Abstract

This work presents theoretical analysis on the cross correlation function (CCF) of partially coherent vortex beam (PCVB), where the relation of the number of the rings of CCF dislocations and orbital angular momentum (OAM) of PCVB is analyzed in detail. It is shown that rings of CCF dislocations do not always exist, and depend on the coherence length, the order of PCVB and location of observation plane, although the CCF indicates topological charge to some degree. Comprehensive analysis of the CCF of PCVB and numerical simulations all validate such phenomenon.

© 2014 Optical Society of America

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  1. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
    [CrossRef]
  2. E. Brasselet, N. Murazawa, H. Misawa, S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
    [CrossRef] [PubMed]
  3. M. Šiler, P. Jákl, O. Brzobohatý, P. Zemánek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express 20(22), 24304–24319 (2012).
    [CrossRef] [PubMed]
  4. K. Y. Bliokh, F. Nori, “Spatiotemporal vortex beams and angular momentum,” Phys. Rev. A 86(3), 033824 (2012).
    [CrossRef]
  5. H. W. Yan, E. T. Zhang, B. Y. Zhao, K. L. Duan, “Free-space propagation of guided optical vortices excited in an annular core fiber,” Opt. Express 20(16), 17904–17915 (2012).
    [CrossRef] [PubMed]
  6. Y. F. Jiang, K. K. Huang, X. H. Lu, “Propagation dynamics of abruptly autofocusing Airy beams with optical vortices,” Opt. Express 20(17), 18579–18584 (2012).
    [CrossRef] [PubMed]
  7. E. Mari, F. Tamburini, G. A. Swartzlander, A. Bianchini, C. Barbieri, F. Romanato, B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express 20(3), 2445–2451 (2012).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  10. S. Prabhakar, A. Kumar, J. Banerji, R. P. Singh, “Revealing the order of a vortex through its intensity record,” Opt. Lett. 36(22), 4398–4400 (2011).
    [CrossRef] [PubMed]
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    [CrossRef]
  12. H. Tao, J. X. Pu, “Measuring the topological charge of vortex beams by using an annular ellipse aperture,” Appl. Phys. B 106(4), 927–932 (2012).
    [CrossRef]
  13. M. E. Anderson, H. Bigman, L. E. E. De Araujo, J. L. Chaloupka, “Measuring the topological charge of ultra-broadband, optical-vortex beams with a triangular aperture,” J. Opt. Soc. Am. B 29(8), 1968–1976 (2012).
  14. 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]
  15. C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
    [CrossRef]
  16. Y. J. 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]
  17. 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]
  18. Z. Y. Chen, J. X. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372(15), 2734–2740 (2008).
    [CrossRef]
  19. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  20. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

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 (10)

K. Y. Bliokh, F. Nori, “Spatiotemporal vortex beams and angular momentum,” Phys. Rev. A 86(3), 033824 (2012).
[CrossRef]

H. Tao, J. X. Pu, “Measuring the topological charge of vortex beams by using an annular ellipse aperture,” Appl. Phys. B 106(4), 927–932 (2012).
[CrossRef]

E. Mari, F. Tamburini, G. A. Swartzlander, A. Bianchini, C. Barbieri, F. Romanato, B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express 20(3), 2445–2451 (2012).
[CrossRef] [PubMed]

M. E. Anderson, H. Bigman, L. E. E. De Araujo, J. L. Chaloupka, “Measuring the topological charge of ultra-broadband, optical-vortex beams with a triangular aperture,” J. Opt. Soc. Am. B 29(8), 1968–1976 (2012).

H. W. Yan, E. T. Zhang, B. Y. Zhao, K. L. Duan, “Free-space propagation of guided optical vortices excited in an annular core fiber,” Opt. Express 20(16), 17904–17915 (2012).
[CrossRef] [PubMed]

Y. F. Jiang, K. K. Huang, X. H. Lu, “Propagation dynamics of abruptly autofocusing Airy beams with optical vortices,” Opt. Express 20(17), 18579–18584 (2012).
[CrossRef] [PubMed]

M. Šiler, P. Jákl, O. Brzobohatý, P. Zemánek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express 20(22), 24304–24319 (2012).
[CrossRef] [PubMed]

Y. J. 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]

L. Shi, L. H. Tian, X. F. Chen, “Characterizing topological charge of optical vortex using non-uniformly distributed multi-pinhole plate,” Chin. Opt. Lett. 10(12), 120501 (2012).
[CrossRef]

C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[CrossRef]

2011 (3)

2009 (1)

E. Brasselet, N. Murazawa, H. Misawa, S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[CrossRef] [PubMed]

2008 (1)

Z. Y. Chen, J. X. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372(15), 2734–2740 (2008).
[CrossRef]

2004 (1)

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]

1974 (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[CrossRef]

Anderson, M. E.

Banerji, J.

Barbieri, C.

Berry, M. V.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[CrossRef]

Bianchini, A.

Bigman, H.

Bliokh, K. Y.

K. Y. Bliokh, F. Nori, “Spatiotemporal vortex beams and angular momentum,” Phys. Rev. A 86(3), 033824 (2012).
[CrossRef]

Brasselet, E.

E. Brasselet, N. Murazawa, H. Misawa, S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[CrossRef] [PubMed]

Brzobohatý, O.

Cai, Y.

C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[CrossRef]

Chaloupka, J. L.

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]

Chen, X. F.

Chen, Z. Y.

Z. Y. Chen, J. X. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372(15), 2734–2740 (2008).
[CrossRef]

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. J. 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]

Dong, Y.

C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[CrossRef]

Duan, K. L.

Han, Y.

C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[CrossRef]

Han, Y. J.

Huang, K. K.

Jákl, P.

Jiang, Y. F.

Juodkazis, S.

E. Brasselet, N. Murazawa, H. Misawa, S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[CrossRef] [PubMed]

Kumar, A.

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]

Liu, Y. X.

Lu, X. H.

Lü, B. D.

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. 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]

Mari, E.

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]

Y. J. 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]

Misawa, H.

E. Brasselet, N. Murazawa, H. Misawa, S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[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]

Murazawa, N.

E. Brasselet, N. Murazawa, H. Misawa, S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[CrossRef] [PubMed]

Nori, F.

K. Y. Bliokh, F. Nori, “Spatiotemporal vortex beams and angular momentum,” Phys. Rev. A 86(3), 033824 (2012).
[CrossRef]

Nye, J. F.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[CrossRef]

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]

Prabhakar, S.

Pu, J. X.

H. Tao, J. X. Pu, “Measuring the topological charge of vortex beams by using an annular ellipse aperture,” Appl. Phys. B 106(4), 927–932 (2012).
[CrossRef]

Y. X. Liu, J. X. Pu, B. D. Lü, “Method for exploring the orbital angular momentum of an optical vortex beam with a triangular multipoint plate,” Appl. Opt. 50(24), 4844–4847 (2011).
[CrossRef] [PubMed]

Z. Y. Chen, J. X. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372(15), 2734–2740 (2008).
[CrossRef]

Romanato, F.

Shi, L.

Šiler, M.

Singh, R. P.

Swartzlander, G. A.

E. Mari, F. Tamburini, G. A. Swartzlander, A. Bianchini, C. Barbieri, F. Romanato, B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express 20(3), 2445–2451 (2012).
[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]

Tamburini, F.

Tao, H.

H. Tao, J. X. Pu, “Measuring the topological charge of vortex beams by using an annular ellipse aperture,” Appl. Phys. B 106(4), 927–932 (2012).
[CrossRef]

Thidé, B.

Tian, L. H.

Wang, F.

C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[CrossRef]

Yan, H. W.

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]

Yang, Y. J.

Zemánek, P.

Zhang, E. T.

Zhao, B. Y.

Zhao, C.

C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[CrossRef]

Zhao, G. H.

Appl. Opt. (1)

Appl. Phys. B (1)

H. Tao, J. X. Pu, “Measuring the topological charge of vortex beams by using an annular ellipse aperture,” Appl. Phys. B 106(4), 927–932 (2012).
[CrossRef]

Appl. Phys. Lett. (1)

C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012).
[CrossRef]

Chin. Opt. Lett. (1)

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

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 (4)

Opt. Lett. (3)

Phys. Lett. A (1)

Z. Y. Chen, J. X. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372(15), 2734–2740 (2008).
[CrossRef]

Phys. Rev. A (1)

K. Y. Bliokh, F. Nori, “Spatiotemporal vortex beams and angular momentum,” Phys. Rev. A 86(3), 033824 (2012).
[CrossRef]

Phys. Rev. Lett. (2)

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]

E. Brasselet, N. Murazawa, H. Misawa, S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[CrossRef]

Other (2)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

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Figures (5)

Fig. 1
Fig. 1

Distributions of normalized CCF of the 1st order PCVB in near and far field for different coherence length. (a) z = 0.01ZR; (b) z = 0.5 ZR; (c) z = ZR; (d) z = 20 ZR.

Fig. 2
Fig. 2

Distributions of normalized CCF of the 2nd order PCVB in near and far field for different coherence length. (a) z = 0.01 ZR; (b) z = 0.5 ZR; (c) z = ZR; (d) z = 20 ZR.

Fig. 3
Fig. 3

Distributions of normalized CCF of higher order PCVB (m = 3, 4, 5) with low coherence length (Lc = 0.2w). (a) z = 0.01 ZR; (b) z = 0.02 ZR; (c) z = 0.04 ZR; (d) z = 0.5 ZR.

Fig. 4
Fig. 4

Distributions of normalized CCF of higher order PCVB (m = 3, 4, 5) with moderate coherence length (Lc = w). (a) z = 0.05 ZR; (b) z = 0.2 ZR; (c) z = 0.5 ZR; (d) z = ZR.

Fig. 5
Fig. 5

Distributions of normalized CCF of higher order PCVB (m = 3, 4, 5) with high coherence length (Lc = 3w). (a) z = 0.05 ZR; (b) z = 0.5 ZR; (c) z = 5 ZR; (d) z = 20 ZR.

Equations (9)

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Γ( r 1 , r 2 ,0)= E 0 2 ( r 1 r 2 w 2 ) m exp[ | r 1 r 2 | 2 L c 2 r 1 2 + r 2 2 w 2 im( ϕ 2 ϕ 1 ) ],
Γ( ρ 1 , ρ 2 ,z)= (k/ 2πz ) 2 Γ( r 1 , r 2 ,0)exp( ik 2z [ ( ρ 1 r 1 ) 2 ( ρ 2 r 2 ) 2 ] ) d r 1 d r 2 ,
χ ( ρ )=Γ( ρ , ρ ,z)= E 0 2 π exp( Δζ ) Δ 1 n=0 m n! ( C m n ) 2 Δ 2 n ( Δ 3 ζ ) mn ,
{ Δ= 2 k 2 w 4 (2 w 2 + L c 2 ) k 2 w 4 L c 2 +4 z 2 (2 w 2 + L c 2 ) , Δ 1 = ( 4z k w 2 ) 2m [ k 2 w 4 L c 2 k 2 w 4 L c 2 +4 z 2 (2 w 2 + L c 2 ) ] m+1 , Δ 2 = 4 z 2 k 2 w 2 L c 2 ,ζ= ρ 2 w 2 , C m n = m! n!(mn)! , Δ 3 = k 2 w 4 L c 4 +4 z 2 (2 w 2 + L c 2 ) 2 k 2 w 4 L c 4 +4 z 2 L c 2 (2 w 2 + L c 2 ) .
χ (ζ)exp( Δζ )( Δ 2 Δ 3 ζ).
ρ 0 = Δ 2 / Δ 3 w.
χ (ζ)exp( Δζ )(2 Δ 2 2 4 Δ 2 Δ 3 ζ+ Δ 3 2 ζ 2 ).
ρ 0 = 2± 2 Δ 2 / Δ 3 w.
χ (ζ)exp( Δζ )(6 Δ 2 3 18 Δ 2 2 Δ 3 ζ+9 Δ 2 Δ 3 2 ζ 2 Δ 3 3 ζ 3 ).

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