View source for Orbital angular momentum of free electrons

{{Short description|Quantised attribute of electrons in free space}}
[[File:Laguerre-gauss modes larger text.svg|alt=|thumb|upright=1.5|Phase (color) and amplitude (brightness) of electron wavefunctions with several values of the orbital [[angular momentum]] [[quantum number]] <math> m </math> and a [[Gaussian beam|Laguerre-Gauss]] amplitude profile. <math> \ell = +1 </math> (top left), <math> \ell = -1 </math> (top right), <math> \ell = 0 </math> (lower left) are all eigenstates of the orbital [[angular momentum operator]], while the superposition of <math> \ell = +1 </math> and <math> \ell = -1 </math> (lower right) is not. Both of the upper wavefunctions have <math> \langle L_z \rangle \neq 0 </math>, while the lower wavefunctions have <math> \langle L_z \rangle = 0 </math>.]]
[[Electron]]s in free space can carry [[angular momentum quantization|quantized]] orbital [[angular momentum]] (OAM) projected along the direction of propagation.<ref>

{{Cite journal| doi = 10.1103/PhysRevLett.99.190404| pmid = 18233051| issn = 0031-9007| volume = 99| issue = 19| pages = 190404| last1 = Bliokh| first1 = Konstantin| last2 = Bliokh| first2 = Yury| last3 = Savel’ev| first3 = Sergey| last4 = Nori| first4 = Franco| title = Semiclassical Dynamics of Electron Wave Packet States with Phase Vortices| journal = Physical Review Letters| date = November 2007| arxiv = 0706.2486| bibcode = 2007PhRvL..99s0404B| s2cid = 17918457}}

</ref> This orbital angular momentum corresponds to [[helix|helical]] wavefronts, or, equivalently, a [[Phase (waves)|phase]] proportional to the [[azimuth]]al angle.<ref name="Bliokh 2017">{{Cite journal| doi = 10.1016/j.physrep.2017.05.006| issn = 0370-1573| volume = 690| pages = 1–70| last1 = Bliokh| first1 = K. Y.| last2 = Ivanov| first2 = I. P.| last3 = Guzzinati| first3 = G.| last4 = Clark| first4 = L.| last5 = Van Boxem| first5 = R.| last6 = Béché| first6 = A.| last7 = Juchtmans| first7 = R.| last8 = Alonso| first8 = M. A.| last9 = Schattschneider| first9 = P.| last10 = Nori| first10 = F.| last11 = Verbeeck| first11 = J.| title = Theory and applications of free-electron vortex states| journal = Physics Reports| date = 2017-05-24| bibcode = 2017PhR...690....1B| arxiv = 1703.06879| s2cid = 119067068}}

{{Cite journal| doi = 10.1103/RevModPhys.89.035004| volume = 89| issue = 3| pages = 035004| last1 = Lloyd| first1 = S. M.| last2 = Babiker| first2 = M.| last3 = Thirunavukkarasu| first3 = G.| last4 = Yuan| first4 = J.| title = Electron vortices: Beams with orbital angular momentum| journal = Reviews of Modern Physics| date = 2017-08-16| bibcode = 2017RvMP...89c5004L| s2cid = 125753983| url = https://eprints.whiterose.ac.uk/133764/1/Lloyd2017_RK10072_preprint.pdf}}

</ref> Electron beams with quantized orbital angular momentum are also called '''electron vortex beams'''.

== Theory ==
An electron in free space travelling at non-[[Special relativity|relativistic]] speeds, follows the [[Schrödinger equation]] for a [[free particle]], that is <math display="block">i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \frac{-\hbar^2}{2m}\nabla^2 \Psi(\mathbf{r},t),</math> where <math display="inline">\hbar</math> is the reduced [[Planck constant]], <math display="inline">\Psi(\mathbf r , t)</math> is the single-electron [[wave function]], <math display="inline">m</math> its mass, <math display="inline">\mathbf r</math> the position vector, and <math display="inline">t</math> is time.
This equation is a type of [[wave equation]] and when written in the [[Cartesian coordinate system]] (<math display="inline">x</math>,<math display="inline">y</math>,<math display="inline">z</math>), the solutions are given by a [[linear combination]] of [[plane wave]]s, in the form of <math display="block">\Psi_{\mathbf p}(\mathbf{r},t)\propto e^{i(\mathbf{p}\cdot\mathbf{r}-E(\mathbf{p})t)/\hbar}</math> where <math display="inline">\mathbf{p}</math> is the [[Momentum|linear momentum]] and <math display="inline">E(\mathbf{p})</math> is the electron energy, given by the usual [[dispersion relation]] <math display="inline">E(\mathbf{p})=\frac{p^2}{2m}</math>. By measuring the momentum of the electron, its [[Wave function collapse|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]] (<math display="inline">\rho</math>,<math display="inline">\theta</math>,<math display="inline">z</math>), the solutions are no longer plane waves, but instead are given by [[Bessel beam|Bessel beams]],<ref name="Bliokh 2017" /> solutions that are a linear combination of <math display="block">\Psi_{p_\rho,\,p_z,\,\ell}(\rho,\theta,z)\propto J_{|\ell|}\left(\frac{p_\rho \rho}{\hbar} \right)e^{i(p_zz-Et)/\hbar}e^{i\ell\theta},</math> that is, the product of three types of functions: a plane wave with momentum <math display="inline">p_z</math>  in the <math display="inline">z</math>-direction, a radial component written as a [[Bessel function of the first kind]] <math display="inline">J_{|\ell|}</math>, where <math display="inline">p_\rho</math> is the linear momentum in the radial direction, and finally an azimuthal component written as <math display="inline">e^{i\ell\theta}</math> where <math display="inline">\ell</math> (sometimes written <math display="inline">m_z</math>) is the [[magnetic quantum number]] related to the angular momentum <math display="inline">L_z</math> in the <math display="inline">z</math>-direction. Thus, the dispersion relation reads  <math display="inline">E=(p_z^2+p_\rho^2)/2m</math>. By azimuthal symmetry, the wave function has the property that  <math display="inline">\ell=0,\pm 1,\pm2,\cdots</math> is necessarily an [[integer]], thus <math display="inline">L_z = \hbar\ell</math> is quantized. If a measurement of <math display="inline">L_z</math> is performed on an electron with selected energy, as <math display="inline">E</math> does not depend on <math display="inline">\ell</math>, it can give any integer value. It is possible to experimentally [[#Preparation|prepare]] states with non-zero <math display="inline">\ell</math> by adding an azimuthal phase to an initial state with <math display="inline">\ell = 0</math>; experimental techniques designed to [[#Measurement|measure]] the orbital angular momentum of a single electron are under development. Simultaneous [[Measurement in quantum mechanics|measurement]] of electron energy and orbital angular momentum is allowed because the [[Hamiltonian (quantum mechanics)|Hamiltonian]] [[Commutator|commutes]] with the [[angular momentum operator]] related to <math display="inline">L_z</math>.

Note that the equations above follow for any free quantum particle with mass, not necessarily electrons. The quantization of <math>L_z</math> can also be shown in the [[spherical coordinate system]], where the wave function reduces to a product of [[spherical Bessel function]]s 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 [[electron optics|optical element]] such that the electron acquires an azimuthal phase. The optical element can be material,<ref>

{{Cite journal| doi = 10.1038/nature08904| pmid = 20360737| volume = 464| issue = 7289| pages = 737–9| last1 = Uchida| first1 = Masaya| last2 = Tonomura| first2 = Akira| title = Generation of electron beams carrying orbital angular momentum| journal = Nature| date = 2010-04-01| bibcode = 2010Natur.464..737U| s2cid = 4423382}}

</ref><ref name='production and application'>

{{Cite journal| doi = 10.1038/nature09366| pmid = 20844532| volume = 467| issue = 7313| pages = 301–4| last1 = Verbeeck| first1 = J.| last2 = Tian| first2 = H.| last3 = Schattschneider| first3 = P.| title = Production and application of electron vortex beams| journal = Nature| date = 2010| bibcode = 2010Natur.467..301V| s2cid = 2970408}}

</ref><ref name="sciencemag.org">{{Cite journal| doi = 10.1126/science.1198804| pmid = 21233382| volume = 331| issue = 6014| pages = 192–195| last1 = McMorran| first1 = Benjamin J.| last2 = Agrawal| first2 = Amit| last3 = Anderson| first3 = Ian M.| last4 = Herzing| first4 = Andrew A.| last5 = Lezec| first5 = Henri J.| last6 = McClelland| first6 = Jabez J. |author6-link=Jabez Jenkins McClelland| last7 = Unguris| first7 = John| title = Electron Vortex Beams with High Quanta of Orbital Angular Momentum| journal = Science| date = 2011-01-14| bibcode = 2011Sci...331..192M| s2cid = 37753036}}</ref> magnetostatic,<ref>

{{Cite journal| doi = 10.1016/j.ultramic.2013.08.009| pmid = 24128851| volume = 136| pages = 127–143| last1 = Blackburn| first1 = A. M.| last2 = Loudon| first2 = J. C.| title = Vortex beam production and contrast enhancement from a magnetic spiral phase plate| journal = Ultramicroscopy| date = January 2014}}
{{Cite journal| doi = 10.1038/nphys2816| volume = 10| issue = 1| pages = 26–29| last1 = Béché| first1 = Armand| last2 = Van Boxem| first2 = Ruben| last3 = Van Tendeloo| first3 = Gustaaf| last4 = Verbeeck| first4 = Jo| title = Magnetic monopole field exposed by electrons| journal = Nature Physics| date = January 2014| arxiv = 1305.0570| bibcode = 2014NatPh..10...26B| s2cid = 17919153}}

</ref> or electrostatic.<ref>

{{Cite journal| doi = 10.1016/j.ultramic.2017.06.001| pmid = 28609665| volume = 181| pages = 191–196| last1 = Pozzi| first1 = Giulio| last2 = Lu| first2 = Peng-Han| last3 = Tavabi| first3 = Amir H.| last4 = Duchamp| first4 = Martial| last5 = Dunin-Borkowski| first5 = Rafal E.| title = Generation of electron vortex beams using line charges via the electrostatic Aharonov-Bohm effect| journal = Ultramicroscopy| date = 2017-10-01| doi-access = free}}

</ref> 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<ref>

{{Cite journal| doi = 10.1063/1.4863564 | volume = 104| issue = 4| pages = 043109| last1 = Grillo| first1 = Vincenzo| last2 = Gazzadi| first2 = Gian Carlo| last3 = Karimi| first3 = Ebrahim| last4 = Mafakheri| first4 = Erfan| last5 = Boyd| first5 = Robert W.| last6 = Frabboni| first6 = Stefano| title = Highly efficient electron vortex beams generated by nanofabricated phase holograms| journal = Applied Physics Letters| date = 2014-01-30| bibcode = 2014ApPhL.104d3109G| s2cid = 142215}}

{{Cite journal| doi = 10.1088/1367-2630/16/9/093039| volume = 16| issue = 9| pages = 093039| last1 = Harvey| first1 = Tyler R.| last2 = Pierce| first2 = Jordan S.| last3 = Agrawal| first3 = Amit K.| last4 = Ercius| first4 = Peter| last5 = Linck| first5 = Martin| last6 = McMorran| first6 = Benjamin J.| title = Efficient diffractive phase optics for electrons| journal = New Journal of Physics| date = 2014-09-01| bibcode = 2014NJPh...16i3039H| doi-access = free}}

</ref> or spherical<ref>
{{Cite journal| doi = 10.1093/jmicro/dfs036 | pmid = 22394576| volume = 61| issue = 3| pages = 171–177| last1 = Saitoh| first1 = Koh| last2 = Hasegawa| first2 = Yuya| last3 = Tanaka| first3 = Nobuo| last4 = Uchida| first4 = Masaya| title = Production of electron vortex beams carrying large orbital angular momentum using spiral zone plates| journal = Journal of Electron Microscopy| date = 2012-06-01| doi-access = free}}

{{Cite journal| doi = 10.1016/j.ultramic.2011.10.008| volume = 113| pages = 83–87| last1 = Verbeeck| first1 = J.| last2 = Tian| first2 = H.| last3 = Béché| first3 = A.| title = A new way of producing electron vortex probes for STEM| journal = Ultramicroscopy| date = February 2012| arxiv = 1405.7222| s2cid = 54728013}}

</ref> carrier wave.

== Applications ==

Electron vortex beams have a variety of proposed and demonstrated applications, including for [[electron magnetic circular dichroism|mapping magnetization]],<ref name='production and application' /><ref>

{{Cite journal| doi = 10.1093/jmicro/dfr069| pmid = 21949052| volume = 60| issue = 5| pages = 295–300| last1 = Idrobo| first1 = Juan C.| last2 = Pennycook| first2 = Stephen J.| title = Vortex beams for atomic resolution dichroism| journal = Journal of Electron Microscopy| date = 2011-10-01| bibcode = 2011MiMic..17S1296I| doi-access = free}}

</ref><ref>

{{Cite journal| doi = 10.1103/PhysRevLett.108.074802| pmid = 22401214| volume = 108| issue = 7| pages = 074802| last1 = Lloyd| first1 = Sophia| last2 = Babiker| first2 = Mohamed| last3 = Yuan| first3 = Jun| title = Quantized Orbital Angular Momentum Transfer and Magnetic Dichroism in the Interaction of Electron Vortices with Matter| journal = Physical Review Letters| date = 2012-02-15| arxiv = 1111.3259| bibcode = 2012PhRvL.108g4802L| s2cid = 14016354}}

</ref><ref>

{{Cite journal| doi = 10.1103/PhysRevLett.111.105504| pmid = 25166681| volume = 111| issue = 10| pages = 105504| last1 = Rusz| first1 = Ján| last2 = Bhowmick| first2 = Somnath| title = Boundaries for Efficient Use of Electron Vortex Beams to Measure Magnetic Properties| journal = Physical Review Letters| date = 2013-09-06| arxiv = 1304.5461| bibcode = 2013PhRvL.111j5504R| s2cid = 42498494}}

</ref> studying chiral molecules and chiral plasmon resonances,<ref name='plasmon dichroism'>

{{Cite journal| doi = 10.1103/PhysRevLett.113.066102| pmid = 25148337| volume = 113| issue = 6| pages = 066102| last1 = Asenjo-Garcia| first1 = A.| last2 = García de Abajo| first2 = F. J.| title = Dichroism in the Interaction between Vortex Electron Beams, Plasmons, and Molecules| journal = Physical Review Letters| date = 2014-08-06| bibcode = 2014PhRvL.113f6102A}}

{{cite arXiv| last1 = Harvey| first1 = Tyler R.| last2 = Pierce| first2 = Jordan S.| last3 = Chess| first3 = Jordan J.| last4 = McMorran| first4 = Benjamin J.| title = Demonstration of electron helical dichroism as a local probe of chirality| date = 2015-07-05| eprint = 1507.01810| class = cond-mat.mtrl-sci}}

{{Cite journal| doi = 10.1038/ncomms14999| pmid = 28401942| pmc = 5394338| volume = 8| pages = 14999| last1 = Guzzinati| first1 = Giulio| last2 = Béché| first2 = Armand| last3 = Lourenço-Martins| first3 = Hugo| last4 = Martin| first4 = Jérôme| last5 = Kociak| first5 = Mathieu| last6 = Verbeeck| first6 = Jo| title = Probing the symmetry of the potential of localized surface plasmon resonances with phase-shaped electron beams| journal = Nature Communications| date = 2017-04-12| arxiv = 1608.07449| bibcode = 2017NatCo...814999G}}

</ref> and identification of crystal chirality.<ref>

{{Cite journal| doi = 10.1103/PhysRevB.91.094112| volume = 91| issue = 9| pages = 094112| last1 = Juchtmans| first1 = Roeland| last2 = Béché| first2 = Armand| last3 = Abakumov| first3 = Artem| last4 = Batuk| first4 = Maria| last5 = Verbeeck| first5 = Jo| title = Using electron vortex beams to determine chirality of crystals in transmission electron microscopy| journal = Physical Review B| date = 2015-03-26| arxiv = 1410.2155| bibcode = 2015PhRvB..91i4112J| s2cid = 19753751}}
</ref>

== Measurement ==
[[Interferometric]] methods borrowed from [[orbital angular momentum of light|light optics]] also work to determine the orbital angular momentum of free electrons in pure states. Interference with a planar reference wave,<ref name="sciencemag.org"/> diffractive filtering and self-interference<ref>

{{Cite journal| doi = 10.1103/PhysRevLett.114.096102| pmid = 25793830| volume = 114| issue = 9| pages = 096102| last1 = Shiloh| first1 = Roy| last2 = Tsur| first2 = Yuval| last3 = Remez| first3 = Roei| last4 = Lereah| first4 = Yossi| last5 = Malomed| first5 = Boris A.| last6 = Shvedov| first6 = Vladlen| last7 = Hnatovsky| first7 = Cyril| last8 = Krolikowski| first8 = Wieslaw| last9 = Arie| first9 = Ady| title = Unveiling the Orbital Angular Momentum and Acceleration of Electron Beams| journal = Physical Review Letters| date = 2015-03-04| arxiv = 1402.3133| bibcode = 2015PhRvL.114i6102S| s2cid = 6396731}}

</ref><ref>

{{Cite journal| doi = 10.1103/PhysRevA.89.053818| volume = 89| issue = 5| pages = 053818| last1 = Clark| first1 = L.| last2 = Béché| first2 = A.| last3 = Guzzinati| first3 = G.| last4 = Verbeeck| first4 = J.| title = Quantitative measurement of orbital angular momentum in electron microscopy| journal = Physical Review A| date = 2014-05-13| arxiv = 1403.4398| bibcode = 2014PhRvA..89e3818C| s2cid = 45042167}}

</ref><ref name='meas. diff. and wf'>

{{Cite journal| doi = 10.1103/PhysRevA.89.025803| volume = 89| issue = 2| pages = 025803| last1 = Guzzinati| first1 = Giulio| last2 = Clark| first2 = Laura| last3 = Béché| first3 = Armand| last4 = Verbeeck| first4 = Jo| title = Measuring the orbital angular momentum of electron beams| journal = Physical Review A| date = 2014-02-13| arxiv = 1401.7211| bibcode = 2014PhRvA..89b5803G| s2cid = 19593282}}

</ref> 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,<ref name='meas. diff. and wf' /><ref>

{{Cite journal| doi = 10.1103/PhysRevLett.111.074801| pmid = 23992070| volume = 111| issue = 7| pages = 074801| last1 = Saitoh| first1 = Koh| last2 = Hasegawa| first2 = Yuya| last3 = Hirakawa| first3 = Kazuma| last4 = Tanaka| first4 = Nobuo| last5 = Uchida| first5 = Masaya| title = Measuring the Orbital Angular Momentum of Electron Vortex Beams Using a Forked Grating| journal = Physical Review Letters| date = 2013-08-14| arxiv = 1307.6304| bibcode = 2013PhRvL.111g4801S| s2cid = 37702862}}

</ref> transformation of an orbital angular momentum state into a planar wave,<ref>

{{Cite journal| doi = 10.1088/1367-2630/aa5f6f| volume = 19| issue = 2| pages = 023053| last1 = McMorran| first1 = Benjamin J.| last2 = Harvey| first2 = Tyler R.| last3 = Lavery| first3 = Martin P. J.| title = Efficient sorting of free electron orbital angular momentum| journal = New Journal of Physics| date = 2017| arxiv = 1609.09124| bibcode = 2017NJPh...19b3053M| s2cid = 119192171}}

{{Cite journal| doi = 10.1038/ncomms15536| pmid = 28537248| pmc = 5458084| volume = 8| pages = 15536| last1 = Grillo| first1 = Vincenzo| last2 = Tavabi| first2 = Amir H.| last3 = Venturi| first3 = Federico| last4 = Larocque| first4 = Hugo| last5 = Balboni| first5 = Roberto| last6 = Gazzadi| first6 = Gian Carlo| last7 = Frabboni| first7 = Stefano| last8 = Lu| first8 = Peng-Han| last9 = Mafakheri| first9 = Erfan| last10 = Bouchard| first10 = Frédéric| last11 = Dunin-Borkowski| first11 = Rafal E.| last12 = Boyd| first12 = Robert W.| last13 = Lavery| first13 = Martin P. J.| last14 = Padgett| first14 = Miles J.| last15 = Karimi| first15 = Ebrahim| title = Measuring the orbital angular momentum spectrum of an electron beam| journal = Nature Communications| date = 2017-05-24| bibcode = 2017NatCo...815536G}}

</ref> or cylindrically symmetric Stern-Gerlach-like measurement<ref>

{{Cite journal| doi = 10.1103/PhysRevA.95.021801| volume = 95| issue = 2| pages = 021801| last1 = Harvey| first1 = Tyler R.| last2 = Grillo| first2 = Vincenzo| last3 = McMorran| first3 = Benjamin J.| title = Stern-Gerlach-like approach to electron orbital angular momentum measurement| journal = Physical Review A| date = 2017-02-28| arxiv = 1606.03631| bibcode = 2017PhRvA..95b1801H| s2cid = 119086719}}

</ref> is necessary to measure the orbital angular momentum mixed or superposition state.

== See also ==

* [[Matter wave]]

==References==
{{reflist}}

[[Category:Angular momentum]]
[[Category:Electron beam]]
[[Category:Electron microscopy]]
[[Category:Orbital angular momentum of waves]]