Orbital angular momentum of an astigmatic Gaussian laser beam
Kotlyar V.V., Kovalev A.A.


Image Processing Systems Institute оf RAS – Branch of the FSRC “Crystallography and Photonics” RAS, Samara, Russia,
Samara National Research University, Samara, Russia

Full text of article: Russian language.


We show that an elliptic Gaussian beam focused by a cylindrical lens can be represented as a linear combination of a countable number of only even angular harmonics with both positive and negative topological charge. For the orbital angular momentum of an astigmatic Gaussian beam, an exact expression is obtained in the form of a converging series of the Legendre functions of the second kind. It is shown that at some conditions only terms with the positive or negative topological charge are retained in this series.

elliptic Gaussian beam, cylindrical lens, orbital angular momentum, topological charge.

Kotlyar VV, Kovalev AA. Orbital angular momentum of an astigmatic Gaussian laser beam. Computer Optics 2017; 41(5): 609-616. DOI: 10.18287/2412-6179-2017-41-5-609-616.


  1. Grier D. A revolution in optical manipulation. Nature 2003; 424(6950): 810-816. DOI: 10.1038/nature01935.
  2. Kuga T, Torii Y, Shiokawa N, Hirano T, Shimizu Y, Sasada H. Novel optical trap of atoms with a doughnut beam. Phys Rev Lett 1997; 78(25): 4713-4716. DOI: 10.1103/PhysRevLett.78.4713.
  3. Bernet S, Jesacher A, Furhapter S, Maurer C, Ritsch-Marte M. Quantitative imaging of complex samples by spiral phase contrast microscopy. Opt Express 2006; 14(9): 3792-3805. DOI: 10.1364/OE.14.003792.
  4. Willig KI, Rizzoli SO, Westphal V, Jahn R, Hell SW. STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis. Nature 2006; 440(7086): 935-939. DOI: 10.1038/nature04592.
  5. Wang J, Yang J, Fazal IM, Ahmed N, Yan Y, Huang H, Ren Y, Yue Y, Dolinar S, Tur M, Willner AE. Terabit free-space data transmission employing orbital angular momentum multiplexing. Nature Photon 2012; 6(7): 488-496. DOI: 10.1038/nphoton.2012.138.
  6. Mair A, Vaziri A, Weihs G, Zeilinger A. Entanglement of the orbital angular momentum states of photons. Nature 2001; 412(6844): 313-316. DOI: 10.1038/35085529.
  7. Courtial J, Dholakia K, Allen L, Padgett MJ. Gaussian beams with very high orbital angular momentum. Opt Commun 1997; 144(4-6): 210-3. DOI: 10.1016/S0030-4018(97)00376-3.
  8. Abramochkin EG, Volostnikov VG. Beam transformations and nontransformed beams. Opt Commun 1991; 83(1-2): 123-135. DOI: 10.1016/0030-4018(91)90534-K.
  9. Izdebskaya Y, Fadeyeva T, Shvedov V, Volyar A. Vortex-bearing array of singular beams with very high orbital angular momentum. Opt Lett 2006; 31(17): 2523-2525. DOI: 10.1364/OL.31.002523.
  10. Li Z, Zhang M, Liang G, Li X, Chen X, Cheng C. Generation of high-order optical vortices with asymmetrical pinhole plates under plane wave illumination. Opt Express 2012; 21(13): 15755-64. DOI: 10.1364/OE.21.015755.
  11. Krenn M, Tischler N, Zeilinger A. On small beams with large topological charge. New J Phys 2016; 18: 033012. DOI: 10.1088/1367-2630/18/3/033012.
  12. Zheng S, Wang J. Measuring orbital angular momentum (OAM) states of vortex beams with annular gratings. Sci Rep 2017; 7: 40781. DOI: 10.1038/srep40781.
  13. Vieira J, Trines RMGM, Alves EP, Fonseca RA, Mendonca JT, Bingham R, Norreys P, Silva LO. High orbital angular momentum harmonic generation. Phys Rev Lett 2016; 117(26): 265001. DOI: 10.1103/PhysRevLett.117.265001.
  14. Chen Y, Fang ZX, Ren YX, Gong L, Lu RD. Generation and characterization of a perfect vortex beam with a large topological charge through a digital micromirror device. Appl Opt 2015; 54(27): 8030-8035. DOI: 10.1364/AO.54.008030.
  15. Jesacher A, Furhapter S, Maurer C, Bernet S, Ritsch-Marte M. Holographic optical tweezers for object manipulations at an air-liquid surface. Opt Express 2006; 14(13): 6342-6352.
  16. Fickler R, Lapkiewicz R, Plick WN, Krenn M, Schaeff C, Ramelow S, Zeilinger A. Quantum entanglement of high angular momenta. Science 2012; 338(6107): 640-643. DOI: 10.1126/science.1227193.
  17. Campbell G, Hage B, Buchler B, Lam PK. Generation of high-order optical vortices using directly machined spiral phase mirrors. Appl Opt 2012; 51(7): 873-876. DOI: 10.1364/AO.51.000873.
  18. Shen Y, Campbell GT, Hage B, Zou H, Buchler BC, Lam PK. Generation and interferometric analysis of high charge optical vortices. J Opt 2013; 15(4): 044005. DOI: 10.1088/2040-8978/15/4/044005.
  19. Mafakheri E, Tavabi AH, Lu P, Balboni R, Venturi F, Menozzi C, Gazzadi GC, Frabboni S, Sit A, Dunin-Borkowski RE, Karimi E. Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography. Appl Phys Lett 2017; 110: 093113. DOI: 10.1063/1.4977879.
  20. Fickler R, Campbell G, Buchler B, Lam PK, Zeilinger A. Quantum entanglement of angular momentum states with quantum number up to 10010. Proc Natl Acad Sci USA 2016; 113(48): 13642-13647. DOI: 10.1073/pnas.1616889113.
  21. Prudnikov AP, Brychkov YA, Marichev OI. Integrals and Series. Vol. 2. Special Functions. New York: Gordon and Breach; 1992. ISBN: 978-2-881240973.
  22. Abramovitz M, Stegun IA. Handbook of Mathematical Functions. National Bureau of Standards, Applied Mathematics Series; 1964.

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