Electron tomography of electromagnetic fields, potentials and sources

The exit phase of an electron moving through electromagnetic potentials can, as is well known, be written as line integrals of the potentials. The use of vector tomography to reconstruct electromagnetic potentials from such phase measurements in a transmission electron microscope is proposed. Electromagnetic fields and source distributions can similarly be reconstructed from phase gradients and Laplacians, respectively. The method can recover electric and magnetic fields independently, and requires only that the fields be magnetostatic, that the projection approximation applies and that there not be excessive inelastic scattering. Propagation-based phase retrieval is considered as the method of measuring electron phase, giving as a useful special case the reconstruction of current densities directly from defocused electron micrographs.