Production and application of electron vortex beams

Compton Scattered X-Gamma Rays with Orbital Momentum

We study the possibility of producing x-gamma rays with orbital angular momentum by means of the inverse Compton backscattering between a high brightness electron beam and a twisted laser pulse. We use the classical electrodynamics retarded fields for evaluating the orbital angular momentum of the radiation and connecting it to that of the primary laser pulse. We then propose the dimensioning of a linearly polarized x-ray source with orbital angular momentum, starting from the parameters of operating Thomson setups.

Magnetic measurements with atomic-plane resolution

Rapid development of magnetic nanotechnologies calls for experimental techniques capable of providing magnetic information with subnanometre spatial resolution. Available probes of magnetism either detect only surface properties, such as spin-polarized scanning tunnelling microscopy, magnetic force microscopy or spin-polarized low-energy electron microscopy, or they are bulk probes with limited spatial resolution or quantitativeness, such as X-ray magnetic circular dichroism or classical electron magnetic circular dichroism (EMCD). Atomic resolution EMCD methods have been proposed, although not yet experimentally realized. Here, we demonstrate an EMCD technique with an atomic size electron probe utilizing a probe-corrected scanning transmission electron microscope in its standard operation mode. The crucial element of the method is a ramp in the phase of the electron beam wavefunction, introduced by a controlled beam displacement. We detect EMCD signals with atomic-plane resolution, thereby bringing near-atomic resolution magnetic circular dichroism spectroscopy to hundreds of laboratories worldwide.

It has been predicted that electron beam probes may allow for the imaging of magnetism with atomic-scale resolution. Here, the authors demonstrate a scanning transmission electron microscopy method capable of resolving magnetic contrast from individual atomic planes.

Synthesis and characterization of attosecond light vortices in the extreme ultraviolet

Infrared and visible light beams carrying orbital angular momentum (OAM) are currently thoroughly studied for their extremely broad applicative prospects, among which are quantum information, micromachining and diagnostic tools. Here we extend these prospects, presenting a comprehensive study for the synthesis and full characterization of optical vortices carrying OAM in the extreme ultraviolet (XUV) domain. We confirm the upconversion rules of a femtosecond infrared helically phased beam into its high-order harmonics, showing that each harmonic order carries the total number of OAM units absorbed in the process up to very high orders (57). This allows us to synthesize and characterize helically shaped XUV trains of attosecond pulses. To demonstrate a typical use of these new XUV light beams, we show our ability to generate and control, through photoionization, attosecond electron beams carrying OAM. These breakthroughs pave the route for the study of a series of fundamental phenomena and the development of new ultrafast diagnosis tools using either photonic or electronic vortices.

Twisted light beams have found several applications in the infrared and visible regime, but reaching the extreme ultraviolet has been difficult due to lack of sources. Here the authors report generation of helically shaped extreme ultraviolet trains of attosecond pulses via high harmonic generation.

Experimental study of diode pumped rubidium amplifier for single higher-order Laguerre-Gaussian modes

In this paper, we have set up a diode laser pumped rubidium amplifier for higher-order Laguerre-Gauss (LG) modes. We experimentally realized amplification of higher-order LG modes including helical and sinusoidal LG₀₃, LG₁₃, LG₂₃, and LG₃₃ modes with their high purity held. This novel scheme of generating high-purity higher-order LG beams at high laser power is preferred to the second-generation gravitational wave interferometers. To the best of our knowledge, it is the first time this scheme is formulated.

Detecting magnetic ordering with atomic size electron probes

Although magnetism originates at the atomic scale, the existing spectroscopic techniques sensitive to magnetic signals only produce spectra with spatial resolution on a larger scale. However, recently, it has been theoretically argued that atomic size electron probes with customized phase distributions can detect magnetic circular dichroism. Here, we report a direct experimental real-space detection of magnetic circular dichroism in aberration-corrected scanning transmission electron microscopy (STEM). Using an atomic size-aberrated electron probe with a customized phase distribution, we reveal the checkerboard antiferromagnetic ordering of Mn moments in LaMnAsO by observing a dichroic signal in the Mn L-edge. The novel experimental setup presented here, which can easily be implemented in aberration-corrected STEM, opens new paths for probing dichroic signals in materials with unprecedented spatial resolution.

Unified Approach towards the Dynamics of Optical and Electron Vortex Beams

We have proposed a unified framework towards the dynamics of optical and electron vortex beams from the perspective of the geometric phase and the associated Hall effects. The unification is attributed to the notion that the spin degrees of freedom of a relativistic particle, either massive or massless, are associated with a vortex line. Based on a cylindrical coordinate formulation, which leads to a local vortex structure related to orbital angular momentum (OAM), it can be shown that, when electron vortex beams (EVBs) move in an external electric field, paraxial beams give rise to an OAM Hall effect, and nonparaxial beams with tilted vortices initiate a spin Hall effect in free space as well as in an external field. A similar analysis reveals that the paraxial optical vortex beams (OVBs) in an inhomogeneous medium induce an OAM Hall effect, whereas nonparaxial beams with tilted vortices drive the spin Hall effect. Moreover, both OVBs and EVBs with tilted vortices give rise to OAM states with an arbitrary fractional value.

Generation and application of bessel beams in electron microscopy

We report a systematic treatment of the holographic generation of electron Bessel beams, with a view to applications in electron microscopy. We describe in detail the theory underlying hologram patterning, as well as the actual electron-optical configuration used experimentally. We show that by optimizing our nanofabrication recipe, electron Bessel beams can be generated with relative efficiencies reaching 37±3%. We also demonstrate by tuning various hologram parameters that electron Bessel beams can be produced with many visible rings, making them ideal for interferometric applications, or in more highly localized forms with fewer rings, more suitable for imaging. We describe the settings required to tune beam localization in this way, and explore beam and hologram configurations that allow the convergences and topological charges of electron Bessel beams to be controlled. We also characterize the phase structure of the Bessel beams generated with our technique, using a simulation procedure that accounts for imperfections in the hologram manufacturing process.

Elastic Scattering of Electron Vortex Beams in Magnetic Matter

Elastic scattering of electron vortex beams on magnetic materials leads to a weak magnetic contrast due to Zeeman interaction of orbital angular momentum of the beam with magnetic fields in the sample. The magnetic signal manifests itself as a redistribution of intensity in diffraction patterns due to a change of sign of the orbital angular momentum of the electron vortex beam. While in the atomic resolution regime the magnetic signal is most likely under the detection limits of present transmission electron microscopes, for electron probes with high orbital angular momenta, and correspondingly larger spatial extent, its detection is predicted to be feasible.

Prospects for electron beam aberration correction using sculpted phase masks

Technological advances in fabrication methods allowed the microscopy community to take incremental steps towards perfecting the electron microscope, and magnetic lens design in particular. Still, state of the art aberration-corrected microscopes are yet 20-30 times shy of the theoretical electron diffraction limit. Moreover, these microscopes consume significant physical space and are very expensive. Here, we show how a thin, sculpted membrane is used as a phase-mask to induce specific aberrations into an electron beam probe in a standard high resolution TEM. In particular, we experimentally demonstrate beam splitting, two-fold astigmatism, three-fold astigmatism, and spherical aberration.

Efficient linear phase contrast in scanning transmission electron microscopy with matched illumination and detector interferometry

The ability to image light elements in soft matter at atomic resolution enables unprecedented insight into the structure and properties of molecular heterostructures and beam-sensitive nanomaterials. In this study, we introduce a scanning transmission electron microscopy technique combining a pre-specimen phase plate designed to produce a probe with structured phase with a high-speed direct electron detector to generate nearly linear contrast images with high efficiency. We demonstrate this method by using both experiment and simulation to simultaneously image the atomic-scale structure of weakly scattering amorphous carbon and strongly scattering gold nanoparticles. Our method demonstrates strong contrast for both materials, making it a promising candidate for structural determination of heterogeneous soft/hard matter samples even at low electron doses comparable to traditional phase-contrast transmission electron microscopy. Simulated images demonstrate the extension of this technique to the challenging problem of structural determination of biological material at the surface of inorganic crystals.

Scanning transmission electron microscopy is a powerful material probe, but constrained to large atomic number samples due to the issues of beam damage and weak scattering. Here, Ophus et al. propose a method that produces linear phase contrast in a focused electron beam to image dose-sensitive objects.

Bragg–von Laue diffraction generalized to twisted X-rays

A pervasive limitation of nearly all practical X-ray methods for the determination of the atomic scale structure of matter is the need to crystallize the molecule, compound or alloy in a sufficiently large (∼10 × 10 × 10 µm) periodic array. In this paper an X-ray method applicable to structure determination of some important noncrystalline structures is proposed. It is designed according to a strict mathematical analog of von Laue's method, but replacing the translation group by another symmetry group, and simultaneously replacing plane waves by different exact closed-form solutions of Maxwell's equations. Details are presented for helical structures like carbon nanotubes or filamentous viruses. In computer simulations the accuracy of the determination of structure is shown to be comparable to the periodic case.

Relativistic Electron Vortex Beams in a Laser Field

The orbital angular momentum Hall effect and the spin Hall effect of electron vortex beams (EVBs) have been studied for the EVBs interacting with a laser field. In the scenario of a paraxial beam, the cumulative effect of the orbit-orbit interaction of EVBs and laser fields drives the orbital Hall effect, which in turn produces a shift of the center of the beam from that of the field-free case towards the polarization axis of the photons. In addition, for nonparaxial beams one can also perceive a similar shift of the center of the beam owing to the spin Hall effect involving spin-orbit interaction. Our analysis suggests that the shift in the paraxial beams will always be larger than that in the nonparaxial beams.

Focused electron beam induced deposition as a tool to create electron vortices

Focused electron beam induced deposition (FEBID) is a microscopic technique that allows geometrically controlled material deposition with very high spatial resolution. This technique was used to create a spiral aperture capable of generating electron vortex beams in a transmission electron microscope (TEM). The vortex was then fully characterized using different TEM techniques, estimating the average orbital angular momentum to be ∼0.8ℏ per electron with almost 60% of the beam ending up in the ℓ=1 state.

A multislice theory of electron scattering in crystals including backscattering and inelastic effects

In the framework of the slice transition operator technique, a general multislice theory for electron scattering in crystals is developed. To achieve this generalization, we combine the approaches for inelastic scattering derived by Yoshioka [J. Phys. Soc. Jpn. 12, 6 (1957)] and backscattering based on the formalism of Chen and Van Dyck [Ultramicroscopy 70, 29-44 (1997)]. A computational realization of the obtained equations is suggested. The proposed computational scheme is tested on elastic backscattering of electrons, where we consider single backscattering in analogy to the computational scheme proposed by Chen and Van Dyck.

Peculiar rotation of electron vortex beams

Standard electron optics predicts Larmor image rotation in the magnetic lens field of a TEM. Introducing the possibility to produce electron vortex beams with quantized orbital angular momentum brought up the question of their rotational dynamics in the presence of a magnetic field. Recently, it has been shown that electron vortex beams can be prepared as free electron Landau states showing peculiar rotational dynamics, including no and cyclotron (double-Larmor) rotation. Additionally very fast Gouy rotation of electron vortex beams has been observed. In this work a model is developed which reveals that the rotational dynamics of electron vortices are a combination of slow Larmor and fast Gouy rotations and that the Landau states naturally occur in the transition region in between the two regimes. This more general picture is confirmed by experimental data showing an extended set of peculiar rotations, including no, cyclotron, Larmor and rapid Gouy rotations all present in one single convergent electron vortex beam.

Classification of coherent vortices creation and distance of topological charge conservation in non-Kolmogorov atmospheric turbulence

The analytical expressions for the cross-spectral density function of partially coherent sinh-Gaussian (ShG) vortex beams propagating through free space and non-Kolmogorov atmospheric turbulence are derived, and used to study the classification of coherent vortices creation and distance of topological charge conservation. With the increment of the general structure constant and the waist width, as well as the decrement of the general exponent, the inner scale of turbulence and spatial correlation length, the distance of topological charge conservation will decrease, whereas the outer scale of turbulence and the Sh-part parameter have no effect on the distance of topological charge conservation. According to the creation, the coherent vortices are grouped into three classes: the first is the inherent coherent vortices of the vortex beams, the second is created by the vortex beams when propagating through free space, and the third is created by the atmospheric turbulence inducing the vortex beams.

Unveiling the orbital angular momentum and acceleration of electron beams

New forms of electron beams have been intensively investigated recently, including vortex beams carrying orbital angular momentum, as well as Airy beams propagating along a parabolic trajectory. Their traits may be harnessed for applications in materials science, electron microscopy, and interferometry, and so it is important to measure their properties with ease. Here, we show how one may immediately quantify these beams' parameters without need for additional fabrication or nonstandard microscopic tools. Our experimental results are backed by numerical simulations and analytic derivation.

Creating electron vortex beams with light

We propose an all-optical method of creating electron vortices utilizing the Kapitza-Dirac effect. This technique uses the transfer of orbital angular momentum from photons to free electrons creating electron vortex beams in the process. The laser intensities needed for this experiment can be obtained with available pulsed lasers and the resulting electron beams carrying orbital angular momentum will be particularly useful in the study of magnetic materials and chiral plasmonic structures in ultrafast electron microscopy.

Holographic generation of highly twisted electron beams

Free electrons can possess an intrinsic orbital angular momentum, similar to those in an electron cloud, upon free-space propagation. The wave front corresponding to the electron's wave function forms a helical structure with a number of twists given by the angular speed. Beams with a high number of twists are of particular interest because they carry a high magnetic moment about the propagation axis. Among several different techniques, electron holography seems to be a promising approach to shape a conventional electron beam into a helical form with large values of angular momentum. Here, we propose and manufacture a nanofabricated phase hologram for generating a beam of this kind with an orbital angular momentum up to 200ℏ. Based on a novel technique the value of orbital angular momentum of the generated beam is measured and then compared with simulations. Our work, apart from the technological achievements, may lead to a way of generating electron beams with a high quanta of magnetic moment along the propagation direction and, thus, may be used in the study of the magnetic properties of materials and for manipulating nanoparticles.

Is the angular momentum of an electron conserved in a uniform magnetic field?

We show that an electron moving in a uniform magnetic field possesses a time-varying "diamagnetic" angular momentum. Surprisingly this means that the kinetic angular momentum of the electron may vary with time, despite the rotational symmetry of the system. This apparent violation of angular momentum conservation is resolved by including the angular momentum of the surrounding fields.

Light's twist

That light travels in straight lines is a statement of the obvious. However, the energy and momentum flow within light beams can twist to form vortices such as eddies in a stream. These twists carry angular momentum, which can make microscopic objects spin, be used to encode extra information in communication systems, enable the design of novel imaging systems and allow new tests of quantum mechanics.

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 the 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.

Achieving atomic resolution magnetic dichroism by controlling the phase symmetry of an electron probe

The calculations presented here reveal that an electron probe carrying orbital angular momentum is just a particular case of a wider class of electron beams that can be used to measure electron magnetic circular dichroism (EMCD) with atomic resolution. It is possible to obtain an EMCD signal with atomic resolution by simply breaking the symmetry of the electron probe phase distribution using the aberration-corrected optics of a scanning transmission electron microscope. The required phase distribution of the probe depends on the magnetic symmetry and crystal structure of the sample. The calculations indicate that EMCD signals utilizing the phase of the electron probe are as strong as those obtained by nanodiffraction methods.

Imaging the dynamics of free-electron Landau states

Landau levels and states of electrons in a magnetic field are fundamental quantum entities underlying the quantum Hall and related effects in condensed matter physics. However, the real-space properties and observation of Landau wave functions remain elusive. Here we report the real-space observation of Landau states and the internal rotational dynamics of free electrons. States with different quantum numbers are produced using nanometre-sized electron vortex beams, with a radius chosen to match the waist of the Landau states, in a quasi-uniform magnetic field. Scanning the beams along the propagation direction, we reconstruct the rotational dynamics of the Landau wave functions with angular frequency ~100 GHz. We observe that Landau modes with different azimuthal quantum numbers belong to three classes, which are characterized by rotations with zero, Larmor and cyclotron frequencies, respectively. This is in sharp contrast to the uniform cyclotron rotation of classical electrons, and in perfect agreement with recent theoretical predictions.

Landau states are associated with the quantised orbits of charged particles in magnetic fields. By manipulating electron vortex beams in a magnetic field, this study reconstructs the internal quantum dynamics of free-electron Landau states, which differs strongly from the classical cyclotron rotation.

Dichroism in the interaction between vortex electron beams, plasmons, and molecules

We study the transfer of orbital angular momentum between vortex electron beams and chiral samples, such as staircase plasmonic nanostructures and biomolecules. Inelastic electron scattering from these samples produces large dichroism in the momentum-resolved electron energy-loss spectra. We illustrate this phenomenon with calculations for chiral and nonchiral clusters of silver spheres using both focused and extended electron beams, which exhibit ∼10% difference between channels of opposite angular momentum. In addition to its fundamental interest, this remarkably high dichroism suggests a way of spatially resolving chiral optical excitations, including dark plasmons. We also predict a dichroic response when probing a chiral biomolecule, which suggests the use of these electron beams for resolving different enantiomers.

Two-dimensional skyrmions and other solitonic structures in confinement-frustrated chiral nematics

We explore spatially localized solitonic configurations of a director field, generated using optical realignment and laser-induced heating, in frustrated chiral nematic liquid crystals confined between substrates with perpendicular surface anchoring. We demonstrate that, in addition to recently studied torons and Hopf-fibration solitonic structures (hopfions), one can generate a host of other axially symmetric stable and metastable director field configurations where local twist is matched to the surface boundary conditions through introduction of point defects and loops of singular and nonsingular disclinations. The experimentally demonstrated structures include the so-called "baby-skyrmions" in the form of double twist cylinders oriented perpendicular to the confining substrates where their double twist field configuration is matched to the perpendicular boundary conditions by loops of twist disclinations. We also generate complex textures with arbitrarily large skyrmion numbers. A simple back-of-the-envelope theoretical analysis based on free energy considerations and the nonpolar nature of chiral nematics provides insights into the long-term stability and diversity of these inter-related solitonic field configurations, including different types of torons, cholestric-finger loops, two-dimensional skyrmions, and more complex structures comprised of torons, hopfions, and various disclination loops that are experimentally observed in a confinement-frustrated chiral nematic system.

Toward single mode, atomic size electron vortex beams

We propose a practical method of producing a single mode electron vortex beam suitable for use in a scanning transmission electron microscope (STEM). The method involves using a holographic "fork" aperture to produce a row of beams of different orbital angular momenta, as is now well established, magnifying the row so that neighboring beams are separated by about 1 µm, selecting the desired beam with a narrow slit, and demagnifying the selected beam down to 1-2 Å in size. We show that the method can be implemented by adding two condenser lenses plus a selection slit to a straight-column cold-field emission STEM. It can also be carried out in an existing instrument, the monochromated Nion high-energy-resolution monochromated electron energy-loss spectroscopy-STEM, by using its monochromator in a novel way. We estimate that atom-sized vortex beams with ≥ 20 pA of current should be attainable at 100-200 keV in either instrument.

Interaction of relativistic electron-vortex beams with few-cycle laser pulses

We study the interaction of relativistic electron-vortex beams (EVBs) with laser light. Exact analytical solutions for this problem are obtained by employing the Dirac-Volkov wave functions to describe the (monoenergetic) distribution of the electrons in vortex beams with well-defined orbital angular momentum. Our new solutions explicitly show that the orbital angular momentum components of the laser field couple to the total angular momentum of the electrons. When the field is switched off, it is shown that the laser-driven EVB coincides with the field-free EVB as reported by Bliokh et al. [Phys. Rev. Lett. 107, 174802 (2011)]. Moreover, we calculate the probability density for finding an electron in the beam profile and demonstrate that the center of the beam is shifted with respect to the center of the field-free EVB.

A perturbation theory study of electron vortices in electromagnetic fields: the case of infinitely long line charge and magnetic dipole

The novel discovery of electron vortices carrying quantized orbital angular momentum motivated intensive research of their basic properties as well as applications, e.g. structural characterization of magnetic materials. In this paper, the fundamental interactions of electron vortices within infinitely long atomic-column-like electromagnetic fields are studied based on the relativistically corrected Pauli-Schrödinger equation and the perturbation theory. The relative strengths of three fundamental interactions, i.e. the electron-electric potential interaction, the electron-magnetic potential/field interaction and the spin-orbit coupling are discussed. The results suggest that the perturbation energies of the last two interactions are in an order of 10(3)-10(4) smaller than that of the first one for electron vortices. In addition, it is also found that the strengths of these interactions are strongly dependant on the spatial distributions of the electromagnetic field as well as the electron vortices.

Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation

Plasmonic metasurfaces have recently attracted much attention due to their ability to abruptly change the phase of light, allowing subwavelength optical elements for polarization and wavefront control. However, most previously demonstrated metasurface designs suffer from low coupling efficiency and are based on metallic resonators, leading to ohmic loss. Here, we present an alternative approach to plasmonic metasurfaces by replacing the metallic resonators with high-refractive-index silicon cut-wires in combination with a silver ground plane. We experimentally demonstrate that this meta-reflectarray can be used to realize linear polarization conversion with more than 98% conversion efficiency over a 200 nm bandwidth in the short-wavelength infrared band. We also demonstrate optical vortex beam generation using a meta-reflectarray with an azimuthally varied phase profile. The vortex beam generation is shown to have high efficiency over a wavelength range from 1500 to 1600 nm. The use of dielectric resonators in place of their plasmonic counterparts could pave the way for ultraefficient metasurface-based devices at high frequencies.

The geometric phase and the geometrodynamics of relativistic electron vortex beams

We have studied here the geometrodynamics of relativistic electron vortex beams from the perspective of the geometric phase associated with the scalar electron encircling the vortex line. It is pointed out that the electron vortex beam carrying orbital angular momentum is a natural consequence of the skyrmion model of a fermion. This follows from the quantization procedure of a fermion in the framework of Nelson's stochastic mechanics when a direction vector (vortex line) is introduced to depict the spin degrees of freedom. In this formalism, a fermion is depicted as a scalar particle encircling a vortex line. It is here shown that when the Berry phase acquired by the scalar electron encircling the vortex line involves quantized Dirac monopole, we have paraxial (non-paraxial) beam when the vortex line is parallel (orthogonal) to the wavefront propagation direction. Non-paraxial beams incorporate spin–orbit interaction. When the vortex line is tilted with respect to the propagation direction, the Berry phase involves non-quantized monopole. The temporal variation of the direction of the tilted vortices is studied here taking into account the renormalization group flow of the monopole charge and it is predicted that this gives rise to the spin Hall effect.

Electromagnetically induced torque on a large ring in the microwave range

We report on the exchange of orbital angular momentum between an electromagnetic wave and a 30 cm diameter ring. Using a turnstile antenna in the GHz range, we induce a torque on a suspended copper strip of the order of 10(-8)  N m. Rotations of a few degrees and accelerations up to 4×10(-4) °/s2 are observed. A linear dependence of the acceleration as a function of the applied power is found. There are many applications in the detection of angular momentum in electromagnetics, in acoustics, and also in the magnetization of nanostructures.

Vector spherical quasi-Gaussian vortex beams

Model equations for describing and efficiently computing the radiation profiles of tightly spherically focused higher-order electromagnetic beams of vortex nature are derived stemming from a vectorial analysis with the complex-source-point method. This solution, termed as a high-order quasi-Gaussian (qG) vortex beam, exactly satisfies the vector Helmholtz and Maxwell's equations. It is characterized by a nonzero integer degree and order (n,m), respectively, an arbitrary waist w(0), a diffraction convergence length known as the Rayleigh range z(R), and an azimuthal phase dependency in the form of a complex exponential corresponding to a vortex beam. An attractive feature of the high-order solution is the rigorous description of strongly focused (or strongly divergent) vortex wave fields without the need of either the higher-order corrections or the numerically intensive methods. Closed-form expressions and computational results illustrate the analysis and some properties of the high-order qG vortex beams based on the axial and transverse polarization schemes of the vector potentials with emphasis on the beam waist.

Is magnetic chiral dichroism feasible with electron vortices?

We discuss the feasibility of detecting magnetic transitions with focused electron vortex probes, suggested by selection rules for the magnetic quantum number. We theoretically estimate the dichroic signal strength in the L₂,₃ edge of ferromagnetic d metals. It is shown that under realistic conditions, the dichroic signal is undetectable for nanoparticles larger than ∼1 nm. This is confirmed by a key experiment with nanometer-sized vortices.

Generation of a spin-polarized electron beam by multipole magnetic fields

The propagation of an electron beam in the presence of transverse magnetic fields possessing integer topological charges is presented. The spin-magnetic interaction introduces a nonuniform spin precession of the electrons that gains a space-variant geometrical phase in the transverse plane proportional to the field's topological charge, whose handedness depends on the input electron's spin state. A combination of our proposed device with an electron orbital angular momentum sorter can be utilized as a spin-filter of electron beams in a mid-energy range. We examine these two different configurations of a partial spin-filter generator numerically. The results of this analysis could prove useful in the design of an improved electron microscope.

Invited review article: Methods for imaging weak-phase objects in electron microscopy

Contrast has traditionally been produced in electron-microscopy of weak phase objects by simply defocusing the objective lens. There now is renewed interest, however, in using devices that apply a uniform quarter-wave phase shift to the scattered electrons relative to the unscattered beam, or that generate in-focus image contrast in some other way. Renewed activity in making an electron-optical equivalent of the familiar "phase-contrast" light microscope is based in part on the improved possibilities that are now available for device microfabrication. There is also a better understanding that it is important to take full advantage of contrast that can be had at low spatial frequency when imaging large, macromolecular objects. In addition, a number of conceptually new phase-plate designs have been proposed, thus increasing the number of options that are available for development. The advantages, disadvantages, and current status of each of these options is now compared and contrasted. Experimental results that are, indeed, superior to what can be accomplished with defocus-based phase contrast have been obtained recently with two different designs of phase-contrast aperture. Nevertheless, extensive work also has shown that fabrication of such devices is inconsistent, and that their working lifetime is short. The main limitation, in fact, appears to be electrostatic charging of any device that is placed into the electron diffraction pattern. The challenge in fabricating phase plates that are practical to use for routine work in electron microscopy thus may be more in the area of materials science than in the area of electron optics.

Transport of intensity phase retrieval of arbitrary wave fields including vortices

The phase problem can be considered as one of the cornerstones of quantum mechanics intimately connected to the detection process and the uncertainty relation. The latter impose fundamental limits on the manifold phase reconstruction schemes invented to date, in particular, at small magnitudes of the quantum wave. Here, we show that a rigorous solution of the transport of intensity reconstruction (TIE) scheme in terms of a linear elliptic partial differential equation for the phase provides reconstructions even in the presence of wave zeros if particular boundary conditions are given. We furthermore discuss how partial coherence hampers phase reconstruction and show that a modified version of the TIE reconstructs the curl-free current density at arbitrary (in)coherence. Our results open the way for TIE-based phase retrieval of arbitrary wave fields, eventually containing zeros such as phase vortices.

Generation of arbitrary complex quasi-non-diffracting optical patterns

Due to their unique ability to maintain an intensity distribution upon propagation, non-diffracting light fields are used extensively in various areas of science, including optical tweezers, nonlinear optics and quantum optics, in applications where complex transverse field distributions are required. However, the number and type of rigorously non-diffracting beams is severely limited because their symmetry is dictated by one of the coordinate system where the Helmholtz equation governing beam propagation is separable. Here, we demonstrate a powerful technique that allows the generation of a rich variety of quasi-non-diffracting optical beams featuring nearly arbitrary intensity distributions in the transverse plane. These can be readily engineered via modifications of the angular spectrum of the beam in order to meet the requirements of particular applications. Such beams are not rigorously non-diffracting but they maintain their shape over large distances, which may be tuned by varying the width of the angular spectrum. We report the generation of unique spiral patterns and patterns involving arbitrary combinations of truncated harmonic, Bessel, Mathieu, or parabolic beams occupying different spatial domains. Optical trapping experiments illustrate the opto-mechanical properties of such beams.