Reconstruction of the Wigner function of electron beams based on coherence measurements

We developed a reconstruction method for the density matrix and Wigner function of electron beams through analysis of the Airy pattern intensity profile. The density matrix in a transmission electron microscope object plane was calculated using the coherence function and the electron wave amplitude and phase distributions. The Wigner function was then reconstructed using the matrix elements. Based on the Wigner function at the origin of the phase space, we derived a formula to calculate the axial brightness and then determined the axial brightness of a Schottky field emission gun, which reflects the emitter performance more accurately and precisely than the conventional mean brightness measurements.