Phase measurement error in summation of electron holography series.

Off-axis electron holography is a method for the transmission electron microscope (TEM) that measures the electric and magnetic properties of a specimen. The electrostatic and magnetic potentials modulate the electron wavefront phase. The error in measurement of the phase therefore determines the smallest observable changes in electric and magnetic properties. Here we explore the summation of a hologram series to reduce the phase error and thereby improve the sensitivity of electron holography. Summation of hologram series requires independent registration and correction of image drift and phase wavefront drift, the consequences of which are discussed. Optimization of the electro-optical configuration of the TEM for the double biprism configuration is examined. An analytical model of image and phase drift, composed of a combination of linear drift and Brownian random-walk, is derived and experimentally verified. The accuracy of image registration via cross-correlation and phase registration is characterized by simulated hologram series. The model of series summation errors allows the optimization of phase error as a function of exposure time and fringe carrier frequency for a target spatial resolution. An experimental example of hologram series summation is provided on WS2 fullerenes. A metric is provided to measure the object phase error from experimental results and compared to analytical predictions. The ultimate experimental object root-mean-square phase error is 0.006 rad (2π/1050) at a spatial resolution less than 0.615 nm and a total exposure time of 900 s. The ultimate phase error in vacuum adjacent to the specimen is 0.0037 rad (2π/1700). The analytical prediction of phase error differs with the experimental metrics by +7% inside the object and -5% in the vacuum, indicating that the model can provide reliable quantitative predictions.