Abstract

The spatial correlation singularity of a partially coherent vortex beam was demonstrated recently [Phys. Rev. Lett. 92, 143905 (2004)], and it was shown that the cross-correlation singularity disappears when the spatial coherence is high. In this paper, we demonstrate that the spatial autocorrelation function of a fully coherent beam in the far-field is equivalent to the Fourier transform of its intensity in the source plane. Our theoretical and experimental results show that, depending on both the radial and azimuthal mode indices (p, λ) of the incident light beam, the distribution of the far-field autocorrelation function displays a series of concentric, alternate bright and dark rings. This phenomenon may be used to determine the topological charge (the azimuthal index) of light beam with a nonzero radial index.

© 2014 Optical Society of America

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  1. B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
    [CrossRef] [PubMed]
  2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2013 (1)

Y. Yang, M. Chen, M. Mazilu, A. Mourka, Y. Liu, K. Dholakia, “Effect of the radial and azimuthal mode indices of a partially coherent vortex field upon a spatial correlation singularity,” New J. Phys. 15(11), 113053 (2013).
[CrossRef]

2012 (3)

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100(23), 231115 (2012).
[CrossRef]

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

Y. Yang, M. Mazilu, K. Dholakia, “Measuring the orbital angular momentum of partially coherent optical vortices through singularities in their cross-spectral density functions,” Opt. Lett. 37(23), 4949–4951 (2012).
[PubMed]

2011 (7)

2010 (1)

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[CrossRef] [PubMed]

2009 (1)

C.-S. Guo, L. Lu, S.-J. Yue, G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. 94(23), 231104 (2009).
[CrossRef]

2007 (1)

G. Molina-Terriza, J. P. Torres, L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[CrossRef]

2004 (4)

D. M. Palacios, I. D. Maleev, A. S. Marathay, G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[CrossRef] [PubMed]

G. Molina-Terriza, A. Vaziri, J. Rehácek, Z. Hradil, A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92(16), 167903 (2004).
[CrossRef] [PubMed]

I. Maleev, D. Palacios, A. Marathay, G. Swartzlander., “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B 21(11), 1895–1900 (2004).
[CrossRef]

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[CrossRef] [PubMed]

2003 (2)

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[CrossRef] [PubMed]

H. F. Schouten, G. Gbur, T. D. Visser, E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28(12), 968–970 (2003).
[CrossRef] [PubMed]

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Agrawal, A.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[CrossRef] [PubMed]

Ahmed, N.

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

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Anderson, I. M.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[CrossRef] [PubMed]

Anderson, M. E.

Banerji, J.

Barnett, S.

Baumgartl, J.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Chávez-Cerda, S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[CrossRef] [PubMed]

Chen, M.

Y. Yang, M. Chen, M. Mazilu, A. Mourka, Y. Liu, K. Dholakia, “Effect of the radial and azimuthal mode indices of a partially coherent vortex field upon a spatial correlation singularity,” New J. Phys. 15(11), 113053 (2013).
[CrossRef]

Cizmar, T.

K. Dholakia, T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[CrossRef]

Courtial, J.

de Araujo, L. E. E.

Dholakia, K.

Y. Yang, M. Chen, M. Mazilu, A. Mourka, Y. Liu, K. Dholakia, “Effect of the radial and azimuthal mode indices of a partially coherent vortex field upon a spatial correlation singularity,” New J. Phys. 15(11), 113053 (2013).
[CrossRef]

Y. Yang, M. Mazilu, K. Dholakia, “Measuring the orbital angular momentum of partially coherent optical vortices through singularities in their cross-spectral density functions,” Opt. Lett. 37(23), 4949–4951 (2012).
[PubMed]

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100(23), 231115 (2012).
[CrossRef]

A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express 19(7), 5760–5771 (2011).
[CrossRef] [PubMed]

K. Dholakia, T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[CrossRef]

Djordjevic, I. B.

Dolinar, S.

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

Fazal, I.

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

Ferreira, Q. S.

Fonseca, E. J. S.

Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett. 36(16), 3106–3108 (2011).
[CrossRef] [PubMed]

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[CrossRef] [PubMed]

Franke-Arnold, S.

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

Gbur, G.

Gibson, G.

Guo, C.-S.

C.-S. Guo, L. Lu, S.-J. Yue, G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. 94(23), 231104 (2009).
[CrossRef]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

Herzing, A. A.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[CrossRef] [PubMed]

Hickmann, J. M.

Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett. 36(16), 3106–3108 (2011).
[CrossRef] [PubMed]

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[CrossRef] [PubMed]

Hradil, Z.

G. Molina-Terriza, A. Vaziri, J. Rehácek, Z. Hradil, A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92(16), 167903 (2004).
[CrossRef] [PubMed]

Huang, H.

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

Jennewein, T.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[CrossRef] [PubMed]

Jesus-Silva, A. J.

Kumar, A.

Lezec, H. J.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[CrossRef] [PubMed]

Liu, Y.

Y. Yang, M. Chen, M. Mazilu, A. Mourka, Y. Liu, K. Dholakia, “Effect of the radial and azimuthal mode indices of a partially coherent vortex field upon a spatial correlation singularity,” New J. Phys. 15(11), 113053 (2013).
[CrossRef]

Lu, L.

C.-S. Guo, L. Lu, S.-J. Yue, G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. 94(23), 231104 (2009).
[CrossRef]

Maleev, I.

Maleev, I. D.

D. M. Palacios, I. D. Maleev, A. S. Marathay, G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[CrossRef] [PubMed]

Marathay, A.

Marathay, A. S.

D. M. Palacios, I. D. Maleev, A. S. Marathay, G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[CrossRef] [PubMed]

Mazilu, M.

Y. Yang, M. Chen, M. Mazilu, A. Mourka, Y. Liu, K. Dholakia, “Effect of the radial and azimuthal mode indices of a partially coherent vortex field upon a spatial correlation singularity,” New J. Phys. 15(11), 113053 (2013).
[CrossRef]

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100(23), 231115 (2012).
[CrossRef]

Y. Yang, M. Mazilu, K. Dholakia, “Measuring the orbital angular momentum of partially coherent optical vortices through singularities in their cross-spectral density functions,” Opt. Lett. 37(23), 4949–4951 (2012).
[PubMed]

McClelland, J. J.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[CrossRef] [PubMed]

McMorran, B. J.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[CrossRef] [PubMed]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[CrossRef]

G. Molina-Terriza, A. Vaziri, J. Rehácek, Z. Hradil, A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92(16), 167903 (2004).
[CrossRef] [PubMed]

Mourka, A.

Y. Yang, M. Chen, M. Mazilu, A. Mourka, Y. Liu, K. Dholakia, “Effect of the radial and azimuthal mode indices of a partially coherent vortex field upon a spatial correlation singularity,” New J. Phys. 15(11), 113053 (2013).
[CrossRef]

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100(23), 231115 (2012).
[CrossRef]

A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express 19(7), 5760–5771 (2011).
[CrossRef] [PubMed]

Padgett, M.

Palacios, D.

Palacios, D. M.

D. M. Palacios, I. D. Maleev, A. S. Marathay, G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[CrossRef] [PubMed]

Pan, J. W.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[CrossRef] [PubMed]

Pas’ko, V.

Prabhakar, S.

Rehácek, J.

G. Molina-Terriza, A. Vaziri, J. Rehácek, Z. Hradil, A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92(16), 167903 (2004).
[CrossRef] [PubMed]

Ren, Y.

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

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

Schouten, H. F.

Shanor, C.

Singh, R. P.

Soares, W. C.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[CrossRef] [PubMed]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Swartzlander, G.

Swartzlander, G. A.

D. M. Palacios, I. D. Maleev, A. S. Marathay, G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[CrossRef] [PubMed]

Torner, L.

G. Molina-Terriza, J. P. Torres, L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[CrossRef]

Tur, M.

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

Unguris, J.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[CrossRef] [PubMed]

Vasnetsov, M.

Vaziri, A.

G. Molina-Terriza, A. Vaziri, J. Rehácek, Z. Hradil, A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92(16), 167903 (2004).
[CrossRef] [PubMed]

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[CrossRef] [PubMed]

Vettenburg, T.

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100(23), 231115 (2012).
[CrossRef]

Visser, T. D.

Wang, J.

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

Wei, G.-X.

C.-S. Guo, L. Lu, S.-J. Yue, G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. 94(23), 231104 (2009).
[CrossRef]

Weihs, G.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[CrossRef] [PubMed]

Willner, A.

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

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Wolf, E.

Wright, E. M.

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100(23), 231115 (2012).
[CrossRef]

A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express 19(7), 5760–5771 (2011).
[CrossRef] [PubMed]

Yan, Y.

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

Yang, Y.

Y. Yang, M. Chen, M. Mazilu, A. Mourka, Y. Liu, K. Dholakia, “Effect of the radial and azimuthal mode indices of a partially coherent vortex field upon a spatial correlation singularity,” New J. Phys. 15(11), 113053 (2013).
[CrossRef]

Y. Yang, M. Mazilu, K. Dholakia, “Measuring the orbital angular momentum of partially coherent optical vortices through singularities in their cross-spectral density functions,” Opt. Lett. 37(23), 4949–4951 (2012).
[PubMed]

Yue, S.-J.

C.-S. Guo, L. Lu, S.-J. Yue, G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. 94(23), 231104 (2009).
[CrossRef]

Yue, Y.

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

Zeilinger, A.

G. Molina-Terriza, A. Vaziri, J. Rehácek, Z. Hradil, A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92(16), 167903 (2004).
[CrossRef] [PubMed]

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100(23), 231115 (2012).
[CrossRef]

C.-S. Guo, L. Lu, S.-J. Yue, G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. 94(23), 231104 (2009).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nat. Photonics (2)

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

K. Dholakia, T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[CrossRef]

Nat. Phys. (1)

G. Molina-Terriza, J. P. Torres, L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[CrossRef]

New J. Phys. (1)

Y. Yang, M. Chen, M. Mazilu, A. Mourka, Y. Liu, K. Dholakia, “Effect of the radial and azimuthal mode indices of a partially coherent vortex field upon a spatial correlation singularity,” New J. Phys. 15(11), 113053 (2013).
[CrossRef]

Opt. Express (3)

Opt. Lett. (5)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (5)

H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[CrossRef] [PubMed]

G. Molina-Terriza, A. Vaziri, J. Rehácek, Z. Hradil, A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92(16), 167903 (2004).
[CrossRef] [PubMed]

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[CrossRef] [PubMed]

D. M. Palacios, I. D. Maleev, A. S. Marathay, G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[CrossRef] [PubMed]

Science (1)

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[CrossRef] [PubMed]

Other (2)

J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, 1996).

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic Press, 2007).

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Figures (4)

Fig. 1
Fig. 1

Contour graph of the far-field autocorrelation function with different azimuthal index λ and radial index p. (a) p = 1, λ = 1; (b) p = 1, λ = 2; (c) p = 1, λ = 3; (d) p = 2, λ = 1; (e) p = 2, λ = 2; (f) p = 2, λ = 3;

Fig. 2
Fig. 2

Contour graph of the far-field autocorrelation function, or FT, for non-vortex beams with (a) p = 2, and (b) p = 5.

Fig. 3
Fig. 3

Schematic of the experimental setup to generate a LG beam and measure its topological charge (the azimuthal index). DPSSL, diode-pumped solid-state laser ( λ=532nm ); NDF, neutral density filter; RM, reflective mirror; BE, beam expander; SLM, spatial light modulator; CGH, computer-generated hologram; CA, circular aperture; L, thin lens; BPA, beam profile analyzer; PC1 and PC2, personal computer.

Fig. 4
Fig. 4

Experimental results of the focused intensity profile (a) and its FT pattern (b), and the corresponding theoretical results (c) and (d) for a LG beam with p = 1 and λ = 2.

Equations (11)

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U( X,Y )=FT{ u( x,y ) }.
χ( X,Y )= U ( X,Y )U( X,Y ),
χ( X,Y )=FT{ u ( x,y )u( x,y ) }=FT{ I( x,y ) }.
u ( ρ,ϕ,0 ) p, ρ | | L p | | ( 2 ρ 2 w 0 2 )exp( ρ 2 w 0 2 )exp( iϕ )exp( iθ ),
χ( ξ )=2π 0 | u( ρ,φ,0 ) | 2 J 0 ( 2πξρ )ρdρ.
χ( ξ )=2π 0 ρ 2| |+1 exp( 2 ρ 2 w 0 2 ) ( L p | | ( 2 ρ 2 w 0 2 ) ) 2 J 0 ( 2πξρ )dρ = π w 0 2| |+2 2 | |+1 exp( π 2 w 0 2 ξ 2 2 ) s=0 p t=0 p ( 1 ) s+t ( s+t+ )! s!t! ×( p+| | ps )( p+| | pt ) L s+t+| | ( π 2 w 0 2 ξ 2 2 ) = π w 0 2| |+2 2 | |+1 exp( π 2 w 0 2 ξ 2 2 ) s=0 p t=0 p ( 1 ) s+t ( s+t+| | )! s!t! × ( p+| | )! ( s+| | )!( ps )! ( p+| | )! ( t+| | )!( pt )! L s+t+| | ( π 2 w 0 2 ξ 2 2 ) = π w 0 2| |+2 2 | |+1 exp( π 2 w 0 2 ξ 2 2 ) s=0 p t=0 p+| | ( 1 ) s+t ( s+t+| | )! s!t! × ( p+| | )! ( s+t+| | )!s!( ps )! ( p+| | )! t!( p+| |t )! ( π 2 w 0 2 ξ 2 2 ) s+t = π w 0 2| |+2 2 | |+1 ( p+| | )! p! exp( π 2 w 0 2 ξ 2 2 ) L p ( π 2 w 0 2 ξ 2 2 ) L p+| | ( π 2 w 0 2 ξ 2 2 ).
L n α ( x )= m=0 n ( 1 ) m ( n+α nm ) x m m! ,
0 x μ e α x 2 J v ( xy )dx = y v Γ( 1 2 ( μv+1 ) ) 2 v+1 α 1 2 ( μ+v+1 ) exp( y 2 4α ) L 1 2 ( μv1 ) v ( y 2 4α ). [Reα>0,Re( μ+v )>1]
χ l ( ξ )= π w 0 2| |+2 2 | |+1 | |!exp( π 2 w 0 2 ξ 2 2 ) L | | ( π 2 w 0 2 ξ 2 2 ).
χ p ( ξ )= π w 0 2 2 exp( π 2 w 0 2 ξ 2 2 ) ( L p ( π 2 w 0 2 ξ 2 2 ) ) 2 .
N={ 2p+| | p , 0 =0 .

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