Abstract

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 (OAM) of the astigmatic Gaussian beam, an exact expression is obtained in a form of a converging series of the Legendre functions of the second kind. It is shown that at some conditions only the terms with the positive or negative topological charge are remained in this series. Using a hybrid numeric-experimental approach, we obtained the normalized OAM of the astigmatic beam, equal to 109, which is just 6% different from the exact OAM of 116, calculated by the equation. To generate such laser beams, there is no need in special optical elements such as spiral phase plates. The OAM of such beams can be adjusted by varying the waist radius of the Gaussian beam and the focal length of the cylindrical lens. The OAM of such beams can reach large values.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (4)

S. Zheng and J. Wang, “Measuring orbital angular momentum (OAM) states of vortex beams with annular gratings,” Sci. Rep. 7, 40781 (2017).
[Crossref] [PubMed]

E. Mafakheri, A. H. Tavabi, P. Lu, R. Balboni, F. Venturi, C. Menozzi, G. C. Gazzadi, S. Frabboni, A. Sit, R. E. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

S. Fu, T. Wang, Z. Zhang, Y. Zhai, and C. Gao, “Non-diffractive Bessel-Gauss beams for the detection of rotating object free of obstructions,” Opt. Express 25(17), 20098–20108 (2017).
[Crossref] [PubMed]

A. D’Errico, R. D’Amelio, B. Piccirillo, F. Cardano, and L. Marrucci, “Measuring the complex orbital angular momentum spectrum and spatial mode decomposition of structured light beams,” Optica 4(11), 1350–1357 (2017).
[Crossref]

2016 (3)

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10010,” Proc. Natl. Acad. Sci. U.S.A. 113(48), 13642–13647 (2016).
[Crossref] [PubMed]

M. Krenn, N. Tischler, and A. Zeilinger, “On small beams with large topological charge,” New J. Phys. 18(3), 033012 (2016).
[Crossref]

J. Vieira, R. M. G. M. Trines, E. P. Alves, R. A. Fonseca, J. T. Mendonça, R. Bingham, P. Norreys, and L. O. Silva, “High orbital angular momentum harmonic generation,” Phys. Rev. Lett. 117(26), 265001 (2016).
[Crossref] [PubMed]

2015 (1)

2013 (3)

Z. Li, M. Zhang, G. Liang, X. Li, X. Chen, and C. Cheng, “Generation of high-order optical vortices with asymmetrical pinhole plates under plane wave illumination,” Opt. Express 21(13), 15755–15764 (2013).
[Crossref] [PubMed]

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15(4), 044005 (2013).
[Crossref]

2012 (3)

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[Crossref] [PubMed]

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

G. Campbell, B. Hage, B. Buchler, and P. K. Lam, “Generation of high-order optical vortices using directly machined spiral phase mirrors,” Appl. Opt. 51(7), 873–876 (2012).
[Crossref] [PubMed]

2006 (4)

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

2001 (2)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref] [PubMed]

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87(2), 023902 (2001).
[Crossref] [PubMed]

1997 (2)

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4-6), 210–213 (1997).
[Crossref]

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

1991 (1)

E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83(1–2), 123–135 (1991).
[Crossref]

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83(1–2), 123–135 (1991).
[Crossref]

Ahmed, N.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Allen, L.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4-6), 210–213 (1997).
[Crossref]

Alves, E. P.

J. Vieira, R. M. G. M. Trines, E. P. Alves, R. A. Fonseca, J. T. Mendonça, R. Bingham, P. Norreys, and L. O. Silva, “High orbital angular momentum harmonic generation,” Phys. Rev. Lett. 117(26), 265001 (2016).
[Crossref] [PubMed]

Balboni, R.

E. Mafakheri, A. H. Tavabi, P. Lu, R. Balboni, F. Venturi, C. Menozzi, G. C. Gazzadi, S. Frabboni, A. Sit, R. E. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

Barnett, S. M.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

Bernet, S.

Bingham, R.

J. Vieira, R. M. G. M. Trines, E. P. Alves, R. A. Fonseca, J. T. Mendonça, R. Bingham, P. Norreys, and L. O. Silva, “High orbital angular momentum harmonic generation,” Phys. Rev. Lett. 117(26), 265001 (2016).
[Crossref] [PubMed]

Buchler, B.

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10010,” Proc. Natl. Acad. Sci. U.S.A. 113(48), 13642–13647 (2016).
[Crossref] [PubMed]

G. Campbell, B. Hage, B. Buchler, and P. K. Lam, “Generation of high-order optical vortices using directly machined spiral phase mirrors,” Appl. Opt. 51(7), 873–876 (2012).
[Crossref] [PubMed]

Buchler, B. C.

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15(4), 044005 (2013).
[Crossref]

Campbell, G.

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10010,” Proc. Natl. Acad. Sci. U.S.A. 113(48), 13642–13647 (2016).
[Crossref] [PubMed]

G. Campbell, B. Hage, B. Buchler, and P. K. Lam, “Generation of high-order optical vortices using directly machined spiral phase mirrors,” Appl. Opt. 51(7), 873–876 (2012).
[Crossref] [PubMed]

Campbell, G. T.

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15(4), 044005 (2013).
[Crossref]

Cardano, F.

Chen, X.

Chen, Y.

Cheng, C.

Courtial, J.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4-6), 210–213 (1997).
[Crossref]

D’Amelio, R.

D’Errico, A.

Dholakia, K.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4-6), 210–213 (1997).
[Crossref]

Dolinar, S.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Dunin-Borkowski, R. E.

E. Mafakheri, A. H. Tavabi, P. Lu, R. Balboni, F. Venturi, C. Menozzi, G. C. Gazzadi, S. Frabboni, A. Sit, R. E. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

Fadeyeva, T.

Fang, Z. X.

Fazal, I. M.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Fickler, R.

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10010,” Proc. Natl. Acad. Sci. U.S.A. 113(48), 13642–13647 (2016).
[Crossref] [PubMed]

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[Crossref] [PubMed]

Fonseca, R. A.

J. Vieira, R. M. G. M. Trines, E. P. Alves, R. A. Fonseca, J. T. Mendonça, R. Bingham, P. Norreys, and L. O. Silva, “High orbital angular momentum harmonic generation,” Phys. Rev. Lett. 117(26), 265001 (2016).
[Crossref] [PubMed]

Frabboni, S.

E. Mafakheri, A. H. Tavabi, P. Lu, R. Balboni, F. Venturi, C. Menozzi, G. C. Gazzadi, S. Frabboni, A. Sit, R. E. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

Fu, S.

Fürhapter, S.

Gao, C.

Gazzadi, G. C.

E. Mafakheri, A. H. Tavabi, P. Lu, R. Balboni, F. Venturi, C. Menozzi, G. C. Gazzadi, S. Frabboni, A. Sit, R. E. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

Gong, L.

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

Grillo, V.

E. Mafakheri, A. H. Tavabi, P. Lu, R. Balboni, F. Venturi, C. Menozzi, G. C. Gazzadi, S. Frabboni, A. Sit, R. E. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

Hage, B.

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15(4), 044005 (2013).
[Crossref]

G. Campbell, B. Hage, B. Buchler, and P. K. Lam, “Generation of high-order optical vortices using directly machined spiral phase mirrors,” Appl. Opt. 51(7), 873–876 (2012).
[Crossref] [PubMed]

Hell, S. W.

K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440(7086), 935–939 (2006).
[Crossref] [PubMed]

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Huang, H.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Izdebskaya, Y.

Jahn, R.

K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440(7086), 935–939 (2006).
[Crossref] [PubMed]

Jesacher, A.

Karimi, E.

E. Mafakheri, A. H. Tavabi, P. Lu, R. Balboni, F. Venturi, C. Menozzi, G. C. Gazzadi, S. Frabboni, A. Sit, R. E. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

Krenn, M.

M. Krenn, N. Tischler, and A. Zeilinger, “On small beams with large topological charge,” New J. Phys. 18(3), 033012 (2016).
[Crossref]

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[Crossref] [PubMed]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Lam, P. K.

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10010,” Proc. Natl. Acad. Sci. U.S.A. 113(48), 13642–13647 (2016).
[Crossref] [PubMed]

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15(4), 044005 (2013).
[Crossref]

G. Campbell, B. Hage, B. Buchler, and P. K. Lam, “Generation of high-order optical vortices using directly machined spiral phase mirrors,” Appl. Opt. 51(7), 873–876 (2012).
[Crossref] [PubMed]

Lapkiewicz, R.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[Crossref] [PubMed]

Lavery, M. P. J.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

Li, X.

Li, Z.

Liang, G.

Lu, P.

E. Mafakheri, A. H. Tavabi, P. Lu, R. Balboni, F. Venturi, C. Menozzi, G. C. Gazzadi, S. Frabboni, A. Sit, R. E. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

Lu, R. D.

Mafakheri, E.

E. Mafakheri, A. H. Tavabi, P. Lu, R. Balboni, F. Venturi, C. Menozzi, G. C. Gazzadi, S. Frabboni, A. Sit, R. E. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref] [PubMed]

Marrucci, L.

Maurer, C.

Mendonça, J. T.

J. Vieira, R. M. G. M. Trines, E. P. Alves, R. A. Fonseca, J. T. Mendonça, R. Bingham, P. Norreys, and L. O. Silva, “High orbital angular momentum harmonic generation,” Phys. Rev. Lett. 117(26), 265001 (2016).
[Crossref] [PubMed]

Menozzi, C.

E. Mafakheri, A. H. Tavabi, P. Lu, R. Balboni, F. Venturi, C. Menozzi, G. C. Gazzadi, S. Frabboni, A. Sit, R. E. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

Molina-Terriza, G.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87(2), 023902 (2001).
[Crossref] [PubMed]

Norreys, P.

J. Vieira, R. M. G. M. Trines, E. P. Alves, R. A. Fonseca, J. T. Mendonça, R. Bingham, P. Norreys, and L. O. Silva, “High orbital angular momentum harmonic generation,” Phys. Rev. Lett. 117(26), 265001 (2016).
[Crossref] [PubMed]

Padgett, M. J.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4-6), 210–213 (1997).
[Crossref]

Piccirillo, B.

Plick, W. N.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[Crossref] [PubMed]

Ramelow, S.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[Crossref] [PubMed]

Recolons, J.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87(2), 023902 (2001).
[Crossref] [PubMed]

Ren, Y.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Ren, Y. X.

Ritsch-Marte, M.

Rizzoli, S. O.

K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440(7086), 935–939 (2006).
[Crossref] [PubMed]

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Schaeff, C.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[Crossref] [PubMed]

Shen, Y.

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15(4), 044005 (2013).
[Crossref]

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Shvedov, V.

Silva, L. O.

J. Vieira, R. M. G. M. Trines, E. P. Alves, R. A. Fonseca, J. T. Mendonça, R. Bingham, P. Norreys, and L. O. Silva, “High orbital angular momentum harmonic generation,” Phys. Rev. Lett. 117(26), 265001 (2016).
[Crossref] [PubMed]

Sit, A.

E. Mafakheri, A. H. Tavabi, P. Lu, R. Balboni, F. Venturi, C. Menozzi, G. C. Gazzadi, S. Frabboni, A. Sit, R. E. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

Speirits, F. C.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

Tavabi, A. H.

E. Mafakheri, A. H. Tavabi, P. Lu, R. Balboni, F. Venturi, C. Menozzi, G. C. Gazzadi, S. Frabboni, A. Sit, R. E. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

Tischler, N.

M. Krenn, N. Tischler, and A. Zeilinger, “On small beams with large topological charge,” New J. Phys. 18(3), 033012 (2016).
[Crossref]

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Torner, L.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87(2), 023902 (2001).
[Crossref] [PubMed]

Torres, J. P.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87(2), 023902 (2001).
[Crossref] [PubMed]

Trines, R. M. G. M.

J. Vieira, R. M. G. M. Trines, E. P. Alves, R. A. Fonseca, J. T. Mendonça, R. Bingham, P. Norreys, and L. O. Silva, “High orbital angular momentum harmonic generation,” Phys. Rev. Lett. 117(26), 265001 (2016).
[Crossref] [PubMed]

Tur, M.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref] [PubMed]

Venturi, F.

E. Mafakheri, A. H. Tavabi, P. Lu, R. Balboni, F. Venturi, C. Menozzi, G. C. Gazzadi, S. Frabboni, A. Sit, R. E. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

Vieira, J.

J. Vieira, R. M. G. M. Trines, E. P. Alves, R. A. Fonseca, J. T. Mendonça, R. Bingham, P. Norreys, and L. O. Silva, “High orbital angular momentum harmonic generation,” Phys. Rev. Lett. 117(26), 265001 (2016).
[Crossref] [PubMed]

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83(1–2), 123–135 (1991).
[Crossref]

Volyar, A.

Wang, J.

S. Zheng and J. Wang, “Measuring orbital angular momentum (OAM) states of vortex beams with annular gratings,” Sci. Rep. 7, 40781 (2017).
[Crossref] [PubMed]

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Wang, T.

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref] [PubMed]

Westphal, V.

K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440(7086), 935–939 (2006).
[Crossref] [PubMed]

Willig, K. I.

K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440(7086), 935–939 (2006).
[Crossref] [PubMed]

Willner, A. E.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Wright, E. M.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87(2), 023902 (2001).
[Crossref] [PubMed]

Yan, Y.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yang, J.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yue, Y.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Zeilinger, A.

M. Krenn, N. Tischler, and A. Zeilinger, “On small beams with large topological charge,” New J. Phys. 18(3), 033012 (2016).
[Crossref]

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10010,” Proc. Natl. Acad. Sci. U.S.A. 113(48), 13642–13647 (2016).
[Crossref] [PubMed]

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[Crossref] [PubMed]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref] [PubMed]

Zhai, Y.

Zhang, M.

Zhang, Z.

Zheng, S.

S. Zheng and J. Wang, “Measuring orbital angular momentum (OAM) states of vortex beams with annular gratings,” Sci. Rep. 7, 40781 (2017).
[Crossref] [PubMed]

Zou, H.

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15(4), 044005 (2013).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

E. Mafakheri, A. H. Tavabi, P. Lu, R. Balboni, F. Venturi, C. Menozzi, G. C. Gazzadi, S. Frabboni, A. Sit, R. E. Dunin-Borkowski, E. Karimi, and V. Grillo, “Realization of electron vortices with large orbital angular momentum using miniature holograms fabricated by electron beam lithography,” Appl. Phys. Lett. 110(9), 093113 (2017).
[Crossref]

J. Opt. (1)

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15(4), 044005 (2013).
[Crossref]

Nat. Photonics (1)

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Nature (3)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref] [PubMed]

K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440(7086), 935–939 (2006).
[Crossref] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

New J. Phys. (1)

M. Krenn, N. Tischler, and A. Zeilinger, “On small beams with large topological charge,” New J. Phys. 18(3), 033012 (2016).
[Crossref]

Opt. Commun. (2)

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4-6), 210–213 (1997).
[Crossref]

E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83(1–2), 123–135 (1991).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Optica (1)

Phys. Rev. Lett. (3)

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87(2), 023902 (2001).
[Crossref] [PubMed]

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

J. Vieira, R. M. G. M. Trines, E. P. Alves, R. A. Fonseca, J. T. Mendonça, R. Bingham, P. Norreys, and L. O. Silva, “High orbital angular momentum harmonic generation,” Phys. Rev. Lett. 117(26), 265001 (2016).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10010,” Proc. Natl. Acad. Sci. U.S.A. 113(48), 13642–13647 (2016).
[Crossref] [PubMed]

Sci. Rep. (1)

S. Zheng and J. Wang, “Measuring orbital angular momentum (OAM) states of vortex beams with annular gratings,” Sci. Rep. 7, 40781 (2017).
[Crossref] [PubMed]

Science (2)

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[Crossref] [PubMed]

Other (2)

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series, Special Functions (New York: Gordon and Breach, 1981).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Applied Math. Series, 1979).

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Figures (6)

Fig. 1
Fig. 1

Distributions of intensity (left column), phase (middle column) and OAM density (right column) of the field (1) at different distances from the initial plane: (a) z = λ (R = 40λ); (b) z = f (R = 40λ); (c) z = 2f (R = 80λ). Positive and negative OAM density is shown by the red and blue color respectively.

Fig. 2
Fig. 2

Distribution of the coefficients | C ¯ n | (only even numbers, since for the odd numbers they are equal to zero) for two different beams and focal lengths of the cylindrical lens: wx = 20λ, wy = 400λ, f = 1260λ (continuous curve); wx = 10λ, wy = 100λ, f = 317λ (dashed curve). The inset shows a magnified fragment for n = −10, ..., 50 (each dot is depicted for integer value of n).

Fig. 3
Fig. 3

Experimental setup for generation and analysis of the elliptical Gaussian beams: Laser is a solid-state laser (λ = 532 nm), PH is a pinhole (hole size 40 μm), L is a spherical lens (f = 150 mm), BS1 and BS2 are beam splitting cubes, F is a neutral density filter, M1 and M2 are mirrors, CL1 and CL2 are cylindrical lenses (f1 = 500 mm, f2 = 100 mm), CCD is a video-camera.

Fig. 4
Fig. 4

Intensity distributions of the astigmatic Gaussian beam, registered at a distance z from the cylindrical lens, equal to 0 (a), 2f (b), 3f (d), as well as fringe patterns of this beam with a plane wave at the distances 2f (c) and 3f (e). The size of all frames is 5608 × 4255 μm.

Fig. 5
Fig. 5

Two interferograms (a, b), registered in the optical setup from Fig. 3 at a distance of 2f from the second cylindrical lens. The second interferogram was registered by using a reference beam with the phase delayed by π. Phase of the astigmatic Gaussian beam reconstructed by using the interferograms (c). The size of all frames is 5608 × 4255 μm.

Fig. 6
Fig. 6

Computed intensity distributions (a, b) of the interferograms, registered in the optical setup from Fig. 3 at a distance 2f. The second interferogram was calculated by using a reference beam with the phase delayed by π. Phase of the astigmatic Gaussian beam, reconstructed by using the computed interferograms (c). The size of all frames is 5608 × 4255 μm.

Equations (54)

Equations on this page are rendered with MathJax. Learn more.

E(x,y)=exp( x 2 w x 2 y 2 w y 2 )exp( ik x 2 cos 2 α 2f ik y 2 sin 2 α 2f ikxysin2α 2f ),
J z =Im E ¯ ( x,y )[ x E( x,y ) y y E( x,y ) x ]dxdy ,
W= E ¯ ( x,y )E( x,y )dxdy ,
J z W =( ksin2α 8f )( w y 2 w x 2 ).
E( ξ,η,z )= ik z p( z )q( z ) exp[ A( z ) ξ 2 +B( z ) η 2 +C( z )ξη ],
A(z)= ik 2z k 2 4 z 2 p(z) + k 4 sin 2 2α 64 f 2 z 2 p 2 (z)q(z) , B(z)= ik 2z k 2 4 z 2 q(z) ,C(z)= i k 3 sin2α 16f z 2 p(z)q(z) , p(z)= 1 w x 2 + ik 2 z x ,q(z)= 1 w y 2 + ik 2 z y + k 2 sin 2 2α 16 f 2 p(z) , z x = zf z cos 2 αf , z y = zf z sin 2 αf .
E( ξ,η,z=2f )=2i γ 1 exp[ ik 4f ( ξ 2 + η 2 ) ξ 2 w y 2 γ 2 η 2 w x 2 γ 2 + ikξη 2f γ 2 ],
γ= ( 1+ 16 f 2 k 2 w x 2 w y 2 ) 1/2 .
E( x,y,z )= 1 q 0 ( z ) q 1 ( z ) exp( x 2 w 2 q 2 ( z ) y 2 w 2 q 0 ( z ) ),
q 0 (z)=1+ iz z 0 , q 1 (z)= q 0 (z) z f 1 , q 2 (z)= q 1 (z) ( 1+ i z 0 f 1 ) 1 , z 0 = k w 2 2 .
E(x,y,z)= ( q 0 (z) q 1 (z) ) 1/2 exp( x 2 w 2 q 2 (z) y 2 w 2 q 0 (z) )× ×exp( ik x 2 cos 2 α 2f ik y 2 sin 2 α 2f ikxysin2α 2f ).
J z W =( k w 2 sin2α 8f )[ | q 0 | 2 Re( q 0 ) | q 2 | 2 Re( q 2 ) ]=( k w 2 sin2α 8f )( z f 1 )( 2 z f 1 ).
J z W =( k w 2 sin2α 8f ).
E(r,φ)=exp( a r 2 cos 2 φb r 2 sin 2 φic r 2 sin2φ ),
E(r,φ)= n= C n (r)exp(inφ) ,
C n ( r )= ( 2π ) 1 0 2π E( r,φ )exp( inφ )dφ ,
a= w x 2 +ik ( 2f ) 1 cos 2 α, b= w y 2 +ik ( 2f ) 1 sin 2 α, c=k ( 4f ) 1 sin2α.
C n ( r )= ( 2π ) 1 e D 0 2π exp( inφAcos2φiBsin2φ )dφ ,
C n ( r )= (2π) 1 e D 0 2π exp( imtAcostiBsint )dt = = (i) m e D+imθ J m (F)= e D ( F AB ) m J m (F),
2π C 2m+1 ( r )= e D 0 2π exp[ i( 2m+1 )φAcos2φiBsin2φ ]dφ = = e D { 0 π exp[ i( 2m+1 )φAcos2φiBsin2φ ]dφ+ + 0 π exp[ i( 2m+1 )( φ+π )Acos2( φ+π )iBsin2( φ+π ) ]dφ }= = e D 0 π [ 1+ ( 1 ) 2m+1 ]exp[ i( 2m+1 )φAcos2φiBsin2φ ]dφ =0.
C n (r)={ e D ( F AB ) n/2 J n/2 ( F ),n=2m, 0,n=2m+1.
| C 2m ( r ) | 2 = e 2ReD | A+B AB | m | J m ( F ) | 2 , | C 2m ( r ) | 2 = e 2ReD | AB A+B | m | J m ( F ) | 2 .
J z W = n= n C ¯ n n= C ¯ n , C ¯ n = 0 | C n (r) | 2 rdr .
C ¯ 2m = | ab+2c ab2c | m 0 exp[ Re( a+b ) r 2 ] J m 2 ( r 2 2 4 c 2 ( ab ) 2 ) rdr.
C ¯ 2m = 0 e 2ReD | A+B AB | m | J m ( F ) | 2 rdr = = | ab+2c ab2c | m 0 e Re( a+b ) r 2 J m ( G r 2 ) J m ( G * r 2 )rdr = = 1 2 | ab+2c ab2c | m 0 e Re( a+b )u J m ( Gu ) J m ( G * u )du .
0 e p r 2 J m 2 ( q r 2 )rdr = ( 2πq ) 1 Q | m |1/2 ( 1+ p 2 2 q 2 ).
0 e px J ν ( bx ) J ν ( cx )dx = 1 π bc Q ν1/2 ( p 2 + b 2 + c 2 2bc ),
Q ν ( x )= π ( 2x ) ν1 Г( v+1 ) Г 1 ( v+3/2 ) 2 F 1 ( ν+1 2 , ν+2 2 ,ν+ 3 2 , x 2 ),
C ¯ 2m = π 1 ( c+ab сa+b ) m [ c 2 ( ab ) 2 ] 1/2 Q | m |1/2 [ 1+ 2 ( a+b ) 2 c 2 ( ab ) 2 ],
C ¯ 2m = 1 2π| G | | ab+2c ab2c | m Q | m |1/2 ( Re 2 ( a+b )+2Re G 2 2 | G | 2 ).
Q | ν | ( x ) π ( 2x ) | ν |1 Г( | ν |+1 ) Г 1 ( | ν |+3/2 )
C ¯ 2m = 1 2 2| m |+1 π Г( | m |+1/2 ) Г( | m |+1 ) | ab+2c | | m |+m | ab2c | | m |m [ Re( a+b ) ] 2| m |+1 .
C ¯ 2m ={ 0,m>0, f k π Г( | m |+1/2 ) Г( | m |+1 ) ( w y 2 + w x 2 w x 2 w y 2 ) 2| m |1 ,m0.
C ¯ 2m ={ f k π Г( m+1/2 ) Г( m+1 ) ( w x 2 + w y 2 w y 2 w x 2 ) 2m1 ,m0, 0,m<0.
C ¯ 2m f k π ( w y 2 w x 2 w y 2 + w x 2 ) 2m+1 .
C ¯ 0 = 1 π c 2 (ab) 2 Q 1/2 [ 1+ 2 (a+b) 2 c 2 (ab) 2 ].
Q 1/2 (x)= 2 1+x K( 2 1+x ),K(t)= 0 1 [ (1 x 2 )(1t x 2 ) ] 1/2 dx .
j z =kI(x,y) (2f) 1 ( y 2 x 2 ).
J z W =[ I( x,y )( x y y x )φ( x,y )dxdy ]× × [ I(x,y)dxdy ] 1 109.63.
J z W =( k w 2 sin2α 8f )( z f 1 )( 2 z f 1 )116.
E( x,y )=exp( x 2 w x 2 y 2 w y 2 ik x 2 2 f x ik y 2 2 f y ),
{ x =xcosα+ysinα, y =ycosαxsinα,
J z W = k 8 ( 1 f y 1 f x )( w x 2 w y 2 )sin2α.
E 2 ( ξ,η )= ik B G exp[ ikD 2B ( ξ 2 + η 2 ) k 2 B 2 P xx η 2 + P yy ξ 2 P xy ξη G ],
P xx = 1 w x 2 + ik 2 f x cos 2 α+ ik 2 f y sin 2 α ikA 2B , P yy = 1 w y 2 + ik 2 f x sin 2 α+ ik 2 f y cos 2 α ikA 2B , P xy = ik 2 ( 1 f x 1 f y )sin2α,G=4 P xx P yy P xy 2 .
I 2 ( ξ,η )= k 2 B 2 | G | exp[ 2 k 2 B 2 | G | 2 Ψ( ξ,η ) ],
Ψ( ξ,η )=Re{ P yy G * } ξ 2 +Re{ P xx G * } η 2 Re{ P xy G * }ξη.
j z | E 2 | 2 =k r 2 2 z 0x z 0y [ f x z 2 + z 0x z 0y ( z f x ) ]cos2φ+ f x ( z 0y 2 z 0x 2 )z( z2 f x )sin2φ 4 [ f x z 2 + z 0x z 0y ( z f x ) ] 2 + ( z 0x + z 0y ) 2 z 2 ( z2 f x ) 2 ,
j z | E 2 | 2 = k 2 f x z 0x z 0y 4 f x 2 + z 0x z 0y r 2 cos2φ.
E( x,y,z )= [ q x ( z ) q y ( z ) ] 1/2 exp( x 2 w x 2 q x ( z ) y 2 w y 2 q y ( z ) ),
q x ( z )=1+ iz z 0x , q y ( z )=1+ iz z 0y ,
E(x,y,z)= [ q x ( z ) q y ( z ) ] 1/2 exp( x 2 w x 2 q x ( z ) y 2 w y 2 q y ( z ) )× ×exp( ik x 2 cos 2 α 2f ik y 2 sin 2 α 2f ikxysin2α 2f ).
J z W =( ksin2α 8f )( w y 2 | q y | 2 w x 2 | q x | 2 ).
J z W =( ksin2α 8f )( w y 2 w x 2 )( 1 4 z 2 k 2 w x 2 w y 2 ).