In the transmission electron microscope (TEM) almost all specimens are very thin and create mainly phase shifts of the electron beam, hardly any absorption. The phase shift gives a lot of information about the specimen like e.g. thickness modulations, electric fields or magnetic fields. Measuring the phase by conventional imaging techniques is quite difficult as in general only changes in intensity are detected.

One way to get the phase difference between object wave and a reference wave is the transport of intensity equation (TIE) which was initially developed for light microscopes [1]. The most common use of the TIE is to input a linear approximation for the intensity changes which occur when the image is taken at different values of defocus along the z direction and then to solve the equation for the phase. Thus it is possible to calculate the phase in good approximation. From the phase it is e.g. possible to evaluate direction and strength of intrinsic magnetic and electric fields of the specimen.

Recent publications use mainly a commercially available software to perform TIE calculations [2-4]. This software uses different filter algorithms which are not all known to the user. Therefore it turns out to be difficult to interpret the TIE reconstructions as the the filtering may create measurement artifacts.

Here we present a home written TIE Matlab code to calculate the phase. Our aim was to create a transparent code where the user is aware of all parameters. We use as little filtering as possible to minimize the room for misinterpretations and clearly show where modifications have been made to the original data. We are even able to calculate useful TIE images without any filtering as shown in figure 1.

Our software is capable of calculating magnetic or electric field maps from focal series (in-focus, over-focus, under-focus). Further, it is possible to create automated multiple TIE images for videos to visualize e.g. the temporal development of a magnetic structure. We show one example of a fluctuating skyrmion lattice in Cu₂OSeO₃.

We will provide the code as a Matlab applet for everybody interested for free.

References

[1] Michael Reed Teague, Deterministic phase retrieval: a Green’s function solution, J. Opt. Soc. Am. 73, 1434-1441 (1983).

[2] Kazuo Ishizuka, and Brendan Allman, Phase Measurement in Electron Microscopy Using the Transport of Intensity Equation, Microscopy Today 13, 22-24 (2005).

[3] Yu, X. Z. et al. Real-space observation of a two-dimensional skyrmion crystal. Nature 465, 901–904 (2010).

[4] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y. Matsui, Y. Tokura, Near room-temperature formation of a skyrmion crystal in thin-films of the helimagnet FeGe, Nature Materials 10, 106–109 (2011).

Figures:

In-focus (A) and over-focus (B) micrographs of magnetic stripes in Mn-Pt-Sn. C shows the reconstructed phase without filtering and D with band-pass filtered. The reconstructed magnetic induction is depicted in E. F show the reconstructed TIE image of a skyrmion lattice.

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EMC Abstracts - https://emc-proceedings.com/abstract/transport-of-intensity-equation-tie-without-filtering-and-tie-videos/