Abstract text (incl. figure legends and references)
Electron Tomography allows retrieval of the 3D structure of nanoscopic objects inside a TEM, by acquisition of a tilt series of 2D projections [1]. Electron holographic tomography [2] in particular makes use of the phase, which apart from the electrostatic potential distribution (e.g. mean inner potential) also probes the magnetic field via the Aharonov–Bohm effect [3]. After separation of both using opposing projections, this allows vector-field tomography [3, 4]. Each 2D projection contains two independent projected vector components of the 3D field. However, only the component parallel to a single tilt axis is always referring to the same vector component of the 3D field, whereas the other is constantly changing in accordance to the reference frame of the tilted object. Even though the information of the latter is for dual axis tomography redundant to the third component as retrieved via Maxwell's laws [4], by directly incorporating it in the reconstruction scheme, a variety of new tilt geometries becomes possible to use for vector field tomography. Therefore, a back-projection in 3D vector space, throwing back the 2D vector projection from arbitrary projection directions, has been implemented in a WSIRT algorithm [2] adaptation in python, enabling reconstruction for any given tilt geometry. The according weighting in 3D Fourier space accounts for over representation of lower spacial frequencies and possibly for non-uniform representation of the 3D vector components. As a proof of principle, a double conical tilt geometry (incl. zero tilt) has been simulated. Fig. 1, shows all projection directions with their corresponding 2D vector projections. As a reference object (with 128x128x128 pixels), a homogeneously (in direction of the knot) magnetized torus knot (3, -4) was chosen, as it exhibits features in multiple different directions. Fig. 2 shows the reconstruction after one iteration in comparison to the original reference object. The simulation proves, that such tilt geometries can indeed be used for 3D vector field reconstruction and suggests a high fidelity of this reconstruction algorithm with only small variations of the magnetization directions within the object. Fig 1. Double Conical Setup: Tilt geometry used for the simulation in Fig. 2. Each 3D vector is color coded according to its direction (x green, y blue, z red), while each 2D vector projection is color coded according to its own reference frame (x green, y blue, no z axis). Please note, that experimentally, it would still be required to additionally record each opposing projection, in order to separate electric and magnetic field. Fig. 2 Reconstruction of a simulated reference object: Comparison reference object (top row, with 3 different views) with WSIRT reconstruction after 1 iteration (bottom row, with the corresponding 3 views). Each 3D vector is color coded according to its direction (x green, y blue, z red). For visualization, a threshold was used to limit the number of plotted arrows and only every 23rd vector has been plotted for both. References: [1] M WEYLAND, et al. Electron tomography. Materials Today, 2004 [2] D Wolf, et al. Towards automated electron holographic tomography [...], Ultramicroscopy, 2010 [3] G LAI, et al. Three‐dimensional reconstruction of magnetic vector fields [..]. Journal of Applied Physics, 1994 [4] D WOLF, et al. Holographic vector field electron tomography [...]. Communications Physics, 2019 [5] K. M.-C. acknowledges funding from the DFG, contract EXC 2089/1 – 390776260 (e-conversion). A. L. received funding from the European Research Council (ERC) under the Horizon 2020 Research and Innovation Program of the European Union (grant agreement number 715620).