Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy

We discuss phase retrieval and the correction of images for aberrations, in particular defocus and spherical aberration, in high-resolution transmission electron microscopy. Non-interferometric phase retrieval requires at least two intensity measurements in different planes. Vortices in the phase may occur in the image plane or the other planes involved in the phase retrieval. We discuss the performance of various methods of phase retrieval in that case. After retrieval of the phase, the aberrations can be corrected in the Fraunhofer diffraction space (the wave function in the diffraction space is related to that in the image space by a Fourier transform). The aberration-corrected image is obtained from the aberration-corrected wave function in the diffraction plane by inverse Fourier transformation.