Boundary-artifact-free determination of potential distribution from differential phase contrast signals

We develop the numerical procedures to obtain the object phase from the differential phase contrast signals. Contrary to the conventional FFT, the discrete cosine transform and the extended FFT give a boundary-artifact-free solution. We also develop the real-time integration scheme that updates the result with the progress of the scan.The differential phase contrast (DPC) imaging in STEM was mainly used for a study of magnetic material in a medium resolution. An ideal DPC signals give the center of mass of the diffraction pattern, which is proportional to an electric field. Recently, the possibility of the DPC imaging at atomic resolution was demonstrated. Thus, the DPC imaging opens up the possibility to observe the object phase that is proportional to the electrostatic potential. In this report we investigate the numerical procedures to obtain the object phase from the two perpendicular DPC signals. Specifically, we demonstrate that the discrete cosine transform (DCT) is the method to solve the Poisson equation, since we can use the Neumann boundary condition directly specified by the DPC signals. Furthermore, based on the fast Fourier transform (FFT) of an extended DPC signal we introduce the scheme that gives an equivalent result that is obtained with the DCT. The results obtained with the DCT and extended FFT method are superior to the results obtained with commonly used FFT. In addition, we develop real-time integration schemes that update the result with the progress of the scan. Our real-time integration gives the reasonable result, and can be used in a view mode. We demonstrate that our numerical procedures work excellently with the experimental DPC signals obtained from SrTiO3 single crystal.