Abstract text (incl. figure legends and references)

In transmission electron microscopy, the evaluation of Thon ring patterns in diffractograms of defocused amorphous objects is the most often applied method to determine the present aberrations of the TEM: an individual diffractogram provides access to defocus C1 and two-fold astigmatism A1 and a tilt tableau of diffractograms enables the measurement of the existing higher order aberration coefficients by analysing tilt-induced C1A1 contributions [1].

Complementary, the cross-correlation (XC) analysis is a valuable tool to determine image displacements in between two or more subsequent images. This is often used for drift analysis but can also be used in a systematic tilt tableau to assess the existing aberration function by analysis of tilt-induced image displacements [2].

In the case of weak phase objects (WPO), it turns out that the phase contrast transfer function (PCTF) can have a considerable effect on the quality of the XC analysis. This is mainly due to the fact that the PCTF can cause contrast reversals between the two images which leads to mixed partial correlation and anti-correlation of the two respective images. As an example, Fig. 1 shows the XC of an over-/ underfocus image pair. Fortunately, this partial anti-correlation effect, which corresponds to a frequency-dependend phase jump by pi in the XC's Fourier phase, can quite simply be corrected for. As a consequence, the resulting PCTF-corrected XC becomes much more robust and more easy to interpret in terms of image shift.

In the WPO approximation antisymmetric aberrations such as coma can be considered as frequency-depenend image shifts (point-spread function = coma tail). Therefore, it becomes possible to recover relative changes of 2nd order aberrations, i.e. three-fold astigmatism A2 and axial coma B2 in between two images, if PCTF artifacts are properly corrected. Fig. 2 shows the PCTF-corrected XC of two images comprising a hexapole change in between. Subsequently, the relative A2B2 changes can be recovered from the Fourier phase of the XC image.

The PCTF-corrected XC can help a lot to improve the precision of image shift determination, and moreover reduces the required number of images for calibrations on second-order aberration changes. Alternatively, the analysis of the existing tilt tableau method can benefit from additional information on relative A2B2-changes during tilted illumination. [€]

References:
[1] F. Zemlin et al., Ultramicroscopy 3 (1978), 49-60.
[2] A.J. Koster et al., Ultramicroscopy 38 (1991), 235-240.
[€] CEOS GmbH has received funding from the European Union"s Horizon 2020 research and innovation program under grant agreement No. 823717 – ESTEEM3.

Fig.1: (a) The XC of two images in under-/overfocus is not a peak any more because of partial contrast reversals between the images. These show up as pi phase jumps in the XC's Fourier phase. (b) Since C1A1 of both images can be determined at the same time, the sign of the contrast contributions can be properly corrected to obtain a sharp XC peak with a uniform wedge-shaped Fourier phase.

Fig.2: (a) The pure XC of two images taken with different hexapole excitations is difficult to analyse. (b) Using the PCTF-corrected XC, however, further analysis in terms of three-fold astigmatism is straight forward. (inlay: robust phase contours by Hanning-window)