Abstract

The optical coherent lattice (OCL) with periodic reciprocity has been previously proposed for free-space information transfer and optical communications. Here, a new class of partially coherent radially and azimuthally polarized rotating elliptical Gaussian optical coherent lattice (PCRPREGOCL and PCAPREGOCL) is introduced. Based on the extended Huygens-Fresnel principle and the spatial power spectrum of the anisotropic ocean turbulence, the analytical expressions of the average intensity of the PCRPREGOCL and the PCAPREGOCL through the anisotropic ocean turbulence are obtained. The effects of elliptical coefficients, lattice constants, the number of lattice lobes, wavelengths and anisotropic ocean turbulence parameters on the statistical properties of the PCRPREGOCL and the PCAPREGOCL are studied in detail. It is found that each sub-pattern in the PCRPREGOCL maintains a controllable rotation within a certain distance, which plays an important role in resisting the influence of turbulence. When the propagation distance increases, the PCRPREGOCL and the PCAPREGOCL gradually change from two elliptical Gaussian patterns into a coherent array with periodic reciprocity and eventually evolves into a Gaussian-like pattern. Our work provides new thoughts in applying OCL to overcome turbulence influence in underwater optical communication and underwater laser radar.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

2018 (8)

X. Huang, Z. Deng, X. Shi, Y. Bai, and X. Fu, “Average intensity and beam quality of optical coherence lattices in oceanic turbulence with anisotropy,” Opt. Express 26(4), 4786–4797 (2018).
[Crossref]

J. Zhang, J. Xie, and D. Deng, “Second-order statistics of a partially coherent electromagnetic rotating elliptical Gaussian vortex beam through non-Kolmogorov turbulence,” Opt. Express 26(16), 21249–21257 (2018).
[Crossref]

C. Clivati, A. Tampellini, A. Mura, F. Levi, G. Marra, P. Galea, A. Xuereb, and D. Calonico, “Optical frequency transfer over submarine fiber links,” Optica 5(8), 893–901 (2018).
[Crossref]

J. B. Zhang, J. T. Xie, F. Ye, K. Z. Zhou, X. Y. Chen, D. M. Deng, and X. B. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B: Lasers Opt. 124(8), 168 (2018).
[Crossref]

C. Wu, J. Ko, J. R. Rzasa, D. A. Paulson, and C. C. Davis, “Phase and amplitude beam shaping with two deformable mirrors implementing input plane and Fourier plane phase modifications,” Appl. Opt. 57(9), 2337–2345 (2018).
[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

2017 (3)

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]

X. Huang, Y. Bai, and X. Fu, “Propagation factors of partially coherent model beams in oceanic turbulence,” IEEE Photonics J. 9(5), 1–11 (2017).
[Crossref]

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

2016 (4)

Y. Li, Y. Zhang, Y. Zhu, and M. Chen, “Effects of anisotropic turbulence on average polarizability of Gaussian Schell-model quantized beams through ocean link,” Appl. Opt. 55(19), 5234–5239 (2016).
[Crossref]

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
[Crossref]

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

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

2015 (5)

2013 (2)

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

J. M. Aunñón and M. Nieto-Vesperinas, “Partially coherent fluctuating sources that produce the same optical force as a laser beam,” Opt. Lett. 38(15), 2869–2872 (2013).
[Crossref]

2012 (2)

J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3(1), 993 (2012).
[Crossref]

S. Yu, Z. Chen, T. Wang, G. Wu, H. Guo, and W. Gu, “Beam wander of electromagnetic Gaussian-Schell model beams propagating in atmospheric turbulence,” Appl. Opt. 51(31), 7581–7585 (2012).
[Crossref]

2011 (1)

Z. Chen and Q. Guo, “Rotation of elliptic optical beams in anisotropic media,” Opt. Commun. 284(13), 3183–3191 (2011).
[Crossref]

2010 (3)

2009 (1)

2008 (2)

M. H. Mahdieh, “Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor,” Opt. Commun. 281(13), 3395–3402 (2008).
[Crossref]

X. Xiao, X. Ji, and B. Lü, “The influence of turbulence on propagation properties of partially coherent sinh-Gaussian beams and their beam quality in the far field,” Opt. Laser Technol. 40(1), 129–136 (2008).
[Crossref]

2007 (1)

2006 (2)

2005 (2)

M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005).
[Crossref]

T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, “Coherence controlled soliton interactions,” Phys. Rev. Lett. 94(6), 063904 (2005).
[Crossref]

2003 (2)

A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28(1), 10–12 (2003).
[Crossref]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref]

2002 (2)

C. Monroe, “Quantum information processing with atoms and photons,” Nature 416(6877), 238–246 (2002).
[Crossref]

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

2000 (2)

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the seawater refraction index,” Inter J. Fluid Mech. Res. 27(1), 82–98 (2000).
[Crossref]

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[Crossref]

1993 (1)

1991 (1)

A. M. Goncharenko, Y. A. Logvin, A. M. Samson, and P. S. Shapovalov, “Rotating elliptical gaussian beams in nonlinear media,” Opt. Commun. 81(3-4), 225–230 (1991).
[Crossref]

1990 (1)

1978 (1)

R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. A 68(8), 1067–1072 (1978).
[Crossref]

Abramochkin, E.

Alieva, T.

Amarande, S.

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Medium, 2nd ed. (SPIE, Washington, 2005).

Asenjo-Garcia, A.

Aunñón, J. M.

Bai, Y.

X. Huang, Z. Deng, X. Shi, Y. Bai, and X. Fu, “Average intensity and beam quality of optical coherence lattices in oceanic turbulence with anisotropy,” Opt. Express 26(4), 4786–4797 (2018).
[Crossref]

X. Huang, Y. Bai, and X. Fu, “Propagation factors of partially coherent model beams in oceanic turbulence,” IEEE Photonics J. 9(5), 1–11 (2017).
[Crossref]

Baykal, Y.

Bekshaev, A.

Cai, Y.

Calonico, D.

Chandrasekharan, H. K.

S. Mukherjee, H. K. Chandrasekharan, P. Öhberg, N. Goldman, and R. R. Thomson, “State-recycling and time-resolved imaging in topological photonic lattices,” Nat. Commun. 9(1), 4209 (2018).
[Crossref]

Charnotskii, M. I.

Chen, M.

Chen, X. Y.

J. B. Zhang, J. T. Xie, F. Ye, K. Z. Zhou, X. Y. Chen, D. M. Deng, and X. B. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B: Lasers Opt. 124(8), 168 (2018).
[Crossref]

Chen, Y.

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

Chen, Z.

Clark, J.

J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3(1), 993 (2012).
[Crossref]

Clivati, C.

Davis, C. C.

Deng, D.

Deng, D. M.

C. Sun, X. Lv, B. B. Ma, J. B. Zhang, D. M. Deng, and W. Y. Hong, “Statistical properties of partially coherent radially and azimuthally polarized rotating elliptical Gaussian beams in oceanic turbulence with anisotropy,” Opt. Express 27(8), A245–A256 (2019).
[Crossref]

J. B. Zhang, J. T. Xie, F. Ye, K. Z. Zhou, X. Y. Chen, D. M. Deng, and X. B. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B: Lasers Opt. 124(8), 168 (2018).
[Crossref]

Deng, Z.

Dholakia, K.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref]

Dogariu, A.

Duan, Z.

Durst, F.

Eyyuboglu, H. T.

Fu, W.

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

Fu, X.

X. Huang, Z. Deng, X. Shi, Y. Bai, and X. Fu, “Average intensity and beam quality of optical coherence lattices in oceanic turbulence with anisotropy,” Opt. Express 26(4), 4786–4797 (2018).
[Crossref]

X. Huang, Y. Bai, and X. Fu, “Propagation factors of partially coherent model beams in oceanic turbulence,” IEEE Photonics J. 9(5), 1–11 (2017).
[Crossref]

Galea, P.

Gao, Z.

Gbur, G.

Goldman, N.

S. Mukherjee, H. K. Chandrasekharan, P. Öhberg, N. Goldman, and R. R. Thomson, “State-recycling and time-resolved imaging in topological photonic lattices,” Nat. Commun. 9(1), 4209 (2018).
[Crossref]

Golmohammady, S.

Goncharenko, A. M.

A. M. Goncharenko, Y. A. Logvin, A. M. Samson, and P. S. Shapovalov, “Rotating elliptical gaussian beams in nonlinear media,” Opt. Commun. 81(3-4), 225–230 (1991).
[Crossref]

Gu, W.

Gu, Y.

Guo, H.

Guo, Q.

Z. Chen and Q. Guo, “Rotation of elliptic optical beams in anisotropic media,” Opt. Commun. 284(13), 3183–3191 (2011).
[Crossref]

Harder, R.

J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3(1), 993 (2012).
[Crossref]

He, J.

He, S.

Hecht, B.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[Crossref]

Heller, E. J.

S. N. Sanders, F. Mintert, and E. J. Heller, “Matter-wave scattering from ultracold atoms in an optical lattice,” Phys. Rev. Lett. 105(3), 035301 (2010).
[Crossref]

Hill, R. J.

R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. A 68(8), 1067–1072 (1978).
[Crossref]

Hong, W. Y.

Hu, M.

Hu, Z.

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

Huang, X.

X. Huang, Z. Deng, X. Shi, Y. Bai, and X. Fu, “Average intensity and beam quality of optical coherence lattices in oceanic turbulence with anisotropy,” Opt. Express 26(4), 4786–4797 (2018).
[Crossref]

X. Huang, Y. Bai, and X. Fu, “Propagation factors of partially coherent model beams in oceanic turbulence,” IEEE Photonics J. 9(5), 1–11 (2017).
[Crossref]

J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3(1), 993 (2012).
[Crossref]

Huang, Y.

Ji, X.

X. Ji, H. T. Eyyuboglu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial gaussian array beams,” Opt. Express 18(7), 6922–6928 (2010).
[Crossref]

X. Xiao, X. Ji, and B. Lü, “The influence of turbulence on propagation properties of partially coherent sinh-Gaussian beams and their beam quality in the far field,” Opt. Laser Technol. 40(1), 129–136 (2008).
[Crossref]

Jin, Y.

Kahn, J. M.

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

Kashani, F. D.

Kivshar, Y. S.

T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, “Coherence controlled soliton interactions,” Phys. Rev. Lett. 94(6), 063904 (2005).
[Crossref]

Ko, J.

Korotkova, O.

Ku, T. S.

T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, “Coherence controlled soliton interactions,” Phys. Rev. Lett. 94(6), 063904 (2005).
[Crossref]

Levi, F.

Li, Y.

Y. Li, Y. Zhang, Y. Zhu, and M. Chen, “Effects of anisotropic turbulence on average polarizability of Gaussian Schell-model quantized beams through ocean link,” Appl. Opt. 55(19), 5234–5239 (2016).
[Crossref]

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

Liang, C.

Liu, D.

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

Liu, X.

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
[Crossref]

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]

Logvin, Y. A.

A. M. Goncharenko, Y. A. Logvin, A. M. Samson, and P. S. Shapovalov, “Rotating elliptical gaussian beams in nonlinear media,” Opt. Commun. 81(3-4), 225–230 (1991).
[Crossref]

Lü, B.

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B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
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S. Mukherjee, H. K. Chandrasekharan, P. Öhberg, N. Goldman, and R. R. Thomson, “State-recycling and time-resolved imaging in topological photonic lattices,” Nat. Commun. 9(1), 4209 (2018).
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Sanders, S. N.

S. N. Sanders, F. Mintert, and E. J. Heller, “Matter-wave scattering from ultracold atoms in an optical lattice,” Phys. Rev. Lett. 105(3), 035301 (2010).
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T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, “Coherence controlled soliton interactions,” Phys. Rev. Lett. 94(6), 063904 (2005).
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B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
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M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
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T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, “Coherence controlled soliton interactions,” Phys. Rev. Lett. 94(6), 063904 (2005).
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S. Mukherjee, H. K. Chandrasekharan, P. Öhberg, N. Goldman, and R. R. Thomson, “State-recycling and time-resolved imaging in topological photonic lattices,” Nat. Commun. 9(1), 4209 (2018).
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J. B. Zhang, J. T. Xie, F. Ye, K. Z. Zhou, X. Y. Chen, D. M. Deng, and X. B. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B: Lasers Opt. 124(8), 168 (2018).
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Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
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Zhou, K. Z.

J. B. Zhang, J. T. Xie, F. Ye, K. Z. Zhou, X. Y. Chen, D. M. Deng, and X. B. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B: Lasers Opt. 124(8), 168 (2018).
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Appl. Opt. (4)

Appl. Phys. B: Lasers Opt. (1)

J. B. Zhang, J. T. Xie, F. Ye, K. Z. Zhou, X. Y. Chen, D. M. Deng, and X. B. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B: Lasers Opt. 124(8), 168 (2018).
[Crossref]

Appl. Phys. Lett. (1)

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

IEEE Photonics J. (1)

X. Huang, Y. Bai, and X. Fu, “Propagation factors of partially coherent model beams in oceanic turbulence,” IEEE Photonics J. 9(5), 1–11 (2017).
[Crossref]

IEEE Trans. Commun. (1)

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

Inter J. Fluid Mech. Res. (1)

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the seawater refraction index,” Inter J. Fluid Mech. Res. 27(1), 82–98 (2000).
[Crossref]

J. Opt. Soc. Am. A (5)

Laser Phys. (1)

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

Nat. Commun. (2)

J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3(1), 993 (2012).
[Crossref]

S. Mukherjee, H. K. Chandrasekharan, P. Öhberg, N. Goldman, and R. R. Thomson, “State-recycling and time-resolved imaging in topological photonic lattices,” Nat. Commun. 9(1), 4209 (2018).
[Crossref]

Nature (2)

C. Monroe, “Quantum information processing with atoms and photons,” Nature 416(6877), 238–246 (2002).
[Crossref]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref]

Opt. Commun. (5)

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

Z. Chen and Q. Guo, “Rotation of elliptic optical beams in anisotropic media,” Opt. Commun. 284(13), 3183–3191 (2011).
[Crossref]

A. M. Goncharenko, Y. A. Logvin, A. M. Samson, and P. S. Shapovalov, “Rotating elliptical gaussian beams in nonlinear media,” Opt. Commun. 81(3-4), 225–230 (1991).
[Crossref]

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

M. H. Mahdieh, “Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor,” Opt. Commun. 281(13), 3395–3402 (2008).
[Crossref]

Opt. Express (9)

M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005).
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Y. Huang, F. Wang, Z. Gao, and B. Zhang, “Propagation properties of partially coherent electromagnetic hyperbolic-sine-Gaussian vortex beams through non-Kolmogorov turbulence,” Opt. Express 23(2), 1088–1102 (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]

T. Alieva, E. Abramochkin, A. Asenjo-Garcia, and E. Razueva, “Rotating beams in isotropic optical system,” Opt. Express 18(4), 3568–3573 (2010).
[Crossref]

X. Ji, H. T. Eyyuboglu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial gaussian array beams,” Opt. Express 18(7), 6922–6928 (2010).
[Crossref]

J. Zhang, J. Xie, and D. Deng, “Second-order statistics of a partially coherent electromagnetic rotating elliptical Gaussian vortex beam through non-Kolmogorov turbulence,” Opt. Express 26(16), 21249–21257 (2018).
[Crossref]

C. Sun, X. Lv, B. B. Ma, J. B. Zhang, D. M. Deng, and W. Y. Hong, “Statistical properties of partially coherent radially and azimuthally polarized rotating elliptical Gaussian beams in oceanic turbulence with anisotropy,” Opt. Express 27(8), A245–A256 (2019).
[Crossref]

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]

X. Huang, Z. Deng, X. Shi, Y. Bai, and X. Fu, “Average intensity and beam quality of optical coherence lattices in oceanic turbulence with anisotropy,” Opt. Express 26(4), 4786–4797 (2018).
[Crossref]

Opt. Laser Technol. (1)

X. Xiao, X. Ji, and B. Lü, “The influence of turbulence on propagation properties of partially coherent sinh-Gaussian beams and their beam quality in the far field,” Opt. Laser Technol. 40(1), 129–136 (2008).
[Crossref]

Opt. Lett. (7)

Optica (2)

Phys. Rev. Lett. (3)

T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, “Coherence controlled soliton interactions,” Phys. Rev. Lett. 94(6), 063904 (2005).
[Crossref]

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000).
[Crossref]

S. N. Sanders, F. Mintert, and E. J. Heller, “Matter-wave scattering from ultracold atoms in an optical lattice,” Phys. Rev. Lett. 105(3), 035301 (2010).
[Crossref]

Prog. Electromagnetics Res. (1)

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]

Other (3)

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

G. Z. Xu, Free Space Optical Communication (SXET, 2002).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Medium, 2nd ed. (SPIE, Washington, 2005).

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Figures (9)

Fig. 1.
Fig. 1. The spectral intensity distribution of the PCRPREGOCL for different propagation distances: (a) $z=0Z_R$, (b) $z=0.003Z_R$, (c) $z=0.02Z_R$, (d) $z=0.3Z_R$, (e) $z=1.5Z_R$, (f) $z=6Z_R$.
Fig. 2.
Fig. 2. The evolution of the spectral intensity for the PCRPREGOCL with $d=0.1$ at different propagation distances in the anisotropic ocean turbulence: (a) $z=0Z_R$, (b) $z=0.02Z_R$, (c) $z=1Z_R$, (d) $z=5Z_R$.
Fig. 3.
Fig. 3. The evolution of the spectral intensity of the PCRPREGOCL with different elliptical coefficients $c$ and lattice constants $d$ in the anisotropic ocean turbulence at different propagation distances: (a1)-(d1) $z=0.01Z_R$, (a2)-(d2) $z=1.2Z_R$ and with different elliptical coefficients: (a1) and (a2) $c=0.01$, (b1) and (b2) $c=0.06$, (c1) and (c2) $c=0.1$, (d1) and (d2) $c=0.5$. (a3)-(d3) $z=0.01Z_R$, (a4)-(d4) $z=0.5Z_R$ and with different lattice constants: (a3) and (a4) $c=0.005$, (b3) and (b4) $d=0.01$, (c3) and (c4) $d=0.05$, (d3) and (d4) $d=0.2$.
Fig. 4.
Fig. 4. The spectral intensity of the PCRPREGOCL in the anisotropic ocean turbulence with different $\varepsilon$ in (a), $\chi _{T}$ in (b), $\eta$ in (c), $\omega$ in (d) and $\xi$ in (e) at $z=4Z_R$ .
Fig. 5.
Fig. 5. The normalized spectral density distribution of the PCRPREGOCL as a function of $x$ through the anisotropic oceanic turbulence with different $z$ in (a), $\chi _{T}$ in (b), $w_{0}$ in (c), $\lambda$ in (d), $\sigma _{0}$ in (e) and $\varepsilon$ in (f) at $z=100m$ except (a).
Fig. 6.
Fig. 6. The spectral DOC of the PCRPREGOCL through the anisotropic oceanic turbulence, (a) versus $x$ with different $\lambda$ at $z=50m$, (b) versus $z$ with different $\lambda$, (c) versus $\omega$ with different $\xi$ at $z=50m$, (d) versus $\omega$ with different $\varepsilon$ at $z=50m$, (e) versus $\omega$ with different $z$ and (f) versus $z$ with different $\xi$.
Fig. 7.
Fig. 7. The spectral DOP of the PCRPREGOCL and the corresponding cross line $(y'=0)$ with different the number of lattice lobes $N_{0}$ in the anisotropic ocean turbulence at different propagation distances: (a1)-(f1) the corresponding cross line $(y'=0)$ with the number of lattice lobes $N_{0}=0$ for different values of $z$, (a2)-(f2) the spectral DOP with $N_{0}=0$ for different values of $z$, (a3)-(f3) the spectral DOP with $N_{0}=5$ for different values of $z$.
Fig. 8.
Fig. 8. Cross lines $(y'=0)$ of the DOP of the PCRPREGOCL for difference values of $\omega$, $\lambda$ and $\varepsilon$: (a)-(c) with $N_{0}=0$, (d)-(f) with $N_{0}=5$.
Fig. 9.
Fig. 9. The SR of the PCRPREGOCL through the anisotropic ocean turbulence from $0$ to $100m$ with different $c$ in (a), $\chi _{T}$ in (b), $\omega$ in (c) and $N_{0}$ in (d).

Equations (25)

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W r ( r 1 , r 2 , 0 ) = A 0 2 V n x V n y π n x = 0 N 0 n y = 0 N 0 exp [ 2 π i n x ( x 1 x 2 ) d w 0 2 π i n y ( y 1 y 2 ) d w 0 x 1 2 + x 2 2 a 2 w 0 2 y 1 2 + y 2 2 b 2 w 0 2 + i ( x 2 y 2 x 1 y 1 ) c 2 w 0 2 ] ( x 1 x 2 x 1 y 2 y 1 x 2 y 1 y 2 ) exp [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ 0 2 ] ,
W θ ( r 1 , r 2 , 0 ) = A 0 2 V n x V n y π n x = 0 N 0 n y = 0 N 0 exp [ 2 π i n x ( x 1 x 2 ) d w 0 2 π i n y ( y 1 y 2 ) d w 0 x 1 2 + x 2 2 a 2 w 0 2 y 1 2 + y 2 2 b 2 w 0 2 + i ( x 2 y 2 x 1 y 1 ) c 2 w 0 2 ] ( y 1 y 2 y 1 x 2 x 1 y 2 x 1 x 2 ) exp [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ 0 2 ] ,
W α β ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 W α β ( r 1 , r 2 , 0 ) ψ ( r 1 , ρ 1 ) + ψ ( r 2 , ρ 2 ) × exp { i k 2 z [ ( r 1 ρ 1 ) 2 ( r 2 ρ 2 ) 2 ] } d 2 r 1 d 2 r 2 ,
ψ ( r 1 , ρ 1 ) + ψ ( r 2 , ρ 2 ) = exp [ ( r 1 r 2 ) 2 + ( r 1 r 2 ) ( ρ 1 ρ 2 ) + ( ρ 1 ρ 2 ) 2 ρ o c ξ 2 ] .
ρ o c ξ 2 = π 2 k 2 z ξ 4 3 0 κ 3 ψ ~ a n ( κ ) d κ ,
ψ ~ a n ( κ ) = 0.388 × 10 8 ε 1 / 3 χ T ξ 2 ( κ ) 11 / 3 [ 1 + 2.35 ( κ η ) 2 / 3 ] × [ exp ( A T δ ) + ω 2 exp ( A S δ ) 2 ω 1 exp ( A T S δ ) ] ,
ρ o c ξ = ξ | ω | [ 1.802 × 10 7 k 2 z ( ε η ) 1 / 3 χ T ( 0.483 ω 2 0.835 ω + 3.380 ) ] 1 / 2 ,
x n exp ( p x 2 + q x ) d x = n ! exp ( q 2 p ) ( q p ) n π p l = 0 n / 2 1 l ! ( n 2 l ) ! ( q 2 4 p ) l ,
W r x x ( ρ 1 , ρ 2 , z ) = n x = 0 N 0 n y = 0 N 0 G 2 L 4 ( q 3 + q 3 Q 2 2 L 4 + q 4 Q + 2 q 5 L 4 ) exp ( M 2 4 L 1 + N 2 4 L 2 + P 2 4 L 3 + Q 2 4 L 4 ) ,
W r x y ( ρ 1 , ρ 2 , z ) = n x = 0 N 0 n y = 0 N 0 G 2 L 4 ( q 6 + q 6 Q 2 2 L 4 + q 7 Q ) exp ( M 2 4 L 1 + N 2 4 L 2 + P 2 4 L 3 + Q 2 4 L 4 ) ,
W r y x ( ρ 1 , ρ 2 , z ) = W r x y ( ρ 1 , ρ 2 , z ) ,
W r y y ( ρ 1 , ρ 2 , z ) = n x = 0 N 0 n y = 0 N 0 G 2 L 4 ( q 8 + q 8 Q 2 2 L 4 + q 9 Q ) exp ( M 2 4 L 1 + N 2 4 L 2 + P 2 4 L 3 + Q 2 4 L 4 ) ,
T = 1 ρ o c ξ 2 , G = A 0 2 k 2 V n x V n y 4 π z 2 w 0 2 L 1 L 2 L 3 L 4 exp [ i k 2 z ( ρ 1 2 ρ 2 2 ) T ( ρ 1 ρ 2 ) 2 ] ,
u = T + 1 2 σ 0 2 , v = i c 2 w 0 2 , L 1 = i k 2 z + 1 a 2 w 2 + u , M = 2 i π n x d w + i k x 1 z + T ( x 2 x 1 ) ,
L 2 = 1 b 2 w 2 + i k 2 z + u v 2 4 L 1 , N = 2 i π n y d w + i k y 1 z + T ( y 2 y 1 ) v M 2 L 1 , R = 1 u 2 L 1 L 2 ,
L 3 = 1 a 2 w 2 i k 2 z + u u 2 L 1 v 2 u 2 4 L 1 2 L 2 , P = 2 i π n x d w i k x 2 z + T ( x 1 x 2 ) + u M L 1 u v N 2 L 1 L 2 ,
L 4 = 1 b 2 w 2 i k 2 z + u u 2 L 2 v 2 R 2 4 L 3 , Q = 2 i π n y d w i k y 2 z + T ( y 1 y 2 ) + u N L 2 + v P R 2 L 3 ,
q 1 = M 2 L 1 , q 4 = q 2 v P R 2 L 3 2 + q 1 v R 2 L 3 v u P 4 L 1 L 2 L 3 , q 7 = q 2 P 2 L 3 + q 1 ,
q 2 = u L 1 + v 2 u 4 L 1 2 L 2 , q 5 = q 2 2 L 3 + q 2 P 2 4 L 3 2 + q 1 P 2 L 3 , q 8 = u L 2 v 2 u R 4 L 1 L 2 L 3 ,
q 3 = q 2 v 2 R 2 4 L 3 2 v 2 u R 4 L 1 L 2 L 3 , q 6 = q 2 v R 2 L 3 v u 2 L 1 L 2 , q 9 = N 2 L 2 v u P 4 L 1 L 2 L 3 .
W θ x x ( ρ 1 , ρ 2 , z ) = W r y y ( ρ 1 , ρ 2 , z ) , W θ y y ( ρ 1 , ρ 2 , z ) = W r x x ( ρ 1 , ρ 2 , z ) , W θ x y ( ρ 1 , ρ 2 , z ) = W θ y x ( ρ 1 , ρ 2 , z ) = W r x y ( ρ 1 , ρ 2 , z ) .
I ( ρ , z ) = Tr W ( ρ , ρ ; z ) = W r x x ( ρ , ρ , z ) + W r y y ( ρ , ρ , z ) ,
μ ( ρ 1 , ρ 2 , z ) = Tr W ( ρ 1 , ρ 2 ; z ) I ( ρ 1 , z ) I ( ρ 2 , z ) ,
P 0 ( ρ , z ) = 1 4 Det W ( ρ , ρ ; z ) [ Tr W ( ρ , ρ ; z ) ] 2 = 1 4 ( W r x x W r y y W r x y W r y x ) ( W r x x + W r y y ) 2 ,
S R = I max I 0 max ,

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