The problem of phase retrieval in electron microscopy is generally related to the characterization of electric and magnetic fields in materials, to the retrieval of crystallographic structure or to the imaging of very weakly scattering objects. Recently, increasing attention is payed to the phase retrieval of electron vortex beams (EVB) i.e. beams carrying orbital angular momentum (OAM)   . The difficulty in this case arises due to the presence of an inherent phase singularity. There exist various kinds of phase retrieval schemes, and they mainly divide into off- and on-axis, whether the electron beam is displaced from the electro-optical axis or not. Furthermore, they can be computational (iterative or deterministic) or interferential. In this work, we use interferometry with synthetic beam shaping  to retrieve the phase of an EVB. In particular, we use an off-axis interferential method, where a reference beam interferes with a vortex beam in the diffraction plane. From the interference pattern it is then possible to retrieve the phase with Fourier methods. This method overcomes the difficulties on in-line methods and can be applied to the diffraction of many nanometer-sized features.
For this experiment, two holograms have been closely spaced and imprinted with focused ion beam (FIB) on a Si3N4 membrane . The two holograms are fabricated close to each other in the same membrane window, and their diffraction patterns superimpose in the diffraction plane. The first produces the aimed EVB in the form of a Laguerre Gauss with topological charge 10, and the other one is a hologram with a parabolic modulation that produces the reference wave.
A scanning electron microscope (SEM) image of the two holograms can be observed in figure 1a (the parabolic hologram is on top and the LG hologram is at the bottom). Their separate diffractions are shown in figure 1b and 1c.The EVB shows the expected circular symmetry and the dark region in the central region. Conversely, the parabolic beam is characterized by a fully circular diffraction. The visible set of fringes here come from the interference with the 0th order background. An image of superposition diffraction pattern is shown in figure 2. The parabolic wave hologram has the effect of adding a uniform phase ramp to the phase of the LG hologram, resulting in a pitchfork pattern typical of the superposition of beams with an azimuthal and linear phase ramp.
The phase reconstruction then proceeds as in conventional holography: the interference (figure 3a) is Fourier transformed (figure 3b), a sideband is isolated and back Fourier transformed to obtain the phase shift as in figure 3c. Here the spiralling phase is visible, winding up by 10 x 2π in a cycle as expected for this EVB. The parabolic phase effect can be easily removed but does not alter the topologic charge consideration.
This case study opens the way to a reliable solution of the phase problem in low angle electron diffraction.
 M. Uchida, A. Tonomura, Nature 464, 737-739 (2010)
 J. Verbeeck, H. Tian, and P. Schattschneider, Nature 467, 301 (2010)
 J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, Science 331, 192 (2011)
 J. Harris, V. Grillo, E. Mafakheri, G. C. Gazzadi, S. Frabboni, R. W. Boyd, E. Karimi, Nature Phys. 11, 629-633 (2015)
 V. Grillo, G. C. Gazzadi, E. Karimi, E. Mafakheri, R. W. Boyd, and S. Frabboni, Appl. Phys. Lett. 104, 043109 (2014)
To cite this abstract:Federico Venturi, Vincenzo Grillo, Ebrahim Karimi, Roberto Balboni, Gian Carlo Gazzadi, Marco Campanini, Stefano Frabboni, Robert W Boyd; Diffraction holography for the phase retrieval of vortex beams. The 16th European Microscopy Congress, Lyon, France. https://emc-proceedings.com/abstract/diffraction-holography-for-the-phase-retrieval-of-vortex-beams/. Accessed: July 13, 2020
EMC Abstracts - https://emc-proceedings.com/abstract/diffraction-holography-for-the-phase-retrieval-of-vortex-beams/