Please wait a minute...
Chin. Phys. B, 2020, Vol. 29(8): 084102    DOI: 10.1088/1674-1056/ab96a9

Electromagnetic field of a relativistic electron vortex beam

Changyong Lei(雷长勇)1, Guangjiong Dong(董光炯)1,2
1 State Key Laboratory of Precision Spectroscopy, School of Physics and Electronics, East China Normal University, Shanghai 200241, China;
2 Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
Abstract  Electron vortex beams (EVBs) have potential applications in nanoscale magnetic probes of condensed matter and nanoparticle manipulation as well as radiation physics. Recently, a relativistic electron vortex beam (REVB) has been proposed[Phys. Rev. Lett. 107 174802 (2011)]. Compared with EVBs, except for orbital angular momentum, an REVB has intrinsic relativistic effect, i.e., spin angular momentum and spin-orbit coupling. We study the electromagnetic field of an REVB analytically. We show that the electromagnetic field can be separated into two parts, one is only related to orbital quantum number, and the other is related to spin-orbit coupling effect. Exploiting this separation property, the difference between the electromagnetic fields of the REVB in spin-up and spin-down states can be used as a demonstration of the relativistic quantum effect. The linear momentum and angular momentum of the generated electromagnetic field have been further studied and it is shown that the linear momentum is weakly dependent on the spin state; while the angular momentum is evidently dependent on the spin state and linearly increases with the topological charge of electron vortex beam. The electromagnetic and mechanical properties of the REVB could be useful for studying the interaction between REVBs and materials.
Keywords:  relativistic electron vortex beam      electromagnetic vortex field      spin-orbit coupling      orbital angular momentum  
Received:  25 February 2020      Revised:  05 April 2020      Accepted manuscript online: 
PACS:  41.85.-p (Beam optics)  
  42.50.Tx (Optical angular momentum and its quantum aspects)  
  03.65.Pm (Relativistic wave equations)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11574085, 91536218, and 11834003), the 111 Project, China (Grant No. B12024), the National Key Research and Development Program of China (Grant No. 2017YFA0304201), and the Innovation Program of Shanghai Municipal Education Commission, China (Grant No. 2019-01-07-00-05-E00079).
Corresponding Authors:  Guangjiong Dong     E-mail:

Cite this article: 

Changyong Lei(雷长勇), Guangjiong Dong(董光炯) Electromagnetic field of a relativistic electron vortex beam 2020 Chin. Phys. B 29 084102

[1] Bliokh K Y, Bliokh Y P, Savelev S and Nori F 2007 Phys. Rev. Lett. 99 190404
[2] Uchida M and Tonomura A 2010 Nature 464 737
[3] Verbeeck J, Tian H and Schattschneider P 2010 Nature 467 301
[4] McMorran B J, Agrawal A, Anderson I M, Herzing A A, Lezec H J, McClelland J J and Unguris J 2011 Science 331 192
[5] Grillo V, Gazzadi G C, Mafakheri E, Frabboni S, Karimi E and Boyd R W 2015 Phys. Rev. Lett. 114 034801
[6] Rusz J, Bhowmick S, Eriksson M and Karlsson N 2014 Phys. Rev. B 89 134428
[7] Rusz J, Idrobo J C and Bhowmick S 2014 Phys. Rev. Lett. 113 145501
[8] Schattschneider P, Löffler S, Stöger-Pollach M and Verbeeck J 2014 Ultramicroscopy 136 81
[9] Karimi E, Marrucci L, Grillo V and Santamato E 2012 Phys. Rev. Lett. 108 044801
[10] Lloyd S M, Babiker M and Yuan J 2013 Phys. Rev. A 88 031802(R)
[11] Gnanavel T, Yuan J and Babiker M 2012 Proc. 15th European Microscopy Congresss, September 16-21, 2012 Manchester, UK, p. PS2.9
[12] Verbeeck J, Tian H and Tendeloo G V 2013 Adv. Mater. 25 1114
[13] Lloyd S, Babiker M and Yuan J 2012 Phys. Rev. Lett. 108 074802
[14] Konkov A S, Potylitsyn A P and Polonskaya M S 2014 JETP Lett. 100 421
[15] Ivanov I P and Karlovers D V 2013 Phys. Rev. A 88 043840
[16] Ivanov I P and Karlovers D V 2013 Phys. Rev. Lett. 110 264801
[17] Ivanov I P, Serbo V G and Zaytsev V A 2016 Phys. Rev. A 93 053825
[18] Kaminer I, Mutzafi M, Levy A, Harari G, Sheinfux H H, Skirlo S, Nemirovsky J, Joannopoulos J D, Segev M and Soljačić M 2016 Phys. Rev. X 6 011006
[19] Larocque H, Bouchard F, Grillo V, Sit A, Frabboni S, Dunin-Borkowski R E, Padgett M J, Boyd R W and Karimi E 2016 Phys. Rev. Lett. 117 154801
[20] Galyamin S N and Tyukhtin A V 2017 Nucl. Ins. Met. Phy. Res. B 402 185
[21] Galyamin S N, Vorobev V V and Tyukhtin A V 2019 Phys. Rev. Accel. Beams. 22 083001
[22] Han Y J, Liao G Q, Chen L M, Li Y T, Wang W M and Zhang J 2015 Chin. Phys. B 24 065202
[23] Wu X F, Deng D M and Guo Q 2016 Chin. Phys. B 25 030701
[24] Cheng K, Liu P S and Lü B D 2008 Chin. Phys. B 17 1743
[25] Zhou Z H and Zhu L Q 2011 Chin. Phys. B 20 084201
[26] Katoh M, Fujimoto M, Kawaguchi H, Tsuchiya K, Ohmi K, Kaneyasu T, Taira Y, Hosaka M, Mochihashi A and Takashima Y 2017 Phys. Rev. Lett. 118 094801
[27] Katoh M, Fujimoto, Mirian N S, Konomi T, Taira Y, Kaneyasu T, Hosaka M, Yamamoto N, Mochihashi A, Takashima Y, Kuroda K, Miyamoto A, Miyamoto K and Sasaki S 2017 Sci. Rep. 7 6130
[28] Hébert C and Schattschneider P 2003 Ultramicroscopy 96 463
[29] Schattschneider P, Rubino S, Hébert C, Rusz J, Kuneš J, Novák P, Carlino E, Fabrizioli M, Panaccione G and Rossi G 2006 Nature 441 486
[30] Schattschneider P, Rubino S, Stöger-Pollach M, Hébert C, Rusz J, Calmels L and Snoeck E 2008 J. Appl. Phys. 103 07D931
[31] Rubino S, Schattschneider P, Stöger-Pollach M, Hébert C, Rusz J, Calmels L, Warot-Fonrose B, Houdellier F, Serin V and Novak P 2008 J. Mater. Res. 23 2582
[32] Schüler M and Berakdar J 2016 Phys. Rev. A 94 052710
[33] Weissbluth M 1978 Atoms and Molecules (New York:Academic Press) p. 313
[34] Bliokh K Y, Dennis M R and Nori F 2011 Phys. Rev. Lett. 107 174802
[35] Jentschura U D and Serbo V G 2011 Eur. Phys. J. C 71 1571
[36] For a nonrelativistic electron, spin-orbit coupling as a relativistic correction term is proportional to the orbital angular momentum quantum number l. However, for a relativistic electron, the spin-orbit coupling has a nonlinear realtion to the angular momentum number l.
[37] Jeffrey A and Dai H 2008 Handbook of Mathematical Formulas and Integrals 4th Edn. (Boston:Academic Press) p. 294
[38] Lloyd S M, Babiker M, Yuan J and Kerr-Edwards C 2012 Phys. Rev. Lett. 109 254801
[1] Dynamics of bright soliton in a spin-orbit coupled spin-1 Bose-Einstein condensate
Hui Guo(郭慧), Xu Qiu(邱旭), Yan Ma(马燕), Hai-Feng Jiang(姜海峰), and Xiao-Fei Zhang(张晓斐). Chin. Phys. B, 2021, 30(6): 060310.
[2] Tunable valley filter efficiency by spin-orbit coupling in silicene nanoconstrictions
Yi-Jian Shi(施一剑), Yuan-Chun Wang(王园春), and Peng-Jun Wang(汪鹏君). Chin. Phys. B, 2021, 30(5): 057201.
[3] Configuration interaction study on low-lying states of AlCl molecule
Xiao-Ying Ren(任笑影), Zhi-Yu Xiao(肖志宇), Yong Liu(刘勇), and Bing Yan(闫冰). Chin. Phys. B, 2021, 30(5): 053101.
[4] Spin-orbit-coupled spin-1 Bose-Einstein condensates confined in radially periodic potential
Ji Li(李吉), Tianchen He(何天琛), Jing Bai(白晶), Bin Liu(刘斌), and Huan-Yu Wang(王寰宇). Chin. Phys. B, 2021, 30(3): 030302.
[5] Efficient manipulation of terahertz waves by multi-bit coding metasurfaces and further applications of such metasurfaces
Yunping Qi(祁云平) Baohe Zhang(张宝和), Jinghui Ding(丁京徽), Ting Zhang(张婷), Xiangxian Wang(王向贤), and Zao Yi(易早). Chin. Phys. B, 2021, 30(2): 024211.
[6] Recent advances in generation of terahertz vortex beams andtheir applications
Honggeng Wang(王弘耿), Qiying Song(宋其迎), Yi Cai(蔡懿), Qinggang Lin(林庆钢), Xiaowei Lu(陆小微), Huangcheng Shangguan(上官煌城), Yuexia Ai(艾月霞), Shixiang Xu(徐世祥). Chin. Phys. B, 2020, 29(9): 097404.
[7] Hybrid vector beams with non-uniform orbital angular momentum density induced by designed azimuthal polarization gradient
Lei Han(韩磊), Shuxia Qi(齐淑霞), Sheng Liu(刘圣), Peng Li(李鹏), Huachao Cheng(程华超), Jianlin Zhao(赵建林). Chin. Phys. B, 2020, 29(9): 094203.
[8] Giant interface spin-orbit torque in NiFe/Pt bilayers
Shu-Fa Li(李树发), Tao Zhu(朱涛). Chin. Phys. B, 2020, 29(8): 087102.
[9] Transparently manipulating spin-orbit qubit via exact degenerate ground states
Kuo Hai(海阔), Wenhua Zhu(朱文华), Qiong Chen(陈琼), Wenhua Hai(海文华). Chin. Phys. B, 2020, 29(8): 083203.
[10] Two-dimensional hexagonal Zn3Si2 monolayer: Dirac cone material and Dirac half-metallic manipulation
Yurou Guan(官雨柔), Lingling Song(宋玲玲), Hui Zhao(赵慧), Renjun Du(杜仁君), Liming Liu(刘力铭), Cuixia Yan(闫翠霞), Jinming Cai(蔡金明). Chin. Phys. B, 2020, 29(8): 087103.
[11] Optical spin-to-orbital angular momentum conversion instructured optical fields
Yang Zhao(赵阳), Cheng-Xi Yang(阳成熙), Jia-Xi Zhu(朱家玺), Feng Lin(林峰), Zhe-Yu Fang(方哲宇), Xing Zhu(朱星). Chin. Phys. B, 2020, 29(6): 067301.
[12] Ferromagnetic transition of a spin–orbit coupled dipolar Fermi gas at finite temperature
Xue-Jing Feng(冯雪景) and Lan Yin(尹澜). Chin. Phys. B, 2020, 29(11): 110306.
[13] Ground-state phases and spin textures of spin–orbit-coupled dipolar Bose–Einstein condensates in a rotating toroidal trap
Qing-Bo Wang(王庆波), Hui Yang(杨慧), Ning Su(苏宁), and Ling-Hua Wen(文灵华). Chin. Phys. B, 2020, 29(11): 116701.
[14] Lattice configurations in spin-1 Bose–Einstein condensates with the SU(3) spin–orbit coupling
Ji-Guo Wang(王继国)†, Yue-Qing Li(李月晴), and Yu-Fei Dong(董雨菲). Chin. Phys. B, 2020, 29(10): 100304.
[15] Generation of orbital angular momentum and focused beams with tri-layer medium metamaterial
Zhi-Chao Sun(孙志超), Meng-Yao Yan(闫梦瑶), and Bi-Jun Xu(徐弼军)†. Chin. Phys. B, 2020, 29(10): 104101.
No Suggested Reading articles found!