Abstract
Lorentz transmission electron microscopy is an advanced characterization technique that enables the simultaneous imaging of both the microstructure and functional properties of materials. Information such as magnetization and electric potentials is carried by the phase of the electron wave, and is lost during image acquisition. Various methods have been proposed to retrieve the phase of the electron wavefunction using intensities of the acquired images, most of which work only in the small defocus limit. Imaging at strong defoci not only carries more quantitative phase information, but is essential to the study of weak magnetic and electrostatic fields at the nanoscale. In this work we develop a method based on differentiable programming to solve the inverse problem of phase retrieval. We show that our method maintains a high spatial resolution and robustness against noise even at the upper defocus limit of the microscope. More importantly, our proposed method can go beyond recovering just the phase information. We demonstrate this by retrieving the electron-optical parameters of the contrast transfer function alongside the electron exit wavefunction.
Highlights
The design and development of new and improved engineered materials requires a fundamental understanding of the structureproperty relationship at the nanoscale
We initialize the guess amplitude as the square root of the intensity of the infocus image and keep it fixed at the beginning of the phase retrieval process. (For reasons why this is a good estimation of the true amplitude, the reader is referred to the Supplementary Notes.) The starting guess φ0 for the phase is set to a constant
The wavefunction at the image plane is calculated by the convolution of the exit wavefunction with the microscope contrast transfer function (TF)
Summary
The design and development of new and improved engineered materials requires a fundamental understanding of the structureproperty relationship at the nanoscale. The quest to understand how these excitations interact locally with material inhomogeneities requires the capability to map the weak magnetization with both high spatial resolution and high phase accuracy. 2D electron gas at the interface of heterostructures plays an important role in the emergence of unique properties such as interfacial magnetism, and high interfacial conductivity[9,10]. Their functional characterization requires high spatial resolution and high phase sensitivity owing to their confined nature
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