Phase measurement of atomic resolution image using transport of intensity equation

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.