Abstract

We have investigated the correlation singularities, coherence vortices of two-point correlation function in a partially coherent vector beam with initially radial polarization, i.e., partially coherent radially polarized (PCRP) beam. It is found that these singularities generally occur during free space propagation. Analytical formulae for characterizing the dynamics of the correlation singularities on propagation are derived. The influence of the spatial coherence length of the beam on the evolution properties of the correlation singularities and the conditions for creation and annihilation of the correlation singularities during propagation have been studied in detail based on the derived formulae. Some interesting results are illustrated. These correlation singularities have implication for interference experiments with a PCRP beam.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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2014 (2)

2013 (4)

S. B. Raghunathan, H. F. Schouten, and T. D. Visser, “Topological reactions of correlation functions in partially coherent electromagnetic beams,” J. Opt. Soc. Am. A 30(4), 582–588 (2013).
[Crossref] [PubMed]

W. Fu and H. Zhang, “Propagation properties of partially coherent radially polarized doughnut beam in turbulent ocean,” Opt. Commun. 304(1), 11–18 (2013).
[Crossref]

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B 112(2), 247–259 (2013).
[Crossref]

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).

2012 (2)

S. B. Raghunathan, H. F. Schouten, and T. D. Visser, “Correlation singularities in partially coherent electromagnetic beams,” Opt. Lett. 37(20), 4179–4181 (2012).
[Crossref] [PubMed]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

2011 (2)

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
[Crossref] [PubMed]

H. Wang, D. Liu, Z. Zhou, S. Tong, and Y. Song, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 49(1–2), 1238–1244 (2011).

2010 (4)

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Radially polarized partially coherent beams propagation in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 52(3), 31301 (2010).
[Crossref]

V. Ramírez-Sánchez, G. Piquero, and M. Santarsiero, “Synthesis and characterization of partially coherent beams with propagation-invariant transverse polarization pattern,” Opt. Commun. 283(22), 4484–4489 (2010).
[Crossref]

P. Liu, H. Yang, J. Rong, G. Wang, and Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. 42(1), 99–104 (2010).
[Crossref]

M. L. Marasinghe, M. Premaratne, and D. M. Paganin, “Coherence vortices in Mie scattering of statistically stationary partially coherent fields,” Opt. Express 18(7), 6628–6641 (2010).
[Crossref] [PubMed]

2009 (7)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

Y. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282(5), 709–716 (2009).
[Crossref]

T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 (2009).
[PubMed]

T. van Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the field generated by partially coherent sources,” Phys. Rev. A 79(3), 033805 (2009).
[Crossref]

H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
[Crossref]

W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009).
[Crossref] [PubMed]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009).
[Crossref]

2008 (2)

2006 (4)

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006).
[Crossref] [PubMed]

W. Wang and M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96(22), 223904 (2006).
[Crossref] [PubMed]

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259(2), 428–435 (2006).
[Crossref]

R. W. Schoonover and T. D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express 14(12), 5733–5745 (2006).
[Crossref] [PubMed]

2005 (1)

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast Microscopy,” Phys. Rev. Lett. 94(23), 233902 (2005).
[Crossref] [PubMed]

2004 (4)

2003 (5)

2002 (1)

2001 (2)

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457(2005), 141–155 (2001).
[Crossref]

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[Crossref]

1997 (1)

1974 (1)

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

1967 (1)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in aplanatic systems,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Allen, L.

Angelsky, O. V.

Barnett, S.

Bernet, S.

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast Microscopy,” Phys. Rev. Lett. 94(23), 233902 (2005).
[Crossref] [PubMed]

Berry, M. V.

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457(2005), 141–155 (2001).
[Crossref]

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

Blok, H.

Bogatyryova, G. V.

Boivin, A.

Cai, Y.

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B 112(2), 247–259 (2013).
[Crossref]

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
[Crossref] [PubMed]

Chen, R.

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B 112(2), 247–259 (2013).
[Crossref]

Cheng, W.

Courtial, J.

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[Crossref] [PubMed]

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457(2005), 141–155 (2001).
[Crossref]

Dholakia, K.

Dong, Y.

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B 112(2), 247–259 (2013).
[Crossref]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
[Crossref] [PubMed]

Dow, J.

Duan, Z.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006).
[Crossref] [PubMed]

Fel’de, C. V.

Franke-Arnold, S.

Freund, I.

Fu, W.

W. Fu and H. Zhang, “Propagation properties of partially coherent radially polarized doughnut beam in turbulent ocean,” Opt. Commun. 304(1), 11–18 (2013).
[Crossref]

Fürhapter, S.

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast Microscopy,” Phys. Rev. Lett. 94(23), 233902 (2005).
[Crossref] [PubMed]

Gbur, G.

Gibson, G.

Gu, Y.

Y. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282(5), 709–716 (2009).
[Crossref]

Guo, L.

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Radially polarized partially coherent beams propagation in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 52(3), 31301 (2010).
[Crossref]

Hanson, S. G.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006).
[Crossref] [PubMed]

Haus, J. W.

Jesacher, A.

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast Microscopy,” Phys. Rev. Lett. 94(23), 233902 (2005).
[Crossref] [PubMed]

Korotkova, O.

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

Leach, J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[Crossref] [PubMed]

Lenstra, D.

Liang, C.

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Radially polarized partially coherent beams propagation in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 52(3), 31301 (2010).
[Crossref]

Lin, H.

H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
[Crossref]

Liu, D.

H. Wang, D. Liu, Z. Zhou, S. Tong, and Y. Song, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 49(1–2), 1238–1244 (2011).

Liu, L.

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).

Liu, P.

P. Liu, H. Yang, J. Rong, G. Wang, and Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. 42(1), 99–104 (2010).
[Crossref]

Liu, X.

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).

Maleev, I. D.

Marasinghe, M. L.

Marathay, A. S.

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

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

Miyamoto, Y.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006).
[Crossref] [PubMed]

Mokhun, A. I.

Mokhun, I. I.

Nye, J. F.

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

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

Padgett, M.

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[Crossref] [PubMed]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997).
[Crossref] [PubMed]

Paganin, D. M.

Palacios, D. M.

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

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

Pas’ko, V.

Piquero, G.

V. Ramírez-Sánchez, G. Piquero, and M. Santarsiero, “Synthesis and characterization of partially coherent beams with propagation-invariant transverse polarization pattern,” Opt. Commun. 283(22), 4484–4489 (2010).
[Crossref]

Polyanskii, P. V.

Ponomarenko, S. A.

Premaratne, M.

Pu, J.

H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
[Crossref]

Raghunathan, S. B.

Ramírez-Sánchez, V.

V. Ramírez-Sánchez, G. Piquero, and M. Santarsiero, “Synthesis and characterization of partially coherent beams with propagation-invariant transverse polarization pattern,” Opt. Commun. 283(22), 4484–4489 (2010).
[Crossref]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in aplanatic systems,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Ritsch-Marte, M.

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast Microscopy,” Phys. Rev. Lett. 94(23), 233902 (2005).
[Crossref] [PubMed]

Rong, J.

P. Liu, H. Yang, J. Rong, G. Wang, and Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. 42(1), 99–104 (2010).
[Crossref]

Santarsiero, M.

V. Ramírez-Sánchez, G. Piquero, and M. Santarsiero, “Synthesis and characterization of partially coherent beams with propagation-invariant transverse polarization pattern,” Opt. Commun. 283(22), 4484–4489 (2010).
[Crossref]

Schoonover, R. W.

Schouten, H.

Schouten, H. F.

Simpson, N. B.

Song, Y.

H. Wang, D. Liu, Z. Zhou, S. Tong, and Y. Song, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 49(1–2), 1238–1244 (2011).

Soskin, M. S.

Stahl, C. S. D.

Swartzlander, G. A.

Takeda, M.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006).
[Crossref] [PubMed]

W. Wang and M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96(22), 223904 (2006).
[Crossref] [PubMed]

Tan, Z.

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Radially polarized partially coherent beams propagation in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 52(3), 31301 (2010).
[Crossref]

Tang, Z.

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Radially polarized partially coherent beams propagation in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 52(3), 31301 (2010).
[Crossref]

Tong, S.

H. Wang, D. Liu, Z. Zhou, S. Tong, and Y. Song, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 49(1–2), 1238–1244 (2011).

van Dijk, T.

T. van Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the field generated by partially coherent sources,” Phys. Rev. A 79(3), 033805 (2009).
[Crossref]

T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 (2009).
[PubMed]

Vasnetsov, M.

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[Crossref]

Visser, T.

Visser, T. D.

Wang, F.

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B 112(2), 247–259 (2013).
[Crossref]

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

Wang, G.

P. Liu, H. Yang, J. Rong, G. Wang, and Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. 42(1), 99–104 (2010).
[Crossref]

Wang, H.

H. Wang, D. Liu, Z. Zhou, S. Tong, and Y. Song, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 49(1–2), 1238–1244 (2011).

Wang, W.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006).
[Crossref] [PubMed]

W. Wang and M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96(22), 223904 (2006).
[Crossref] [PubMed]

Wolf, E.

Yan, Y.

P. Liu, H. Yang, J. Rong, G. Wang, and Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. 42(1), 99–104 (2010).
[Crossref]

Yang, H.

P. Liu, H. Yang, J. Rong, G. Wang, and Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. 42(1), 99–104 (2010).
[Crossref]

Yao, M.

Yuan, Y.

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).

Zhan, Q.

Zhang, H.

W. Fu and H. Zhang, “Propagation properties of partially coherent radially polarized doughnut beam in turbulent ocean,” Opt. Commun. 304(1), 11–18 (2013).
[Crossref]

Zhao, C.

Zhou, Z.

H. Wang, D. Liu, Z. Zhou, S. Tong, and Y. Song, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 49(1–2), 1238–1244 (2011).

Adv. Opt. Photon. (1)

Appl. Phys. B (2)

H. Wang, D. Liu, Z. Zhou, S. Tong, and Y. Song, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 49(1–2), 1238–1244 (2011).

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B 112(2), 247–259 (2013).
[Crossref]

Appl. Phys. Lett. (2)

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

Eur. Phys. J. Appl. Phys. (1)

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Radially polarized partially coherent beams propagation in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 52(3), 31301 (2010).
[Crossref]

J. Mod. Opt. (1)

H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
[Crossref]

J. Opt. Soc. Am. (1)

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

J. Opt. Soc. Am. B (3)

Nature (1)

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004).
[Crossref] [PubMed]

Opt. Commun. (5)

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1–6), 117–125 (2003).
[Crossref]

Y. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282(5), 709–716 (2009).
[Crossref]

V. Ramírez-Sánchez, G. Piquero, and M. Santarsiero, “Synthesis and characterization of partially coherent beams with propagation-invariant transverse polarization pattern,” Opt. Commun. 283(22), 4484–4489 (2010).
[Crossref]

W. Fu and H. Zhang, “Propagation properties of partially coherent radially polarized doughnut beam in turbulent ocean,” Opt. Commun. 304(1), 11–18 (2013).
[Crossref]

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259(2), 428–435 (2006).
[Crossref]

Opt. Express (6)

Opt. Laser Technol. (1)

P. Liu, H. Yang, J. Rong, G. Wang, and Y. Yan, “Coherence vortex evolution of partially coherent vortex beams in the focal region,” Opt. Laser Technol. 42(1), 99–104 (2010).
[Crossref]

Opt. Lett. (7)

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5–6), 263–267 (2003).
[Crossref]

Phys. Rev. A (1)

T. van Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the field generated by partially coherent sources,” Phys. Rev. A 79(3), 033805 (2009).
[Crossref]

Phys. Rev. Lett. (4)

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

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006).
[Crossref] [PubMed]

W. Wang and M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96(22), 223904 (2006).
[Crossref] [PubMed]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast Microscopy,” Phys. Rev. Lett. 94(23), 233902 (2005).
[Crossref] [PubMed]

Proc. R. Soc. Lond. A (1)

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457(2005), 141–155 (2001).
[Crossref]

Proc. R. Soc. Lond. A Math. Phys. Sci. (2)

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

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in aplanatic systems,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Prog. Opt. (2)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[Crossref]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

Other (2)

J. F. Nye, Natural Focusing and Fine Structure of Light (IOP Publishing, 999).

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

Supplementary Material (2)

» Media 1: MOV (288 KB)     
» Media 2: MOV (176 KB)     

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Figures (6)

Fig. 1
Fig. 1 Illustration of the zeros of real and imaginary part of SDOC η(ρ1, ρ2, z) at several propagation distances z with ρ2 = (1.0,1.0)mm (green point) kept fixed (Media 1). The intersections (red points) ρA and ρB are correlation singularities. The background is the normalized spectral density of the PCRP beam at that propagation distance. The black arrows denote the moving directions of the singularities. The beam parameters are σ0 = 1.0mm, δ0 = 0.5mm, λ = 632.8nm.
Fig. 2
Fig. 2 Phase contours of η(ρ1, ρ2, z) for several propagation distances z. The parameters used in calculation are same with those in Fig. 1.
Fig. 3
Fig. 3 Variation of the distance d with the propagation distance z for different values of spatial coherence length δ0. (a) R = 1.414mm. (b) R = 0.707mm. The green points denote the propagation distance where two singularities merge together. The beam parameters in calculation are λ = 632.8nm, σ0 = 1.0mm.
Fig. 4
Fig. 4 Propagation dynamics of correlation singularities ρA and ρB of the PCRP beam at several propagation distances z with ρ2 = (0.5,0.5)mm (green point) kept fixed (Media 2). The background is the normalized spectral density of the PCRP beam at that propagation distance. The black arrows denote the moving directions of the singularities. The beam parameters are λ = 632.8nm, σ0 = 1.0mm, δ0 = 1.0mm.
Fig. 5
Fig. 5 Three-dimensional plot of the evolution of two singularities as a function of z for different spatial coherence length. (a) ρ2 = (1.0,1.0)mm; (b) ρ2 = (0.5,0.5)mm. The beam parameters in calculation are σ0 = 1.0mm, λ = 632.8nm.
Fig. 6
Fig. 6 Dependence of the propagation distance zm where two singularities merge together on the spatial coherent length of the PCRP beam for three different reference points ρ2. (a) ρ2 = (0.5,0.5)mm, (b) ρ2 = (1.0,1.0)mm, (c) ρ2 = (1.0,2.0)mm. The beam parameters in calculation are σ0 = 1.0mm, λ = 632.8nm.

Equations (15)

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W ^ (s) ( r 1 , r 2 ,ω )=( W xx (s) ( r 1 , r 2 ,ω ) W xy (s) ( r 1 , r 2 ,ω ) W yx (s) ( r 1 , r 2 ,ω ) W yy (s) ( r 1 , r 2 ,ω ) ),
W αβ (s) ( r 1 , r 2 ,ω )= E α * ( r 1 ,ω ) E β ( r 2 ,ω ) , ( α,β=x,y )
η( ρ 1 , ρ 2 ,z)= Tr W ^ ( ρ 1 , ρ 2 ,z) [Tr W ^ ( ρ 1 , ρ 1 ,z)Tr W ^ ( ρ 2 , ρ 2 ,z)] 1/2 ,
W αβ ( ρ 1 , ρ 2 ,z)= k 2 4 π 2 z 2 W αβ (s) ( r 1 , r 2 )exp[ ik 2z ( r 1 ρ 1 ) 2 + ik 2z ( r 2 ρ 2 ) 2 ] d 2 r 1 d 2 r 2 ,(α,β=x,y),
Re[ Tr W ^ ( ρ 1 , ρ 2 ,z) ]=0,
Im[ Tr W ^ ( ρ 1 , ρ 2 ,z) ]=0.
W ^ RP (s) ( r 1 , r 2 )= 1 4 σ 0 2 exp[ r 1 2 + r 2 2 4 σ 0 2 ( r 1 r 2 ) 2 2 δ 0 2 ]( x 1 x 2 x 1 y 2 x 2 y 1 y 1 y 2 ),
W αα ( ρ 1 , ρ 2 ,z)= 1 4 σ 0 2 Δ exp[ iz( ρ 2 2 ρ 1 2 ) 2k σ 0 2 Δ M 2 ]exp[ ( ρ 1 + ρ 2 ) 2 8 σ 0 2 Δ ]exp[ ( ρ 2 ρ 1 ) 2 2Δ M 2 ] [ 1 4 Δ 2 ( 1+ z 2 4 k 2 σ 0 4 ) ( ρ 1α + ρ 2α ) 2 1 4 Δ 2 ( 1+ 4 z 2 M 4 k 2 ) ( ρ 2α ρ 1α ) 2 + z 2 k 2 δ 0 2 Δ 2iz 2k δ 0 2 Δ 2 ( ρ 2α 2 ρ 1α 2 ) ],(α=x,y)
S(ρ,z)=Tr[ W ^ (ρ,ρ,z)]= 1 4 σ 0 2 Δ exp( ρ 2 2 σ 0 2 Δ )[ 1 Δ 2 ( 1+ z 2 4 k 2 σ 0 4 ) ρ 2 + 2 z 2 k 2 δ 0 2 Δ ].
( 1+ z 2 4 k 2 σ 0 4 ) ( ρ 1 + ρ 2 ) 2 ( 1+ 4 z 2 M 4 k 2 ) ( ρ 2 ρ 1 ) 2 + 8Δ z 2 k 2 δ 0 2 =0,
ρ 1 2 ρ 2 2 =0.
( Δ+ 2 z 2 k 2 δ 0 4 )( ρ 1x ρ 2x + ρ 1y ρ 2y ) 2 z 2 k 2 δ 0 2 [ ( 1 2 σ 0 2 + 1 δ 0 2 ) R 2 Δ ]=0.
d= 2 z 2 k 2 δ 0 2 R( Δ+ 2 z 2 k 2 δ 0 4 ) | R 2 ( 1 2 σ 0 2 + 1 δ 0 2 )Δ |.
R 2 /2 σ 0 2 + R 2 / δ 0 2 >1,
z m = Mk σ 0 2 ( b 2 + 8 δ 0 2 R 2 M 2 σ 0 2 b ) 1/2 ,

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