Production and application of electron vortex beams

Vortex beam production and contrast enhancement from a magnetic spiral phase plate

Electron vortex beam probes offer the possibility of mapping magnetic moments with atomic resolution. In this work we consider using the stray magnetic field produced from a narrow ferromagnetic rod magnetised along its long axis to produce a vortex beam probe, as an alternative to the currently used holographic apertures or gratings. We show through numerical modelling, electron holography observations and direct imaging of the electron probe, that a long narrow ferromagnetic rod induces a phase shift in the wave-function of passing electrons that approximately describes a helix in the regions near its ends. Directing this rod towards the optical axis of a charged-particle beam probe forming system at a limiting aperture position, with the free-end sufficiently close to the axis, is shown to offer a point spread function composed of vortex modes, with evidence of this appearing in observations of the electron probe formed from inserting a micro-fabricated CoFe rod into the beam path of a 300 keV transmission electron microscope (TEM). If the rod is arranged to contain the magnetic flux of h/e, thus producing a maximum phase shift of 2π, it produces a simple spiral-like phase contrast transfer function for weak phase objects. In this arrangement the ferromagnetic rod can be used as a phase plate, positioned at the objective aperture position of a TEM, yielding enhanced image contrast which is simulated to be intermediate between comparable Zernike and Hilbert phase plates. Though this aspect of the phase plate performance is not demonstrated here, agreement between our observations and models for the probe formed from an example rod containing a magnetic flux of ~2.35h/e, indicate this phase plate arrangement could be a simple means of enhancing contrast and gaining additional information from TEM imaged weak phase samples, while also offering the capability to produce vortex beam probes. However, steps still need to be taken to either remove or improve the support membrane for the rod in our experiments to reduce any effects from charging in the phase plate.

Measuring the orbital angular momentum of electron vortex beams using a forked grating

The present study experimentally examines how an electron vortex beam with orbital angular momentum (OAM) undergoes diffraction through a forked grating. The nth-order diffracted electron vortex beam after passing through a forked grating with a Burgers vector of 1 shows an OAM transfer of nℏ. Hence, the diffraction patterns become mirror asymmetric owing to the size difference between the electron beams. Such a forked grating, when used in combination with a pinhole located at the diffraction plane, could act as an analyzer to measure the OAM of input electrons.

Exploiting lens aberrations to create electron-vortex beams

A model for a new electron-vortex beam production method is proposed and experimentally demonstrated. The technique calls on the controlled manipulation of the degrees of freedom of the lens aberrations to achieve a helical phase front. These degrees of freedom are accessible by using the corrector lenses of a transmission electron microscope. The vortex beam is produced through a particular alignment of these lenses into a specifically designed astigmatic state and applying an annular aperture in the condenser plane. Experimental results are found to be in good agreement with simulations.

Detecting transition radiation from a magnetic moment

Electromagnetic radiation can be emitted not only by particle charges but also by magnetic moments and higher electric and magnetic multipoles. However, experimental proofs of this fundamental fact are extremely scarce. In particular, the magnetic moment contribution has never been observed in any form of polarization radiation. Here, we propose to detect it using vortex electrons carrying large orbital angular momentum ℓ. The relative contribution of the orbital angular momentum-induced magnetic moment, ℓℏω/Ee, becomes much larger than the spin-induced contribution ℏω/E and it can be observed experimentally. As a particular example, we consider transition radiation from vortex electrons obliquely incident on an interface between a vacuum and a dispersive medium, in which the magnetic moment contribution manifests itself via a left-right angular asymmetry. For electrons with Ee=300  keV and ℓ=100-1000, we predict an asymmetry of the order of 0.1%-1%, which could be measured with existing technology. Thus, vortex electrons emerge as a new tool in the physics of electromagnetic radiation.

Chemical mapping and quantification at the atomic scale by scanning transmission electron microscopy

With innovative modern material-growth methods, a broad spectrum of fascinating materials with reduced dimensions-ranging from single-atom catalysts, nanoplasmonic and nanophotonic materials to two-dimensional heterostructural interfaces-is continually emerging and extending the new frontiers of materials research. A persistent central challenge in this grand scientific context has been the detailed characterization of the individual objects in these materials with the highest spatial resolution, a problem prompting the need for experimental techniques that integrate both microscopic and spectroscopic capabilities. To date, several representative microscopy-spectroscopy combinations have become available, such as scanning tunneling microscopy, tip-enhanced scanning optical microscopy, atom probe tomography, scanning transmission X-ray microscopy, and scanning transmission electron microscopy (STEM). Among these tools, STEM boasts unique chemical and electronic sensitivity at unparalleled resolution. In this Perspective, we elucidate the advances in STEM and chemical mapping applications at the atomic scale by energy-dispersive X-ray spectroscopy and electron energy loss spectroscopy with a focus on the ultimate challenge of chemical quantification with atomic accuracy.

Electron holography for fields in solids: problems and progress

Electron holography initially was invented by Dennis Gabor for solving the problems raised by the aberrations of electron lenses in Transmission Electron Microscopy. Nowadays, after hardware correction of aberrations allows true atomic resolution of the structure, for comprehensive understanding of solids, determination of electric and magnetic nanofields is the most challenging task. Since fields are phase objects in the TEM, electron holography is the unrivaled method of choice. After more than 40 years of experimental realization and steady improvement, holography is increasingly contributing to these highly sophisticated and essential questions in materials science, as well to the understanding of electron waves and their interaction with matter.

Observation of the Larmor and Gouy rotations with electron vortex beams

Electron vortex beams carrying intrinsic orbital angular momentum (OAM) are produced in electron microscopes where they are controlled and focused by using magnetic lenses. We observe various rotational phenomena arising from the interaction between the OAM and magnetic lenses. First, the Zeeman coupling, proportional to the OAM and magnetic field strength, produces an OAM-independent Larmor rotation of a mode superposition inside the lens. Second, when passing through the focal plane, the electron beam acquires an additional Gouy phase dependent on the absolute value of the OAM. This brings about the Gouy rotation of the superposition image proportional to the sign of the OAM. A combination of the Larmor and Gouy effects can result in the addition (or subtraction) of rotations, depending on the OAM sign. This behavior is unique to electron vortex beams and has no optical counterpart, as Larmor rotation occurs only for charged particles. Our experimental results are in agreement with recent theoretical predictions.

Study on power coupling of annular vortex beam propagating through a two-Cassegrain-telescope optical system in turbulent atmosphere

As a new attractive application of the vortex beams, power coupling of annular vortex beam propagating through a two- Cassegrain-telescope optical system in turbulent atmosphere has been investigated. A typical model of annular vortex beam propagating through a two-Cassegrain-telescope optical system is established, the general analytical expression of vortex beams with limited apertures and the analytical formulas for the average intensity distribution at the receiver plane are derived. Under the H-V 5/7 turbulence model, the average intensity distribution at the receiver plane and power coupling efficiency of the optical system are numerically calculated, and the influences of the optical topological charge, the laser wavelength, the propagation path and the receiver apertures on the power coupling efficiency are analyzed. These studies reveal that the average intensity distribution at the receiver plane presents a central dark hollow profile, which is suitable for power coupling by the Cassegrain telescope receiver. In the optical system with optimized parameters, power coupling efficiency can keep in high values with the increase of the propagation distance. Under the atmospheric turbulent conditions, great advantages of vortex beam in power coupling of the two-Cassegrain-telescope optical system are shown in comparison with beam without vortex.

Electron vortex production and control using aberration induced diffraction catastrophes

An aberration corrected electron microscope is used to create electron diffraction catastrophes, containing arrays of intensity zeros threading vortex cores. Vortices are ascribed to these arrays using catastrophe theory, scalar diffraction integrals, and experimentally retrieved phase maps. From measured wave function phases, obtained using focal-series phase retrieval, the orbital angular momentum density is mapped for highly astigmatic electron probes. We observe vortex rings and topological reconnections of nodal lines by tracking the vortex cores using the retrieved phases.

Electromagnetic vortex fields, spin, and spin-orbit interactions in electron vortices

Electron vortices are shown to possess electric and magnetic fields by virtue of their quantized orbital angular momentum and their charge and current density sources. The spatial distributions of these fields are determined for a Bessel electron vortex. It is shown how these fields lead naturally to interactions involving coupling to the spin magnetic moment and spin-orbit interactions which are absent for ordinary electron beams. The orders of magnitude of the effects are estimated here for ȧngström scale electron vortices generated within a typical electron microscope.

Dispersionless phase discontinuities for controlling light propagation

Ultrathin metasurfaces consisting of a monolayer of subwavelength plasmonic resonators are capable of generating local abrupt phase changes and can be used for controlling the wavefront of electromagnetic waves. The phase change occurs for transmitted or reflected wave components whose polarization is orthogonal to that of a linearly polarized (LP) incident wave. As the phase shift relies on the resonant features of the plasmonic structures, it is in general wavelength-dependent. Here, we investigate the interaction of circularly polarized (CP) light at an interface composed of a dipole antenna array to create spatially varying abrupt phase discontinuities. The phase discontinuity is dispersionless, that is, it solely depends on the orientation of dipole antennas, but not their spectral response and the wavelength of incident light. By arranging the antennas in an array with a constant phase gradient along the interface, the phenomenon of broadband anomalous refraction is observed ranging from visible to near-infrared wavelengths. We further design and experimentally demonstrate an ultrathin phase gradient interface to generate a broadband optical vortex beam based on the above principle.

STEM_CELL: a software tool for electron microscopy: part 1--simulations

The software STEM_CELL, here presented, is a useful tool for (S) TEM simulation. In particular innovative solutions are presented in (1) the supercell manipulation and parameters setting (2) simulation execution through the modified Kirkland routines (3) simulation post-processing with extended output and comprehensive graphic tools (4) image contrast interpretation through a strain channeling equation accounting for strain effects in STEM-ADF.

Novel vortex generator and mode converter for electron beams

A mode converter for electron vortex beams is described. Numerical simulations, confirmed by experiment, show that the converter transforms a vortex beam with a topological charge m=±1 into beams closely resembling Hermite-Gaussian HG(10) and HG(01) modes. The converter can be used as a mode discriminator or filter for electron vortex beams. Combining the converter with a phase plate turns a plane wave into modes with topological charge m=±1. This combination serves as a generator of electron vortex beams of high brilliance.

Optical generation of crystalline, quasicrystalline, and arbitrary arrays of torons in confined cholesteric liquid crystals for patterning of optical vortices in laser beams

Condensed matter systems with topological defects in the ground states range from the Abrikosov phases in superconductors, to various blue phases and twist grain boundary phases in liquid crystals, and to phases of skyrmion lattices in chiral ferromagnets and Bose-Einstein condensates. In nematic and chiral nematic liquid crystals, which are true fluids with long-range orientational ordering of constituent molecules, point and line defects spontaneously occur as a result of symmetry-breaking phase transitions or due to flow, but they are unstable, hard to control, and typically annihilate with time. Here we describe the optical generation of two-dimensional crystalline, quasicrystalline, and arbitrary ensembles of particlelike structures manifesting both skyrmionlike and Hopf fibration features--dubbed "torons"--composed of looped double twist cylinders and point defects embedded in a uniform director field. In these two-dimensional lattices, we then introduce various dislocations, defects in positional ordering of the torons. We show that the periodic defect lattices with and without dislocation are light- and voltage-tunable reconfigurable two-dimensional diffraction gratings and can be used to generate various controlled phase singularities in the diffracted laser beams. The results of computer simulations of optical images, diffraction patterns, and phase distributions with optical vortices are in a good agreement with the corresponding experimental findings.

On-column 2p bound state with topological charge ±1 excited by an atomic-size vortex beam in an aberration-corrected scanning transmission electron microscope

Atomic-size vortex beams have great potential in probing the magnetic moment of materials at atomic scales. However, the limited depth of field of vortex beams constrains the probing depth in which the helical phase front is preserved. On the other hand, electron channeling in crystals can counteract beam divergence and extend the vortex beam without disrupting its topological charge. Specifically, in this article, we report that atomic vortex beams with topological charge ±1 can be coupled to the 2p columnar bound states and propagate for more than 50 nm without being dispersed and losing its helical phase front. We give numerical solutions to the 2p columnar orbitals and tabulate the characteristic size of the 2p states of two typical elements, Co and Dy, for various incident beam energies and various atomic densities. The tabulated numbers allow estimates of the optimal convergence angle for maximal coupling to 2p columnar orbital. We have also developed analytic formulae for beam energy, convergence angle, and hologram-dependent scaling for various characteristic sizes. These length scales are useful for the design of pitch-fork apertures and operations of microscopes in the vortex-beam imaging mode.

Vortex electron energy loss spectroscopy for near-field mapping of magnetic plasmons

The theory of vortex electron beam electron energy loss spectroscopy (EELS), or vortex-EELS for short, is presented. This theory is applied, using Green function calculations within the finite-difference time-domain method, to calculate spatially resolved vortex-EELS maps of a metal split ring resonator (SRR). The vortex-EELS scattering cross section for the SRR structure is within an order of magnitude of conventional EELS typically for metal nanoparticles. This is promising in terms of feasibility for future measurements to map out the local magnetic response of metal nanostructures and to characterize their magnetic plasmon response in applications, including metamaterials.

Phase measurement using an optical vortex lattice produced with a three-beam interferometer

A new phase-measurement technique is proposed, which utilizes a three-beam interferometer. Three-wave interference in the interferometer generates a uniform lattice of optical vortices, which is distorted by the presence of an object inserted in one arm of the interferometer. The transverse displacement of the vortices is proportional to the phase shift in the object wave. Tracking the vortices permits the phase of the object to be reconstructed. We demonstrate the method experimentally using a simple lens and a more complex object, namely the wing of a common house fly. Since the technique is implemented in real space, it is capable of reconstructing the phase locally.

Twisting light by nonlinear photonic crystals

We report the observation of nonlinear interactions in quadratic nonlinear crystals having a geometrically twisted susceptibility pattern. The quasi-angular-momentum of these crystals is imprinted on the interacting photons during the nonlinear process so that the total angular momentum is conserved. These crystals affect three basic physical quantities of the output photons: energy, translational momentum, and angular momentum. Here we study the case of second-order harmonic vortex beams, generated from a gaussian pump beam. These crystals can be used to produce multidimensional entanglement of photons by angular momentum states or for shaping the vortex's structure and polarization.

Laser-directed hierarchical assembly of liquid crystal defects and control of optical phase singularities

Topological defect lines are ubiquitous and important in a wide variety of fascinating phenomena and theories in many fields ranging from materials science to early-universe cosmology, and to engineering of laser beams. However, they are typically hard to control in a reliable manner. Here we describe facile erasable “optical drawing” of self-assembled defect clusters in liquid crystals. These quadrupolar defect clusters, stabilized by the medium's chirality and the tendency to form twisted configurations, are shaped into arbitrary two-dimensional patterns, including reconfigurable phase gratings capable of generating and controlling optical phase singularities in laser beams. Our findings bridge the studies of defects in condensed matter physics and optics and may enable applications in data storage, singular optics, displays, electro-optic devices, diffraction gratings, as well as in both optically- and electrically-addressed pixel-free spatial light modulators.

Production of electron vortex beams carrying large orbital angular momentum using spiral zone plates

We report the production of electron vortex beams carrying large orbital angular momentum (OAM) using micro-fabricated spiral zone plates. A series of the spherical waves, focussing onto different positions along the propagating direction of the electron beam, were observed. The nth order vortex beam has an OAM n times larger than that of the first-order vortex beam. We observed an electron vortex with an OAM up to in a high-order diffracted wave. A linear dependence of the diameter of the vortex beam on the OAM was observed, being consistent to numerical simulations.

Relativistic Hall effect

We consider the relativistic deformation of quantum waves and mechanical bodies carrying intrinsic angular momentum (AM). When observed in a moving reference frame, the centroid of the object undergoes an AM-dependent transverse shift. This is the relativistic analogue of the spin-Hall effect, which occurs in free space without any external fields. Remarkably, the shifts of the geometric and energy centroids differ by a factor of 2, and both centroids are crucial for the Lorentz transformations of the AM tensor. We examine manifestations of the relativistic Hall effect in quantum vortices and mechanical flywheels and also discuss various fundamental aspects of this phenomenon. The perfect agreement of quantum and relativistic approaches allows applications at strikingly different scales, from elementary spinning particles, through classical light, to rotating black holes.

Quantized orbital angular momentum transfer and magnetic dichroism in the interaction of electron vortices with matter

Following the very recent experimental realization of electron vortices, we consider their interaction with matter, in particular, the transfer of orbital angular momentum in the context of electron energy-loss spectroscopy, and the recently observed dichroism in thin film magnetized iron samples. We show here that orbital angular momentum exchange does indeed occur between electron vortices and the internal electronic-type motion, as well as center-of-mass motion of atoms in the electric dipole approximation. This contrasts with the case of optical vortices where such transfer only occurs in transitions involving multipoles higher than the dipole. The physical basis of the observed dichroism is explained.

Spin-to-orbital angular momentum conversion and spin-polarization filtering in electron beams

We propose the design of a space-variant Wien filter for electron beams that induces a spin half-turn and converts the corresponding spin angular momentum variation into orbital angular momentum of the beam itself by exploiting a geometrical phase arising in the spin manipulation. When applied to a spatially coherent input spin-polarized electron beam, such a device can generate an electron vortex beam, carrying orbital angular momentum. When applied to an unpolarized input beam, the proposed device, in combination with a suitable diffraction element, can act as a very effective spin-polarization filter. The same approach can also be applied to neutron or atom beams.

Angular momentum flux of nonparaxial acoustic vortex beams and torques on axisymmetric objects

An acoustic vortex in an inviscid fluid and its radiation torque on an axisymmetric absorbing object are analyzed beyond the paraxial approximation to clarify an analogy with an optical vortex. The angular momentum flux density tensor from the conservation of angular momentum is used as an efficient description of the transport of angular momentum. Analysis of a monochromatic nonparaxial acoustic vortex beam indicates that the local ratio of the axial (or radial) flux density of axial angular momentum to the axial (or radial) flux density of energy is exactly equal to the ratio of the beam's topological charge l to the acoustic frequency ω. The axial radiation torque exerted by the beam on an axisymmetric object centered on the beam's axis due to the transfer of angular momentum is proportional to the power absorbed by the object with a factor l/ω, which can be understood as a result of phonon absorption from the beam. Depending on the vortex's helicity, the torque is parallel or antiparallel to the beam's axis.

Relativistic electron vortex beams: angular momentum and spin-orbit interaction

Motivated by the recent discovery of electron vortex beams carrying orbital angular momentum (AM), we construct exact Bessel-beam solutions of the Dirac equation. They describe relativistic and nonparaxial corrections to the scalar electron beams. We describe the spin and orbital AM of the electron with Berry-phase corrections and predict the intrinsic spin-orbit coupling in free space. This can be observed as a spin-dependent probability distribution of the focused electron vortex beams. Moreover, the magnetic moment is calculated, which shows different g factors for spin and orbital AM and also contains the Berry-phase correction.

Theory of free electron vortices

The recent creation of electron vortex beams and their first practical application motivates a better understanding of their properties. Here, we develop the theory of free electron vortices with quantized angular momentum, based on solutions of the Schrödinger equation for cylindrical boundary conditions. The principle of transformation of a plane wave into vortices with quantized angular momentum, their paraxial propagation through round magnetic lenses, and the effect of partial coherence are discussed.

EMCD real space maps of Magnetospirillum magnetotacticum

In this study we combine energy loss magnetic circular dichroism (EMCD) and energy filtered transmission electron microscopy (EFTEM) to map magnetic properties of nanoparticles. We show that it is a functional tool for investigating the magnetic behaviour of bio-mineralized magnetite crystals of Magnetospirillum magnetotacticum. We find that the spatial resolution of our experimental set-up is in the range of less than 2 nm. The results are compared with EMCD studies of abiogenic magnetite.