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

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 how, 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, our findings are of particular interest for electron microscopy where versatile programmable phase plates do not yet exist.

Figure 1

Sketch of the experimental setup to measure local OAM components of a wave function. In the back focal plane, a SPP is inserted adding a TC $\mathrm{\Delta }m$ to the wave in Fourier space after which an image is taken.

Figure 2

$T_m^{k_{\max }}\left(r\right)$ in arbitrary units, $r$ in units of $k_{\text{max}}^{-1}$.

Figure 3

Illustration of how the intensity in each point of a SPP image is linked to the local OAM component of the wave function. The upper image shows the amplitude and phase of a Bessel beam with TC $m=1$; see Eq. (9). In the center of the beam, the only nonzero OAM component is $a_1$ and the intensity in the center will be zero in each but the $\mathrm{\Delta }m=-1$ SPP image. In any other point of the wave, all components can be present and the intensity in the SPP images varies as such.

Figure 4

Sum of two Bessel beams with TC $m=-1$ positioned at the green X and $m=1$ positioned at the red X. (a) Phase-amplitude plot. (b) Intensity plot of panel (a). (c) Image obtained with SPP with TC $\mathrm{\Delta }m=-1$ showing the position of the $m=1$ Bessel beam. (d) Image obtained with SPP with TC $\mathrm{\Delta }m=1$ indicating the position of the $m=-1$ Bessel beam.

Figure 5

(a) Aharonov–Bohm-induced phase map of a plane wave passing a magnetic dipole. (b) Image obtained without SPP, only blocking central pixel. (c) and (d) Images obtained with $\mathrm{\Delta }m=1$ and $\mathrm{\Delta }m=-1$ phase plate, respectively, from which the “magnetic charge” of the poles can be deduced.