  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.
