Abstract

Fractional vortex beams (FVBs) with non-integer topological charges attract much attention due to unique features of propagations, but different viewpoints still exist on the change of their total vortex strength. Here we have experimentally demonstrated the distribution and number of vortices contained in FVBs at the Fraunhofer diffraction region. We have verified that the jumps of total vortex strength for FVBs happen only when non-integer topological charge is before and after (but very close to) any even integer number that originates from two different mechanisms for generation and movement of vortices on focal plane. Meanwhile, we have also measured the beam propagation factor (BPF) of such FVBs and have found that their BPF values almost increase linearly in the x component (along the initial edge dislocation) and oscillate increasingly in the y component (vertical to the initial edge dislocation). Our experimental results are in good agreement with numerical results.

© 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 (3)

2017 (4)

P. Bandyopadhyay, B. Basu, and D. Chowdhury, “Geometric phase and fractional orbital-angular-momentum states in electron vortex beams,” Phys. Rev. A 95, 013821 (2017).
[Crossref]

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

G. Tkachenko, M. Z. Chen, K. Dholakia, and M. Mazilu, “Is it possible to create a perfect fractional vortex beam?” Optica 4, 330–333 (2017).
[Crossref]

Y. Q. Fang, Q. H. Lu, X. L. Wang, W. H. Zhang, and L. X. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95, 023821 (2017).
[Crossref]

2016 (2)

2015 (1)

Z. Y. Hong, J. Zhang, and B. W. Drinkwater, “On the radiation force fields of fractional-order acoustic vortices,” EPL 110, 14002 (2015).
[Crossref]

2014 (1)

L. X. Chen, J. J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light: Sci. Appl. 3, e153(2014).
[Crossref]

2012 (2)

2011 (1)

2010 (1)

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

2009 (1)

M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009).
[Crossref]

2007 (1)

2005 (2)

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[Crossref] [PubMed]

S. H. Tao, X. C. Yuan, J. Lin, X. Peng, and H. B. Niu, “Fractional optical vortex beam induced rotation of particles,” Opt. Express 13, 7726–7731 (2005).
[Crossref] [PubMed]

2004 (5)

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

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[Crossref] [PubMed]

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239, 129–135 (2004).
[Crossref]

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. 6, S166–S169 (2004).

2001 (1)

2000 (1)

1998 (1)

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[Crossref]

1997 (1)

1995 (1)

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8819 (1992).
[Crossref] [PubMed]

1990 (1)

A. E. Siegman,“New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[Crossref]

1970 (1)

’t Hooft, G. W.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[Crossref] [PubMed]

Aiello, A.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[Crossref] [PubMed]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[Crossref] [PubMed]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8819 (1992).
[Crossref] [PubMed]

Alperin, S. N.

Anderson, M. E.

Assmann, M.

Bandyopadhyay, P.

P. Bandyopadhyay, B. Basu, and D. Chowdhury, “Geometric phase and fractional orbital-angular-momentum states in electron vortex beams,” Phys. Rev. A 95, 013821 (2017).
[Crossref]

Basistiy, I. V.

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. 6, S166–S169 (2004).

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

Basu, B.

P. Bandyopadhyay, B. Basu, and D. Chowdhury, “Geometric phase and fractional orbital-angular-momentum states in electron vortex beams,” Phys. Rev. A 95, 013821 (2017).
[Crossref]

Baumgartl, J.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8819 (1992).
[Crossref] [PubMed]

Berger, B.

Berry, M. V.

M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009).
[Crossref]

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

Bigman, H.

Borghi, R.

Cai, Y. J.

Chaloupka, J. L.

Chávez-Cerda, S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

Chen, L. X.

Y. Q. Fang, Q. H. Lu, X. L. Wang, W. H. Zhang, and L. X. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95, 023821 (2017).
[Crossref]

L. X. Chen, J. J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light: Sci. Appl. 3, e153(2014).
[Crossref]

Chen, M. Z.

Chowdhury, D.

P. Bandyopadhyay, B. Basu, and D. Chowdhury, “Geometric phase and fractional orbital-angular-momentum states in electron vortex beams,” Phys. Rev. A 95, 013821 (2017).
[Crossref]

Collins, S. A.

de Araujo, L. E. E.

Dholakia, K.

Drinkwater, B. W.

Z. Y. Hong, J. Zhang, and B. W. Drinkwater, “On the radiation force fields of fractional-order acoustic vortices,” EPL 110, 14002 (2015).
[Crossref]

Eliel, E. R.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[Crossref] [PubMed]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[Crossref] [PubMed]

Fang, Y. Q.

Y. Q. Fang, Q. H. Lu, X. L. Wang, W. H. Zhang, and L. X. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95, 023821 (2017).
[Crossref]

Fonseca, E. J. S.

A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Study of the birth of a vortex at Fraunhofer zone,” Opt. Lett. 37, 4552–4554 (2012).
[Crossref] [PubMed]

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

Gbur, G.

Gopinath, J. T.

Gutiérrez-Vega, J. C.

Hickmann, J. M.

A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Study of the birth of a vortex at Fraunhofer zone,” Opt. Lett. 37, 4552–4554 (2012).
[Crossref] [PubMed]

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

Hong, Z. Y.

Z. Y. Hong, J. Zhang, and B. W. Drinkwater, “On the radiation force fields of fractional-order acoustic vortices,” EPL 110, 14002 (2015).
[Crossref]

Jesus-Silva, A. J.

Jia, Y. R.

Y. R. Jia, Q. Wei, D. J. Wu, Z. Xu, and X. J. Liu, “Generation of fractional acoustic vortex with a discrete Archimedean spiral structure plate,” Appl. Phys. Lett. 112, 173501 (2018).
[Crossref]

Kahlert, M.

Leach, J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Lee, W. M.

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239, 129–135 (2004).
[Crossref]

Lei, J. J.

L. X. Chen, J. J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light: Sci. Appl. 3, e153(2014).
[Crossref]

Lin, J.

Liu, X. J.

Y. R. Jia, Q. Wei, D. J. Wu, Z. Xu, and X. J. Liu, “Generation of fractional acoustic vortex with a discrete Archimedean spiral structure plate,” Appl. Phys. Lett. 112, 173501 (2018).
[Crossref]

Liu, X. L.

Lu, Q. H.

Y. Q. Fang, Q. H. Lu, X. L. Wang, W. H. Zhang, and L. X. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95, 023821 (2017).
[Crossref]

Ma, X.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[Crossref] [PubMed]

Mazilu, M.

Mourka, A.

Niederriter, R. D.

Nienhuis, G.

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[Crossref] [PubMed]

Niu, H. B.

Oemrawsingh, S. S. R.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[Crossref] [PubMed]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[Crossref] [PubMed]

Padgett, M. J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Pas’ko, V. A.

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. 6, S166–S169 (2004).

Peng, X.

Porras, M. A.

Ramee, S.

Romero, J.

L. X. Chen, J. J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light: Sci. Appl. 3, e153(2014).
[Crossref]

Saghafi, S.

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[Crossref]

Santarsiero, M.

Schmidt, D.

Shanor, C.

Sheppard, C. J. R.

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[Crossref]

Siegman, A. E.

A. E. Siegman,“New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[Crossref]

Siemens, M. E.

Simon, R.

Slyusar, V. V.

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. 6, S166–S169 (2004).

Soares, W. C.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

Soskin, M. S.

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. 6, S166–S169 (2004).

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8819 (1992).
[Crossref] [PubMed]

Tao, S. H.

Tkachenko, G.

Vasnetsov, M. V.

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. 6, S166–S169 (2004).

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

Voigt, D.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[Crossref] [PubMed]

Wang, F.

Wang, S.

S. Wang and D. Zhao, Matrix Optics (Springer, 2000).

Wang, X. L.

Y. Q. Fang, Q. H. Lu, X. L. Wang, W. H. Zhang, and L. X. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95, 023821 (2017).
[Crossref]

Wei, Q.

Y. R. Jia, Q. Wei, D. J. Wu, Z. Xu, and X. J. Liu, “Generation of fractional acoustic vortex with a discrete Archimedean spiral structure plate,” Appl. Phys. Lett. 112, 173501 (2018).
[Crossref]

Woerdman, J. P.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[Crossref] [PubMed]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[Crossref] [PubMed]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8819 (1992).
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Wu, D. J.

Y. R. Jia, Q. Wei, D. J. Wu, Z. Xu, and X. J. Liu, “Generation of fractional acoustic vortex with a discrete Archimedean spiral structure plate,” Appl. Phys. Lett. 112, 173501 (2018).
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Xu, Z.

Y. R. Jia, Q. Wei, D. J. Wu, Z. Xu, and X. J. Liu, “Generation of fractional acoustic vortex with a discrete Archimedean spiral structure plate,” Appl. Phys. Lett. 112, 173501 (2018).
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Appl. Phys. Lett. (1)

Y. R. Jia, Q. Wei, D. J. Wu, Z. Xu, and X. J. Liu, “Generation of fractional acoustic vortex with a discrete Archimedean spiral structure plate,” Appl. Phys. Lett. 112, 173501 (2018).
[Crossref]

EPL (1)

Z. Y. Hong, J. Zhang, and B. W. Drinkwater, “On the radiation force fields of fractional-order acoustic vortices,” EPL 110, 14002 (2015).
[Crossref]

J. Opt. (1)

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. 6, S166–S169 (2004).

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New J. Phys. (1)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

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W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239, 129–135 (2004).
[Crossref]

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[Crossref]

Opt. Express (5)

Opt. Lett. (3)

Optica (2)

Phys. Rev. A (3)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8819 (1992).
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P. Bandyopadhyay, B. Basu, and D. Chowdhury, “Geometric phase and fractional orbital-angular-momentum states in electron vortex beams,” Phys. Rev. A 95, 013821 (2017).
[Crossref]

Y. Q. Fang, Q. H. Lu, X. L. Wang, W. H. Zhang, and L. X. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95, 023821 (2017).
[Crossref]

Phys. Rev. Lett. (4)

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[Crossref] [PubMed]

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
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Figures (5)

Fig. 1
Fig. 1 Experimental setup for measuring intensity distributions of FVBs near the focal plane of a 2-f lens system. When a beam splitter (BS) is inserted after the lens L1, the reflected light is used for verifying their topological charges by using a triangle aperture placed on a two-axis translation stage. Left insert figure is experimental intensity measurement of FVBs with α = 1.5 and 3.5, respectively, locating at 40 mm before (top), at (middle), and at 40 mm after (bottom) the focal plane. The lens L1 has a focal length f = 300 mm. Other notations are: HWP, half-wave plate; PBS, polarized beam splitter; PM, power meter; SLM, spatial light modulator; AP, Aperture.
Fig. 2
Fig. 2 Numerical results of the vortex strength of FVBs at focal plane as a function of topological charge α with steps of 0.1. The inset figures are the schematic diagrams of vortex dynamics for FVBs at focal plane as α increases within each interval between any two integers, where red-solid and blue-open circles, respectively, denote the vortices with +1 and -1 charge. The arrows indicate the movement directions of new vortices, and the cross symbol is the geometric center on the focal plane.
Fig. 3
Fig. 3 (a) Experimental and (b) numerical results of typical intensity distributions for FVBs with different values α from 1.2 (left) to 2.8 (right). The corresponding numerical phase structures are plotted in (c). In (a) and (b), the first and second rows are, respectively, in normal and logarithmic scales. Here we use the lens L1 with f = 1000 mm. In (c), the positive unit vortex is signed by "+", the negative unit vortex is signed by "-", and "+2" represents positive two charges.
Fig. 4
Fig. 4 Experimental verification on the topological charge of each vortex in FVBs at the focal plane of a 2-f lens system. (a) The vortex distributions in different FVBs (left), and the corresponding interference patterns (right) of individual vortices by the diffraction method of a tiny triangle aperture. (b) The interference patterns diffracted by the straight blade via a 4-f lens system. Here the blade edge is moved from up to down, and is located at different positions a, b, or c [see the left sketch of each case in (b)], for detecting the fork-like interference patterns. The measurement system by the straight blade is also plotted in the bottom of (b). Each vortex and its interference patten in (a) and (b) are correspondingly marked with a number, and here the topological charge α = 1.5 , 2.5 , 3.5 .
Fig. 5
Fig. 5 Experimental results (black and red dots), and matched numerical results (black and red dashed lines) of the BPF values in (a) x-direction, (b) y-direction, and (c) total BPF values of FVBs as functions of α with steps of 0.1. The error bars are obtained from multiple measurements and the focal length of the lens L1 is chosen to be f = 300 mm and f = 1000 mm for two groups of multiple measurements.

Equations (13)

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E i ( u , v , 0 ) = exp  ( i α ϕ ) ,
E o ( x , y , z ) = exp  ( i k L ) i λ B E i ( u , v , 0 ) exp  { i k 2 B [ A ( u 2 + v 2 ) 2 ( x u + y v ) + D ( x 2 + y 2 ) ] } d u d v ,
E o , n ( ρ , φ , z ) = ( i ) | n | 2 A 3 / 2 ( π k ρ 2 8 B ) 1 / 2 exp   [ i ( k L + n φ ) ] × exp   [ i k ρ 2 2 B ( D 1 2 A ) ] × [ J | n | 1 2 ( k ρ 2 4 A B ) i J | n | + 1 2 ( k ρ 2 4 A B ) ] ,
E i ( u , v , 0 ) = exp  ( u 2 + v 2 w 0 2 ) exp  ( i α ϕ ) ,
E o , n ( ρ , φ , z ) = ( i ) | n | + 1 z R 2 ( B i A z R ) 3 / 2 ( π ρ 2 4 B w 0 2 ) 1 / 2 exp   [ i ( k L + n φ ) ] × exp   [ i k ρ 2 2 B ( D + i z R 2 ( B i A z R ) ) ] × [ I | n | 1 2 ( z R 2 ρ 2 / w 0 2 2 B ( B i A z R ) ) I | n | + 1 2 ( z R 2 ρ 2 / w 0 2 2 B ( B i A z R ) ) ] ,
exp  ( i α ϕ ) = exp  ( i π α ) sin  ( π α ) π n = exp  ( i n ϕ ) α n .
E o , α ( ρ , φ , z ) = exp  ( i π α ) sin  ( π α ) π n = E o , n ( ρ , φ , z ) α n .
M x 2 = 4 π w x Δ θ x ,
M y 2 = 4 π w y Δ θ y ,
w q 2 = ( q q ¯ ) 2 | E i ( u , v , 0 ) | 2 d u d v | E i ( u , v , 0 ) | 2 d u d v ,
( Δ θ q ) 2 = ( θ q θ ¯ q ) 2 | E o ( θ x , θ y , f ) | 2 d θ x d θ y | E o ( θ x , θ y , f ) | 2 d θ x d θ y ,
S α = lim   ρ 1 2 π 0 2 π d φ [ arg   E o , α ( ρ , φ , f ) ] / φ = lim   ρ 1 2 π 0 2 π d φ Re [ ( i ) E o , α 1 ( ρ , φ , f ) E o , α ( ρ , φ , f ) / φ ] .
S α = { m , for α = m 2 m 1 , for 2 ( m 1 ) < α < 2 m

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