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 is 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, shows great potential 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 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 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.Comment: 7 pages, 5 figure

Extension of Friedel's law to Vortex Beam Diffraction

Friedel's law states that the modulus of the Fourier transform of real functions is centrosymmetric, while the phase is antisymmetric. As a consequence of this, elastic scattering of plane wave photons or electrons within the first-order Born-approximation as well as Fraunhofer diffraction on any aperture, is bound to result in centrosymmetric diffraction patterns. Friedel's law, however, does not apply for vortex beams, and centrosymmetry in general is not present in their diffraction patterns. In this work we extend Friedel's law for vortex beams by showing that the diffraction patterns of vortex beams with opposite topological charge, scattered on the same two dimensional potential, always are centrosymmetric to one another, regardless of the symmetry of the scattering object. We verify our statement by means of numerical simulations and experimental data. Our research provides deeper understanding in vortex beam diffraction and can be used to design new experiments to measure the topological charge of vortex beams with diffraction gratings, or study general vortex beam diffraction.Comment: 7 pages, 3 figure

Using electron vortex beams to determine chirality of crystals in transmission electron microscopy

We investigate electron vortex beams elastically scattered on chiral crystals. After deriving a general expression for the scattering amplitude of a vortex electron, we study its diffraction on point scatterers arranged on a helix. We derive a relation between the handedness of the helix and the topological charge of the electron vortex on one hand, and the symmetry of the Higher Order Laue Zones in the diffraction pattern on the other for kinematically and dynamically scattered electrons. We then extend this to atoms arranged on a helix as found in crystals which belong to chiral space groups and propose a new method to determine the handedness of such crystals by looking at the symmetry of the diffraction pattern. Contrary to alternative methods, our technique does not require multiple scattering which makes it possible to also investigate extremely thin samples in which multiple scattering is suppressed. In order to verify the model, elastic scattering simulations are performed and an experimental demonstration on Mn$_2$Sb$_2$O$_7$ is given where we find the sample to belong to the right handed variant of its enantiomorphic pair. This demonstrates the usefulness of electron vortex beams to reveal the chirality of crystals in a transmission electron microscope and provides the required theoretical basis for further developments in this field

Prospects for versatile phase manipulation in the TEM: beyond aberration correction

In this paper we explore the desirability of a transmission electron microscope in which the phase of the electron wave can be freely controlled. We discuss different existing methods to manipulate the phase of the electron wave and their limitations. We show how with the help of current techniques the electron wave can already be crafted into specific classes of waves each having their own peculiar properties. Assuming a versatile phase modulation device is feasible, we explore possible benefits and methods that could come into existence borrowing from light optics where so-called spatial light modulators provide programmable phase plates for quite some time now. We demonstrate that a fully controllable phase plate building on Harald Rose's legacy in aberration correction and electron optics in general would open an exciting field of research and applications.Comment: 9 pages, 4 figures, special Ultramicroscopy issue for PICO2015 conferenc

Shaping electron beams for the generation of innovative measurements in the (S)TEM

In TEM, a typical goal consists of making a small electron probe in the sample plane in order to obtain high spatial resolution in scanning transmission electron microscopy. In order to do so, the phase of the electron wave is corrected to resemble a spherical wave compensating for aberrations in the magnetic lenses. In this contribution we discuss the advantage of changing the phase of an electron wave in a specific way in order to obtain fundamentally different electron probes opening up new application in the (S)TEM. We focus on electron vortex states as a specific family of waves with an azimuthal phase signature and discuss their properties, production and applications. The concepts presented here are rather general and also different classes of probes can be obtained in a similar fashion showing that electron probes can be tuned to optimise a specific measurement or interaction

Orbital angular momentum in electron diffraction and its use to determine chiral crystal symmetries

In this work we present an alternative way to look at electron diffraction in a transmission electron microscope. In stead of writing the scattering amplitude in Fourier space as a set of plane waves, we use the cylindrical Fourier transform to describe the scattering amplitude in a basis of orbital angular momentum (OAM) eigenstates. We show how working in this framework can be very convenient when investigating e.g. rotation and screw axis symmetries. For the latter we find selection rules on the OAM-coefficients that unambiguously reveal the handedness of the screw axis. Detecting the OAM-coefficients of the scattering amplitude thus offers the possibility to detect the handedness of crystals without the need for dynamical simulations, the thickness of the sample nor the exact crystal structure. We propose an experimental setup to measure the OAM-components where an image of the crystal is taken after inserting a spiral phase plate in the diffraction plane and perform mulsti-slice simulations on $\alpha$-quartz to demonstrate how the method indeed reveals the chirality. Experimental feasibility of the technique is discussed together with its main advantages with respect to chirality determination of screw axes. The method shows how the use of a spiral phase plate can be extended from a simple phase imaging technique to a tool to measure the local OAM-decomposition of an electron wave, widening the field of interest well beyond chiral space group determination.Comment: 12 pages, 10 figure