Why Do We Need to Use Three-Dimensional (3D) Fourier Transform (FT) Analysis to Evaluate a High-Performance Transmission Electron Microscope (TEM)?

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.