Orbital_angular_momentum_of_free_electrons Knowpia

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 edit

An electron in free space travelling at non-relativistic speeds, follows the Schrödinger equation for a free particle, that is

i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)={\frac {-\hbar ^{2}}{2m}}\nabla ^{2}\Psi (\mathbf {r} ,t),

where

{\textstyle \hbar }

is the reduced Planck constant,

{\textstyle \Psi (\mathbf {r} ,t)}

is the single-electron wave function,

{\textstyle m}

its mass,

{\textstyle \mathbf {r} }

the position vector, and

{\textstyle t}

is time. This equation is a type of wave equation and when written in the Cartesian coordinate system (

{\textstyle x}

,

{\textstyle y}

,

{\textstyle z}

), the solutions are given by a linear combination of plane waves, in the form of

\Psi _{\mathbf {p} }(\mathbf {r} ,t)\propto e^{i(\mathbf {p} \cdot \mathbf {r} -E(\mathbf {p} )t)/\hbar }

where

{\textstyle \mathbf {p} }

is the linear momentum and

{\textstyle E(\mathbf {p} )}

is the electron energy, given by the usual dispersion relation

{\textstyle E(\mathbf {p} )={\frac {p^{2}}{2m}}}

. 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 (

{\textstyle \rho }

,

{\textstyle \theta }

,

{\textstyle z}

), the solutions are no longer plane waves, but instead are given by Bessel beams,^[2] solutions that are a linear combination of

\Psi _{p_{\rho },\,p_{z},\,\ell }(\rho ,\theta ,z)\propto J_{|\ell |}\left({\frac {p_{\rho }\rho }{\hbar }}\right)e^{i(p_{z}z-Et)/\hbar }e^{i\ell \theta },

that is, the product of three types of functions: a plane wave with momentum

{\textstyle p_{z}}

in the

{\textstyle z}

-direction, a radial component written as a Bessel function of the first kind

{\textstyle J_{|\ell |}}

, where

{\textstyle p_{\rho }}

is the linear momentum in the radial direction, and finally an azimuthal component written as

{\textstyle e^{i\ell \theta }}

where

{\textstyle \ell }

(sometimes written

{\textstyle m_{z}}

) is the magnetic quantum number related to the angular momentum

{\textstyle L_{z}}

in the

{\textstyle z}

-direction. Thus, the dispersion relation reads

{\textstyle E=(p_{z}^{2}+p_{\rho }^{2})/2m}

. By azimuthal symmetry, the wave function has the property that

{\textstyle \ell =0,\pm 1,\pm 2,\cdots }

is necessarily an integer, thus

{\textstyle L_{z}=\hbar \ell }

is quantized. If a measurement of

{\textstyle L_{z}}

is performed on an electron with selected energy, as

{\textstyle E}

does not depend on

{\textstyle \ell }

, it can give any integer value. It is possible to experimentally prepare states with non-zero

{\textstyle \ell }

by adding an azimuthal phase to an initial state with

{\textstyle \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

{\textstyle L_{z}}

.

Note that the equations above follow for any free quantum particle with mass, not necessarily electrons. The quantization of $L_{z}$ 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 edit

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 edit

Electron vortex beams have a variety of proposed and demonstrated applications, including for mapping magnetization,^[4]^[10]^[11]^[12] studying chiral molecules and chiral plasmon resonances,^[13] and identification of crystal chirality.^[14]

Measurement edit

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,^[17]^[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.

References edit

^ Bliokh, Konstantin; Bliokh, Yury; Savel’ev, Sergey; Nori, Franco (November 2007). "Semiclassical Dynamics of Electron Wave Packet States with Phase Vortices". Physical Review Letters. 99 (19): 190404. arXiv:0706.2486. Bibcode:2007PhRvL..99s0404B. doi:10.1103/PhysRevLett.99.190404. ISSN 0031-9007. PMID 18233051. S2CID 17918457.
^ ^a ^b Bliokh, K. Y.; Ivanov, I. P.; Guzzinati, G.; Clark, L.; Van Boxem, R.; Béché, A.; Juchtmans, R.; Alonso, M. A.; Schattschneider, P.; Nori, F.; Verbeeck, J. (2017-05-24). "Theory and applications of free-electron vortex states". Physics Reports. 690: 1–70. arXiv:1703.06879. Bibcode:2017PhR...690....1B. doi:10.1016/j.physrep.2017.05.006. ISSN 0370-1573. S2CID 119067068. Lloyd, S. M.; Babiker, M.; Thirunavukkarasu, G.; Yuan, J. (2017-08-16). "Electron vortices: Beams with orbital angular momentum" (PDF). Reviews of Modern Physics. 89 (3): 035004. Bibcode:2017RvMP...89c5004L. doi:10.1103/RevModPhys.89.035004. S2CID 125753983.
^ Uchida, Masaya; Tonomura, Akira (2010-04-01). "Generation of electron beams carrying orbital angular momentum". Nature. 464 (7289): 737–9. Bibcode:2010Natur.464..737U. doi:10.1038/nature08904. PMID 20360737. S2CID 4423382.
^ ^a ^b Verbeeck, J.; Tian, H.; Schattschneider, P. (2010). "Production and application of electron vortex beams". Nature. 467 (7313): 301–4. Bibcode:2010Natur.467..301V. doi:10.1038/nature09366. PMID 20844532. S2CID 2970408.
^ ^a ^b McMorran, Benjamin J.; Agrawal, Amit; Anderson, Ian M.; Herzing, Andrew A.; Lezec, Henri J.; McClelland, Jabez J.; Unguris, John (2011-01-14). "Electron Vortex Beams with High Quanta of Orbital Angular Momentum". Science. 331 (6014): 192–195. Bibcode:2011Sci...331..192M. doi:10.1126/science.1198804. PMID 21233382. S2CID 37753036.
^ Blackburn, A. M.; Loudon, J. C. (January 2014). "Vortex beam production and contrast enhancement from a magnetic spiral phase plate". Ultramicroscopy. 136: 127–143. doi:10.1016/j.ultramic.2013.08.009. PMID 24128851. Béché, Armand; Van Boxem, Ruben; Van Tendeloo, Gustaaf; Verbeeck, Jo (January 2014). "Magnetic monopole field exposed by electrons". Nature Physics. 10 (1): 26–29. arXiv:1305.0570. Bibcode:2014NatPh..10...26B. doi:10.1038/nphys2816. S2CID 17919153.
^ Pozzi, Giulio; Lu, Peng-Han; Tavabi, Amir H.; Duchamp, Martial; Dunin-Borkowski, Rafal E. (2017-10-01). "Generation of electron vortex beams using line charges via the electrostatic Aharonov-Bohm effect". Ultramicroscopy. 181: 191–196. doi:10.1016/j.ultramic.2017.06.001. PMID 28609665.
^ Grillo, Vincenzo; Gazzadi, Gian Carlo; Karimi, Ebrahim; Mafakheri, Erfan; Boyd, Robert W.; Frabboni, Stefano (2014-01-30). "Highly efficient electron vortex beams generated by nanofabricated phase holograms". Applied Physics Letters. 104 (4): 043109. Bibcode:2014ApPhL.104d3109G. doi:10.1063/1.4863564. S2CID 142215. Harvey, Tyler R.; Pierce, Jordan S.; Agrawal, Amit K.; Ercius, Peter; Linck, Martin; McMorran, Benjamin J. (2014-09-01). "Efficient diffractive phase optics for electrons". New Journal of Physics. 16 (9): 093039. Bibcode:2014NJPh...16i3039H. doi:10.1088/1367-2630/16/9/093039.
^ Saitoh, Koh; Hasegawa, Yuya; Tanaka, Nobuo; Uchida, Masaya (2012-06-01). "Production of electron vortex beams carrying large orbital angular momentum using spiral zone plates". Journal of Electron Microscopy. 61 (3): 171–177. doi:10.1093/jmicro/dfs036. PMID 22394576. Verbeeck, J.; Tian, H.; Béché, A. (February 2012). "A new way of producing electron vortex probes for STEM". Ultramicroscopy. 113: 83–87. arXiv:1405.7222. doi:10.1016/j.ultramic.2011.10.008. S2CID 54728013.
^ Idrobo, Juan C.; Pennycook, Stephen J. (2011-10-01). "Vortex beams for atomic resolution dichroism". Journal of Electron Microscopy. 60 (5): 295–300. Bibcode:2011MiMic..17S1296I. doi:10.1093/jmicro/dfr069. PMID 21949052.
^ Lloyd, Sophia; Babiker, Mohamed; Yuan, Jun (2012-02-15). "Quantized Orbital Angular Momentum Transfer and Magnetic Dichroism in the Interaction of Electron Vortices with Matter". Physical Review Letters. 108 (7): 074802. arXiv:1111.3259. Bibcode:2012PhRvL.108g4802L. doi:10.1103/PhysRevLett.108.074802. PMID 22401214. S2CID 14016354.
^ Rusz, Ján; Bhowmick, Somnath (2013-09-06). "Boundaries for Efficient Use of Electron Vortex Beams to Measure Magnetic Properties". Physical Review Letters. 111 (10): 105504. arXiv:1304.5461. Bibcode:2013PhRvL.111j5504R. doi:10.1103/PhysRevLett.111.105504. PMID 25166681. S2CID 42498494.
^ Asenjo-Garcia, A.; García de Abajo, F. J. (2014-08-06). "Dichroism in the Interaction between Vortex Electron Beams, Plasmons, and Molecules". Physical Review Letters. 113 (6): 066102. Bibcode:2014PhRvL.113f6102A. doi:10.1103/PhysRevLett.113.066102. PMID 25148337. Harvey, Tyler R.; Pierce, Jordan S.; Chess, Jordan J.; McMorran, Benjamin J. (2015-07-05). "Demonstration of electron helical dichroism as a local probe of chirality". arXiv:1507.01810 [cond-mat.mtrl-sci]. Guzzinati, Giulio; Béché, Armand; Lourenço-Martins, Hugo; Martin, Jérôme; Kociak, Mathieu; Verbeeck, Jo (2017-04-12). "Probing the symmetry of the potential of localized surface plasmon resonances with phase-shaped electron beams". Nature Communications. 8: 14999. arXiv:1608.07449. Bibcode:2017NatCo...814999G. doi:10.1038/ncomms14999. PMC 5394338. PMID 28401942.
^ Juchtmans, Roeland; Béché, Armand; Abakumov, Artem; Batuk, Maria; Verbeeck, Jo (2015-03-26). "Using electron vortex beams to determine chirality of crystals in transmission electron microscopy". Physical Review B. 91 (9): 094112. arXiv:1410.2155. Bibcode:2015PhRvB..91i4112J. doi:10.1103/PhysRevB.91.094112. S2CID 19753751.
^ Shiloh, Roy; Tsur, Yuval; Remez, Roei; Lereah, Yossi; Malomed, Boris A.; Shvedov, Vladlen; Hnatovsky, Cyril; Krolikowski, Wieslaw; Arie, Ady (2015-03-04). "Unveiling the Orbital Angular Momentum and Acceleration of Electron Beams". Physical Review Letters. 114 (9): 096102. arXiv:1402.3133. Bibcode:2015PhRvL.114i6102S. doi:10.1103/PhysRevLett.114.096102. PMID 25793830. S2CID 6396731.
^ Clark, L.; Béché, A.; Guzzinati, G.; Verbeeck, J. (2014-05-13). "Quantitative measurement of orbital angular momentum in electron microscopy". Physical Review A. 89 (5): 053818. arXiv:1403.4398. Bibcode:2014PhRvA..89e3818C. doi:10.1103/PhysRevA.89.053818. S2CID 45042167.
^ ^a ^b Guzzinati, Giulio; Clark, Laura; Béché, Armand; Verbeeck, Jo (2014-02-13). "Measuring the orbital angular momentum of electron beams". Physical Review A. 89 (2): 025803. arXiv:1401.7211. Bibcode:2014PhRvA..89b5803G. doi:10.1103/PhysRevA.89.025803. S2CID 19593282.
^ Saitoh, Koh; Hasegawa, Yuya; Hirakawa, Kazuma; Tanaka, Nobuo; Uchida, Masaya (2013-08-14). "Measuring the Orbital Angular Momentum of Electron Vortex Beams Using a Forked Grating". Physical Review Letters. 111 (7): 074801. arXiv:1307.6304. Bibcode:2013PhRvL.111g4801S. doi:10.1103/PhysRevLett.111.074801. PMID 23992070. S2CID 37702862.
^ McMorran, Benjamin J.; Harvey, Tyler R.; Lavery, Martin P. J. (2017). "Efficient sorting of free electron orbital angular momentum". New Journal of Physics. 19 (2): 023053. arXiv:1609.09124. Bibcode:2017NJPh...19b3053M. doi:10.1088/1367-2630/aa5f6f. S2CID 119192171. 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 (2017-05-24). "Measuring the orbital angular momentum spectrum of an electron beam". Nature Communications. 8: 15536. Bibcode:2017NatCo...815536G. doi:10.1038/ncomms15536. PMC 5458084. PMID 28537248.
^ Harvey, Tyler R.; Grillo, Vincenzo; McMorran, Benjamin J. (2017-02-28). "Stern-Gerlach-like approach to electron orbital angular momentum measurement". Physical Review A. 95 (2): 021801. arXiv:1606.03631. Bibcode:2017PhRvA..95b1801H. doi:10.1103/PhysRevA.95.021801. S2CID 119086719.

[1] Bliokh, Konstantin; Bliokh, Yury; Savel’ev, Sergey; Nori, Franco (November 2007). "Semiclassical Dynamics of Electron Wave Packet States with Phase Vortices". Physical Review Letters. 99 (19): 190404. arXiv:0706.2486. Bibcode:2007PhRvL..99s0404B. doi:10.1103/PhysRevLett.99.190404. ISSN 0031-9007. PMID 18233051. S2CID 17918457.

[Bliokh_2017-2] Bliokh, K. Y.; Ivanov, I. P.; Guzzinati, G.; Clark, L.; Van Boxem, R.; Béché, A.; Juchtmans, R.; Alonso, M. A.; Schattschneider, P.; Nori, F.; Verbeeck, J. (2017-05-24). "Theory and applications of free-electron vortex states". Physics Reports. 690: 1–70. arXiv:1703.06879. Bibcode:2017PhR...690....1B. doi:10.1016/j.physrep.2017.05.006. ISSN 0370-1573. S2CID 119067068. Lloyd, S. M.; Babiker, M.; Thirunavukkarasu, G.; Yuan, J. (2017-08-16). "Electron vortices: Beams with orbital angular momentum" (PDF). Reviews of Modern Physics. 89 (3): 035004. Bibcode:2017RvMP...89c5004L. doi:10.1103/RevModPhys.89.035004. S2CID 125753983.

[3] Uchida, Masaya; Tonomura, Akira (2010-04-01). "Generation of electron beams carrying orbital angular momentum". Nature. 464 (7289): 737–9. Bibcode:2010Natur.464..737U. doi:10.1038/nature08904. PMID 20360737. S2CID 4423382.

[production_and_application-4] Verbeeck, J.; Tian, H.; Schattschneider, P. (2010). "Production and application of electron vortex beams". Nature. 467 (7313): 301–4. Bibcode:2010Natur.467..301V. doi:10.1038/nature09366. PMID 20844532. S2CID 2970408.

[sciencemag.org-5] McMorran, Benjamin J.; Agrawal, Amit; Anderson, Ian M.; Herzing, Andrew A.; Lezec, Henri J.; McClelland, Jabez J.; Unguris, John (2011-01-14). "Electron Vortex Beams with High Quanta of Orbital Angular Momentum". Science. 331 (6014): 192–195. Bibcode:2011Sci...331..192M. doi:10.1126/science.1198804. PMID 21233382. S2CID 37753036.

[6] Blackburn, A. M.; Loudon, J. C. (January 2014). "Vortex beam production and contrast enhancement from a magnetic spiral phase plate". Ultramicroscopy. 136: 127–143. doi:10.1016/j.ultramic.2013.08.009. PMID 24128851. Béché, Armand; Van Boxem, Ruben; Van Tendeloo, Gustaaf; Verbeeck, Jo (January 2014). "Magnetic monopole field exposed by electrons". Nature Physics. 10 (1): 26–29. arXiv:1305.0570. Bibcode:2014NatPh..10...26B. doi:10.1038/nphys2816. S2CID 17919153.

[7] Pozzi, Giulio; Lu, Peng-Han; Tavabi, Amir H.; Duchamp, Martial; Dunin-Borkowski, Rafal E. (2017-10-01). "Generation of electron vortex beams using line charges via the electrostatic Aharonov-Bohm effect". Ultramicroscopy. 181: 191–196. doi:10.1016/j.ultramic.2017.06.001. PMID 28609665.

[8] Grillo, Vincenzo; Gazzadi, Gian Carlo; Karimi, Ebrahim; Mafakheri, Erfan; Boyd, Robert W.; Frabboni, Stefano (2014-01-30). "Highly efficient electron vortex beams generated by nanofabricated phase holograms". Applied Physics Letters. 104 (4): 043109. Bibcode:2014ApPhL.104d3109G. doi:10.1063/1.4863564. S2CID 142215. Harvey, Tyler R.; Pierce, Jordan S.; Agrawal, Amit K.; Ercius, Peter; Linck, Martin; McMorran, Benjamin J. (2014-09-01). "Efficient diffractive phase optics for electrons". New Journal of Physics. 16 (9): 093039. Bibcode:2014NJPh...16i3039H. doi:10.1088/1367-2630/16/9/093039.

[9] Saitoh, Koh; Hasegawa, Yuya; Tanaka, Nobuo; Uchida, Masaya (2012-06-01). "Production of electron vortex beams carrying large orbital angular momentum using spiral zone plates". Journal of Electron Microscopy. 61 (3): 171–177. doi:10.1093/jmicro/dfs036. PMID 22394576. Verbeeck, J.; Tian, H.; Béché, A. (February 2012). "A new way of producing electron vortex probes for STEM". Ultramicroscopy. 113: 83–87. arXiv:1405.7222. doi:10.1016/j.ultramic.2011.10.008. S2CID 54728013.

[10] Idrobo, Juan C.; Pennycook, Stephen J. (2011-10-01). "Vortex beams for atomic resolution dichroism". Journal of Electron Microscopy. 60 (5): 295–300. Bibcode:2011MiMic..17S1296I. doi:10.1093/jmicro/dfr069. PMID 21949052.

[11] Lloyd, Sophia; Babiker, Mohamed; Yuan, Jun (2012-02-15). "Quantized Orbital Angular Momentum Transfer and Magnetic Dichroism in the Interaction of Electron Vortices with Matter". Physical Review Letters. 108 (7): 074802. arXiv:1111.3259. Bibcode:2012PhRvL.108g4802L. doi:10.1103/PhysRevLett.108.074802. PMID 22401214. S2CID 14016354.

[12] Rusz, Ján; Bhowmick, Somnath (2013-09-06). "Boundaries for Efficient Use of Electron Vortex Beams to Measure Magnetic Properties". Physical Review Letters. 111 (10): 105504. arXiv:1304.5461. Bibcode:2013PhRvL.111j5504R. doi:10.1103/PhysRevLett.111.105504. PMID 25166681. S2CID 42498494.

[plasmon_dichroism-13] Asenjo-Garcia, A.; García de Abajo, F. J. (2014-08-06). "Dichroism in the Interaction between Vortex Electron Beams, Plasmons, and Molecules". Physical Review Letters. 113 (6): 066102. Bibcode:2014PhRvL.113f6102A. doi:10.1103/PhysRevLett.113.066102. PMID 25148337. Harvey, Tyler R.; Pierce, Jordan S.; Chess, Jordan J.; McMorran, Benjamin J. (2015-07-05). "Demonstration of electron helical dichroism as a local probe of chirality". arXiv:1507.01810 [cond-mat.mtrl-sci]. Guzzinati, Giulio; Béché, Armand; Lourenço-Martins, Hugo; Martin, Jérôme; Kociak, Mathieu; Verbeeck, Jo (2017-04-12). "Probing the symmetry of the potential of localized surface plasmon resonances with phase-shaped electron beams". Nature Communications. 8: 14999. arXiv:1608.07449. Bibcode:2017NatCo...814999G. doi:10.1038/ncomms14999. PMC 5394338. PMID 28401942.

[14] Juchtmans, Roeland; Béché, Armand; Abakumov, Artem; Batuk, Maria; Verbeeck, Jo (2015-03-26). "Using electron vortex beams to determine chirality of crystals in transmission electron microscopy". Physical Review B. 91 (9): 094112. arXiv:1410.2155. Bibcode:2015PhRvB..91i4112J. doi:10.1103/PhysRevB.91.094112. S2CID 19753751.

[15] Shiloh, Roy; Tsur, Yuval; Remez, Roei; Lereah, Yossi; Malomed, Boris A.; Shvedov, Vladlen; Hnatovsky, Cyril; Krolikowski, Wieslaw; Arie, Ady (2015-03-04). "Unveiling the Orbital Angular Momentum and Acceleration of Electron Beams". Physical Review Letters. 114 (9): 096102. arXiv:1402.3133. Bibcode:2015PhRvL.114i6102S. doi:10.1103/PhysRevLett.114.096102. PMID 25793830. S2CID 6396731.

[16] Clark, L.; Béché, A.; Guzzinati, G.; Verbeeck, J. (2014-05-13). "Quantitative measurement of orbital angular momentum in electron microscopy". Physical Review A. 89 (5): 053818. arXiv:1403.4398. Bibcode:2014PhRvA..89e3818C. doi:10.1103/PhysRevA.89.053818. S2CID 45042167.

[meas._diff._and_wf-17] Guzzinati, Giulio; Clark, Laura; Béché, Armand; Verbeeck, Jo (2014-02-13). "Measuring the orbital angular momentum of electron beams". Physical Review A. 89 (2): 025803. arXiv:1401.7211. Bibcode:2014PhRvA..89b5803G. doi:10.1103/PhysRevA.89.025803. S2CID 19593282.

[18] Saitoh, Koh; Hasegawa, Yuya; Hirakawa, Kazuma; Tanaka, Nobuo; Uchida, Masaya (2013-08-14). "Measuring the Orbital Angular Momentum of Electron Vortex Beams Using a Forked Grating". Physical Review Letters. 111 (7): 074801. arXiv:1307.6304. Bibcode:2013PhRvL.111g4801S. doi:10.1103/PhysRevLett.111.074801. PMID 23992070. S2CID 37702862.

[19] McMorran, Benjamin J.; Harvey, Tyler R.; Lavery, Martin P. J. (2017). "Efficient sorting of free electron orbital angular momentum". New Journal of Physics. 19 (2): 023053. arXiv:1609.09124. Bibcode:2017NJPh...19b3053M. doi:10.1088/1367-2630/aa5f6f. S2CID 119192171. 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 (2017-05-24). "Measuring the orbital angular momentum spectrum of an electron beam". Nature Communications. 8: 15536. Bibcode:2017NatCo...815536G. doi:10.1038/ncomms15536. PMC 5458084. PMID 28537248.

[20] Harvey, Tyler R.; Grillo, Vincenzo; McMorran, Benjamin J. (2017-02-28). "Stern-Gerlach-like approach to electron orbital angular momentum measurement". Physical Review A. 95 (2): 021801. arXiv:1606.03631. Bibcode:2017PhRvA..95b1801H. doi:10.1103/PhysRevA.95.021801. S2CID 119086719.

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Summary

Theory edit

Preparation edit

Applications edit

Measurement edit

See also edit

References edit