Fundamentals of Focal Series Inline Electron Holography

Abstract

Abstract The loss of the quantum phase information in the measurement process constitutes a fundamental limitation to transmission electron microscopy as the electron wave's phase often contains valuable information about the studied specimen. Phase retrieval from focal series is a holographic technique that seeks to recover the lost information from a set of images recorded at different defoci. In spite of its widespread use, especially for wave reconstruction at atomic resolution, a number of fundamental properties (e.g., regarding the conditions for a unique wave function reconstruction) the magnitude of the reconstruction error and the influence of inconsistencies or incomplete data are not well understood. Here, we elaborate on the fundamentals of the technique, making extensive use of the tomographic representation of a focal series as tilt series in phase space. Using this perspective, we discuss, among others, requirements for the focal series for a unique reconstruction, such as the focus interval ranging from the far field at underfocus to the far field at overfocus or the focus step size. We reveal that the prominent Gerchberg–Saxton iterative projection algorithm corresponds to a numerical integration of the quantum Hamilton–Jacobi equation in the small focus step limit. Moreover, we show that the topology of the starting guess divides the solution space of the Gerchberg–Saxton algorithm into equivalence classes, which mitigates the impact of the incompleteness of typical focal series data. To facilitate a focal series reconstruction meeting these theoretical requirements, such as the long range focus interval, we develop a dedicated calibration procedure facilitating the determination of unknown electron optical parameters such as the focal length of the principal imaging lens or the position of object and image planes. The findings are demonstrated with an example of a focal series reconstruction of an electron vortex beam.