Electron Holography
Electron Holography (EH) is an interferometric technique that allows the measurement of the phase shift of the electron wave having interacted with static electrostatic and magnetic fields. This phase shift gives quantitative information about these electromagnetic fields.
In offline EH the phase shift measurement is performed by overlapping a socalled “reference” beam, which has not interacted with any field and just travelled through the vacuum, and a beam that has been phase shifted after interaction with any electromagnetic field inside the sample and with possible stray fields out of it. The overlapping of the two beams is realized thanks to an electrostatic biprism (Möllenstedt biprism) located in the SA image plane.
The resulting overlap creates an interference pattern (i.e. the hologram) between the two beams:
where ψ_{ref} is the reference electron wave; ψ_{s} the electron wave that has been phase shifted, A_{s} the amplitude of ψ_{s}, φ_{s}(x,y) its phase shift after interaction with the sample and surrounding stray fields, R_{o} the periodicity of the hologram fringes (which depend on the polarization of the biprism and the accelerating voltage).
The phase shift φ_{s}(x,y) is given by the AharonovBohm effect:
where V(x,y,z) is the electrostatic potential, B_{⊥} (x,y,z) the component of the magnetic induction perpendicular to the beam ; C_{E} a constant depending on the beam energy, and e and ℏ the electron charge and Planck constant, respectively.
The phase shift then contains two terms:

The first term represents the electrostatic contribution to the phase shift of the wave.

The second term carries the magnetic contribution to the phase shift (note that only the component of the magnetic induction perpendicular to the electron beam B_{⊥}(x,y,z), contributes to the phase shift).