With the ability to form sub-Ångstrom sized beams and the electron beam's sensitivity to very small amounts of material, the electron microscope is a powerful tool for materials scientists and biologists.With recent developments in detectors, direct and efficient visualization of specimen features such as long range electric and magnetic fields and light atomic columns are now routine in STEM using differential phase contrast (DPC) and ptychographic reconstruction.

Received 17 December 2018DOI:https://doi.org/10.1103/PhysRevLett.122.139402© 2019 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasSpecial relativityTechniquesElectromagnetic wave theoryCondensed Matter, Materials & Applied Physics

A definitive statement of the model used to describe orbital angular momentum is essentially now available. Its early history, and the interaction of those who played key roles in its development over 20 years ago in its development, is outlined in this Memoir. This article is part of the themed issue ‘Optical orbital angular momentum’.

The modern field of optical angular momentum began with the realisation by Allen et al in 1992 that, in addition to the spin associated with polarisation, light beams with helical phase fronts carry orbital angular momentum. There has been much confusion and debate, however, surrounding the intricacies of the field and, in particular, the separation of the angular momentum into its spin and orbital parts. Here we take the opportunity to state the current position as we understand it, which we present as six perspectives: (i) we start with a reprise of the 1992 paper in which it was pointed out that the Laguerre–Gaussian modes, familiar from laser physics, carry orbital angular momentum. (ii) The total angular momentum may be separated into spin and orbital parts, but neither alone is a true angular momentum. (iii) The spin and orbital parts, although not themselves true angular momenta, are distinct and physically meaningful, as has been demonstrated clearly in a range of experiments. (iv) The orbital part of the angular momentum in the direction of propagation of a beam is not simply the azimuthal component of the linear momentum. (v) The component of spin in the direction of propagation is not the helicity, although these are related quantities. (vi) Finally, the spin and orbital parts of the angular momentum correspond to distinct symmetries of the free electromagnetic field and hence are separately conserved quantities.

We review the formulation of the operator for rotation angles and the corresponding uncertainty relation. The orbital angular momentum of light allows us to test these ideas and also to explore angular entanglement.

When fast electrons are used to study matter at subnanometer length scales, it is often necessary to model the inelastic cross section and the absorptive effect of phonon excitation on elastically scattered electrons. The inelastic cross section for the excitation of a phonon in a crystal by a fast electron is well modeled by using an effective absorptive potential. In this paper, the absorption potential for phonon excitation by fast electrons is rigorously derived from many-body quantum mechanics taking into account correlated atomic motion. This potential is calculated for a silicon crystal at room temperature from the force constants and dispersion curves for the crystal. It is shown that the total absorption for a crystal at room temperature predicted by a phonon model with correlated atomic motion agrees with the Einstein-model potential, based on independent atomic motions. This suggests that ignoring correlated atomic motion is not likely to contribute to the well-known quantitative discrepancy in contrast between simulated and experimental transmission electron microscopy images (the so-called "Stobbs factor"). The quantum-mechanical formulation allows us to further investigate the form of the inelastically scattered waves and the nonlocality of the absorption potential in directions both perpendicular and parallel to the direction of propagation, providing deeper insight into underlying physics of phonon excitation by fast electrons.

Extended abstract of a paper presented at Microscopy and Microanalysis 2009 in Richmond, Virginia, USA, July 26 – July 30, 2009

Extended abstract of a paper presented at Microscopy and Microanalysis 2009 in Richmond, Virginia, USA, July 26 – July 30, 2009

Some 16 years ago, Allen et al. [Phys. Rev. A 45, 8185 (1992)] recognised that laser beams which carried an angular momentum additional to photon spin, could be realized in the laboratory. Such beams-have helical phase fronts and so have an azimuthal component to the Poynting vector, which results in angular momentum along the beam axis. This orbital angular momentum, very often combined with spin to make optical angular momentum, has given rise to many developments. These range from optical spanners for driving micro-machines to high dimensional quantum entanglement and new opportunities in quantum information processing. The concept of orbital angular momentum is now leading to new understanding of a wide range of phenomena, including fundamental processes in Bose-Einstein condensates, while the associated technologies have led to new applications in optical tweezing and microscopy.

We contrast the two situations in which either a light beam is incident on a moving medium or a moving optical image is incident on a stationary medium. The principle of relativity suggests that the effects on the light of propagating through the medium should be similar. We find, however, that there are subtle differences which we can understand in terms of the relative alignment of the Poynting and wave vectors. Our analysis and experiments investigate both translational motion and rotation.

In the 1970s Jones demonstrated a mechanical Faraday effect where a spinning window rotated linearly polarized light through a small angle. A similar treatment applied to orbital angular momentum predicts a rotation of an image through the identical angle. However, this latter effect can be interpreted as a transverse photon drag, also observed by Jones, acting around a rotation axis. Rather than translating the medium, the speed of which is limited by mechanical considerations, we translate the image and measure its lateral delay with respect to a similar image that has not passed through the window. Our initial results are not what we expected.

In the 1970s Jones demonstrated a photon drag by showing that the translation of a window caused a slight displacement of a transmitted light beam. Similarly he showed that a spinning medium slightly rotated the polarization state. Rather than translating the medium, the speed of which is limited by mechanical considerations, we translate the image and measure its lateral delay with respect to a similar image that has not passed through the window. The equivalence, or lack of it, of the two frames is subtle and great care needs to be taken in determining whether or not similar results are to be obtained.

We consider the analogous geometric transformations for spin and orbital angular momentum states of light. Spin angular momentum is manifested as polarization and its possible transformations are typified by those introduced by waveplates and the rotation associated with optical activity. Orbital angular momentum is associated with the mode structure of the beam and, while the action of a waveplate is similar to that of a mode converter, the equivalent analogue of optical activity is not obvious. We reason that the equivalent is a rotation of the transmitted image. We consider the extent to which an image orientation of this type might be achieved by a coherent fibre bundle, twisted around its central axis. The possibility of equivalents to the Kerr, Pockels and Faraday electro-optic effects for orbital angular momentum is raised.

The realization that light beams can have quantized orbital angular momentum in addition to spin angular momentum has led, in recent years, to novel experiments in quantum mechanics and new methods for manipulating microparticles

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation L. Allen, M. J. Padgett; Response to Question #79. Does a plane wave carry spin angular momentum?. Am. J. Phys. 1 June 2002; 70 (6): 567–568. https://doi.org/10.1119/1.1456075 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAmerican Association of Physics TeachersAmerican Journal of Physics Search Advanced Search |Citation Search

We investigate the orbital angular momentum correlation of a photon pair created in a spontaneous parametric down-conversion process. We show how the conservation of the orbital angular momentum in this process results from phase matching in the nonlinear crystal.

We explain that, unlike the spin angular momentum of a light beam which is always intrinsic, the orbital angular momentum may be either extrinsic or intrinsic. Numerical calculations of both spin and orbital angular momentum are confirmed by means of experiments with particles trapped off axis in optical tweezers, where the size of the particle means it interacts with only a fraction of the beam profile. Orbital angular momentum is intrinsic only when the interaction with matter is about an axis where there is no net transverse momentum.