  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.
