Linear image transfer down to a few tens of pm can be attained by a modern Cs-corrected transmission electron microscope. However, it is difficult to accurately evaluate such a high-performance microscope. We examine three-dimensional (3D) Fourier transform (FT) analysis in comparison with diffractogram (2D FT) analysis to evaluate aberration-corrected electron microscopes. The 3D FT can analyze information transfer on the Ewald sphere up to high-angles using a thick sample or a sample containing strong scattering elements. Therefore, the 3D FT analysis is necessary to evaluate Cs-corrected microscopes, especially those equipped with a Cc-corrector, or a monochromator, or microscopes operated at lower voltages.

The resolution of high-resolution transmission electron microscopes (TEM) has been improved down to subangstrom levels by correcting the spherical aberration (Cs) of the objective lens, and the information limit is thus determined mainly by partial temporal coherence. As a traditional Young’s fringe test does not reveal the true information limit for an ultra-high-resolution electron microscope, new methods to evaluate temporal coherence have been proposed based on a tilted-beam diffractogram. However, the diffractogram analysis cannot be applied when the nonlinear contribution becomes significant. Therefore, we have proposed a method based on the three-dimensional (3D) Fourier transform (FT) of through-focus TEM images, and evaluated the performance of some Cs-corrected TEMs at lower voltages. In this report, we generalize the 3D FT analysis and derive the 3D transmission cross-coefficient. The profound difference of the 3D FT analysis from the diffractogram analysis is its capability to extract linear image information from the image intensity, and further to evaluate two linear image contributions separately on the Ewald sphere envelopes. Therefore, contrary to the diffractogram analysis the 3D FT analysis can work with a strong scattering object. This is the necessary condition if we want to directly observe the linear image transfer down to a few tens of picometer.

Most of the specimens for high-resolution electron microscopy have amorphous surface layers due to contamination during observation and/or damaged surface layers during specimen preparation. Moreover, many specimens are radiation sensitive, and a part of the specimen easily becomes amorphous during the observation. These amorphous materials make clear observation of crystal structure difficult. A periodic structure may be extracted by simply using a periodic mask in Fourier space. However, this kind of mask often introduces a periodic feature in addition to the crystal structure. To reduce such artifacts a Wiener filter or an average background subtraction filter has been discussed. However, these filters do not work for non-ideal crystals, such as cylindrical crystals and nano-crystals, where a translational periodicity is limited to the order of nano-meter. In this report we improve these filters by introducing new ways to estimate a contribution from the amorphous materials.

Although many samples in electron microscopy are phase objects, as in the case of optical microscopy, we cannot directly measure phase modulation by microscopy. Dennis Gabor proposed a technique called in-line holography to record both amplitude and phase information at the rather early stage of electron microscopy (in 1947). With the development of highly coherent field emission electron sources, another type of holography, off-axis holography, became available for electrons. However, holography cannot be applied to general cases, since there is a requirement for a vacuum region where the reference wave passes through.

The first paper on the FFT multislice method was published in 1977, a quarter of a century ago. The formula was extended in 1982 to include a large tilt of an incident beam relative to the specimen surface. Since then, with advances of computing power, the FFT multislice method has been successfully applied to coherent CBED and HAADF-STEM simulations. However, because the multislice formula is built on some physical approximations and approximations in numerical procedure, there seem to be controversial conclusions in the literature on the multislice method. In this report, the physical implication of the multislice method is reviewed based on the formula for the tilted illumination. Then, some results on the coherent CBED and the HAADF-STEM simulations are presented.