Phase measurement of atomic resolution image using transport of intensity equation

Abstract

Since the Transport Intensity Equation (TIE) has been applied to electron microscopy only recently, there are controversial discussions in the literature regarding the theoretical concepts underlying the equation and the practical techniques to solve the equation. In this report we explored some of the issues regarding the TIE, especially bearing electron microscopy in mind, and clarified that: (i) the TIE for electrons exactly corresponds to the Schrödinger equation for high-energy electrons in free space, and thus the TIE does not assume weak scattering; (ii) the TIE can give phase information at any distance from the specimen, not limited to a new field; (iii) information transfer in the TIE for each spatial frequency g will be multiplied by g2 and thus low frequency components will be dumped more with respect to high frequency components; (vi) the intensity derivative with respect to the direction of wave propagation is well approximated by using a set of three symmetric images; and (v) a substantially larger defocus distance than expected before can be used for high-resolution electron microscopy. In the second part of this report we applied the TIE down to atomic resolution images to obtain phase information and verified the following points experimentally: (i) although low frequency components are attenuated in the TIE, all frequencies will be recovered satisfactorily except the very low frequencies; and (ii) using a reconstructed phase and the measured image intensity we can correct effectively the defects of imaging, such as spherical aberrations as well as partial coherence.

Skyrmion flow near room temperature in an ultralow current density

“…The inversion of such magnetic contrast can be observed between the under-and overfocussed images 21 . To perform quantitative in-plane magnetizations in Lorentz TEM configurations we analysed the Lorentz TEM images by using a software package QPt 22 , which is based on the transport-of-intensity equation 23 …”

Biskyrmion states and their current-driven motion in a layered manganite

“…To perform the quantitative in-plane magnetizations in Lorentz TEM configurations, TIE 32 was adopted with use of a software package QPt 33 .…”

Generation of high-density biskyrmions by electric current

“…3,7 The magnetic structures could be examined directly in the electron microscope by three sets of images with under-, over-, and just (or zero) focal-lengths and then the high-resolution in-plane magnetization distribution map is obtained using the commercial QPt software based on the transport of the intensity equation. 28 The colors depict the magnitude and direction of the lateral magnetization, which can be deduced from the arrows and the color wheel.…”