3D Fourier Transform Analysis

DigitalMicrograph or a versatile free plug-in [ 10 ]. When your microscope is stable enough, you can take a 3D FT just after image acquisition. If the image drift is constant during the experiment, you will obtain a 3D FT that is simply distorted by image drift. When the image drift is time-dependent during a focal-series acquisition, however, you have to remove such image drift using the cross- correlation between the images.

Other aspects of the 3D FT analysis are the defocus step and the number of images in the through-focus series. In a 2D FT of an image, the pixel size determines the highest special frequency of the FT, and the image size (number of pixels × pixel size) determines the finest step of the 2D FT. Analogously, the defocus step determines the highest spatial frequency of the 3D FT along the w -axis, and the defocus range (number of images × defocus step) determines the finest step along the w -axis. Therefore, if we use the same defocus step, we can calculate the same Fourier space with a coarse step as shown in Figure 7 . Here, all the pixels (1018 × 1018) after the drift correction are used to improve the signal-to-noise ratio. The 3D FT calculated using 64 images shown in Figure 7b does not show significant degradation from the FT calculated using all 129 images in Figure 7a . Even the 3D FT calculated using 32 images shown in Figure 7c gives an approximate estimation of the highest w E , and thus the defocus spread.

Finally, we would like to discuss the implication of 3D FT analysis on an electron microscope with a C c -corrector. The C c -corrector reduces an apparent chromatic aberration due to different velocity (kinetic energy) of incident electrons by introducing a negative chromatic aberration. Namely, the C c corrector creates a fine probe by changing the beam path but does not modify the velocity of electrons. Importantly, the C c -corrector does not reduce chromatic aberration caused by the current fluctuation of the objective lens. Furthermore, a C c -corrector will increase total chromatic aberration due to its own current fluctuations. Thus, it is advisable to preform a 3D FT analysis to evaluate the total defocus spread of C s -corrected microscopes with a Cc-corrector.


We have compared 3D FT analysis with diffractogram analysis (2D FT analysis) for evaluating the performance of a high-resolution TEM. The significant difference of the 3D FT analysis from the diffractogram analysis is its ability to separately analyze two linear image contribu- tions on two spheres in Fourier space that we call Ewald spheres. Then, we don’t need to assume a weak phase object approximation, and we can use a thick sample or a sample made from strong scattering elements. In short, we need the 3D FT analysis to directly observe the linear image transfer down to a few tens of pm that will be attained by a C s -corrected microscope equipped with a C c -corrector or a monochromator. T e tilted-beam 3D FT analysis of our microscope gives the information limit of 90 pm at 80 kV when the monochromator


Figure 6 : Information limit measured by 3D FT analysis of our FEI TITAN 3 operated at 80 kV when the monochromator is ON. Each point on the fi gure is a ratio of two measurements of the Ewald envelopes for the spatial frequency g on the vw- and uv-sections. The horizontal axis corresponds to w E , which is the distance to the Ewald sphere for the spatial frequency g. Here, the ratio H decreases to 1/e 2 (13.5%) at w E of 0.26 nm -1 , which gives a defocus spread ∆ of 1.73 nm (see text).

is on, while the nominal information limit of 73 pm is obtained from an energy spread of incident electrons. T is suggests that other instabilities in the lens currents and/or the C s -corrector becomes dominant when the monochromator is used at 80 kV, and that performance improvement may be obtained by reducing those instabilities. T e 3D FT analysis will give an unbiased means to evaluate the performance of a TEM. T erefore, we strongly recommend that the owners of a Cs-corrected electron microscope at fi rst perform 3D FT analyses of normal-incidence images

Figure 7 : The 3D FTs calculated from a smaller number of though-focus images. Plots (a) to (c) shown in pseudo-color the 3D FTs are calculated from 129, 64, and 32 images, respectively. Since the focal step does not change, we can calculate the same range of Fourier space along the w-direction, although the resolution becomes poor. Note that the FT in (b) does not show signifi cant degradation from the FT in (a). Even from the FT in (c) we can estimate the highest w E and thus the defocus spread. • 2018 March

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