#### Jan Caron^{1}, Patrick Diehle^{1}, Andras Kovács^{1}, Jörn Ungermann^{2}, Rafal E. Dunin-Borkowski^{1}

^{1}Ernst Ruska-Centre for Microscopy and Spectroscopy with Electrons, Forschungszentrum Jülich, 52425 Jülich, Germany^{2}Institute for Energy and Climate Research, Forschungszentrum Jülich, 52425 Jülich, Germany

The development of an experimental technique that allows the three-dimensional magnetic state of a nanoscale object to be measured quantitatively is of great importance for fundamental and applied research in nanomagnetism. Off-axis electron holography is a powerful technique that can be used to record the phase shift of an electron wave that has passed through a specimen in the transmission electron microscope (TEM). The phase shift is, in turn sensitive to the in-plane component of the magnetic induction within and around the specimen projected in the electron beam direction.

We are currently developing a model-based approach to magnetic vector field tomography, which involves using an iterative reconstruction algorithm to recover the three-dimensional magnetization distribution in a specimen from a series of magnetic phase images recorded using electron holography as a function of sample tilt angle. The advantage of using such a model-based approach is that each trial solution is guaranteed to satisfy known physical laws. In order to optimise the speed of the algorithm, pre-computed analytical solutions for the phase contributions of simple geometries are used to avoid full matrix calculations. Together with an efficient use of Fourier space convolution, this approach significantly speeds up the reconstruction process, while minimising memory consumption.

The initially ill-posed problem is first replaced by a least-squares minimisation problem. First order Tikhonov regularisation is applied, motivated by minimisation of the exchange energy of the magnetisation distribution. A mask is introduced to localise magnetised objects and to decrease the number of unknowns. All measures are combined into a cost function, whose minimisation, facilitated by conjugate gradient methods, replaces the original problem. Diagnostic tools are used to assess the quality of the reconstruction result. The regularisation strength and the choice of mask are found to significantly influence the resolution of the reconstruction. Sources of magnetisation outside the field of view are accounted for by introducing buffer pixels. Perturbation of the hologram reference wave, which originates from long-reaching stray fields from the magnetised object, are addressed by modifying the convolution kernels. A confidence array is used to exclude other identifiable artefacts from the reconstruction process.

Examples of the application of the algorithm to the recovery of selected three-dimensional magnetization distributions will be presented [1].

[1] The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ ERC grant agreement number 320832.