Noise correlation and phase resolution in off-axis electron holography

Off-axis electron holography is a quantitative phase contrast technique that allows the measurement of electromagnetic fields both within and around the sample. The sensitivity of the electric and magnetic fields (and sources) is thus largely dependent on the quality of the measured phase. Phase retrieval from holograms is usually achieved by a standard Fourier transform method. The aperture used in the reciprocal-space reconstruction process alters the nature of the noise from independence to correlated among pixels. This correlation is further enhanced if additional operator is used, for example, in the calculation of the charge density using the Laplacian operator, and of the magnetic induction and electric field using the gradient operator on the phase. Therefore, it is of importance to trace the noise and correlation of the phase in the processing of experimental holograms, in order to understand the limit of the noise (phase resolution), and further detect infinitesimal tiny electromagnetic signals. In this work, we systematically analyze the noise and correlation of the phase as a function of a variety of parameters, including the detector, electron dose, reconstruction aperture size, Gaussian smoothing and binning. Furthermore, we also assess the effect of the Laplacian and gradient operators on the phase. In particular, we provide detailed analyses on the possibility of detection of a single electric charge and Bohr magnetron. This finding would provide a solid basis for experimental measurements of very weak electromagnetic signals using off-axis electron holography and other phase contrast techniques in transmission electron microscopy.