Transfer and reconstruction of the density matrix in off-axis electron holography

The reduced density matrix completely describes the quantum state of an electron scattered by an object in transmission electron microscopy. However, the detection process restricts access to the diagonal elements only. The off-diagonal elements, determining the coherence of the scattered electron, may be obtained from electron holography. In order to extract the influence of the object from the off-diagonals, however, a rigorous consideration of the electron microscope influences like aberrations of the objective lens and the Mِllenstedt biprism in the presence of partial coherence is required. Here, we derive a holographic transfer theory based on the generalization of the transmission cross-coefficient including all known holographic phenomena. We furthermore apply a particular simplification of the theory to the experimental analysis of aloof beam electrons scattered by plane silicon surfaces.