Abstract

Two simple and high-efficiency techniques for measuring the orbital angular momentum (OAM) of paraxial laser beams are proposed and studied numerically and experimentally. One technique relies on measuring the intensity in the Fresnel zone, followed by calculating the intensity that is numerically averaged over angle at discrete radii and deriving squared modules of the light field expansion coefficients via solving a linear set of equations. With the other technique, two intensity distributions are measured in the Fourier plane of a pair of cylindrical lenses positioned perpendicularly, before calculating the first-order moments of the measured intensities. The experimental error grows almost linearly from ~1% for small fractional OAM (up to 4) to ~10% for large fractional OAM (up to 34).

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (8)

X. Wang, Z. Nie, Y. Liang, J. Wang, T. Li, and B. Jia, “Recent advances on optical vortex generation,” Nanophotonics 7(9), 1533–1556 (2018).
[Crossref]

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8(1), 6370 (2018).
[Crossref] [PubMed]

A. V. Volyar, M. V. Bretsko, Ya. E. Akimova, and Yu. A. Egorov, “Beyond the light intensity or intensity moments and measurements of the vortex spectrum in complex light beams,” Comput. Opt. 42(5), 736–743 (2018).
[Crossref]

H. Gao, Y. Han, Y. Li, D. Zhu, M. Sun, and S. Yu, “Topological charge measurement of concentric OAM states using the phase-shift method,” J. Opt. Soc. Am. A 35(1), A40–A44 (2018).
[Crossref] [PubMed]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Astigmatic laser beams with a large orbital angular momentum,” Opt. Express 26(1), 141–156 (2018).
[Crossref] [PubMed]

Z. Liu, S. Gao, W. Xiao, J. Yang, X. Huang, Y. Feng, J. Li, W. Liu, and Z. Li, “Measuring high-order optical orbital angular momentum with a hyperbolic gradually changing period pure-phase grating,” Opt. Lett. 43(13), 3076–3079 (2018).
[Crossref] [PubMed]

Z. Xie, S. Gao, T. Lei, S. Feng, Y. Zhang, F. Li, J. Zhang, Z. Li, and X. Yuan, “Integrated (de)multiplexer for orbital angular momentum fiber communication,” Photon. Res. 6(7), 743–749 (2018).
[Crossref]

A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Measurement of the vortex spectrum in a vortex-beam array without cuts and gluing of the wavefront,” Opt. Lett. 43(22), 5635–5638 (2018).
[Crossref] [PubMed]

2017 (7)

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]

Y. Li, X. Li, L. W. Chen, M. B. Pu, J. J. Jin, M. H. Hong, and X. G. Luo, “Orbital angular momentum multiplexing and demultiplexing by a single metasurface,” Adv. Opt. Mater. 5(2), 1600502 (2017).
[Crossref]

S. N. Alperin and M. E. Siemens, “Angular momentum of topologically structured darkness,” Phys. Rev. Lett. 119(20), 203902 (2017).
[Crossref] [PubMed]

G. Ruffato, M. Massari, G. Parisi, and F. Romanato, “Test of mode-division multiplexing and demultiplexing in free-space with diffractive transformation optics,” Opt. Express 25(7), 7859–7868 (2017).
[Crossref] [PubMed]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Astigmatic transforms of an optical vortex for measurement of its topological charge,” Appl. Opt. 56(14), 4095–4104 (2017).
[Crossref] [PubMed]

S. Maji and M. M. Brundavanam, “Controlled noncanonical vortices from higher-order fractional screw dislocations,” Opt. Lett. 42(12), 2322–2325 (2017).
[Crossref] [PubMed]

Z. Wang, Y. Yan, A. Arbabi, G. Xie, C. Liu, Z. Zhao, Y. Ren, L. Li, N. Ahmed, A. J. Willner, E. Arbabi, A. Faraon, R. Bock, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital angular momentum beams generated by passive dielectric phase masks and their performance in a communication link,” Opt. Lett. 42(14), 2746–2749 (2017).
[Crossref] [PubMed]

2016 (3)

2014 (1)

2009 (1)

2007 (1)

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

2004 (1)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

2003 (1)

2001 (1)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).
[Crossref]

2000 (1)

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, J. Turunen, and Y. Wang, “Experimental selection of spatial Gauss-Laguerre Modes,” Opt. Mem. Neural. Networks 9(1), 73–82 (2000).

1987 (1)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Ahmed, N.

Akimova, Y.

Akimova, Ya. E.

A. V. Volyar, M. V. Bretsko, Ya. E. Akimova, and Yu. A. Egorov, “Beyond the light intensity or intensity moments and measurements of the vortex spectrum in complex light beams,” Comput. Opt. 42(5), 736–743 (2018).
[Crossref]

Alperin, S. N.

Arbabi, A.

Arbabi, E.

Ashrafi, S.

Barnett, S. M.

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

Bekshaev, A. Ya.

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

Bock, R.

Bretsko, M.

Bretsko, M. V.

A. V. Volyar, M. V. Bretsko, Ya. E. Akimova, and Yu. A. Egorov, “Beyond the light intensity or intensity moments and measurements of the vortex spectrum in complex light beams,” Comput. Opt. 42(5), 736–743 (2018).
[Crossref]

Brundavanam, M. M.

Cardano, F.

Chávez-Cerda, S.

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8(1), 6370 (2018).
[Crossref] [PubMed]

Chen, L. W.

Y. Li, X. Li, L. W. Chen, M. B. Pu, J. J. Jin, M. H. Hong, and X. G. Luo, “Orbital angular momentum multiplexing and demultiplexing by a single metasurface,” Adv. Opt. Mater. 5(2), 1600502 (2017).
[Crossref]

D’Amelio, R.

D’Errico, A.

Denisenko, V.

Desyatnikov, A. S.

Egorov, Y.

Egorov, Yu. A.

A. V. Volyar, M. V. Bretsko, Ya. E. Akimova, and Yu. A. Egorov, “Beyond the light intensity or intensity moments and measurements of the vortex spectrum in complex light beams,” Comput. Opt. 42(5), 736–743 (2018).
[Crossref]

Faraon, A.

Feng, S.

Feng, Y.

Franke-Arnold, S.

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

Gao, H.

Gao, S.

Gopinath, J. T.

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Götte, J. B.

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Han, Y.

Hong, M. H.

Y. Li, X. Li, L. W. Chen, M. B. Pu, J. J. Jin, M. H. Hong, and X. G. Luo, “Orbital angular momentum multiplexing and demultiplexing by a single metasurface,” Adv. Opt. Mater. 5(2), 1600502 (2017).
[Crossref]

Huang, X.

Jesus-Silva, A. J.

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8(1), 6370 (2018).
[Crossref] [PubMed]

Jia, B.

X. Wang, Z. Nie, Y. Liang, J. Wang, T. Li, and B. Jia, “Recent advances on optical vortex generation,” Nanophotonics 7(9), 1533–1556 (2018).
[Crossref]

Jin, J. J.

Y. Li, X. Li, L. W. Chen, M. B. Pu, J. J. Jin, M. H. Hong, and X. G. Luo, “Orbital angular momentum multiplexing and demultiplexing by a single metasurface,” Adv. Opt. Mater. 5(2), 1600502 (2017).
[Crossref]

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).
[Crossref]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, J. Turunen, and Y. Wang, “Experimental selection of spatial Gauss-Laguerre Modes,” Opt. Mem. Neural. Networks 9(1), 73–82 (2000).

Kivshar, Y. S.

Kotlyar, V. V.

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Astigmatic laser beams with a large orbital angular momentum,” Opt. Express 26(1), 141–156 (2018).
[Crossref] [PubMed]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Astigmatic transforms of an optical vortex for measurement of its topological charge,” Appl. Opt. 56(14), 4095–4104 (2017).
[Crossref] [PubMed]

A. A. Kovalev, V. V. Kotlyar, and A. P. Porfirev, “Asymmetric Laguerre-Gaussian beams,” Phys. Rev. A (Coll. Park) 93(6), 063858 (2016).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Asymmetric Bessel modes,” Opt. Lett. 39(8), 2395–2398 (2014).
[Crossref] [PubMed]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).
[Crossref]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, J. Turunen, and Y. Wang, “Experimental selection of spatial Gauss-Laguerre Modes,” Opt. Mem. Neural. Networks 9(1), 73–82 (2000).

Kovalev, A. A.

Krolikowski, W.

Laakkonen, P.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, J. Turunen, and Y. Wang, “Experimental selection of spatial Gauss-Laguerre Modes,” Opt. Mem. Neural. Networks 9(1), 73–82 (2000).

Lei, T.

Li, F.

Li, J.

Li, L.

Li, T.

X. Wang, Z. Nie, Y. Liang, J. Wang, T. Li, and B. Jia, “Recent advances on optical vortex generation,” Nanophotonics 7(9), 1533–1556 (2018).
[Crossref]

Li, X.

Y. Li, X. Li, L. W. Chen, M. B. Pu, J. J. Jin, M. H. Hong, and X. G. Luo, “Orbital angular momentum multiplexing and demultiplexing by a single metasurface,” Adv. Opt. Mater. 5(2), 1600502 (2017).
[Crossref]

Li, Y.

H. Gao, Y. Han, Y. Li, D. Zhu, M. Sun, and S. Yu, “Topological charge measurement of concentric OAM states using the phase-shift method,” J. Opt. Soc. Am. A 35(1), A40–A44 (2018).
[Crossref] [PubMed]

Y. Li, X. Li, L. W. Chen, M. B. Pu, J. J. Jin, M. H. Hong, and X. G. Luo, “Orbital angular momentum multiplexing and demultiplexing by a single metasurface,” Adv. Opt. Mater. 5(2), 1600502 (2017).
[Crossref]

Li, Z.

Liang, Y.

X. Wang, Z. Nie, Y. Liang, J. Wang, T. Li, and B. Jia, “Recent advances on optical vortex generation,” Nanophotonics 7(9), 1533–1556 (2018).
[Crossref]

Liu, C.

Liu, W.

Liu, Z.

Luo, X. G.

Y. Li, X. Li, L. W. Chen, M. B. Pu, J. J. Jin, M. H. Hong, and X. G. Luo, “Orbital angular momentum multiplexing and demultiplexing by a single metasurface,” Adv. Opt. Mater. 5(2), 1600502 (2017).
[Crossref]

Maji, S.

Marrucci, L.

Massari, M.

Melo, L. A.

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8(1), 6370 (2018).
[Crossref] [PubMed]

Neshev, D. N.

Nie, Z.

X. Wang, Z. Nie, Y. Liang, J. Wang, T. Li, and B. Jia, “Recent advances on optical vortex generation,” Nanophotonics 7(9), 1533–1556 (2018).
[Crossref]

Niederriter, R. D.

Paakkonen, P.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).
[Crossref]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Parisi, G.

Piccirillo, B.

Porfirev, A. P.

Pu, M. B.

Y. Li, X. Li, L. W. Chen, M. B. Pu, J. J. Jin, M. H. Hong, and X. G. Luo, “Orbital angular momentum multiplexing and demultiplexing by a single metasurface,” Adv. Opt. Mater. 5(2), 1600502 (2017).
[Crossref]

Ren, Y.

Ribeiro, P. H. S.

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8(1), 6370 (2018).
[Crossref] [PubMed]

Romanato, F.

Ruffato, G.

Shvedov, V.

Siemens, M. E.

Simonen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).
[Crossref]

Skidanov, R. V.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, J. Turunen, and Y. Wang, “Experimental selection of spatial Gauss-Laguerre Modes,” Opt. Mem. Neural. Networks 9(1), 73–82 (2000).

Soares, W. C.

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8(1), 6370 (2018).
[Crossref] [PubMed]

Soifer, V. A.

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Asymmetric Bessel modes,” Opt. Lett. 39(8), 2395–2398 (2014).
[Crossref] [PubMed]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).
[Crossref]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, J. Turunen, and Y. Wang, “Experimental selection of spatial Gauss-Laguerre Modes,” Opt. Mem. Neural. Networks 9(1), 73–82 (2000).

Soskin, M.

Soskin, M. S.

Sun, M.

Tur, M.

Turunen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).
[Crossref]

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, J. Turunen, and Y. Wang, “Experimental selection of spatial Gauss-Laguerre Modes,” Opt. Mem. Neural. Networks 9(1), 73–82 (2000).

Vasnetsov, M. V.

Volyar, A.

Volyar, A. V.

A. V. Volyar, M. V. Bretsko, Ya. E. Akimova, and Yu. A. Egorov, “Beyond the light intensity or intensity moments and measurements of the vortex spectrum in complex light beams,” Comput. Opt. 42(5), 736–743 (2018).
[Crossref]

Wang, J.

X. Wang, Z. Nie, Y. Liang, J. Wang, T. Li, and B. Jia, “Recent advances on optical vortex generation,” Nanophotonics 7(9), 1533–1556 (2018).
[Crossref]

J. Wang, “Advances in communications using optical vortices,” Photon. Res. 4(5), B14–B28 (2016).
[Crossref]

Wang, X.

X. Wang, Z. Nie, Y. Liang, J. Wang, T. Li, and B. Jia, “Recent advances on optical vortex generation,” Nanophotonics 7(9), 1533–1556 (2018).
[Crossref]

Wang, Y.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, J. Turunen, and Y. Wang, “Experimental selection of spatial Gauss-Laguerre Modes,” Opt. Mem. Neural. Networks 9(1), 73–82 (2000).

Wang, Z.

Willner, A. E.

Willner, A. J.

Xiao, W.

Xie, G.

Xie, Z.

Yan, Y.

Yang, J.

Yu, S.

Yuan, X.

Zambrini, R.

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

Zhang, J.

Zhang, Y.

Zhao, Z.

Zhu, D.

Adv. Opt. Mater. (1)

Y. Li, X. Li, L. W. Chen, M. B. Pu, J. J. Jin, M. H. Hong, and X. G. Luo, “Orbital angular momentum multiplexing and demultiplexing by a single metasurface,” Adv. Opt. Mater. 5(2), 1600502 (2017).
[Crossref]

Appl. Opt. (1)

Comput. Opt. (1)

A. V. Volyar, M. V. Bretsko, Ya. E. Akimova, and Yu. A. Egorov, “Beyond the light intensity or intensity moments and measurements of the vortex spectrum in complex light beams,” Comput. Opt. 42(5), 736–743 (2018).
[Crossref]

J. Mod. Opt. (2)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001).
[Crossref]

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

J. Opt. Soc. Am. A (2)

Nanophotonics (1)

X. Wang, Z. Nie, Y. Liang, J. Wang, T. Li, and B. Jia, “Recent advances on optical vortex generation,” Nanophotonics 7(9), 1533–1556 (2018).
[Crossref]

Opt. Commun. (1)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Opt. Express (3)

Opt. Lett. (6)

Opt. Mem. Neural. Networks (1)

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, J. Turunen, and Y. Wang, “Experimental selection of spatial Gauss-Laguerre Modes,” Opt. Mem. Neural. Networks 9(1), 73–82 (2000).

Optica (1)

Photon. Res. (2)

Phys. Rev. A (Coll. Park) (1)

A. A. Kovalev, V. V. Kotlyar, and A. P. Porfirev, “Asymmetric Laguerre-Gaussian beams,” Phys. Rev. A (Coll. Park) 93(6), 063858 (2016).
[Crossref]

Phys. Rev. Lett. (1)

S. N. Alperin and M. E. Siemens, “Angular momentum of topologically structured darkness,” Phys. Rev. Lett. 119(20), 203902 (2017).
[Crossref] [PubMed]

Sci. Rep. (1)

L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. Soares, “Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8(1), 6370 (2018).
[Crossref] [PubMed]

Other (4)

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G. J. Gbur, Singular Optics (Boca Raton: CRC Press, 545 p, 2016).

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, Vortex laser beams (Boca Raton: CRC Press, 404 p, 2018).

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Volume 2: Special functions (New York: Gordon and Breach, 1986).

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Figures (11)

Fig. 1
Fig. 1 Two distributions of (a,c) intensity and (b,d) phase in the initial plane z = 0 for superposition of LG modes whose normalized OAM equals (a,b) 3 and (c,d) 3.5.
Fig. 2
Fig. 2 Five radii of the rings that were utilized to calculate the intensity for solving the set of Eqs. (6) for the beam in (a) Figs. 1(a, b) (a) and (b) Figs. 1(c, d).
Fig. 3
Fig. 3 Optical setup for the experimental determination of OAM of superpositions of the LG modes: Laser is a solid state laser (λ = 532 nm), MO is a microobjective (8 × , NA = 0.3), PH is a pinhole (the hole diameter is 40 μm), L1, L2 and L3 are spherical lenses (f1 = 350 mm, f2 = 500 mm, f3 = 150 mm), SLM is a spatial light modulator HOLOEYE PLUTO-VIS, M is a mirror, D is a diaphragm, Cam is a camera ToupCam U3CMOS08500KPA.
Fig. 4
Fig. 4 Experimentally generated intensity distributions of various superpositions of LG modes registered by the camera (Fig. 3): LG0,2 + LG0,4 (a), LG0,2 + 1/31/2LG0,4 (b), LG0,2 + 1/21/2LG0,5 (c), LG0,1 + LG0,5 (d). Image size is 1500 × 1500 μm.
Fig. 5
Fig. 5 Optical setup for measuring the OAM with a couple of cylindrical lenses: a Laser, PH - an opaque screen with circular aperture, L1, L2, L3 - spherical lenses, SLM - a spatial light modulator, F - a spatial filter to cut off the non-modulated zero diffraction order, BS - a beam-splitter to divide the beam in two identical beams, CL1, CL2 - perpendicularly positioned cylindrical lenses, CCD1, CCD2 - CCD-arrays, and PC - a computer.
Fig. 6
Fig. 6 Phase patterns (a,d,g,j) in the initial plane z = 0 and intensity patterns in the focal plane of the cylindrical lens whose axis is parallel to the x-axis (b,e, h,k) and y-axis (c,f,i,l) for a Gaussian beam having passed through a SPP with the topological charge 3.00 (a-c), 3.25 (d-f), 3.50 (g-i), and 3.75 (j-l).
Fig. 7
Fig. 7 (a) Phase pattern in the initial plane z = 0 and intensity pattern in the focus of the cylindrical lens whose axis is parallel to the (b) x-axis and (c) y-axis for a Gaussian beam having passed through a SPP with topological charge 30.3.
Fig. 8
Fig. 8 Normalized power of the angular harmonics against the topological charge of (a) a Gaussian beam having passed through a SPP with topological charge μ = 3.5 and (b) superposition of the LG modes (0, 2) and (0, 4), with its normalized OAM equal to 3.5.
Fig. 9
Fig. 9 Intensity distribution of the symmetric (a = 0) (a) and asymmetric (a = 0.25) (b) Laguerre-Gaussian beams, intensity distributions of the asymmetric Laguerre-Gaussian beam in the foci of the cylindrical lens (c, d), and the dependence of the OAM vs the asymmetry parameter (e). Red circles are the values obtained by using Eqs. (14)-(16).
Fig. 10
Fig. 10 Intensity distribution in the foci of the cylindrical lenses CL1 and CL2, given that their axes are parallel to the (a-e) x-axis and (f-j) y-axis for a Gaussian beam having passed through a SPP whose topological charge equals (a,f) 3.00, (b,g) 3.25, (c,h) 3.50, (d,i) 3.75, and (e,j) 30.3.
Fig. 11
Fig. 11 Experimentally measured OAM (red rods) for several fractional topological charges and the theoretical dependence of the OAM on the topological charge (green curve). Length of the rods corresponds to the tolerance of the experiments. The numbers near the plot show the topological charges for which the OAM was measured.

Tables (2)

Tables Icon

Table 1 Comparison of OAM values obtained theoretically by Eq. (9), by solving the system in Eq. (6) and experimentally by using the intensity distributions in Fig. 4

Tables Icon

Table 2 OAM values calculated theoretically using Eq. (20) and numerically using Eq. (16) based on numerically and experimentally derived intensity distributions

Equations (33)

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E(r,φ,z)= n=N N C n exp( inφ ) Ψ n (r,z),
Ψ n (r,z)= i w 0 w(z) ( r 2 w(z) ) n exp[ r 2 w 2 (z) + ik r 2 2R(z) i(2n+1)arctan( z z 0 ) ],
Ψ 1n (r,z)= q 1 (z) J n ( αr q(z) )exp[ r 2 w 0 2 q(z) +ikz i α 2 z 2kq(z) ],
I(r,φ,z)= | E(r,φ,z) | 2 = n=N N m=N N C n C ¯ m exp[ i(nm)φ ] Ψ n (r,z) Ψ ¯ m (r,z),
I ˜ (r,z)= 0 2π I(r,φ,z)dφ=2π n=N N | C n | 2 | Ψ n (r,z) | 2 .
I ˜ m = n=N N M nm x n ,m=N,( N1 ),...,( N1 ),N,
Ψ n ( r,z=0 )=Anexp[ ( r r n ) 2 / w 0 2 ],
J z =Im 0 0 2π E ¯ (r,φ,z)( E(r,φ,z) φ )rdrdφ ,
W= 0 0 2π E(r,φ,z) E ¯ (r,φ,z)rdrdφ .
J z W = n=N N n | C n | 2 I n n=N N | C n | 2 I n ,
E( r,φ,z=0 )= [ C 2 π w 0 2 exp( 2iφ ) ( r 2 w 0 ) 2 + C 4 12π w 0 2 exp( 4iφ ) ( r 2 w 0 ) 4 ]exp( r 2 w 0 2 ).
E n ( r,φ,z )=exp( inφ )Ψ( r,z ),
E μ (r,φ,z)=exp(iμφ)Ψ(r,z).
J z =Im E ¯ (x,y,z)( x y y x )E(x,y,z)dxdy .
E(x,y)= ik 2πf E ^ x (η,y)exp( ikηx f )dη ,
E(x,y)= ik 2πf E ^ y ( x,ξ )exp( ikξy f )dξ ,
J z = k f ( xξ | E ^ y (x,ξ,z) | 2 dxdξ ηy | E ^ x (η,y,z) | 2 dηdy ),
E μ (r,φ,z)=exp(iμφ)Ψ(r,z)= e iπμ sinπμ π Ψ(r,z) n= e inφ μn .
J z =W sin 2 (πμ) π 2 n= n ( μn ) 2 ,
n=1 n 2 ( n 2 ± a 2 ) 2 = π 4a [ ±{ cothπa cotπa }a{ csch 2 πa csc 2 πa } ] ,
J z W =μ sin2πμ 2π .
E μg ( r,ϕ,z=0 )=exp( iμϕ r 2 w 0 2 ).
E μg (ρ,θ,z)=( i z 0 z q 1 (z) )( e iμπ sinμπ 2π ) e ik ρ 2 2z x e x n= (i) n e inθ μn [ I (n1)/2 (x) I (n+1)/2 (x) ] ,
q 1 (z)=1 i z 0 z ,x= ( z 0 z ) 2 ( ρ w 0 ) 2 ( 1 8 q 1 (z) ),
0 e p r 2 r J n (cr)dr= c π 8 p 3/2 e c 2 8p [ I (n1)/2 ( c 2 8p ) I (n+1)/2 ( c 2 8p ) ].
E μb (r,ϕ,z=0)= r 0 exp( iμϕ )δ(r r 0 ),
E μb (ρ,θ,z)=( ik r 0 2 z )( e iμπ sinμπ π )exp[ ik 2z ( ρ 2 + r 0 2 ) ] n= (i) n e inθ μn J n ( k r 0 ρ 2z ) ,
J z =na+m(1a),a= | C | 2 1.
E( x,y )= ik 2πf E ^ x ( ξ,η ) exp{ ik 2f [ ξ 2 + ( ηy ) 2 ]+i k f xξ }dξdη,
E( x,y )= ik 2πf E ^ y ( ξ,η ) exp{ ik 2f [ ( ξx ) 2 + η 2 ]+i k f yη }dξdη.
J z =i E * ( x,y )( x y y x )E( x,y )dxdy = = k 2 2π f 2 η E ^ y * ( ξ ,η ) E ^ y ( ξ ,η )d ξ d ξ dη × ×exp{ ik 2f ( ξ 2 ξ 2 )i k f x( ξ ξ ) }xdx k 2 2π f 2 ξ E ^ x * ( ξ, η ) E ^ x ( ξ, η )dξd η d η × ×exp{ ik 2f ( η 2 η 2 ) ik f y( η η ) }ydy.
J z =i η E ^ y ( ξ,η ) [ ξ E ^ y * ( ξ,η )+ ikξ f E ^ y * ( ξ,η ) ]dξdη+ +i ξ E ^ x ( ξ,η ) [ η E ^ x * ( ξ,η )+ ikη f E ^ x * ( ξ,η ) ]dξdη.
J z = k f ξη | E ^ y ( ξ,η ) | 2 dξdη k f ξη | E ^ x ( ξ,η ) | 2 dξdη i η E ^ y ( ξ,η ) ξ E ^ y * ( ξ,η )dξdη +i ξ E ^ x ( ξ,η ) η E ^ x * ( ξ,η )dξdη .

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