Even-odd effect in higher-order holographic production of electron vortex beams with nontrivial radial structures

Structured electron beams carrying orbital angular momentum are currently of considerable interest, both from a fundamental point of view and for application in electron microscopy and spectroscopy. Until recently, most studies have focused on the azimuthal structure of electron vortex beams with well-defined orbital angular momentum. To unambiguously define real electron-beam states and realise them in the laboratory, the radial structure must also be specified. Here we use a specific set of orthonormal modes of electron (vortex) beams to describe both the radial and azimuthal structures of arbitrary electron wavefronts. The specific beam states are based on truncated Bessel beams localised within the lens aperture plane of an electron microscope. We show that their Fourier transform set of beams can be realised at the focal planes of the probe-forming lens using a binary computer generated electron hologram. Using astigmatic transformation optics, we demonstrate that the azimuthal indices of the diffracted beams scale with the order of the diffraction through phase amplification. However, their radial indices remain the same as those of the encoding beams for all the odd diffraction orders or are reduced to the zeroth order for the even-order diffracted beams. This simple even-odd rule can also be explained in terms of the phase amplification of the radial profiles. We envisage that the orthonormal cylindrical basis set of states could lead to new possibilities in phase contrast electron microscopy and spectroscopy using structured electron beams.