Abstract

We introduce a class of random stationary, scalar source named as multi-cosine-Laguerre-Gaussian-correlated Schell-model (McLGCSM) source whose spectral degree of coherence (SDOC) is a combination of the Laguerre-Gaussian correlated Schell-model (LGCSM) and multi-cosine-Gaussian correlated Schell-model (McGCSM) sources. The analytical expressions for the spectral density function and the propagation factor of a McLGCSM beam propagating in turbulent atmosphere are derived. The statistical properties, such as the spectral intensity and the propagation factor, of a McLGCSM beam are illustrated numerically. It is shown that a McLGCSM beam exhibits a robust ring-shaped beam array with adjustable number and positions in the far field by directly modulating the spatial structure of its SDOC in the source plane. Moreover, we provide a detailed insight into the theoretical origin and characteristics of such a ring-shaped beam array. It is demonstrated that these peculiar shaping properties are the concentrated manifestation of the individual merits respectively associated with the Laguerre- and multi-cosine-related factors of the whole SDOC. Our results provide a novel scheme to generate robust and controllable ring-shaped beam arrays over large distances, and will widen the potentials for manipulation of multiple particles, free-space optical communications and imaging in the atmosphere.

© 2017 Optical Society of America

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2017 (8)

Y. L. Qiu, Z. X. Chen, and Y. J. He, “Propagation of a Laguerre-Gaussian correlated Schell-model beam in strongly nonlocal nonlinear media,” Opt. Commun. 389, 303–309 (2017).
[Crossref]

M. M. Tang, D. M. Zhao, X. Z. Li, and H. H. Li, “Focusing properties of radially polarized multi-cosine Gaussian correlated Schell-model beams,” Opt. Commun. 396, 249–256 (2017).
[Crossref]

X. Wang, Z. R. Liu, K. L. Huang, and J. B. Sun, “Spectral shifts generated by scattering of Gaussian Schell-model arrays beam from a deterministic medium,” Opt. Commun. 387, 230–234 (2017).
[Crossref]

J. Zhu, H. Q. Tang, Q. Su, and K. C. Zhu, “Cascade self-splitting of a Hermite-cos-Gaussian correlated Schell-model beam,” Europhys. Lett. 118(1), 14001 (2017).
[Crossref]

Z. Mei and O. Korotkova, “Random sources for rotating spectral densities,” Opt. Lett. 42(2), 255–258 (2017).
[Crossref] [PubMed]

F. Wang and O. Korotkova, “Circularly symmetric cusped random beams in free space and atmospheric turbulence,” Opt. Express 25(5), 5057–5067 (2017).
[Crossref] [PubMed]

X. Weng, L. Du, P. Shi, and X. Yuan, “Tunable optical cage array generated by Dammann vector beam,” Opt. Express 25(8), 9039–9048 (2017).
[Crossref] [PubMed]

C. Liang, C. Mi, F. Wang, C. Zhao, Y. Cai, and S. A. Ponomarenko, “Vector optical coherence lattices generating controllable far-field beam profiles,” Opt. Express 25(9), 9872–9885 (2017).
[Crossref] [PubMed]

2016 (11)

Z. R. Mei and O. Korotkova, “Electromagnetic Schell-model sources generating far fields with stable and flexible concentric rings profiles,” Opt. Express 24(5), 5572–5583 (2016).
[Crossref]

Y. Zhou, Y. Yuan, J. Qu, and W. Huang, “Propagation properties of Laguerre-Gaussian correlated Schell-model beam in non-Kolmogorov turbulence,” Opt. Express 24(10), 10682–10693 (2016).
[Crossref] [PubMed]

J. Wang, S. Zhu, H. Wang, Y. Cai, and Z. Li, “Second-order statistics of a radially polarized cosine-Gaussian correlated Schell-model beam in anisotropic turbulence,” Opt. Express 24(11), 11626–11639 (2016).
[Crossref] [PubMed]

J. Li, F. Wang, and O. Korotkova, “Random sources for cusped beams,” Opt. Express 24(16), 17779–17791 (2016).
[Crossref] [PubMed]

Z. Z. Song, Z. J. Liu, K. Y. Zhou, Q. G. Sun, and S. T. Liu, “Propagation properties of Gaussian Schell model array beams in non-Kolmogorov turbulence,” J. Opt. 18(10), 105601 (2016).
[Crossref]

Y. H. Mao, Z. R. Mei, and J. G. Gu, “Propagation of Gaussian Schell-model Array beams in free space and atmospheric turbulence,” Opt. Laser Technol. 86, 14–20 (2016).
[Crossref]

Y. H. Mao and Z. R. Mei, “Random sources generating ring-shaped optical lattice,” Opt. Commun. 381, 222–226 (2016).
[Crossref]

H. F. Xu, Z. Zhang, J. Qu, and W. Huang, “The tight focusing properties of Laguerre–Gaussian-correlated Schell-model beams,” J. Mod. Opt. 63(15), 1429–1437 (2016).
[Crossref]

Y. H. Chen, J. Y. Yu, Y. S. Yuan, F. Wang, and Y. J. Cai, “Theoretical and experimental studies of a rectangular Laguerre-Gaussian-correlated Schell-model beam,” Appl. Phys. B 122(2), 31 (2016).
[Crossref]

Y. H. Chen, S. A. Ponomarenko, and Y. J. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
[Crossref]

H. Wang, J. Lin, D. Zhang, Y. Wang, M. Gu, H. P. Urbach, F. Gan, and S. Zhuang, “Creation of an anti-imaging system using binary optics,” Sci. Rep. 6(1), 33064 (2016).
[Crossref] [PubMed]

2015 (9)

C. L. Ding, L. M. Liao, H. X. Wang, Y. T. Zhang, and L. Z. Pan, “Effect of oceanic turbulence on the propagation of cosine-Gaussian-correlated Schell-model beams,” J. Opt. 17(3), 035615 (2015).
[Crossref]

L. N. Guo, Y. H. Chen, L. Liu, and Y. J. Cai, “Propagation of a Laguerre-Gaussian correlated Schell-model beam beyond the paraxial approximation,” Opt. Commun. 352, 127–134 (2015).
[Crossref]

Y. H. Chen, J. X. Gu, F. Wang, and Y. J. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

M. W. Hyde, S. Basu, D. G. Voelz, and X. F. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015).
[Crossref] [PubMed]

Y. Iketaki, H. Kumagai, K. Jahn, and N. Bokor, “Creation of a three-dimensional spherical fluorescence spot for super-resolution microscopy using a two-color annular hybrid wave plate,” Opt. Lett. 40(6), 1057–1060 (2015).
[Crossref] [PubMed]

J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
[Crossref] [PubMed]

Z. Mei, D. Zhao, O. Korotkova, and Y. Mao, “Gaussian Schell-model arrays,” Opt. Lett. 40(23), 5662–5665 (2015).
[Crossref] [PubMed]

S. Zhu, Y. Chen, J. Wang, H. Wang, Z. Li, and Y. Cai, “Generation and propagation of a vector cosine-Gaussian correlated beam with radial polarization,” Opt. Express 23(26), 33099–33115 (2015).
[Crossref] [PubMed]

2014 (9)

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and 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, and 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, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

Z. Mei, “Light sources generating self-splitting beams and their propagation in non-Kolmogorov turbulence,” Opt. Express 22(11), 13029–13040 (2014).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

H. F. Xu, Z. Zhang, J. Qu, and W. Huang, “Propagation factors of cosine-Gaussian-correlated Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 22(19), 22479–22489 (2014).
[Crossref] [PubMed]

P. Zhang, T. Li, J. Zhu, X. Zhu, S. Yang, Y. Wang, X. Yin, and X. Zhang, “Generation of acoustic self-bending and bottle beams by phase engineering,” Nat. Commun. 5, 4316 (2014).
[PubMed]

C. Wan, K. Huang, T. Han, E. S. P. Leong, W. Ding, L. Zhang, T. Yeo, X. Yu, J. Teng, D. Lei, S. A. Maier, B. Luk’yanchuk, S. Zhang, and C. Qiu, “Three-dimensional visible-light capsule enclosing perfect supersized darkness via antiresolution,” Laser Photonics Rev. 8(5), 743–749 (2014).
[Crossref]

2013 (6)

2012 (2)

J. Li, X. M. Gao, and Y. R. Chen, “Tight focusing of J0-correlated Gaussian Schell-model beam through high numerical aperture,” Opt. Commun. 285(16), 3403–3411 (2012).
[Crossref]

K. Zhu, S. Li, Y. Tang, Y. Yu, and H. Tang, “Study on the propagation parameters of Bessel-Gaussian beams carrying optical vortices through atmospheric turbulence,” J. Opt. Soc. Am. A 29(3), 251–257 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (2)

Y. L. Qiu, Z. X. Chen, and L. Liu, “Partially coherent dark hollow beams propagating through real ABCD optical systems in a turbulent atmosphere,” J. Mod. Opt. 57(8), 662–669 (2010).
[Crossref]

P. Xu, X. He, J. Wang, and M. Zhan, “Trapping a single atom in a blue detuned optical bottle beam trap,” Opt. Lett. 35(13), 2164–2166 (2010).
[Crossref] [PubMed]

2009 (2)

2008 (2)

R. Schmidt, C. A. Wurm, S. Jakobs, J. Engelhardt, A. Egner, and S. W. Hell, “Spherical nanosized focal spot unravels the interior of cells,” Nat. Methods 5(6), 539–544 (2008).
[Crossref] [PubMed]

E. Jafarov, S. Lievens, and J. V. Jeugt, “The Wigner distribution function for the one-dimensional parabose oscillator,” J. Phys. A 41(23), 235301 (2008).
[Crossref]

2007 (1)

2005 (1)

1987 (1)

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[Crossref]

Basu, S.

M. W. Hyde, S. Basu, D. G. Voelz, and X. F. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

Bingen, P.

Bokor, N.

Cai, Y.

C. Liang, C. Mi, F. Wang, C. Zhao, Y. Cai, and S. A. Ponomarenko, “Vector optical coherence lattices generating controllable far-field beam profiles,” Opt. Express 25(9), 9872–9885 (2017).
[Crossref] [PubMed]

J. Wang, S. Zhu, H. Wang, Y. Cai, and Z. Li, “Second-order statistics of a radially polarized cosine-Gaussian correlated Schell-model beam in anisotropic turbulence,” Opt. Express 24(11), 11626–11639 (2016).
[Crossref] [PubMed]

J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
[Crossref] [PubMed]

S. Zhu, Y. Chen, J. Wang, H. Wang, Z. Li, and Y. Cai, “Generation and propagation of a vector cosine-Gaussian correlated beam with radial polarization,” Opt. Express 23(26), 33099–33115 (2015).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and 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, and 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, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

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

Cai, Y. J.

Y. H. Chen, S. A. Ponomarenko, and Y. J. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
[Crossref]

Y. H. Chen, J. Y. Yu, Y. S. Yuan, F. Wang, and Y. J. Cai, “Theoretical and experimental studies of a rectangular Laguerre-Gaussian-correlated Schell-model beam,” Appl. Phys. B 122(2), 31 (2016).
[Crossref]

L. N. Guo, Y. H. Chen, L. Liu, and Y. J. Cai, “Propagation of a Laguerre-Gaussian correlated Schell-model beam beyond the paraxial approximation,” Opt. Commun. 352, 127–134 (2015).
[Crossref]

Y. H. Chen, J. X. Gu, F. Wang, and Y. J. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Cang, J.

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

Chen, R.

Chen, Y.

Chen, Y. H.

Y. H. Chen, J. Y. Yu, Y. S. Yuan, F. Wang, and Y. J. Cai, “Theoretical and experimental studies of a rectangular Laguerre-Gaussian-correlated Schell-model beam,” Appl. Phys. B 122(2), 31 (2016).
[Crossref]

Y. H. Chen, S. A. Ponomarenko, and Y. J. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
[Crossref]

Y. H. Chen, J. X. Gu, F. Wang, and Y. J. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

L. N. Guo, Y. H. Chen, L. Liu, and Y. J. Cai, “Propagation of a Laguerre-Gaussian correlated Schell-model beam beyond the paraxial approximation,” Opt. Commun. 352, 127–134 (2015).
[Crossref]

Chen, Y. R.

J. Li, X. M. Gao, and Y. R. Chen, “Tight focusing of J0-correlated Gaussian Schell-model beam through high numerical aperture,” Opt. Commun. 285(16), 3403–3411 (2012).
[Crossref]

Chen, Z. X.

Y. L. Qiu, Z. X. Chen, and Y. J. He, “Propagation of a Laguerre-Gaussian correlated Schell-model beam in strongly nonlocal nonlinear media,” Opt. Commun. 389, 303–309 (2017).
[Crossref]

Y. L. Qiu, Z. X. Chen, and L. Liu, “Partially coherent dark hollow beams propagating through real ABCD optical systems in a turbulent atmosphere,” J. Mod. Opt. 57(8), 662–669 (2010).
[Crossref]

Dally, A.

Dan, Y.

Ding, C. L.

C. L. Ding, L. M. Liao, H. X. Wang, Y. T. Zhang, and L. Z. Pan, “Effect of oceanic turbulence on the propagation of cosine-Gaussian-correlated Schell-model beams,” J. Opt. 17(3), 035615 (2015).
[Crossref]

Ding, W.

C. Wan, K. Huang, T. Han, E. S. P. Leong, W. Ding, L. Zhang, T. Yeo, X. Yu, J. Teng, D. Lei, S. A. Maier, B. Luk’yanchuk, S. Zhang, and C. Qiu, “Three-dimensional visible-light capsule enclosing perfect supersized darkness via antiresolution,” Laser Photonics Rev. 8(5), 743–749 (2014).
[Crossref]

Du, L.

Egner, A.

R. Schmidt, C. A. Wurm, S. Jakobs, J. Engelhardt, A. Egner, and S. W. Hell, “Spherical nanosized focal spot unravels the interior of cells,” Nat. Methods 5(6), 539–544 (2008).
[Crossref] [PubMed]

Engelhardt, J.

P. Bingen, M. Reuss, J. Engelhardt, and S. W. Hell, “Parallelized STED fluorescence nanoscopy,” Opt. Express 19(24), 23716–23726 (2011).
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R. Schmidt, C. A. Wurm, S. Jakobs, J. Engelhardt, A. Egner, and S. W. Hell, “Spherical nanosized focal spot unravels the interior of cells,” Nat. Methods 5(6), 539–544 (2008).
[Crossref] [PubMed]

Gan, F.

H. Wang, J. Lin, D. Zhang, Y. Wang, M. Gu, H. P. Urbach, F. Gan, and S. Zhuang, “Creation of an anti-imaging system using binary optics,” Sci. Rep. 6(1), 33064 (2016).
[Crossref] [PubMed]

Gao, X. M.

J. Li, X. M. Gao, and Y. R. Chen, “Tight focusing of J0-correlated Gaussian Schell-model beam through high numerical aperture,” Opt. Commun. 285(16), 3403–3411 (2012).
[Crossref]

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

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[Crossref]

Gu, J. G.

Y. H. Mao, Z. R. Mei, and J. G. Gu, “Propagation of Gaussian Schell-model Array beams in free space and atmospheric turbulence,” Opt. Laser Technol. 86, 14–20 (2016).
[Crossref]

Gu, J. X.

Y. H. Chen, J. X. Gu, F. Wang, and Y. J. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Gu, M.

H. Wang, J. Lin, D. Zhang, Y. Wang, M. Gu, H. P. Urbach, F. Gan, and S. Zhuang, “Creation of an anti-imaging system using binary optics,” Sci. Rep. 6(1), 33064 (2016).
[Crossref] [PubMed]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[Crossref]

Guo, L. N.

L. N. Guo, Y. H. Chen, L. Liu, and Y. J. Cai, “Propagation of a Laguerre-Gaussian correlated Schell-model beam beyond the paraxial approximation,” Opt. Commun. 352, 127–134 (2015).
[Crossref]

Han, T.

C. Wan, K. Huang, T. Han, E. S. P. Leong, W. Ding, L. Zhang, T. Yeo, X. Yu, J. Teng, D. Lei, S. A. Maier, B. Luk’yanchuk, S. Zhang, and C. Qiu, “Three-dimensional visible-light capsule enclosing perfect supersized darkness via antiresolution,” Laser Photonics Rev. 8(5), 743–749 (2014).
[Crossref]

He, X.

He, Y. J.

Y. L. Qiu, Z. X. Chen, and Y. J. He, “Propagation of a Laguerre-Gaussian correlated Schell-model beam in strongly nonlocal nonlinear media,” Opt. Commun. 389, 303–309 (2017).
[Crossref]

Hell, S. W.

P. Bingen, M. Reuss, J. Engelhardt, and S. W. Hell, “Parallelized STED fluorescence nanoscopy,” Opt. Express 19(24), 23716–23726 (2011).
[Crossref] [PubMed]

R. Schmidt, C. A. Wurm, S. Jakobs, J. Engelhardt, A. Egner, and S. W. Hell, “Spherical nanosized focal spot unravels the interior of cells,” Nat. Methods 5(6), 539–544 (2008).
[Crossref] [PubMed]

Huang, K.

C. Wan, K. Huang, T. Han, E. S. P. Leong, W. Ding, L. Zhang, T. Yeo, X. Yu, J. Teng, D. Lei, S. A. Maier, B. Luk’yanchuk, S. Zhang, and C. Qiu, “Three-dimensional visible-light capsule enclosing perfect supersized darkness via antiresolution,” Laser Photonics Rev. 8(5), 743–749 (2014).
[Crossref]

Huang, K. L.

X. Wang, Z. R. Liu, K. L. Huang, and J. B. Sun, “Spectral shifts generated by scattering of Gaussian Schell-model arrays beam from a deterministic medium,” Opt. Commun. 387, 230–234 (2017).
[Crossref]

Huang, W.

Hyde, M. W.

M. W. Hyde, S. Basu, D. G. Voelz, and X. F. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
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Iketaki, Y.

Isenhower, L.

Jafarov, E.

E. Jafarov, S. Lievens, and J. V. Jeugt, “The Wigner distribution function for the one-dimensional parabose oscillator,” J. Phys. A 41(23), 235301 (2008).
[Crossref]

Jahn, K.

Jakobs, S.

R. Schmidt, C. A. Wurm, S. Jakobs, J. Engelhardt, A. Egner, and S. W. Hell, “Spherical nanosized focal spot unravels the interior of cells,” Nat. Methods 5(6), 539–544 (2008).
[Crossref] [PubMed]

Jeugt, J. V.

E. Jafarov, S. Lievens, and J. V. Jeugt, “The Wigner distribution function for the one-dimensional parabose oscillator,” J. Phys. A 41(23), 235301 (2008).
[Crossref]

Ji, X. L.

X. Q. Li and X. L. Ji, “Propagation of higher-order intensity moments through an optical system in atmospheric turbulence,” Opt. Commun. 298, 1–7 (2013).
[Crossref]

Korotkova, O.

Kumagai, H.

Lei, D.

C. Wan, K. Huang, T. Han, E. S. P. Leong, W. Ding, L. Zhang, T. Yeo, X. Yu, J. Teng, D. Lei, S. A. Maier, B. Luk’yanchuk, S. Zhang, and C. Qiu, “Three-dimensional visible-light capsule enclosing perfect supersized darkness via antiresolution,” Laser Photonics Rev. 8(5), 743–749 (2014).
[Crossref]

Leong, E. S. P.

C. Wan, K. Huang, T. Han, E. S. P. Leong, W. Ding, L. Zhang, T. Yeo, X. Yu, J. Teng, D. Lei, S. A. Maier, B. Luk’yanchuk, S. Zhang, and C. Qiu, “Three-dimensional visible-light capsule enclosing perfect supersized darkness via antiresolution,” Laser Photonics Rev. 8(5), 743–749 (2014).
[Crossref]

Li, H. H.

M. M. Tang, D. M. Zhao, X. Z. Li, and H. H. Li, “Focusing properties of radially polarized multi-cosine Gaussian correlated Schell-model beams,” Opt. Commun. 396, 249–256 (2017).
[Crossref]

Li, J.

J. Li, F. Wang, and O. Korotkova, “Random sources for cusped beams,” Opt. Express 24(16), 17779–17791 (2016).
[Crossref] [PubMed]

J. Li, X. M. Gao, and Y. R. Chen, “Tight focusing of J0-correlated Gaussian Schell-model beam through high numerical aperture,” Opt. Commun. 285(16), 3403–3411 (2012).
[Crossref]

Li, S.

Li, T.

P. Zhang, T. Li, J. Zhu, X. Zhu, S. Yang, Y. Wang, X. Yin, and X. Zhang, “Generation of acoustic self-bending and bottle beams by phase engineering,” Nat. Commun. 5, 4316 (2014).
[PubMed]

Li, X. Q.

X. Q. Li and X. L. Ji, “Propagation of higher-order intensity moments through an optical system in atmospheric turbulence,” Opt. Commun. 298, 1–7 (2013).
[Crossref]

Li, X. Z.

M. M. Tang, D. M. Zhao, X. Z. Li, and H. H. Li, “Focusing properties of radially polarized multi-cosine Gaussian correlated Schell-model beams,” Opt. Commun. 396, 249–256 (2017).
[Crossref]

Li, Y. P.

Li, Z.

Liang, C.

Liao, L. M.

C. L. Ding, L. M. Liao, H. X. Wang, Y. T. Zhang, and L. Z. Pan, “Effect of oceanic turbulence on the propagation of cosine-Gaussian-correlated Schell-model beams,” J. Opt. 17(3), 035615 (2015).
[Crossref]

Lievens, S.

E. Jafarov, S. Lievens, and J. V. Jeugt, “The Wigner distribution function for the one-dimensional parabose oscillator,” J. Phys. A 41(23), 235301 (2008).
[Crossref]

Lin, J.

H. Wang, J. Lin, D. Zhang, Y. Wang, M. Gu, H. P. Urbach, F. Gan, and S. Zhuang, “Creation of an anti-imaging system using binary optics,” Sci. Rep. 6(1), 33064 (2016).
[Crossref] [PubMed]

Liu, L.

Liu, S. T.

Z. Z. Song, Z. J. Liu, K. Y. Zhou, Q. G. Sun, and S. T. Liu, “Propagation properties of Gaussian Schell model array beams in non-Kolmogorov turbulence,” J. Opt. 18(10), 105601 (2016).
[Crossref]

Liu, X.

Liu, Z. J.

Z. Z. Song, Z. J. Liu, K. Y. Zhou, Q. G. Sun, and S. T. Liu, “Propagation properties of Gaussian Schell model array beams in non-Kolmogorov turbulence,” J. Opt. 18(10), 105601 (2016).
[Crossref]

Liu, Z. R.

X. Wang, Z. R. Liu, K. L. Huang, and J. B. Sun, “Spectral shifts generated by scattering of Gaussian Schell-model arrays beam from a deterministic medium,” Opt. Commun. 387, 230–234 (2017).
[Crossref]

Luk’yanchuk, B.

C. Wan, K. Huang, T. Han, E. S. P. Leong, W. Ding, L. Zhang, T. Yeo, X. Yu, J. Teng, D. Lei, S. A. Maier, B. Luk’yanchuk, S. Zhang, and C. Qiu, “Three-dimensional visible-light capsule enclosing perfect supersized darkness via antiresolution,” Laser Photonics Rev. 8(5), 743–749 (2014).
[Crossref]

Ma, L.

Maier, S. A.

C. Wan, K. Huang, T. Han, E. S. P. Leong, W. Ding, L. Zhang, T. Yeo, X. Yu, J. Teng, D. Lei, S. A. Maier, B. Luk’yanchuk, S. Zhang, and C. Qiu, “Three-dimensional visible-light capsule enclosing perfect supersized darkness via antiresolution,” Laser Photonics Rev. 8(5), 743–749 (2014).
[Crossref]

Mao, Y.

Mao, Y. H.

Y. H. Mao and Z. R. Mei, “Random sources generating ring-shaped optical lattice,” Opt. Commun. 381, 222–226 (2016).
[Crossref]

Y. H. Mao, Z. R. Mei, and J. G. Gu, “Propagation of Gaussian Schell-model Array beams in free space and atmospheric turbulence,” Opt. Laser Technol. 86, 14–20 (2016).
[Crossref]

Mei, Z.

Mei, Z. R.

Y. H. Mao, Z. R. Mei, and J. G. Gu, “Propagation of Gaussian Schell-model Array beams in free space and atmospheric turbulence,” Opt. Laser Technol. 86, 14–20 (2016).
[Crossref]

Y. H. Mao and Z. R. Mei, “Random sources generating ring-shaped optical lattice,” Opt. Commun. 381, 222–226 (2016).
[Crossref]

Z. R. Mei and O. Korotkova, “Electromagnetic Schell-model sources generating far fields with stable and flexible concentric rings profiles,” Opt. Express 24(5), 5572–5583 (2016).
[Crossref]

Mi, C.

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[Crossref]

Pan, L. Z.

C. L. Ding, L. M. Liao, H. X. Wang, Y. T. Zhang, and L. Z. Pan, “Effect of oceanic turbulence on the propagation of cosine-Gaussian-correlated Schell-model beams,” J. Opt. 17(3), 035615 (2015).
[Crossref]

Ponomarenko, S. A.

Qiu, C.

C. Wan, K. Huang, T. Han, E. S. P. Leong, W. Ding, L. Zhang, T. Yeo, X. Yu, J. Teng, D. Lei, S. A. Maier, B. Luk’yanchuk, S. Zhang, and C. Qiu, “Three-dimensional visible-light capsule enclosing perfect supersized darkness via antiresolution,” Laser Photonics Rev. 8(5), 743–749 (2014).
[Crossref]

Qiu, Y. L.

Y. L. Qiu, Z. X. Chen, and Y. J. He, “Propagation of a Laguerre-Gaussian correlated Schell-model beam in strongly nonlocal nonlinear media,” Opt. Commun. 389, 303–309 (2017).
[Crossref]

Y. L. Qiu, Z. X. Chen, and L. Liu, “Partially coherent dark hollow beams propagating through real ABCD optical systems in a turbulent atmosphere,” J. Mod. Opt. 57(8), 662–669 (2010).
[Crossref]

Qu, J.

Reuss, M.

Saffman, M.

Santarsiero, M.

Schchepakina, E.

Schmidt, R.

R. Schmidt, C. A. Wurm, S. Jakobs, J. Engelhardt, A. Egner, and S. W. Hell, “Spherical nanosized focal spot unravels the interior of cells,” Nat. Methods 5(6), 539–544 (2008).
[Crossref] [PubMed]

Shi, P.

Song, Z. Z.

Z. Z. Song, Z. J. Liu, K. Y. Zhou, Q. G. Sun, and S. T. Liu, “Propagation properties of Gaussian Schell model array beams in non-Kolmogorov turbulence,” J. Opt. 18(10), 105601 (2016).
[Crossref]

Su, Q.

J. Zhu, H. Q. Tang, Q. Su, and K. C. Zhu, “Cascade self-splitting of a Hermite-cos-Gaussian correlated Schell-model beam,” Europhys. Lett. 118(1), 14001 (2017).
[Crossref]

Sun, J. B.

X. Wang, Z. R. Liu, K. L. Huang, and J. B. Sun, “Spectral shifts generated by scattering of Gaussian Schell-model arrays beam from a deterministic medium,” Opt. Commun. 387, 230–234 (2017).
[Crossref]

Sun, Q. G.

Z. Z. Song, Z. J. Liu, K. Y. Zhou, Q. G. Sun, and S. T. Liu, “Propagation properties of Gaussian Schell model array beams in non-Kolmogorov turbulence,” J. Opt. 18(10), 105601 (2016).
[Crossref]

Tang, H.

Tang, H. Q.

J. Zhu, H. Q. Tang, Q. Su, and K. C. Zhu, “Cascade self-splitting of a Hermite-cos-Gaussian correlated Schell-model beam,” Europhys. Lett. 118(1), 14001 (2017).
[Crossref]

Tang, M. M.

M. M. Tang, D. M. Zhao, X. Z. Li, and H. H. Li, “Focusing properties of radially polarized multi-cosine Gaussian correlated Schell-model beams,” Opt. Commun. 396, 249–256 (2017).
[Crossref]

Tang, Y.

Teng, J.

C. Wan, K. Huang, T. Han, E. S. P. Leong, W. Ding, L. Zhang, T. Yeo, X. Yu, J. Teng, D. Lei, S. A. Maier, B. Luk’yanchuk, S. Zhang, and C. Qiu, “Three-dimensional visible-light capsule enclosing perfect supersized darkness via antiresolution,” Laser Photonics Rev. 8(5), 743–749 (2014).
[Crossref]

Urbach, H. P.

H. Wang, J. Lin, D. Zhang, Y. Wang, M. Gu, H. P. Urbach, F. Gan, and S. Zhuang, “Creation of an anti-imaging system using binary optics,” Sci. Rep. 6(1), 33064 (2016).
[Crossref] [PubMed]

Voelz, D. G.

M. W. Hyde, S. Basu, D. G. Voelz, and X. F. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

Wan, C.

C. Wan, K. Huang, T. Han, E. S. P. Leong, W. Ding, L. Zhang, T. Yeo, X. Yu, J. Teng, D. Lei, S. A. Maier, B. Luk’yanchuk, S. Zhang, and C. Qiu, “Three-dimensional visible-light capsule enclosing perfect supersized darkness via antiresolution,” Laser Photonics Rev. 8(5), 743–749 (2014).
[Crossref]

Wang, F.

C. Liang, C. Mi, F. Wang, C. Zhao, Y. Cai, and S. A. Ponomarenko, “Vector optical coherence lattices generating controllable far-field beam profiles,” Opt. Express 25(9), 9872–9885 (2017).
[Crossref] [PubMed]

F. Wang and O. Korotkova, “Circularly symmetric cusped random beams in free space and atmospheric turbulence,” Opt. Express 25(5), 5057–5067 (2017).
[Crossref] [PubMed]

J. Li, F. Wang, and O. Korotkova, “Random sources for cusped beams,” Opt. Express 24(16), 17779–17791 (2016).
[Crossref] [PubMed]

Y. H. Chen, J. Y. Yu, Y. S. Yuan, F. Wang, and Y. J. Cai, “Theoretical and experimental studies of a rectangular Laguerre-Gaussian-correlated Schell-model beam,” Appl. Phys. B 122(2), 31 (2016).
[Crossref]

Y. H. Chen, J. X. Gu, F. Wang, and Y. J. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and 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, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

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

Wang, H.

Wang, H. X.

C. L. Ding, L. M. Liao, H. X. Wang, Y. T. Zhang, and L. Z. Pan, “Effect of oceanic turbulence on the propagation of cosine-Gaussian-correlated Schell-model beams,” J. Opt. 17(3), 035615 (2015).
[Crossref]

Wang, J.

Wang, X.

X. Wang, Z. R. Liu, K. L. Huang, and J. B. Sun, “Spectral shifts generated by scattering of Gaussian Schell-model arrays beam from a deterministic medium,” Opt. Commun. 387, 230–234 (2017).
[Crossref]

Wang, Y.

H. Wang, J. Lin, D. Zhang, Y. Wang, M. Gu, H. P. Urbach, F. Gan, and S. Zhuang, “Creation of an anti-imaging system using binary optics,” Sci. Rep. 6(1), 33064 (2016).
[Crossref] [PubMed]

P. Zhang, T. Li, J. Zhu, X. Zhu, S. Yang, Y. Wang, X. Yin, and X. Zhang, “Generation of acoustic self-bending and bottle beams by phase engineering,” Nat. Commun. 5, 4316 (2014).
[PubMed]

Weng, X.

Williams, W.

Wu, G.

Wurm, C. A.

R. Schmidt, C. A. Wurm, S. Jakobs, J. Engelhardt, A. Egner, and S. W. Hell, “Spherical nanosized focal spot unravels the interior of cells,” Nat. Methods 5(6), 539–544 (2008).
[Crossref] [PubMed]

Xiao, X. F.

M. W. Hyde, S. Basu, D. G. Voelz, and X. F. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
[Crossref]

Xiu, P.

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

Xu, H. F.

H. F. Xu, Z. Zhang, J. Qu, and W. Huang, “The tight focusing properties of Laguerre–Gaussian-correlated Schell-model beams,” J. Mod. Opt. 63(15), 1429–1437 (2016).
[Crossref]

H. F. Xu, Z. Zhang, J. Qu, and W. Huang, “Propagation factors of cosine-Gaussian-correlated Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 22(19), 22479–22489 (2014).
[Crossref] [PubMed]

Xu, P.

Yang, S.

P. Zhang, T. Li, J. Zhu, X. Zhu, S. Yang, Y. Wang, X. Yin, and X. Zhang, “Generation of acoustic self-bending and bottle beams by phase engineering,” Nat. Commun. 5, 4316 (2014).
[PubMed]

Yeo, T.

C. Wan, K. Huang, T. Han, E. S. P. Leong, W. Ding, L. Zhang, T. Yeo, X. Yu, J. Teng, D. Lei, S. A. Maier, B. Luk’yanchuk, S. Zhang, and C. Qiu, “Three-dimensional visible-light capsule enclosing perfect supersized darkness via antiresolution,” Laser Photonics Rev. 8(5), 743–749 (2014).
[Crossref]

Yin, X.

P. Zhang, T. Li, J. Zhu, X. Zhu, S. Yang, Y. Wang, X. Yin, and X. Zhang, “Generation of acoustic self-bending and bottle beams by phase engineering,” Nat. Commun. 5, 4316 (2014).
[PubMed]

Yu, J.

Yu, J. Y.

Y. H. Chen, J. Y. Yu, Y. S. Yuan, F. Wang, and Y. J. Cai, “Theoretical and experimental studies of a rectangular Laguerre-Gaussian-correlated Schell-model beam,” Appl. Phys. B 122(2), 31 (2016).
[Crossref]

Yu, X.

C. Wan, K. Huang, T. Han, E. S. P. Leong, W. Ding, L. Zhang, T. Yeo, X. Yu, J. Teng, D. Lei, S. A. Maier, B. Luk’yanchuk, S. Zhang, and C. Qiu, “Three-dimensional visible-light capsule enclosing perfect supersized darkness via antiresolution,” Laser Photonics Rev. 8(5), 743–749 (2014).
[Crossref]

Yu, Y.

Yuan, X.

Yuan, Y.

Yuan, Y. S.

Y. H. Chen, J. Y. Yu, Y. S. Yuan, F. Wang, and Y. J. Cai, “Theoretical and experimental studies of a rectangular Laguerre-Gaussian-correlated Schell-model beam,” Appl. Phys. B 122(2), 31 (2016).
[Crossref]

Zhan, M.

Zhan, Q.

Zhang, B.

Zhang, D.

H. Wang, J. Lin, D. Zhang, Y. Wang, M. Gu, H. P. Urbach, F. Gan, and S. Zhuang, “Creation of an anti-imaging system using binary optics,” Sci. Rep. 6(1), 33064 (2016).
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Zhang, L.

C. Wan, K. Huang, T. Han, E. S. P. Leong, W. Ding, L. Zhang, T. Yeo, X. Yu, J. Teng, D. Lei, S. A. Maier, B. Luk’yanchuk, S. Zhang, and C. Qiu, “Three-dimensional visible-light capsule enclosing perfect supersized darkness via antiresolution,” Laser Photonics Rev. 8(5), 743–749 (2014).
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Zhang, P.

P. Zhang, T. Li, J. Zhu, X. Zhu, S. Yang, Y. Wang, X. Yin, and X. Zhang, “Generation of acoustic self-bending and bottle beams by phase engineering,” Nat. Commun. 5, 4316 (2014).
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Zhang, S.

C. Wan, K. Huang, T. Han, E. S. P. Leong, W. Ding, L. Zhang, T. Yeo, X. Yu, J. Teng, D. Lei, S. A. Maier, B. Luk’yanchuk, S. Zhang, and C. Qiu, “Three-dimensional visible-light capsule enclosing perfect supersized darkness via antiresolution,” Laser Photonics Rev. 8(5), 743–749 (2014).
[Crossref]

Zhang, X.

P. Zhang, T. Li, J. Zhu, X. Zhu, S. Yang, Y. Wang, X. Yin, and X. Zhang, “Generation of acoustic self-bending and bottle beams by phase engineering,” Nat. Commun. 5, 4316 (2014).
[PubMed]

Zhang, Y.

Zhang, Y. T.

C. L. Ding, L. M. Liao, H. X. Wang, Y. T. Zhang, and L. Z. Pan, “Effect of oceanic turbulence on the propagation of cosine-Gaussian-correlated Schell-model beams,” J. Opt. 17(3), 035615 (2015).
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Zhang, Z.

H. F. Xu, Z. Zhang, J. Qu, and W. Huang, “The tight focusing properties of Laguerre–Gaussian-correlated Schell-model beams,” J. Mod. Opt. 63(15), 1429–1437 (2016).
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H. F. Xu, Z. Zhang, J. Qu, and W. Huang, “Propagation factors of cosine-Gaussian-correlated Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 22(19), 22479–22489 (2014).
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Zhao, C.

Zhao, D.

Zhao, D. M.

M. M. Tang, D. M. Zhao, X. Z. Li, and H. H. Li, “Focusing properties of radially polarized multi-cosine Gaussian correlated Schell-model beams,” Opt. Commun. 396, 249–256 (2017).
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Zhao, Y.

Zhou, K. Y.

Z. Z. Song, Z. J. Liu, K. Y. Zhou, Q. G. Sun, and S. T. Liu, “Propagation properties of Gaussian Schell model array beams in non-Kolmogorov turbulence,” J. Opt. 18(10), 105601 (2016).
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Zhou, Y.

Zhu, J.

J. Zhu, H. Q. Tang, Q. Su, and K. C. Zhu, “Cascade self-splitting of a Hermite-cos-Gaussian correlated Schell-model beam,” Europhys. Lett. 118(1), 14001 (2017).
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P. Zhang, T. Li, J. Zhu, X. Zhu, S. Yang, Y. Wang, X. Yin, and X. Zhang, “Generation of acoustic self-bending and bottle beams by phase engineering,” Nat. Commun. 5, 4316 (2014).
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Zhu, K.

Zhu, K. C.

J. Zhu, H. Q. Tang, Q. Su, and K. C. Zhu, “Cascade self-splitting of a Hermite-cos-Gaussian correlated Schell-model beam,” Europhys. Lett. 118(1), 14001 (2017).
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Zhu, S.

Zhu, X.

P. Zhang, T. Li, J. Zhu, X. Zhu, S. Yang, Y. Wang, X. Yin, and X. Zhang, “Generation of acoustic self-bending and bottle beams by phase engineering,” Nat. Commun. 5, 4316 (2014).
[PubMed]

Zhuang, S.

H. Wang, J. Lin, D. Zhang, Y. Wang, M. Gu, H. P. Urbach, F. Gan, and S. Zhuang, “Creation of an anti-imaging system using binary optics,” Sci. Rep. 6(1), 33064 (2016).
[Crossref] [PubMed]

Appl. Phys. B (1)

Y. H. Chen, J. Y. Yu, Y. S. Yuan, F. Wang, and Y. J. Cai, “Theoretical and experimental studies of a rectangular Laguerre-Gaussian-correlated Schell-model beam,” Appl. Phys. B 122(2), 31 (2016).
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Appl. Phys. Lett. (1)

Y. H. Chen, S. A. Ponomarenko, and Y. J. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
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Europhys. Lett. (1)

J. Zhu, H. Q. Tang, Q. Su, and K. C. Zhu, “Cascade self-splitting of a Hermite-cos-Gaussian correlated Schell-model beam,” Europhys. Lett. 118(1), 14001 (2017).
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J. Appl. Phys. (1)

M. W. Hyde, S. Basu, D. G. Voelz, and X. F. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015).
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H. F. Xu, Z. Zhang, J. Qu, and W. Huang, “The tight focusing properties of Laguerre–Gaussian-correlated Schell-model beams,” J. Mod. Opt. 63(15), 1429–1437 (2016).
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J. Opt. (2)

C. L. Ding, L. M. Liao, H. X. Wang, Y. T. Zhang, and L. Z. Pan, “Effect of oceanic turbulence on the propagation of cosine-Gaussian-correlated Schell-model beams,” J. Opt. 17(3), 035615 (2015).
[Crossref]

Z. Z. Song, Z. J. Liu, K. Y. Zhou, Q. G. Sun, and S. T. Liu, “Propagation properties of Gaussian Schell model array beams in non-Kolmogorov turbulence,” J. Opt. 18(10), 105601 (2016).
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C. Wan, K. Huang, T. Han, E. S. P. Leong, W. Ding, L. Zhang, T. Yeo, X. Yu, J. Teng, D. Lei, S. A. Maier, B. Luk’yanchuk, S. Zhang, and C. Qiu, “Three-dimensional visible-light capsule enclosing perfect supersized darkness via antiresolution,” Laser Photonics Rev. 8(5), 743–749 (2014).
[Crossref]

Nat. Commun. (1)

P. Zhang, T. Li, J. Zhu, X. Zhu, S. Yang, Y. Wang, X. Yin, and X. Zhang, “Generation of acoustic self-bending and bottle beams by phase engineering,” Nat. Commun. 5, 4316 (2014).
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Nat. Methods (1)

R. Schmidt, C. A. Wurm, S. Jakobs, J. Engelhardt, A. Egner, and S. W. Hell, “Spherical nanosized focal spot unravels the interior of cells,” Nat. Methods 5(6), 539–544 (2008).
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M. M. Tang, D. M. Zhao, X. Z. Li, and H. H. Li, “Focusing properties of radially polarized multi-cosine Gaussian correlated Schell-model beams,” Opt. Commun. 396, 249–256 (2017).
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X. Wang, Z. R. Liu, K. L. Huang, and J. B. Sun, “Spectral shifts generated by scattering of Gaussian Schell-model arrays beam from a deterministic medium,” Opt. Commun. 387, 230–234 (2017).
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Y. L. Qiu, Z. X. Chen, and Y. J. He, “Propagation of a Laguerre-Gaussian correlated Schell-model beam in strongly nonlocal nonlinear media,” Opt. Commun. 389, 303–309 (2017).
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L. N. Guo, Y. H. Chen, L. Liu, and Y. J. Cai, “Propagation of a Laguerre-Gaussian correlated Schell-model beam beyond the paraxial approximation,” Opt. Commun. 352, 127–134 (2015).
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Opt. Express (17)

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
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Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
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R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
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Y. Zhou, Y. Yuan, J. Qu, and W. Huang, “Propagation properties of Laguerre-Gaussian correlated Schell-model beam in non-Kolmogorov turbulence,” Opt. Express 24(10), 10682–10693 (2016).
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Z. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
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Z. Mei, “Light sources generating self-splitting beams and their propagation in non-Kolmogorov turbulence,” Opt. Express 22(11), 13029–13040 (2014).
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H. F. Xu, Z. Zhang, J. Qu, and W. Huang, “Propagation factors of cosine-Gaussian-correlated Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 22(19), 22479–22489 (2014).
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C. Liang, C. Mi, F. Wang, C. Zhao, Y. Cai, and S. A. Ponomarenko, “Vector optical coherence lattices generating controllable far-field beam profiles,” Opt. Express 25(9), 9872–9885 (2017).
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J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
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L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015).
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J. Li, F. Wang, and O. Korotkova, “Random sources for cusped beams,” Opt. Express 24(16), 17779–17791 (2016).
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F. Wang and O. Korotkova, “Circularly symmetric cusped random beams in free space and atmospheric turbulence,” Opt. Express 25(5), 5057–5067 (2017).
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Z. R. Mei and O. Korotkova, “Electromagnetic Schell-model sources generating far fields with stable and flexible concentric rings profiles,” Opt. Express 24(5), 5572–5583 (2016).
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Y. H. Mao, Z. R. Mei, and J. G. Gu, “Propagation of Gaussian Schell-model Array beams in free space and atmospheric turbulence,” Opt. Laser Technol. 86, 14–20 (2016).
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J. Cang, P. Xiu, and X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical systems in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
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Figures (6)

Fig. 1
Fig. 1 Density plots of the square of the modulus of the SDOC for (a) a McGCSM source with n = 0 and βx = βy = 8, (b) a LGCSM source with n = 5 and βx = βy = 0, and (c) a McLGCSM source with n = 5 and βx = βy = 8. The other parameters are Px = Py = 3.
Fig. 2
Fig. 2 Intensity distribution patterns of a McLGCSM beam (a), McGCSM beam (b), and LGCSM beam (c) at several propagation distances in free space with w0 = 5mm, Px = Py = 3, and δw = 1. The x and y axes taken as the transverse coordinates are in arbitrary units.
Fig. 3
Fig. 3 Intensity distribution patterns of a McLGCSM beam at several propagation distances in free space for different values of β and δw with w0 = 5mm, Px = Py = 3, and n = 4.
Fig. 4
Fig. 4 Average intensity distribution patterns of a McLGCSM beam (a), McGCSM beam (b), and LGCSM beam (c) at several propagation distances in atmospheric turbulence with w0 = 5mm, Px = Py = 3, δw = 1, and C n 2 = 5 × 10−14m−2/3.
Fig. 5
Fig. 5 Average intensity distribution patterns of a McLGCSM beam at several propagation distances in atmospheric turbulence for different values of C n 2 and δw with w0 = 5mm, β = 10, and n = 5.
Fig. 6
Fig. 6 Relative propagation factors M r,n,Q 2 ( z ) versus the propagation distance z in atmospheric turbulence for (G) a GSM beam with n = βx = βy = 0, (L) a LGCSM beam with n = 5 and βx = βy = 0, (Mc) a McGCSM beam with n = 0 and βx = βy = 8, and (McL) a McLGCSM beam with n = 5 and βx = βy = 8. The other parameter are w0 = 10mm, δw = 1, Px = Py = 3, and C n 2 = 5 × 10−15m−2/3.

Equations (44)

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d 2 r 1 d 2 r 2 W ( 0 ) ( r 1 , r 2 )f( r 1 )f( r 2 )0.
W ( 0 ) ( r 1 , r 2 )= d 2 v p( v ) H 0 * ( r 1 ,v ) H 0 ( r 2 ,v ),
H 0 ( r ,v )=τ( r )exp( i r v ),
W ( 0 ) ( r 1 , r 2 )= τ * ( r 1 )τ( r 2 ) d 2 v p( v )exp[ iv( r 1 r 2 ) ]= τ * ( r 1 )τ( r 2 )μ( r 1 r 2 )
p( v x , v x )= N 4 l x = M x M x l y = M y M y [ p L ( v x + l x β x δ , v y + l y β y δ ) + p L ( v x + l x β x δ , v y l y β y δ ) + p L ( v x l x β x δ , v y + l y β y δ )+ p L ( v x l x β x δ , v y l y β y δ ) ],
p L ( v x , v y )= ( v x 2 + v y 2 δ 2 ) n exp[ δ 2 ( v x 2 + v y 2 ) 2 ],
f( z z 0 ) exp( ixz )dz=exp( i z 0 x ) f( y ) exp( ixy )dy,
W ( 0 ) ( r 1 , r 2 )=exp( r s 2 2 w 0 2 r d 2 8 w 0 2 )μ( r d ),
μ( r d )=N L n ( r d 2 δ 2 )exp( r d 2 2 δ 2 ) l x = M x M x l y = M y M y cos( l x β x x d δ )cos( l y β y y d δ ) ,
W( r 1 , r 2 ,z )= 1 λ 2 z 2 exp[ ik 2z ( r 2 2 r 1 2 ) ] d 2 r 1 d 2 r 2 W ( 0 ) ( r 1 , r 2 ) ×exp[ ik 2z ( r 1 2 r 2 2 )+ ik z ( r 1 r 1 r 2 r 2 ) ] exp[ Φ( r 1 , r 1 )+ Φ * ( r 2 , r 2 ) ] ,
exp[ Φ( r 1 , r 1 )+ Φ * ( r 2 , r 2 ) ] =exp[ T α ( r d 2 + r d r d + r d 2 ) ],
W( r 1 , r 2 ,z )= N λ 2 z 2 exp( ik z r r d T α r d 2 ) l x =1 M x l y =1 M y d 2 r s d 2 r d L n ( r d 2 2 δ 2 ) ×cos( Q l x x d δ )cos( Q l y y d δ )exp[ r s 2 2 w 0 2 ( 1 δ 2 + 1 4 w 0 2 +2 T α ) r d 2 2 ] ×exp[ ik z r s r d + ik z ( r d r s +r r d ) T α r d r d ].
W( r 1 , r 2 ,z )= 2π w 0 2 N λ 2 z 2 exp[ 2i z R z w 0 2 r r d ( 2 z R 2 z 2 w 0 2 + T α ) r d 2 ] × l x = M x M x l y = M y M y [ W l,++ ( r 1 , r 2 )+ W l,+ ( r 1 , r 2 )+ W l,+ ( r 1 , r 2 )+ W l, ( r 1 , r 2 ) ] ,
W l,uv ( r 1 , r 2 )= 1 4 d 2 r d L n ( r d 2 2 δ 2 )exp[ Ω r d 2 8 w 0 2 + ik z ( b l x ,xu x d + b l y ,yv y d ) ],
Ω=1+ 4 δ w 2 +8 w 0 2 T α + 16 z R 2 z 2 ,
b q± =q+i( T α z k 2 z R z ) q d ± z w 0 2 z R δ w l q β q ,( q=x,yu,v=± ),
L n ( x )= p=0 n ( n p ) ( 1 ) p p! x p
( x 2 + y 2 ) p = m=0 p ( p m ) x 2( pm ) y 2m ,
z n exp[ ( zβ ) 2 ]dz= ( 2i ) n π H n ( iβ ),
W l,uv ( r 1 , r 2 )= 2π w 0 2 Ω exp[ 2 k 2 w 0 2 ( b l x ,xu 2 + b l y ,yv 2 ) Ω z 2 ] p=0 n ( n p ) 1 p! ( 1 Ω δ w 2 ) p × m=0 p ( p m ) H 2( pm ) ( 2 Ω k w 0 z b l x ,xu ) H 2m ( 2 Ω k w 0 z b l y ,yv ),
m=0 n ( n m ) H 2m ( x ) H 2n2m ( y ) = ( 4 ) n n! L n ( x 2 + y 2 ),
L n ( bx )= ( 1b ) n m=0 n ( b 1b ) m ( n m ) L m ( x ) ,
W l,uv ( r 1 , r 2 )= 2π w 0 2 Ω exp[ 2 k 2 w 0 2 ( b l x ,xu 2 + b l y ,yv 2 ) Ω z 2 ] m=0 n ( n m ) ( 4 Ω δ w 2 ) m L m [ 2 k 2 w 0 2 ( b l x ,xu 2 + b l y ,yv 2 ) Ω z 2 ] = 2π w 0 2 Ω ( Ω δ w 2 Ω δ w 2 4 ) n exp[ 2 k 2 w 0 2 ( b l x ,xu 2 + b l y ,yv 2 ) Ω z 2 ] L m [ 8 k 2 w 0 2 ( b l x ,xu 2 + b l y ,yv 2 ) Ω z 2 ( 4Ω δ w 2 ) ].
I( r,z )=W( r,r,z )= πN w 0 2 z 2 z R 2 l x =1 M x l y =1 M y [ I l,++ ( r,z )+ I l,+ ( r,z )+ I l,+ ( r,z )+ I l, ( r,z ) ] ,
I l,±± ( r,z )= ( Ω δ w 2 Ω δ w 2 4 ) n exp( 8 z R 2 Ω z 2 R l x , l y ) L n [ 32 z R 2 δ w 2 z 2 ( 4Ω δ w 2 ) R l x , l y ]
μ( r 1 , r 2 ,z )= W( r 1 , r 2 ,z ) W( r 1 , r 1 ,z )W( r 2 , r 2 ,z ) .
x n 1 y n 2 θ x m 1 θ y m 2 z = 1 P d 2 ρ d 2 θ x n 1 y n 2 θ x m 1 θ y m 2 h( ρ,θ,z ) = 1 P d 2 ρ d 2 θ G( ρ , θ ,z )h( ρ , θ ,0 ),
h( ρ,θ,0 )= ( k 2π ) 2 W ( 0 ) ( ρ, ρ d ,0 )exp( ik ρ d θ ) d 2 ρ d ,
h( ρ,θ,z )= ( k 2π ) 2 W ( ρ, ρ d ,z )exp( ik ρ d θ ) d 2 ρ d ,
G( ρ , θ ,z )= ( i ) n 1 + n 2 + m 1 + m 2 B n 1 + n 2 k n 1 + n 2 + m 1 + m 2 d 2 ρ d d 2 ρ d δ ( n 1 ) ( x d D x d ) δ ( n 2 ) ( y d D y d ) × δ ( m 1 ) ( x d ) δ ( m 2 ) ( y d )exp[ ik B ρ ( A ρ d ρ d )+ik θ ρ d H( ρ d , ρ d ,z ) ],
x 2 z = 2 z 2 k 2 T α + ρ x 2 0 +2z ρ x θ x 0 + z 2 θ x 2 0 ,
θ x 2 z = 6 k 2 T α + θ x 2 0 ,
x θ x z = 3z k 2 T α + ρ x θ x 0 +z θ x 2 0 ,
Q( r,θ ) 0 = 1 P d 2 r d 2 θ Q( r,θ )h( r,θ,0 ).
x 2 + y 2 z =( 4 k 2 T α + M n,β 2 w 0 2 ) z 2 +2 w 0 2 ,
x θ x +y θ y z =( 6 k 2 T α + M n,β 2 w 0 2 )z,
θ x 2 + θ y 2 z = 12 k 2 T α + M n,β 2 w 0 2 ,
M n,β = 1 k 2 [ 1+ 4 w 0 2 δ 2 ( n+1 )+ 2 w 0 2 δ 2 Q M,β ],
Q M,β = β x 2 P x 1 l x = M x M x l x 2 + β y 2 P y 1 l y = M y M y l y 2 .
M 2 ( z )=k ( ρ 2 z θ 2 z ρθ z 2 ) 1/2 .
M n,Q 2 ( z )= [ M n,β + 2 T α k 2 ( 12 w 0 2 + 6 T α k 2 z 2 + M n,β w 0 2 z 2 ) ] 1/2 .
M r,n,Q 2 ( z )= M n,Q 2 ( z )/ M n,β .
M r,0,0 2 ( z )> M r,n,0 2 ( z )[ M r,0,Q 2 ( z ) ]> M r,n,Q 2 ( z ).
M r,n,Q 2 ( z ) M r,n,0 2 ( z )= 12Ω( M n,0 2 M n,β 2 )( 2 w 0 2 + Ω z 2 / k 2 ) k 2 M n,β M n,0 [ M n,0 M r,n,Q 2 ( z )+ M n,β M r,n,0 2 ( z ) ] <0.

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