Orbital angular momentum of free electrons explained

Electrons in free space can carry quantized orbital angular momentum (OAM) projected along the direction of propagation.[1] This orbital angular momentum corresponds to helical wavefronts, or, equivalently, a phase proportional to the azimuthal angle.[2] Electron beams with quantized orbital angular momentum are also called electron vortex beams.

Theory

An electron in free space travelling at non-relativistic speeds, follows the Schrödinger equation for a free particle, that is i\hbar\frac \Psi(\mathbf,t) = \frac\nabla^2 \Psi(\mathbf,t), where \hbar is the reduced Planck constant, \Psi(\mathbf r, t) is the single-electron wave function, m its mass, \mathbf r the position vector, and t is time.This equation is a type of wave equation and when written in the Cartesian coordinate system (x,y,z), the solutions are given by a linear combination of plane waves, in the form of \Psi_(\mathbf,t)\propto e^ where \mathbf is the linear momentum and E(\mathbf) is the electron energy, given by the usual dispersion relation E(\mathbf)=\frac. By measuring the momentum of the electron, its wave function must collapse and give a particular value. If the energy of the electron beam is selected beforehand, the total momentum (not its directional components) of the electrons is fixed to a certain degree of precision.When the Schrödinger equation is written in the cylindrical coordinate system (\rho,\theta,z), the solutions are no longer plane waves, but instead are given by Bessel beams, solutions that are a linear combination of \Psi_(\rho,\theta,z)\propto J_

\left(\frac \right)e^e^, that is, the product of three types of functions: a plane wave with momentum p_z in the z-direction, a radial component written as a Bessel function of the first kind J_
, where p_\rho is the linear momentum in the radial direction, and finally an azimuthal component written as e^ where \ell (sometimes written m_z) is the magnetic quantum number related to the angular momentum L_z in the z-direction. Thus, the dispersion relation reads E=(p_z^2+p_\rho^2)/2m. By azimuthal symmetry, the wave function has the property that \ell=0,\pm 1,\pm2,\cdots is necessarily an integer, thus L_z = \hbar\ell is quantized. If a measurement of L_z is performed on an electron with selected energy, as E does not depend on \ell, it can give any integer value. It is possible to experimentally prepare states with non-zero \ell by adding an azimuthal phase to an initial state with \ell = 0; experimental techniques designed to measure the orbital angular momentum of a single electron are under development. Simultaneous measurement of electron energy and orbital angular momentum is allowed because the Hamiltonian commutes with the angular momentum operator related to L_z.

Note that the equations above follow for any free quantum particle with mass, not necessarily electrons. The quantization of

Lz

can also be shown in the spherical coordinate system, where the wave function reduces to a product of spherical Bessel functions and spherical harmonics.

Preparation

There are a variety of methods to prepare an electron in an orbital angular momentum state. All methods involve an interaction with an optical element such that the electron acquires an azimuthal phase. The optical element can be material,[3] [4] [5] magnetostatic,[6] or electrostatic.[7] It is possible to either directly imprint an azimuthal phase, or to imprint an azimuthal phase with a holographic diffraction grating, where grating pattern is defined by the interference of the azimuthal phase and a planar[8] or spherical[9] carrier wave.

Applications

Electron vortex beams have a variety of proposed and demonstrated applications, including for mapping magnetization,[10] [11] [12] studying chiral molecules and chiral plasmon resonances,[13] and identification of crystal chirality.[14]

Measurement

Interferometric methods borrowed from light optics also work to determine the orbital angular momentum of free electrons in pure states. Interference with a planar reference wave,[5] diffractive filtering and self-interference[15] [16] [17] can serve to characterize a prepared electron orbital angular momentum state. In order to measure the orbital angular momentum of a superposition or of the mixed state that results from interaction with an atom or material, a non-interferometric method is necessary. Wavefront flattening,[18] transformation of an orbital angular momentum state into a planar wave,[19] or cylindrically symmetric Stern-Gerlach-like measurement[20] is necessary to measure the orbital angular momentum mixed or superposition state.

See also

Notes and References

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    10.1103/RevModPhys.89.035004. 89. 3. 035004. Lloyd. S. M.. Babiker. M.. Thirunavukkarasu. G.. Yuan. J.. Electron vortices: Beams with orbital angular momentum. Reviews of Modern Physics. 2017-08-16. 2017RvMP...89c5004L. 125753983.

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    10.1088/1367-2630/16/9/093039. 16. 9. 093039. Harvey. Tyler R.. Pierce. Jordan S.. Agrawal. Amit K.. Ercius. Peter. Linck. Martin. McMorran. Benjamin J.. Efficient diffractive phase optics for electrons. New Journal of Physics. 2014-09-01. 2014NJPh...16i3039H. free.

  9. 10.1093/jmicro/dfs036 . 22394576. 61. 3. 171–177. Saitoh. Koh. Hasegawa. Yuya. Tanaka. Nobuo. Uchida. Masaya. Production of electron vortex beams carrying large orbital angular momentum using spiral zone plates. Journal of Electron Microscopy. 2012-06-01. free.

    10.1016/j.ultramic.2011.10.008. 113. 83–87. Verbeeck. J.. Tian. H.. Béché. A.. A new way of producing electron vortex probes for STEM. Ultramicroscopy. February 2012. 1405.7222. 54728013.

  10. 10.1093/jmicro/dfr069. 21949052. 60. 5. 295–300. Idrobo. Juan C.. Pennycook. Stephen J.. Vortex beams for atomic resolution dichroism. Journal of Electron Microscopy. 2011-10-01. 2011MiMic..17S1296I. free.
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    Harvey. Tyler R.. Pierce. Jordan S.. Chess. Jordan J.. McMorran. Benjamin J.. Demonstration of electron helical dichroism as a local probe of chirality. 2015-07-05. 1507.01810. cond-mat.mtrl-sci.

    10.1038/ncomms14999. 28401942. 5394338. 8. 14999. Guzzinati. Giulio. Béché. Armand. Lourenço-Martins. Hugo. Martin. Jérôme. Kociak. Mathieu. Verbeeck. Jo. Probing the symmetry of the potential of localized surface plasmon resonances with phase-shaped electron beams. Nature Communications. 2017-04-12. 1608.07449. 2017NatCo...814999G.

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    10.1038/ncomms15536. 28537248. 5458084. 8. 15536. Grillo. Vincenzo. Tavabi. Amir H.. Venturi. Federico. Larocque. Hugo. Balboni. Roberto. Gazzadi. Gian Carlo. Frabboni. Stefano. Lu. Peng-Han. Mafakheri. Erfan. Bouchard. Frédéric. Dunin-Borkowski. Rafal E.. Boyd. Robert W.. Lavery. Martin P. J.. Padgett. Miles J.. Karimi. Ebrahim. Measuring the orbital angular momentum spectrum of an electron beam. Nature Communications. 2017-05-24. 2017NatCo...815536G.

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