase images

Off-axis electron holography is a powerful technique for recording the phase shift of high-energy electron waves that pass through a thin specimen in the transmission electron microscope. Information about the electromagnetic field in and around the specimen is encoded in the phase, according to the Aharonov-Bohm equations. In this thesis, a model-based iterative reconstruction (MBIR) algorithm was developed, which allows the retrieval of the projected in-plane magnetisation distribution from individual magnetic phase images or a complete tomographic reconstruction of the three-dimensional magnetisation distribution from two ideally orthogonal tilt series of phase images. To guarantee efficient model-based reconstructions, an optimised forward model implementation for fast and accurate simulations of magnetic phase images from a given magnetisation distribution was derived. This new approach utilises sparse matrix multiplications and fast convolutions in Fourier space with pre-calculated convolution kernels based on known analytic solutions for the phase contribution of simple geometries. As the inverse problem of retrieving the magnetisation distribution is ill-posed, regularisation techniques had to be applied, that guarantee the existence of a solution and its uniqueness. Modelled after the minimisation of the exchange energy, Tikhonov regularisation of first order is used to apply smoothness constraints to the solution of the reconstruction. In addition, $\textit{a priori}$ knowledge about the position and size of the magnetised regions is utilised in the form of a three-dimensional mask to significantly reduce the number of retrieval targets. Optimal estimation diagnostic tools were adapted for the assessment of the quality of the reconstruction results. The MBIR algorithm was successfully applied to simulated phase images for the reconstruction of two-and three-dimensional magnetisation distributions. External sources of magnetisation outside the field of view were addressed by linear phase ramp and offset fits, as well as with buffer pixels that increase the number of degrees of freedom for the MBIR algorithm. A method to account for the perturbed reference wave of the electron hologram was provided and other artefacts in the magnetic phase images were tackled by excluding them from the reconstruction process. In three dimensions, studies about the influence of the maximum tilt angle and angular sampling were performed. The MBIR algorithm was successfully used to reconstruct a projected in-plane magnetisation distribution from a magnetic phase image of a lithographically patterned cobalt structure. Finally, a three-dimensional magnetisation distribution was reconstructed from a set of simulated phase images with limited angular range under the influence of Gaussian noise and random phase offsets and ramps, proving the feasibility of the algorithm for future three-dimensional experimental studies