Abstract

The analytical expressions for the cross-spectral density, the average intensity and the complex degree of spatial coherence of a radial phased-locked partially coherent flat-topped vortex beam array propagating through non-Kolmogorov medium are obtained by using the extended Huygens–Fresnel principle. The evolution behaviors of a radial phased-locked partially coherent flat-topped vortex beam array propagating through non-Kolmogorov medium are studied in detail. It is shown that the evolution behaviors of average intensity depend on beam parameters including the spatial correlation length, the radius of the beam array, as well as the propagation distance. A radial phased-locked partially coherent flat-topped vortex beam array with high coherence evolves more rapidly than that with low coherence.

© 2016 Optical Society of America

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References

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2015 (6)

2014 (6)

Y. Huang, B. Zhang, Z. Gao, G. Zhao, and Z. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
[Crossref] [PubMed]

K. Wang and C. Zhao, “Propagation properties of a radial phased-locked partially coherent anomalous hollow beam array in turbulent atmosphere,” Opt. Laser Technol. 57, 44–51 (2014).
[Crossref]

M. Tang and D. Zhao, “Spectral changes in stochastic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312(1), 89–93 (2014).
[Crossref]

O. Korotkova and E. Shchepakina, “Rectangular Multi-Gaussian Schell-Model beams in atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
[Crossref]

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. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

2013 (3)

2012 (1)

Y. Yuan, Y. Cai, H. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

2011 (4)

2010 (5)

2009 (2)

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. 282(6), 1088–1094 (2009).
[Crossref]

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[Crossref]

2008 (6)

2007 (3)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
[Crossref]

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
[Crossref] [PubMed]

2006 (2)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[Crossref]

Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31(7), 867–869 (2006).
[Crossref] [PubMed]

2003 (5)

2002 (2)

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109 (2002).
[Crossref]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002).
[Crossref] [PubMed]

1988 (1)

Amarande, S.

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
[Crossref]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109 (2002).
[Crossref]

Baykal, Y.

Y. Yuan, Y. Cai, H. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Cai, Y.

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. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

Y. Yang, Y. Dong, C. Zhao, and Y. Cai, “Generation and propagation of an anomalous vortex beam,” Opt. Lett. 38(24), 5418–5421 (2013).
[Crossref] [PubMed]

Y. Yuan, Y. Cai, H. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[Crossref]

Chen, B.

Chen, C.

Chen, J.

Y. Yuan, Y. Cai, H. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Chen, Y.

Chen, Z.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
[Crossref]

Chu, X.

X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283(18), 3398–3403 (2010).
[Crossref]

Dan, Y.

Dang, A.

Deng, D.

Dogariu, A.

Dong, Y.

Du, X.

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[Crossref]

X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93(4), 901–905 (2008).
[Crossref]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
[Crossref] [PubMed]

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
[Crossref] [PubMed]

Duan, Z.

Eyyuboglu, H.

Y. Yuan, Y. Cai, H. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Feng, S.

Feng, X.

X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283(18), 3398–3403 (2010).
[Crossref]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Friberg, A.

Gao, Z.

Gbur, G.

Guo, H.

He, J.

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[Crossref]

He, X.

X. He and B. Lü, “Propagation properties of partially coherent Hermite-Gaussian beams through non-Kolmogorov turbulence,” Chin. Phys. B 20(9), 244–252 (2011).
[Crossref]

X. He and B. Lü, “Propagation of partially coherent flat-topped vortex beams through non-Kolmogorov, atmospheric turbulence,” J. Opt. Soc. Am. A 28(9), 1941–1948 (2011).
[Crossref]

Hu, D.

Huang, Y.

Kan, Q.

Korotkova, O.

Kotlyar, V. V.

Kovalev, A. A.

Liu, D.

Liu, H.

H. Liu, Y. Lü, J. Xia, X. Pu, and L. Zhang, “Flat-topped vortex hollow beam and its propagation properties,” J. Opt. 17(7), 075606 (2015).
[Crossref]

Liu, L.

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

Liu, Z.

Lü, B.

X. He and B. Lü, “Propagation properties of partially coherent Hermite-Gaussian beams through non-Kolmogorov turbulence,” Chin. Phys. B 20(9), 244–252 (2011).
[Crossref]

X. He and B. Lü, “Propagation of partially coherent flat-topped vortex beams through non-Kolmogorov, atmospheric turbulence,” J. Opt. Soc. Am. A 28(9), 1941–1948 (2011).
[Crossref]

Lü, Y.

H. Liu, Y. Lü, J. Xia, X. Pu, and L. Zhang, “Flat-topped vortex hollow beam and its propagation properties,” J. Opt. 17(7), 075606 (2015).
[Crossref]

Luo, B.

Ma, Y.

Ni, Y.

Y. Ni and G. Zhou, “Propagation of a Lorentz–Gauss vortex beam through a paraxial ABCD optical system,” Opt. Commun. 291(15), 19–25 (2013).
[Crossref]

Ou, B.

Peng, X.

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
[Crossref]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109 (2002).
[Crossref]

Porfirev, A. P.

Pu, J.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
[Crossref]

Pu, X.

H. Liu, Y. Lü, J. Xia, X. Pu, and L. Zhang, “Flat-topped vortex hollow beam and its propagation properties,” J. Opt. 17(7), 075606 (2015).
[Crossref]

Qiao, C.

X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283(18), 3398–3403 (2010).
[Crossref]

Qiu, Z.

Schouten, H. F.

Shchepakina, E.

Shirai, T.

Tang, H.

Tang, M.

M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23(25), 32766–32776 (2015).
[Crossref] [PubMed]

M. Tang and D. Zhao, “Spectral changes in stochastic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312(1), 89–93 (2014).
[Crossref]

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Turunen, J.

Tyson, R. K.

Visser, T.

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

Visser, T. D.

Wang, F.

Wang, K.

K. Wang and C. Zhao, “Propagation properties of a radial phased-locked partially coherent anomalous hollow beam array in turbulent atmosphere,” Opt. Laser Technol. 57, 44–51 (2014).
[Crossref]

Wang, L.

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. 282(6), 1088–1094 (2009).
[Crossref]

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. 282(6), 1088–1094 (2009).
[Crossref]

Wang, T.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
[Crossref]

Wang, X.

Wang, Y.

Wolf, E.

Wu, G.

Xia, J.

H. Liu, Y. Lü, J. Xia, X. Pu, and L. Zhang, “Flat-topped vortex hollow beam and its propagation properties,” J. Opt. 17(7), 075606 (2015).
[Crossref]

Yang, Y.

Yao, M.

Ye, J.

Yi, X.

Yin, H.

Yu, S.

Yuan, Y.

Y. Yuan, Y. Cai, H. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Zhan, Q.

Zhang, B.

Zhang, L.

H. Liu, Y. Lü, J. Xia, X. Pu, and L. Zhang, “Flat-topped vortex hollow beam and its propagation properties,” J. Opt. 17(7), 075606 (2015).
[Crossref]

Zhang, Y.

Zhao, C.

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

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]

K. Wang and C. Zhao, “Propagation properties of a radial phased-locked partially coherent anomalous hollow beam array in turbulent atmosphere,” Opt. Laser Technol. 57, 44–51 (2014).
[Crossref]

Y. Yang, Y. Dong, C. Zhao, and Y. Cai, “Generation and propagation of an anomalous vortex beam,” Opt. Lett. 38(24), 5418–5421 (2013).
[Crossref] [PubMed]

Zhao, D.

M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23(25), 32766–32776 (2015).
[Crossref] [PubMed]

M. Tang and D. Zhao, “Spectral changes in stochastic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312(1), 89–93 (2014).
[Crossref]

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[Crossref]

X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93(4), 901–905 (2008).
[Crossref]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
[Crossref] [PubMed]

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
[Crossref] [PubMed]

Zhao, G.

Zhou, G.

Y. Ni and G. Zhou, “Propagation of a Lorentz–Gauss vortex beam through a paraxial ABCD optical system,” Opt. Commun. 291(15), 19–25 (2013).
[Crossref]

G. Zhou, “Propagation of a radial phased-locked Lorentz beam array in turbulent atmosphere,” Opt. Express 19(24), 24699–24711 (2011).
[Crossref] [PubMed]

Zhou, P.

Zhu, S.

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. 282(6), 1088–1094 (2009).
[Crossref]

Zhu, Y.

Appl. Opt. (1)

Appl. Phys. B (1)

X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93(4), 901–905 (2008).
[Crossref]

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
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Chin. Phys. B (1)

X. He and B. Lü, “Propagation properties of partially coherent Hermite-Gaussian beams through non-Kolmogorov turbulence,” Chin. Phys. B 20(9), 244–252 (2011).
[Crossref]

J. Opt. (2)

H. Liu, Y. Lü, J. Xia, X. Pu, and L. Zhang, “Flat-topped vortex hollow beam and its propagation properties,” J. Opt. 17(7), 075606 (2015).
[Crossref]

O. Korotkova and E. Shchepakina, “Rectangular Multi-Gaussian Schell-Model beams in atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
[Crossref]

J. Opt. Soc. Am. A (8)

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

Opt. Commun. (6)

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

M. Tang and D. Zhao, “Spectral changes in stochastic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312(1), 89–93 (2014).
[Crossref]

Y. Ni and G. Zhou, “Propagation of a Lorentz–Gauss vortex beam through a paraxial ABCD optical system,” Opt. Commun. 291(15), 19–25 (2013).
[Crossref]

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. 282(6), 1088–1094 (2009).
[Crossref]

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[Crossref]

X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283(18), 3398–3403 (2010).
[Crossref]

Opt. Eng. (1)

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
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Opt. Express (10)

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
[Crossref] [PubMed]

G. Zhou, “Propagation of a radial phased-locked Lorentz beam array in turbulent atmosphere,” Opt. Express 19(24), 24699–24711 (2011).
[Crossref] [PubMed]

J. He, X. Wang, D. Hu, J. Ye, S. Feng, Q. Kan, and Y. Zhang, “Generation and evolution of the terahertz vortex beam,” Opt. Express 21(17), 20230–20239 (2013).
[Crossref] [PubMed]

Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008).
[Crossref] [PubMed]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
[Crossref] [PubMed]

M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23(25), 32766–32776 (2015).
[Crossref] [PubMed]

E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010).
[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).
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Y. Huang, B. Zhang, Z. Gao, G. Zhao, and Z. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
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X. Wang, M. Yao, Z. Qiu, X. Yi, and Z. Liu, “Evolution properties of Bessel-Gaussian Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 23(10), 12508–12523 (2015).
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Opt. Laser Technol. (1)

K. Wang and C. Zhao, “Propagation properties of a radial phased-locked partially coherent anomalous hollow beam array in turbulent atmosphere,” Opt. Laser Technol. 57, 44–51 (2014).
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Opt. Lasers Eng. (1)

Y. Yuan, Y. Cai, H. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Opt. Lett. (8)

Phys. Lett. A (1)

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
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Proc. SPIE (3)

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109 (2002).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
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Other (1)

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

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Figures (14)

Fig. 1
Fig. 1 Intensity distribution of a radial phased-locked flat-topped beam vortex array in the source plane.
Fig. 2
Fig. 2 Contour graphs of normalized intensity of a radial phased-locked partially coherent flat-topped vortex beam array with several different propagation distances through non-Kolmogorov atmospheric turbulence. r = 2 cm. (a) z = 0.2 km (b) z = 0.3 km (c) z = 0.4 km (d) z = 0.5 km.
Fig. 3
Fig. 3 Contour graphs of normalized intensity of a radial phased-locked partially coherent flat-topped vortex beam array with several different propagation distances through non-Kolmogorov atmospheric turbulence. r = 3 cm. (a) z = 0.2 km (b) z = 0.3 km (c) z = 0.8 km(d) z = 1 km.
Fig. 4
Fig. 4 Contour graphs of normalized intensity of a radial phased-locked partially coherent flat-topped vortex beam array with several different propagation distances through non-Kolmogorov atmospheric turbulence. r = 5 cm. (a) z = 0.3 km (b) z = 0.5 km (c) z = 0.8 km (d) z = 1.35 km.
Fig. 5
Fig. 5 Normalized intensity in y-direction at different propagation distance z.
Fig. 6
Fig. 6 Contour graphs of normalized intensity of a radial phased-locked partially coherent flat-topped vortex beam array with several different structure constants through non-Kolmogorov atmospheric turbulence. r = 5cm, z = 0.8km (a) Cn2 = 10−12m3-α (b) Cn2 = 5 × 10−12m3-α(c) Cn2 = 10−11m3-α (d) Cn2 = 10−10m3-α.
Fig. 7
Fig. 7 Contour graphs of normalized intensity of a radial phased-locked partially coherent flat-topped vortex beam array with different beamlet numbers through non-Kolmogorov atmospheric turbulence. r = 3cm, z = 0.2km (a) M = 6 (b) M = 8 (c) M = 10 (d) M = 12.
Fig. 8
Fig. 8 Normalized intensity in y-direction for different values of σ0 at the different propagation distance z.(a)z = 500m (b)z = 1000m.
Fig. 9
Fig. 9 On-axis normalized intensity of a radial phased-locked partially coherent flat-topped vortex beam array through non-Kolmogorov turbulence versus generalized exponent factor α for different values of N.
Fig. 10
Fig. 10 K versus α for different propagation distance z.
Fig. 11
Fig. 11 The y-direction intensity of a radial phased-locked partially coherent flat-topped vortex beam array through non-Kolmogorov turbulence for different exponent α, beamlet number M, and propagation distance z.
Fig. 12
Fig. 12 The y-direction intensity of a radial phased-locked partially coherent flat-topped vortex beam array through non-Kolmogorov turbulence for different N and exponent α.
Fig. 13
Fig. 13 The y-direction intensity of a radial phased-locked partially coherent flat-topped vortex beam array through non-Kolmogorov turbulence for different exponent α, outer scale L0, and inner scale l0.
Fig. 14
Fig. 14 The complex degree of spatial coherence, (a) at several z values with δ = 5 mm, r = 3cm, Cn2 = 10−14m3-α, (b) at several δ values with r = 3cm, z = 1 km, Cn2 = 10−14m3-α, (c) at several r values with δ = 5 mm, z = 1 km, Cn2 = 10−14m3-α, (d) at several Cn2 values with δ = 5 mm, z = 1 km, r = 3 cm.

Equations (48)

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E m ( ρ x , ρ y ,0)= [ ρ x a mx +isgn( l )( ρ y a my ) ] | l | n=1 N ( 1 ) n1 N ( N n )exp( n ( ρ x a mx ) 2 w 2 n ( ρ y a my ) 2 w )exp( i φ m ).
W m ( 0 ) ( ρ 1 , ρ 2 ,0 )= I m ( ρ 1x , ρ 1y ,0) I m ( ρ 2x , ρ 2y ,0) exp[ ( n 1 + n 2 ) ( ρ 1 ρ 2 ) 2 4 σ 0 2 ].
W ( 0 ) ( ρ 1 , ρ 2 ,0 )= m=1 M W m ( 0 ) ( ρ 1 , ρ 2 ,0 ) .
W ( 0 ) ( ρ 1 , ρ 2 ,0 )={ ρ 1x ρ 2x + ρ 1y ρ 2y a mx ( ρ 1x + ρ 2x ) a my ( ρ 1y + ρ 2y ) + i[ ρ 1x ρ 2y ρ 2x ρ 1y a my ( ρ 1x ρ 2x )+ a mx ( ρ 1y ρ 2y ) ]+ a mx 2 + a my 2 } × m=1 M n 1 =1 N n 2 =1 N ( 1 ) n 1 + n 2 2 N 2 ( N n 1 )( N n 2 ) exp[ n 1 w 2 ( ρ 1x 2 + ρ 1y 2 ) n 2 w 2 ( ρ 2x 2 + ρ 2y 2 ) ] ×exp[ 2 a mx w 2 ( n 1 ρ 1x + n 2 ρ 2x )+ 2 a my w 2 ( n 1 ρ 1y + n 2 ρ 2y ) ]exp( i φ m ) ×exp[ ( a mx 2 + a my 2 )( n 1 + n 2 ) w 2 ]exp[ ( n 1 + n 2 ) 4 σ 0 2 ( ρ 1 ρ 2 ) 2 ].
W( ρ 1 , ρ 2 ,z )= k 2 4 π 2 z 2 + + + + W ( 0 ) ( ρ 1 , ρ 2 ,0 ) ×exp{ ik 2z [ ( ρ 1 ρ 1 ) 2 ( ρ 2 ρ 2 ) 2 ] } × exp[ ψ * ( ρ 1 , ρ 1 )+ψ( ρ 2 , ρ 2 ) ] d 2 ρ 1 d 2 ρ 2 .
exp[ ψ * ( ρ 1 , ρ 1 )+ψ( ρ 2 , ρ 2 ) ] =exp{ 4 π 2 k 2 z 0 1 0 + dκdξ Φ n ( κ,α )[ 1 J 0 ( κ| ( 1ξ )( ρ 1 ρ 2 )+ξ( ρ 1 ρ 2 ) | ) ] } =exp{ π 2 k 2 z 3 0 + κ 3 Φ n ( κ,α )dκ[ ( ρ 1 ρ 2 ) 2 +( ρ 1 ρ 2 )( ρ 1 ρ 2 )+ ( ρ 1 ρ 2 ) 2 ] }.
Φ n ( κ,α )= A( α ) C ˜ n 2 ( κ 2 + κ 0 2 ) α/2 exp( κ 2 κ m 2 ).
K= π 2 k 2 z 3 0 + κ 3 Φ n ( κ,α )dκ= π 2 k 2 zA( α ) C ˜ n 2 6( α2 ) [ βexp( κ 0 2 κ m 2 )Γ( 4α 2 , κ 0 2 κ m 2 ) κ m 2α 2 κ 0 4α ].
W( ρ 1 , ρ 2 ,z )= k 2 4 π 2 z 2 exp[ ik 2z ( ρ 1 2 ρ 2 2 ) ]exp[ K ( x 1 x 2 ) 2 K ( y 1 y 2 ) 2 ] × m=1 M n 1 =1 N n 2 =1 N ( 1 ) n 1 + n 2 2 N 2 ( N n 1 )( N n 2 ) + + + + d 2 u d 2 v exp[ ( a mx 2 + a my 2 )( n 1 + n 2 ) w 2 ] ×[ u v 4 2 a mx u x 2 a my u y +i( v x u y u x v y a my v x + a mx v y )+ a mx 2 + a my 2 ] ×exp[ 2 a mx ( n 1 + n 2 ) w 2 u x + a mx ( n 1 n 2 ) w 2 v x + 2 a my ( n 1 + n 2 ) w 2 u y + a my ( n 1 n 2 ) w 2 v y ] ×exp[ ik( ρ 1 2 ρ 2 2 ) z u ]exp{ [ ik( ρ 1 2 + ρ 2 2 ) 2z K( ρ 1 ρ 2 ) ]v } ×exp( n 1 + n 2 w 2 u 2 )exp( Q v 2 Puv )exp( i φ m ),
P= n 1 n 2 w 2 + ik z ,
Q= n 1 + n 2 4 w 2 + n 1 + n 2 4 σ 0 2 +K.
x n exp( A x 2 +Bx ) dx=n!exp( A 2 B ) π A ( A B ) n m=0 [ n/2 ] 1 m!( n2m )! ( A 4 B 2 ) m .
W( ρ 1 , ρ 2 ,z )= ( k 2πz ) 2 exp[ ik 2z ( ρ 1 2 ρ 2 2 ) ]exp[ K ( x 1 x 2 ) 2 K ( y 1 y 2 ) 2 ] × m=1 M n 1 =1 N n 2 =1 N ( 1 ) n 1 + n 2 2 N 2 ( N n 1 )( N n 2 ) exp( i φ m )exp[ ( a mx 2 + a my 2 )( n 1 + n 2 ) w 2 ] ×{ H 1 ( 1+ 2 C x 2 A )+ H 1 ( 1+ 2 C y 2 A ) H xy 4 ( 1 E + 2 D x 2 E 2 ) H xy 4 ( 1 E + 2 D y 2 E 2 )4 a mx H 1 C x 4 a my H 1 C y +i[ G yx ( D x C y EA ) G xy ( D y C x EA ) a my H xy ( D x E )+ a mx H xy ( D y E ) ]+ a mx 2 H xy + a my 2 H xy },
H 1 = π 2 2 A 2 Q B x B y exp( C x 2 + C y 2 A ),
H xy = π 2 w 2 2E( n 1 + n 2 ) exp{ [ ik( x 1 x 2 ) 2z + 2 a mx ( n 1 + n 2 ) 2 w 2 ] 2 w 2 n 1 + n 2 } ×exp{ [ ik( y 1 y 2 ) 2z + 2 a my ( n 1 + n 2 ) 2 w 2 ] 2 w 2 n 1 + n 2 }exp( D x 2 + D y 2 E ),
G y x = π 2 w AEQ( n 1 + n 2 ) B y exp{ [ ik( x 1 x 2 ) 2z + 2 a mx ( n 1 + n 2 ) 2 w 2 ] 2 w 2 n 1 + n 2 }exp( D x 2 E + C y 2 A ),
G xy π 2 w AEQ( n 1 + n 2 ) B x exp{ [ ik( y 1 y 2 ) 2z + 2 a my ( n 1 + n 2 ) 2 w 2 ] 2 w 2 n 1 + n 2 }exp( D y 2 E + C x 2 A ),
B j =exp[ a 2 mj ( n 1 n 2 ) 2 4Q w 4 ]exp[ ik a mj ( n 1 n 2 ) 4Q w 2 z ( j 1 + j 2 ) ] ×exp[ k 2 16Q z 2 ( j 1 + j 2 ) 2 ]exp[ ikz 4Qz K( j 1 2 j 2 2 ) ] ×exp[ a mj ( n 1 n 2 ) 2Q w 2 K( j 1 j 2 ) ]exp[ 1 4Q K 2 ( j 1 j 2 ) 2 ],
C j = a mj P( n 1 n 2 ) 4Q w 2 ikP( j 1 + j 2 ) 8Qz + P 4Q K( j 1 j 2 )+ ik( j 1 j 2 ) 2z + 2 a mj ( n 1 + n 2 ) 2 w 2 ,
D j = a mj ( n 1 n 2 ) 2 w 2 + ik( j 1 + j 2 ) 4z 1 2 K( j 1 j 2 ) ikP w 2 ( j 1 j 2 ) 4z( n 1 + n 2 ) a mj P 2 ,
A= n 1 + n 2 w 2 P 2 4Q ,
E=Q P 2 w 2 4( n 1 + n 2 ) .
I( ρ,z )=W( ρ,ρ,z )=W( x,y,x,y,z ) = ( k 2πz ) 2 m=1 M n 1 =1 N n 2 =1 N ( 1 ) n 1 + n 2 2 N 2 ( N n 1 )( N n 2 ) exp( i φ m )exp[ ( a mx 2 + a my 2 )( n 1 + n 2 ) w 2 ] ×{ 2 H 1 ( 1+ C x 2 + C y 2 A ) H xy 2 ( 1 E + D x 2 + D y 2 E 2 )4 H 1 ( a mx C x + a my C y ) + i[ ( G yx D x C y G xy D y C x EA )+ H xy E ( a mx D y a my D x ) ]+ H xy ( a mx 2 + a my 2 ) },
C j = a mj P( n 1 n 2 ) 4Q w 2 ikP 4Qz j+ a mj ( n 1 + n 2 ) w 2 ,
D j = a mj ( n 1 n 2 ) 2 w 2 + ik 2z j a mj P 2 ,
H xy = π 2 w 2 2E( n 1 + n 2 ) exp[ ( n 1 + n 2 ) w 2 ( a mx 2 + a my 2 ) ]exp( D x 2 + D y 2 E ),
G yx = π 2 w AEQ( n 1 + n 2 ) B y exp[ a mx 2 ( n 1 + n 2 ) w 2 ]exp( D x 2 E + C y 2 A ),
G xy = π 2 w AEQ( n 1 + n 2 ) B x exp[ a my 2 ( n 1 + n 2 ) w 2 ]exp( D y 2 E + C x 2 A ),
H 1 = π 2 2 A 2 Q B x B y exp( C x 2 + C y 2 A ),
B j =exp[ a 2 mj ( n 1 n 2 ) 2 4Q w 4 ]exp[ ik a mj ( n 1 n 2 ) 2Q w 2 z j ]exp[ k 2 4Q z 2 j 2 ].
μ( x 1 , x 2 ,z )= W( x 1 , x 2 ,z ) I( x 1 ,z )I( x 2 ,z ) .
μ( x,x,z )= Ω 1 Ω 2 Ω 3 ,
Ω p = m=1 M n 1 =1 N n 2 =1 N ( 1 ) n 1 + n 2 2 N 2 ( N n 1 )( N n 2 ) exp( i φ m )exp[ ( a mx 2 + a my 2 )( n 1 + n 2 ) w 2 ] ×{ 2 H p1 ( 1+ C px 2 + C 2 A ) H pxy 2E ( 1+ D px 2 + D 2 E )4 H p1 ( a mx C px + a my C ) + i[ ( G pyx D x C G pxy D C px EA )+ H pxy E ( a mx D a my D px ) ]+ H pxy ( a mx 2 + a my 2 ) },
H p1 = π 2 2 A 2 Q B px exp[ a 2 my ( n 1 n 2 ) 2 4Q w 4 ]exp( C px 2 + C 2 A ),
H 1xy = π 2 w 2 2E( n 1 + n 2 ) exp{ [ ik z x+ 2 a mx ( n 1 + n 2 ) 2 w 2 ] 2 w 2 n 1 + n 2 }exp[ a my 2 ( n 1 + n 2 ) w 2 ]exp( D 1x 2 + D 2 E ),
G 1y x = π 2 w AEQ( n 1 + n 2 ) exp[ a 2 my ( n 1 n 2 ) 2 4Q w 4 ]exp{ [ ikx z + 2 a mx ( n 1 + n 2 ) 2 w 2 ] 2 w 2 n 1 + n 2 }exp( D 1x 2 E + C 2 A ),
G 1xy π 2 w AEQ( n 1 + n 2 ) B 1x exp[ a my 2 ( n 1 + n 2 ) w 2 ]exp( D 2 E + C 1x 2 A ),
B 1x =exp[ a 2 mx ( n 1 n 2 ) 2 4Q w 4 ]exp[ a mx K( n 1 n 2 ) Q w 2 x ]exp( K 2 Q x 2 ),
C 1x = a mx P( n 1 n 2 ) 4Q w 2 + 2 a mx ( n 1 + n 2 ) 2 w 2 ( PK 2Q + ik z )x,
D 1x = a mx ( n 1 n 2 ) 2 w 2 a mx P 2 +[ K+ ikP w 2 2z( n 1 + n 2 ) ]x,
H qxy = π 2 w 2 2E( n 1 + n 2 ) exp[ ( n 1 + n 2 ) w 2 ( a mx 2 + a my 2 ) ]exp( D qx 2 + D 2 E ),
G qy x = π 2 w AEQ( n 1 + n 2 ) exp[ a 2 my ( n 1 n 2 ) 2 4Q w 4 ]exp[ a mx 2 ( n 1 + n 2 ) w 2 ]exp( D qx 2 E + C 2 A ),
G qxy = π 2 w AEQ( n 1 + n 2 ) B qx exp[ a my 2 ( n 1 + n 2 ) w 2 ]exp( D 2 E + C qx 2 A ),
B qx =exp[ a 2 mx ( n 1 n 2 ) 2 4Q w 4 ]exp[ ik a mx ( n 1 n 2 ) 2Q w 2 z x ]exp[ ( 1 ) q k 2 4Q z 2 x 2 ],
C qx = a mx P( n 1 n 2 ) 4Q w 2 + 2 a mx ( n 1 + n 2 ) 2 w 2 + ( 1 ) q ikP 4Qz x,
D qx = a mx ( n 1 n 2 ) 2 w 2 a mx P 2 + ( 1 ) q+1 ik 2z x,
C= a my P( n 1 n 2 ) 4Q w 2 + 2 a my ( n 1 + n 2 ) 2 w 2 ,
D= a my ( n 1 n 2 ) 2 w 2 a my P 2 .

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