Local orbital angular momentum revealed by spiral-phase-plate imaging in transmission-electron microscopy

Abstract

The orbital angular momentum (OAM) of light and matter waves is a parameter that has been getting increasingly more attention over the past couple of years. Beams with a well-defined OAM, the so-called vortex beams, are applied already in, e.g., telecommunication, astrophysics, nanomanipulation and chiral measurements in optics and electron microscopy. Also, the OAM of a wave induced by the interaction with a sample, has attracted a lot of interest. In all these experiments it is crucial to measure the exact (local) OAM content of the wave, whether it is an incoming vortex beam or an exit wave after interacting with a sample. In this work we investigate the use of spiral phase plates (SPPs) as an alternative to the programmable phase plates used in optics to measure OAM. We derive analytically how these can be used to study the local OAM components of any wave function. By means of numerical simulations we illustrate how the OAM of a pure vortex beam can be measured. We also look at a sum of misaligned vortex beams and show, by using SPPs, the position and the OAM of each individual beam can be detected. Finally, we look at the OAM induced by a magnetic dipole on a free-electron wave and show how the SPP can be used to localize the magnetic poles and measure their "magnetic charge." Although our findings can be applied to study the OAM of any wave function, they are of particular interest for electron microscopy where versatile programmable phase plates do not yet exist.

Measuring the orbital angular momentum spectrum of an electron beam

“…In comparison with other methods used to examine magnetic fields (c.f., refs 21 , 30 , 31 ), the OAM sorter proves its effectiveness by readily providing the beam's OAM spectrum. For an identical number of detected electrons, this content is defined by an image showing 20 OAM coefficients yielding more information about a beam's phase than an image obtained using holographic methods.…”

EMCD with an electron vortex filter: Limitations and possibilities

“…Even though there are proposals to use spiral-phase-plates in the DP, e.g. to determine chiral crystal symmetries and the local OAM content of an electron wave [37,38], to date no successful m for the L 3 -edge (j = 3/2) taken from [6]. The weighting factors for the transitions when the final states are completely up-spin polarized show an asymmetry for m + µ = 0, i.e.…”

“…Even though there are proposals to use spiral-phase-plates in the DP, e.g. to determine chiral crystal symmetries and the local OAM content of an electron wave [37,38], to date no successful …”

Spiral phase plate contrast in optical and electron microscopy

“…Although alternative SPPs for electrons have been investigated, e.g., fork gratings [44] or helically shaped lenses made of a transparent material [45], the magnetic phase plates have the advantage of only blocking a relatively small part of the aperture in the objective plane, such that a maximal amount of scattered electrons can contribute to the signal. In previous work [46], we related the intensity of images obtained with a SPP to the local OAM decomposition of arXiv:1605.07847v2 [physics.optics] 23 Aug 2016 the exit wave. However, in ref.…”