Numerical Procedures to determine Potential Distribution from Electronic Field Vectors observed in Differential Phase Contrast (DPC) imaging

Recently, the differential phase contrast (DPC) imaging in STEM becomes a hot topic, since then the possibility of the DPC imaging at atomic resolution has been demonstrated [1]. The idea of the DPC imaging in STEM was proposed by Dekker and de Lang [2] and mainly used for a study of magnetic material in a medium resolution [3]. Müler et al. showed that the expectation value of the momentum transfer, which equals to a center of mass (COM) of the scattering distribution, is proportional to an electric field [4]. Thus, the COM signal corresponds to the DPC signal from the first moment detector that was discussed by Waddell and Chapman [5]. For a pure phase object the COM signal is proportional to a gradient of the object phase. Thus, the DPC experiment opens up the possibility to observe an electrostatic potential of a sample, since the object phase is proportional to a projected electric potential.