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Lorentz transmission electron microscopy is a unique characterizationtechnique that enables the simultaneous imaging of both the microstructure andfunctional properties of materials at high spatial resolution. The quantitativeinformation such as magnetization and electric potentials is carried by thephase of the electron wave, and is lost during imaging. In order to understandthe local interactions and develop structure-property relationships, it isnecessary to retrieve the complete wavefunction of the electron wave, whichrequires solving for the phase shift of the electrons (phase retrieval). Herewe have developed a method based on differentiable programming to solve theinverse problem of phase retrieval, using a series of defocused microscopeimages. We show that our method is robust and can outperform widely used\textit{transport of intensity equation} in terms of spatial resolution andaccuracy of the retrieved phase under same electron dose conditions.Furthermore, our method shares the same basic structure as advanced machinelearning algorithms, and is easily adaptable to various other forms of phaseretrieval in electron microscopy.

High resolution functional imaging through Lorentz transmission electron microscopy and differentiable programming