Long‐range focal series reconstruction in the
TEM
Authors: Lubk Axel , Vogel Karin , Wolf Daniel , Röder Falk , Clark Laura , Verbeeck Jo 
Published: December 2016

Focal series wave reconstruction in the Transmission Electron Microscope (TEM) is a well established holographic technique employed in both the medium and atomic resolution regime to study electric, magnetic and strain fields in solids as well as atomic configurations at crystal defects or grain boundaries. Focal series reconstruction does not require an undisturbed reference wave as off‐axis holography and may be conducted under relaxed partial coherence provided that the latter is well‐behaved and well‐known in advance [1]. Moreover, focal series holography may be considered as an instance of the more general quantum state tomography (see Fig. 1) that is successfully employed to study mixed (i.e., incoherent) quantum states of matter (e.g., atoms) and light [2].
These advantages are opposed by ambiguities in the reconstructed wave function, e.g., due to inconsistent and incomplete focal series data. In reality every focal series is inconsistent, e.g., due to the presence of partial coherence, shot and detector noise, as well as geometric and chromatic aberrations depending on the defocus. Similarly, every focal series is incomplete because of a limited number of foci, typically limited to the near field regime, and the restriction to isotropic foci, where astigmatic foci are necessary to provide a dataset allowing for an unabiguous reconstruction of an underlying wave function [3]. For instance, the problematic reconstruction of low spatial frequencies can be traced back to missing focal series data in the far field.
Here, we elaborate on focal series reconstruction from the perspective of quantum state tomography and use the obtained results to increase the scope of the technique in terms of convergence and uniqueness in particular for low spatial frequencies. Moreover, we explain a number of previous results by exploiting the above analogy, and open pathways to further improvements.
We particularly report on the recording, preprocessing, calibration and reconstruction of a long range focal series ranging from the near to the far field in a TEM. We calibrate the focal series, including the effective defocus and magnification, by a careful calibration of the proportionality between squared current and reziprocal focal length in a magnetic lens. We derive non‐linear focal sampling schemes from the phase space analogy. Subsequently, we adapt a modified Gerchberg‐Saxton algorithm to the long range focal series by exploiting the link to randomized Kaczmarz (ART) algorithm used in tomography [4]. We use different numerical propagation regimes in the near and far field to take into account the scaling of the wave function and overcome convergence problems by replacing the Kaczmarz iteration with the Landweber (SIRT) iteration as proposed by Allen et al.. [5]. To overcome remaining ambiguities in the reconstruction (e.g, pertaining to a different starting guess in the Gerchberg‐Saxton algorithm) resulting from inconsistencies in combination with the non‐convex nature of the set of wave functions possessing the same modulus, we discuss several additional constraints such imposed by the topology of the starting guess [6].
To illustrate the above reconstruction principles, we perform a case study on a higher‐order vortex beam with topological charge (winding number) 3 truncated by a square aperture (Fig. 2). The beam possesses a non‐trivial topology by design, which is nicely suited to discuss the impact of (implicit) topology constraints, rotation alignment as well as other issues.