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In both light optics and electron optics, the amplitude of a wave scatteredby an object is an observable that is usually recorded in the form of anintensity distribution in a real space image or a diffraction image. Incontrast, retrieval of the phase of a scattered wave is a well-known challenge,which is usually approached by interferometric or numerical methods. Inelectron microscopy, as a result of constraints in the lens setup, it isparticularly difficult to retrieve the phase of a diffraction image. Here, weuse a defocused beam generated by a nanofabricated hologram to form a referencewave that can be interfered with a diffracted beam. This setup provides anextended interference region with the sample wavefunction in the Fraunhoferplane. As a case study, we retrieve the phase of an electron vortex beam.Beyond this specific example, the approach can be used to retrieve thewavefronts of diffracted beams from a wide range of samples.

Phase retrieval of an electron vortex beam using diffraction holography