The resolution of HRTEM has been improved down to sub-angstrom by correcting the spherical aberration (Cs) of the objective lens, and the information limit is thus determined mainly by partial temporal coherence. Thus, a method to measure the partial temporal coherence becomes important more than ever. Since a traditional Young’s fringe test does not reveal the true information limit for an ultra-high resolution electron microscope, new methods to evaluate the focus spread, and thus temporal coherence have been proposed based on a tilted-beam diffractogram [1,2]. However, in order to observe literally an actual information transfer during the image formation down to a few ten pm, we need the strong scattering amorphous object, which will inevitably introduce pronounced non-linear contribution. Since the diffractogram analysis cannot be applied when the non-linear contribution becomes significant, we have proposed the method based on the three-dimensional (3D) Fourier transform (FT) of through-focus TEM images, and evaluated the performances of some Cs-corrected TEMs at lower-voltages [3,4]. In this report we generalize the 3D FT analysis and derive the 3D transmission cross coefficient (TCC). Then, we compare the 3D FT analysis with the tilted-beam diffractogram analysis (2D FT analysis), and clarify the necessity to use the 3D FT analysis to evaluate a high-performance TEM.

     The Fourier transform of the image intensity with a tilted-beam illumination may be written as Eq. (1) in Figure 1 with the (2D) TCC, where E/t and E/s are the temporal and spatial envelopes, respectively [5]. You may note that the 2D FT of the tilted-beam image intensity depends on z only through the 2D TCC. Thus, the 3D FT of the image stack may be written as Eq. (2) in Figure 1 with the 3D TCC, which is a Fourier transform of the 2D TCC along the z-direction, where E/Ewald is a normalized Gaussian, and may be called Ewald sphere envelope. Here, w/E represents the Ewald sphere, and w-w/E is a distance measured from the Ewald sphere along the w-axis to the spatial frequency g2 on the uv-plane.

     Figure 2 illustrates temporal damping of tilted-beam diffractogram, which is a power spectrum (an intensity of the 2D FT) of the image intensity. Here, we show the temporal damping for low-pass filtered diffractograms [1] and diffractogram envelopes [2]. We have to note that the diffractogram analysis cannot extract linear image information out from the image intensity. Furthermore, we have to make use of a weak scattering approximation, since the diffractogram cannot separate two linear image contributions. Moreover, the tilted-beam diffractogram becomes broad for the case of a small defocus spread as shown in Figure 2 (b). Thus, the diffractogram analysis may have difficulty for a Cc-corrected microscope or a microscope with a monocromator.

     Figure 3 shows an example of the Ewald sphere envelopes. Using the Ewald sphere envelopes we can extract linear image information from the image intensity. Thus, we can evaluate the temporal envelope on the sharp Ewald spheres, even when the temporal envelopes become broad for the case of a small defocus spread as shown in Fig. 2. Another profound difference of the 3D FT analysis from the diffractogram analysis is its capability to evaluate two linear image contributions separately on the Ewald sphere envelopes. Therefore, we can use a thick sample or a sample made from strong scattering elements, even when the dynamical/multiple scattering becomes significant. This is the necessary condition if we want to directly observe the linear image transfer down to a few ten pm. Furthermore, our method using 3D FT of the through-focus images gives a possibility to directly observe the distribution of the focus spread via a Fourier transform of the measured temporal envelope for a high-performance microscope.

References:

[1] J. Barthel, A. Thust, Physical Review Letters 101 (2008) p.200801. 

[2] M. Haider, et al, Micros. Microanal. 16 (2010) p.393. 

[3] K Kimoto, et al, Ultramicroscopy 121 (2012) p.31.  

[4] K Kimoto, et al, Ultramicroscopy 134 (2013) p.86. 

[5] K. Ishizuka, Ultramicroscopy 5 (1980) p.55.  

Acknowledgements

This study was partly supported by the JST Research Acceleration Program and the Nano Platform Program of MEXT, Japan.

Figures:

Fig. 1 Equations for 2D FT analysis (Diffractogram) and 3D FT analysis

Fig. 2 Temporal damping of tilted-beam diffractograms for defocus spreads of (a) 5 nm and (b) 0.5 nm with 80 kV electrons. Left and right columns show low-pass filtered diffractograms and diffractogram envelopes. The tilt angles from the top are 1, 3 and 5 degrees.

Fig. 3 The uv and uw sections of Ewald spheres for beam convergences of 0.5 (left) and 1.0 mrad (right) for 80 kV electrons. The beam tilt is along +u direction, and the tilt angles from top to bottom rows are 1, 3 and 5 degrees, respectively. The highest spatial frequency is 10 nm-1 along u- and v-axes, but 1 nm-1 along w-axis.

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