Abstract

Laguerre-Gaussian correlated Schell-model (LGCSM) vortex beam is introduced as an extension of LGCSM beam which was proposed [Opt. Lett. 38, 91 (2013) Opt. Lett. 38, 1814 (2013)] just recently. Explicit formula for a LGCSM vortex beam propagating through a stigmatic ABCD optical system is derived, and the propagation properties of such beam in free space and the focusing properties of such beam are studied numerically. Furthermore, we carry out experimental generation of a LGCSM vortex beam, and studied its focusing properties. It is found that the propagation and focusing properties of a LGCSM vortex beam are different from that of a LGCSM beam, and we can shape the beam profile of a LGCSM vortex at the focal plane (or in the far field) by varying its initial spatial coherence. Our experimental results are consistent with the theoretical predictions, and our results will be useful for particle trapping.

© 2014 Optical Society of America

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2014 (4)

2013 (9)

Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[CrossRef] [PubMed]

Y. Gu, G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[CrossRef] [PubMed]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

Z. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[CrossRef] [PubMed]

Z. Mei, E. Shchepakina, O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
[CrossRef] [PubMed]

X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[CrossRef] [PubMed]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[CrossRef]

2012 (6)

2011 (4)

2010 (3)

Y. Zhang, B. Ding, T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[CrossRef]

Y. Cai, F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (2010).
[CrossRef]

J. Ng, Z. Lin, C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[CrossRef] [PubMed]

2009 (2)

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

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

2007 (2)

2006 (1)

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

2004 (1)

2003 (4)

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003).
[CrossRef] [PubMed]

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

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[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]

2002 (1)

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]

1979 (1)

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

1970 (1)

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]

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]

Bogatyryova, G. V.

Cai, Y.

C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[CrossRef] [PubMed]

Y. Yang, Y. Dong, C. Zhao, Y. Liu, Y. Cai, “Autocorrelation properties of fully coherent beam with and without orbital angular momentum,” Opt. Express 22(3), 2925–2932 (2014).
[CrossRef]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[CrossRef] [PubMed]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[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]

Y. Cai, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE 7924, 792402 (2011).
[CrossRef]

F. Wang, S. Zhu, Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36(16), 3281–3283 (2011).
[CrossRef] [PubMed]

C. Zhao, Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011).
[CrossRef] [PubMed]

Y. Cai, F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (2010).
[CrossRef]

F. Wang, Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007).
[CrossRef] [PubMed]

Q. Lin, Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[CrossRef] [PubMed]

Cang, J.

J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[CrossRef]

Chan, C. T.

J. Ng, Z. Lin, C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[CrossRef] [PubMed]

Chen, R.

Chen, Y.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

Collins, S. A.

De Santis, P.

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Ding, B.

Y. Zhang, B. Ding, T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[CrossRef]

Dong, Y.

Y. Yang, Y. Dong, C. Zhao, Y. Liu, Y. Cai, “Autocorrelation properties of fully coherent beam with and without orbital angular momentum,” Opt. Express 22(3), 2925–2932 (2014).
[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]

Du, S.

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Eyyuboglu, H. T.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

Fel’de, C. V.

Fleischer, J. W.

L. Waller, G. Situ, J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[CrossRef]

Gbur, G.

Y. Gu, G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[CrossRef] [PubMed]

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

Gori, F.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[CrossRef] [PubMed]

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

Gu, Y.

Guattari, G.

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

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]

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]

Korotkova, O.

Lajunen, H.

Liang, C.

C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Lin, Q.

Lin, Z.

J. Ng, Z. Lin, C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[CrossRef] [PubMed]

Liu, L.

Liu, X.

Liu, Y.

Mei, Z.

Ng, J.

J. Ng, Z. Lin, C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[CrossRef] [PubMed]

Palma, C.

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

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]

Polyanskii, P. V.

Ponomarenko, S. A.

Qu, J.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

Ramírez-Sánchez, V.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

Saastamoinen, T.

Sahin, S.

Santarsiero, M.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[CrossRef] [PubMed]

Shchepakina, E.

Shen, Y.

Shirai, T.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

Situ, G.

L. Waller, G. Situ, J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[CrossRef]

Soskin, M. S.

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]

Suyama, T.

Y. Zhang, B. Ding, T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[CrossRef]

Takeda, M.

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

Tong, Z.

van Dijk, T.

Vaziri, A.

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]

Visser, T. D.

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

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

Waller, L.

L. Waller, G. Situ, J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[CrossRef]

Wang, F.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[CrossRef] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[CrossRef] [PubMed]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

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]

F. Wang, S. Zhu, Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36(16), 3281–3283 (2011).
[CrossRef] [PubMed]

Y. Cai, F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (2010).
[CrossRef]

F. Wang, Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007).
[CrossRef] [PubMed]

Wang, W.

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

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]

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.

Wu, G.

Xiu, P.

J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[CrossRef]

Yang, Y.

Yuan, Y.

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

Zeilinger, A.

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]

Zhan, Q.

Zhang, Y.

Y. Zhang, B. Ding, T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[CrossRef]

Zhao, C.

Y. Yang, Y. Dong, C. Zhao, Y. Liu, Y. Cai, “Autocorrelation properties of fully coherent beam with and without orbital angular momentum,” Opt. Express 22(3), 2925–2932 (2014).
[CrossRef]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[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]

C. Zhao, Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011).
[CrossRef] [PubMed]

Zhu, S.

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]

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

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Nat. Photonics (1)

L. Waller, G. Situ, J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[CrossRef]

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

Open Opt. J. (1)

Y. Cai, F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (2010).
[CrossRef]

Opt. Commun. (3)

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

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

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Opt. Express (4)

Opt. Laser Technol. (2)

J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[CrossRef]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Opt. Lett. (14)

Q. Lin, Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[CrossRef] [PubMed]

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003).
[CrossRef] [PubMed]

C. Zhao, Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011).
[CrossRef] [PubMed]

F. Wang, S. Zhu, Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36(16), 3281–3283 (2011).
[CrossRef] [PubMed]

H. Lajunen, T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[CrossRef] [PubMed]

S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[CrossRef] [PubMed]

Z. Tong, O. Korotkova, “Non-uniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[CrossRef] [PubMed]

X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[CrossRef] [PubMed]

Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[CrossRef] [PubMed]

Y. Gu, G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[CrossRef] [PubMed]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

Z. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[CrossRef] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

Phys. Rev. A (3)

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]

Y. Zhang, B. Ding, T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[CrossRef]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Phys. Rev. Lett. (3)

J. Ng, Z. Lin, C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[CrossRef] [PubMed]

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

Proc. SPIE (1)

Y. Cai, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE 7924, 792402 (2011).
[CrossRef]

Other (2)

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

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

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Figures (9)

Fig. 1
Fig. 1

Normalized intensity distribution (cross line ρ y = 0 ) of a LGCSM vortex beam with n = 1 and m = 3 at several propagation distances in free space for different values of the initial coherence width δ 0 .

Fig. 2
Fig. 2

Normalized intensity (cross line ρ y = 0 ) of a LGCSM beam with n = 1 at several propagation distances in free space for different values of the initial coherence width δ 0 .

Fig. 3
Fig. 3

Normalized intensity (cross line ρ y = 0 ) of a focused LGCSM vortex beam with n = 1 and m = 3 at the focal plane for different values of the initial coherence width δ 0 .

Fig. 4
Fig. 4

Normalized intensity (cross line ρ y = 0 ) of a focused LGCSM beam with n = 1 at the geometrical focal plane for different values of the initial coherence width δ 0 .

Fig. 5
Fig. 5

Experimental setup for generating a LGCSM vortex beam and measuring its focused intensity. BE, beam expander; SLM, spatial light modulator; CA, circular aperture; L1, L2, L3, thin lenses; GAF, Gaussian amplitude filter; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; SPP, spiral phase plate; BPA, beam profile analyzer; PC1, PC2, personal computers.

Fig. 6
Fig. 6

Experimental results of (a) the intensity distribution and (b) the corresponding cross line (dotted curve) of the generated LGCSM beam with n = 1 in the source plane. The solid curve is a result of the theoretical fit.

Fig. 7
Fig. 7

Experimental results of the square of the modulus of the generated LGCSM beam for different values of the initial coherence width in the source plane. The solid curve is a result of the theoretical fit.

Fig. 8
Fig. 8

Experimental results of the intensity distribution and the corresponding cross line ( ρ y = 0 ) of the generated LGCSM beam with n = 1 at the geometrical focal plane for different values of the initial coherence width δ 0 . The solid curve denotes the theoretical results calculated by Eq. (18).

Fig. 9
Fig. 9

Experimental results of the intensity distribution and the corresponding cross line ( ρ y = 0 ) of the generated LGCSM vortex beam with n = 1 and m = 3 at the geometrical focal plane for different values of the initial coherence width δ 0 . The solid curve denotes the theoretical results calculated by Eq. (15).

Equations (21)

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Γ ( r 1 , r 2 ) = G 0 exp [ r 1 2 + r 2 2 4 σ 0 2 ( r 1 r 2 ) 2 2 δ 0 2 ] L n 0 [ ( r 1 r 2 ) 2 2 δ 0 2 ] ,
μ ( r 1 , r 2 ) = Γ ( r 1 , r 2 ) Γ ( r 1 , r 1 ) Γ ( r 2 , r 2 ) = exp [ ( r 1 r 2 ) 2 2 δ 0 2 ] L n 0 [ ( r 1 r 2 ) 2 2 δ 0 2 ] .
Γ ( r 1 , r 2 ) = G 0 exp [ r 1 2 + r 2 2 4 σ 0 2 ( r 1 r 2 ) 2 2 δ 0 2 ] L n 0 [ ( r 1 r 2 ) 2 2 δ 0 2 ] exp ( i m φ 1 + i m φ 2 ) .
Γ ( ρ 1 , ρ 2 ) = 1 ( λ B ) 2 exp [ i k D 2 B ( ρ 1 2 ρ 2 2 ) ] × Γ ( r 1 , r 2 ) exp [ i k A 2 B ( r 1 2 r 2 2 ) + i k B ( r 1 ρ 1 r 2 ρ 2 ) ] d 2 r 1 d 2 r 2 ,
r s = r 1 + r 2 2 , Δ r = r 1 r 2 ,
ρ s = ρ 1 + ρ 2 2 , Δ ρ = ρ 1 ρ 2 .
Γ ( ρ s , Δ ρ ) = G 0 ( λ B ) 2 exp [ i k D B ρ s Δ ρ ] × P + * ( r s + Δ r 2 ) P ( r s Δ r 2 ) γ ( Δ r ) exp [ i k B ( Δ r ρ s + r s Δ ρ ) ] d 2 r s d 2 Δ r ,
P + * ( r s + Δ r 2 ) = exp [ ( 1 4 σ 0 2 i k A 2 B ) ( r s + Δ r 2 ) 2 ] exp ( i m φ + ) ,
P ( r s Δ r 2 ) = exp [ ( 1 4 σ 0 2 + i k A 2 B ) ( r s Δ r 2 ) 2 ] exp ( i m φ ) ,
γ ( Δ r ) = exp [ Δ r 2 2 δ 0 2 ] L n 0 [ Δ r 2 2 δ 0 2 ] .
P + * ( r s + Δ r 2 ) = 1 ( λ B ) 2 P + * ˜ ( u 1 λ B ) exp ( i k B ( r s + Δ r 2 ) u 1 ) d 2 u 1 ,
P ( r s Δ r 2 ) = 1 ( λ B ) 2 P ˜ ( u 2 λ B ) exp ( i k B ( r s Δ r 2 ) u 2 ) d 2 u 2 .
Γ ( ρ s , Δ ρ ) = G 0 ( λ B ) 4 exp [ i k B ρ s Δ ρ ] × P ˜ + * ( u 1 λ B ) P ˜ ( u 1 + Δ ρ λ B ) γ ˜ ( u 1 + ρ s + Δ ρ / 2 λ B ) d 2 u 1 ,
γ ˜ ( u ) = γ ( r ) exp ( 2 π i u r ) d 2 r .
I ( ρ ) = Γ ( ρ s , Δ ρ ) ρ 1 = ρ 2 = G 0 ( λ B ) 4 P ˜ + * ( u 1 λ B ) P ˜ ( u 1 λ B ) γ ˜ ( u 1 + ρ λ B ) d 2 u 1 ,
γ ˜ ( u 1 + ρ λ B ) = 4 π δ 0 2 2 2 n 1 n ! [ 8 π 2 δ 0 2 ( u 1 + ρ λ B ) 2 ] n exp [ 2 π 2 δ 0 2 ( u 1 + ρ λ B ) 2 ] ,
P ˜ + * ( u 1 λ B ) P ˜ ( u 1 λ B ) = π 5 | σ ( B ) | 6 u 1 2 4 ( λ B ) 2 × | exp [ [ σ ( B ) π ] 2 2 ( λ B ) 2 u 1 2 ] [ I 1 2 m 1 2 ( [ σ ( B ) π ] 2 2 ( λ B ) 2 u 1 2 ) I 1 2 m + 1 2 ( [ σ ( B ) π ] 2 2 ( λ B ) 2 u 1 2 ) ] | 2 .
I ( ρ , z ) = 2 n 1 G 0 ( k B ) 2 σ * 2 ( B ) σ 2 ( B ) δ 0 2 n + 2 [ ( σ * 2 ( B ) + σ 2 ( B ) + 2 δ 0 2 ) ] n 1 × exp [ ( k 2 B ) 2 2 δ 0 2 ( σ * 2 ( B ) + σ 2 ( B ) ) ( σ * 2 ( B ) + σ 2 ( B ) + 2 δ 0 2 ) ρ 2 ] L n [ ( k 2 B ) 2 ( σ * 2 ( B ) + σ 2 ( B ) ) 2 ( σ * 2 ( B ) + σ 2 ( B ) + 2 δ 0 2 ) ρ 2 ] .
( A B C D ) = ( 1 z 0 1 ) .
( A B C D ) = ( 1 f 0 1 ) ( 1 0 1 / f 1 ) = ( 0 f 1 / f 1 ) .
A = 0 , B = f 3 , C = 1 / f 3 , D = 1.

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