Abstract

A new class of partially coherent radially and azimuthally polarized rotating elliptical Gaussian (PCRPREG and PCAPREG) beams is introduced. The analytical expressions of the PCRPREG and PCAPREG beams propagating through anisotropy oceanic turbulence are derived based on the extended Huygens-Fresnel principle and the spatial power spectrum of oceanic turbulence. The effects of beam waist size w0, coherence width σ0, propagation distance z and oceanic turbulence parameters on the evolution statistics properties of PCRPREG and PCAPREG beams are studied in detail by numerical simulation. Our results indicate that with the increase of the propagation distance in the far field region, the normalized initial profile with a doughnut-like distribution of PCRPREG and PCAPREG beams gradually converts into a flat-topped one, and finally evolves into a Gaussian-like beam profile. We also find that the salinity-induced turbulence fluctuation makes a greater contribution to the decrease of beam quality compared with the temperature-induced turbulence fluctuation. Furthermore, the full width at half maximum becomes wider for the larger propagation distance z and wavelength λ or the smaller dissipation rate ε. Our work will pave the way for the development of underwater optical communication and underwater laser radar in oceanic environment.

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

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References

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    [Crossref]
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2018 (8)

D. Liu, Y. Wang, G. Wang, and H. Yin, “Influences of oceanic turbulence on Lorentz Gaussian beam,” Optik 154, 738–747 (2018).
[Crossref]

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

F. Ye, J. Zhang, J. Xie, and D. Deng, “Propagation properties of the rotating elliptical chirped Gaussian vortex beam in the oceanic turbulence,” Opt. Commun. 426, 456–462 (2018).
[Crossref]

T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

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]

Y. Ata and Y. Baykal, “Effect of anisotropy on bit error rate for an asymmetrical Gaussian beam in a turbulent ocean,” Appl. Opt. 57(9), 2258–2262 (2018).
[Crossref]

Y. Jin, M. Hu, M. Luo, Y. Luo, X. Mi, C. Zou, L. Zhou, C. Shu, X. Zhu, J. He, S. Ouyang, and W. Wen, “Beam wander of a partially coherent Airy beam in oceanic turbulence,” J. Opt. Soc. Am. A 35(8), 1457–1464 (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] [PubMed]

2017 (6)

2016 (5)

2015 (6)

2014 (4)

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]

M. Yao, I. Toselli, and O. Korotkova, “Propagation of electromagnetic stochastic beams in anisotropic turbulence,” Opt. Express 22(26), 31608–31619 (2014).
[Crossref]

X. Ji, Y. Ma, Z. Pu, and X. Li, “Characteristics of tilted and off-axis partially coherent beams reflected back by a cat-eye optical lens in atmospheric turbulence,” Appl. Phys. B 115(3), 379–390 (2014).
[Crossref]

S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
[Crossref]

2013 (2)

H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol 49, 332–336 (2013).
[Crossref]

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

2012 (1)

2011 (2)

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

H. Wang and X. Li, “Spectral properties of partially coherent azimuthally polarized beam in a turbulent atmosphere,” Optik 122, 2164–2171 (2011).
[Crossref]

2010 (1)

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the seawater refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2010).
[Crossref]

2009 (2)

J. Cang and Y. Zhang, “The propagation properties of J0-correlated partially coherent beams in the slant atmosphere,” Acta Physica Sinica 58(4), 2444–2450 (2009).

S. Arnon and D. Kedar, “Non-line-of-sight underwater optical wireless communication network,” J. Opt. Soc. Am. A 26(3), 530–539 (2009).
[Crossref]

2008 (1)

2007 (2)

2006 (1)

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A: Pure Appl. Opt. 8, 1052–1058 (2006).
[Crossref]

2004 (1)

G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004).
[Crossref]

2003 (1)

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

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

2000 (2)

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

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
[Crossref]

1978 (1)

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

Andrews, L. C.

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

Arnon, S.

Ata, Y.

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).

Baykal, Y.

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

Brown, T.

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

Bu, J.

Burge, R. E.

Cai, Y.

X. Peng, L. Liu, Y. Cai, and Y. Baykal, “Statistical properties of a radially polarized twisted Gaussian Schell-model beam in an underwater turbulent medium,” J. Opt. Soc. Am. A 34(1), 133–139 (2017).
[Crossref]

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

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]

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]

Y. Dong, L. Guo, C. Liang, F. Wang, and Y. Cai, “Statistical properties of a partially coherent cylindrical vector beam in oceanic turbulence,” J. Opt. Soc. Am. A 32(5), 894–901 (2015).
[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]

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

Y. Cai, Q. Lin, H. T. Eyyuboǧlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[Crossref] [PubMed]

Cang, J.

J. Cang and Y. Zhang, “The propagation properties of J0-correlated partially coherent beams in the slant atmosphere,” Acta Physica Sinica 58(4), 2444–2450 (2009).

Chen, H.

H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol 49, 332–336 (2013).
[Crossref]

Chen, M.

Chen, X.

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

Chen, Y.

Deng, D.

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

F. Ye, J. Zhang, J. Xie, and D. Deng, “Propagation properties of the rotating elliptical chirped Gaussian vortex beam in the oceanic turbulence,” Opt. Commun. 426, 456–462 (2018).
[Crossref]

T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

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] [PubMed]

Deng, Z.

Dong, Y.

Du, X.

Duan, Z.

Eyyuboglu, H. T.

Feng, L.

T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

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).

Gao, Bruce Z.

Gao, Z.

Golmohammady, S.

Goode, W.

Goodeb, W.

S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
[Crossref]

Guo, L.

He, J.

Hecht, B.

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

Hill, R.

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

Hou, W.

Houb, W.

S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
[Crossref]

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).

Huang, Y.

Jarosz, E.

S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
[Crossref]

W. Hou, S. Woods, E. Jarosz, W. Goode, and A. Weidemann, “Optical turbulence on underwater image degradation in natural environments,” Appl. Opt. 51(14), 2678–2686 (2012).
[Crossref] [PubMed]

Ji, X.

X. Ji, Y. Ma, Z. Pu, and X. Li, “Characteristics of tilted and off-axis partially coherent beams reflected back by a cat-eye optical lens in atmospheric turbulence,” Appl. Phys. B 115(3), 379–390 (2014).
[Crossref]

Jin, Y.

Kashani, F. D.

Kedar, D.

Korotkova, O.

Li, X.

X. Ji, Y. Ma, Z. Pu, and X. Li, “Characteristics of tilted and off-axis partially coherent beams reflected back by a cat-eye optical lens in atmospheric turbulence,” Appl. Phys. B 115(3), 379–390 (2014).
[Crossref]

H. Wang and X. Li, “Spectral properties of partially coherent azimuthally polarized beam in a turbulent atmosphere,” Optik 122, 2164–2171 (2011).
[Crossref]

Li, Y.

Li, Z.

Liang, C.

Lin, Q.

Liu, D.

D. Liu, Y. Wang, G. Wang, and H. Yin, “Influences of oceanic turbulence on Lorentz Gaussian beam,” Optik 154, 738–747 (2018).
[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]

Liu, L.

X. Peng, L. Liu, Y. Cai, and Y. Baykal, “Statistical properties of a radially polarized twisted Gaussian Schell-model beam in an underwater turbulent medium,” J. Opt. Soc. Am. A 34(1), 133–139 (2017).
[Crossref]

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A: Pure Appl. Opt. 8, 1052–1058 (2006).
[Crossref]

Liu, W.

Liu, X.

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]

Lu, C.

Lu, W.

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A: Pure Appl. Opt. 8, 1052–1058 (2006).
[Crossref]

Luo, M.

Luo, X.

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]

Luo, Y.

Ma, Y.

X. Ji, Y. Ma, Z. Pu, and X. Li, “Characteristics of tilted and off-axis partially coherent beams reflected back by a cat-eye optical lens in atmospheric turbulence,” Appl. Phys. B 115(3), 379–390 (2014).
[Crossref]

Mashal, A.

Matta, S.

S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
[Crossref]

Mi, X.

Miyaji, G.

G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004).
[Crossref]

Miyanaga, N.

G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004).
[Crossref]

Moh, K. J.

Nikishov, V. I.

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the seawater refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2010).
[Crossref]

Nikishov, V. V.

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the seawater refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2010).
[Crossref]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

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

Ohbayashi, K.

G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004).
[Crossref]

Ouyang, S.

Pang, Z.

T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

Peng, X.

Phillips, R. L.

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

Ping, C.

Pu, Z.

X. Ji, Y. Ma, Z. Pu, and X. Li, “Characteristics of tilted and off-axis partially coherent beams reflected back by a cat-eye optical lens in atmospheric turbulence,” Appl. Phys. B 115(3), 379–390 (2014).
[Crossref]

Sheng, X.

H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol 49, 332–336 (2013).
[Crossref]

Shi, X.

Shu, C.

Sick, B.

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

Sueda, K.

G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004).
[Crossref]

Sun, J.

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A: Pure Appl. Opt. 8, 1052–1058 (2006).
[Crossref]

Tang, M.

Toselli, I.

Tsubakimoto, K.

G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004).
[Crossref]

Wang, F.

Wang, G.

D. Liu, Y. Wang, G. Wang, and H. Yin, “Influences of oceanic turbulence on Lorentz Gaussian beam,” Optik 154, 738–747 (2018).
[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]

Wang, H.

Wang, J.

Wang, L.

T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

Wang, Y.

D. Liu, Y. Wang, G. Wang, and H. Yin, “Influences of oceanic turbulence on Lorentz Gaussian beam,” Optik 154, 738–747 (2018).
[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]

Weidemann, A.

Weidemannb, A.

S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
[Crossref]

Wen, W.

Wolf, E.

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

Woods, S.

Woodsc, S.

S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
[Crossref]

Wu, Y.

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]

Xie, J.

F. Ye, J. Zhang, J. Xie, and D. Deng, “Propagation properties of the rotating elliptical chirped Gaussian vortex beam in the oceanic turbulence,” Opt. Commun. 426, 456–462 (2018).
[Crossref]

J. Zhang, J. Xie, F. Ye, K. Zhou, X. Chen, D. Deng, and X. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B 124, 168 (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] [PubMed]

Xu, Z.

Yang, L.

Yang, X.

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

Yao, M.

Ye, F.

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

F. Ye, J. Zhang, J. Xie, and D. Deng, “Propagation properties of the rotating elliptical chirped Gaussian vortex beam in the oceanic turbulence,” Opt. Commun. 426, 456–462 (2018).
[Crossref]

Yin, H.

D. Liu, Y. Wang, G. Wang, and H. Yin, “Influences of oceanic turbulence on Lorentz Gaussian beam,” Optik 154, 738–747 (2018).
[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]

Youngworth, K.

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

Yousefi, M.

Yu, L.

Yuan, X. C.

Zhang, B.

Zhang, H.

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

Zhang, J.

F. Ye, J. Zhang, J. Xie, and D. Deng, “Propagation properties of the rotating elliptical chirped Gaussian vortex beam in the oceanic turbulence,” Opt. Commun. 426, 456–462 (2018).
[Crossref]

T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

J. Zhang, J. Xie, F. Ye, K. Zhou, X. Chen, D. Deng, and X. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B 124, 168 (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] [PubMed]

Zhang, Y.

Zhao, C.

Zhao, D.

Zhao, F.

H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol 49, 332–336 (2013).
[Crossref]

Zhao, G.

Zhong, T.

T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

Zhou, K.

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

Zhou, L.

Zhu, S.

Zhu, X.

Zhu, Y.

Zou, C.

Acta Physica Sinica (1)

J. Cang and Y. Zhang, “The propagation properties of J0-correlated partially coherent beams in the slant atmosphere,” Acta Physica Sinica 58(4), 2444–2450 (2009).

Appl. Opt. (6)

Appl. Phys. B (2)

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

X. Ji, Y. Ma, Z. Pu, and X. Li, “Characteristics of tilted and off-axis partially coherent beams reflected back by a cat-eye optical lens in atmospheric turbulence,” Appl. Phys. B 115(3), 379–390 (2014).
[Crossref]

Appl. Phys. Lett. (1)

G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004).
[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).

Int. J. Fluid Mech. Res. (1)

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the seawater refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2010).
[Crossref]

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

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A: Pure Appl. Opt. 8, 1052–1058 (2006).
[Crossref]

J. Opt. Soc. Am. A (7)

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

T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

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]

Methods in Oceanography (1)

S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
[Crossref]

Opt. Commun. (3)

F. Ye, J. Zhang, J. Xie, and D. Deng, “Propagation properties of the rotating elliptical chirped Gaussian vortex beam in the oceanic turbulence,” Opt. Commun. 426, 456–462 (2018).
[Crossref]

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

Opt. Express (13)

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

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]

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]

M. Yao, I. Toselli, and O. Korotkova, “Propagation of electromagnetic stochastic beams in anisotropic turbulence,” Opt. Express 22(26), 31608–31619 (2014).
[Crossref]

L. Yu and Y. Zhang, “Analysis of modal crosstalk for communication in turbulent ocean using Lommel-Gaussian beam,” Opt. Express 25(19), 22565–22574 (2017).
[Crossref]

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]

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]

Y. Li, L. Yu, and Y. Zhang, “Influence of anisotropic turbulence on the orbital angular momentum modes of Hermite-Gaussian vortex beam in the ocean,” Opt. Express 25(11), 12203–12215 (2017).
[Crossref]

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]

Y. Cai, Q. Lin, H. T. Eyyuboǧlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[Crossref] [PubMed]

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] [PubMed]

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
[Crossref]

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

Opt. Laser Technol (1)

H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol 49, 332–336 (2013).
[Crossref]

Optik (2)

H. Wang and X. Li, “Spectral properties of partially coherent azimuthally polarized beam in a turbulent atmosphere,” Optik 122, 2164–2171 (2011).
[Crossref]

D. Liu, Y. Wang, G. Wang, and H. Yin, “Influences of oceanic turbulence on Lorentz Gaussian beam,” Optik 154, 738–747 (2018).
[Crossref]

Photon. Res. (1)

Phys. Lett. A (1)

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

Phys. Rev. Lett. (2)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[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 (1)

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

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Figures (8)

Fig. 1
Fig. 1 The RPREG (a) and APREG (b) beams at the initial plane.
Fig. 2
Fig. 2 The normalized spectral density I ( x , 0 , z ) / I m a x ( x , 0 , 0 ) of the PCRPREG beam at several propagation distances in the anisotropic oceanic turbulence for two different values of χT at (a) z=0m, (b) z=10m, (c) z=30m, (d) z=50m, (e) z=80m and (f) z=100m.
Fig. 3
Fig. 3 The evolutions of the spectrum density distributions and the corresponding cross lines of the PCRPREG beam through the anisotropic oceanic turbulence at different propagation distances (a) z=0m, (b) z=10m, (c) z=40m, (d) z=70m, (e) z=100m and (f) z=130m.
Fig. 4
Fig. 4 The normalized average spectral density distributions of the PCRPREG beam through anisotropic oceanic turbulence with different variables.
Fig. 5
Fig. 5 The spectral DOCs of the PCRPREG beam through the anisotropic oceanic turbulence with different variables.
Fig. 6
Fig. 6 The spectral DOPs of the PCRPREG beam and the corresponding cross line ( v = 0 ) at different propagation distances.
Fig. 7
Fig. 7 Cross lines ( v = 0 ) of the DOPs of the PCRPREG beam for difference values of ω, ε and λ.
Fig. 8
Fig. 8 The orientation angles of the PCRPREG beam in the anisotropic oceanic turbulence at different propagation distances.

Equations (21)

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( E r ( x , y )   E θ ( x , y ) ) = exp  ( x 2 a 2 w 0 2 y 2 b 2 w 0 2 i x y c 2 w 0 2 ) ( x w 0 y w 0     y w 0 x w 0 ) ( e x   e y ) ,
W ( x 1 , y 1 , x 2 , y 2 , 0 ) = ( W x x ( x 1 , y 1 , x 2 , y 2 , 0 ) W x y ( x 1 , y 1 , x 2 , y 2 , 0 ) W y x ( x 1 , y 1 , x 2 , y 2 , 0 ) W y y ( x 1 , y 1 , x 2 , y 2 , 0 ) ) ,
W r ( x 1 , y 1 , x 2 , y 2 , 0 ) = 1 w 0 2 exp  ( 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 ) g α β ( r 1 r 2 ) ,
W θ ( x 1 , y 1 , x 2 , y 2 , 0 ) = 1 w 0 2 exp  ( 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 ) g α β ( r 1 r 2 ) ,
g α β ( r 1 r 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 ) = Q M 1 M 3 { [ ( N 1 i N 2 2 M 2 c 2 w 2 ) N 3 + ( Δ Δ 4 M 1 M 2 c 4 w 4 ) ( 1 + N 3 2 2 M 2 ) ] + [ ( N 1 i N 2 2 M 2 c 2 w 2 ) S i Δ N 3 2 M 2 c 2 w 2 + ( Δ Δ 4 M 1 M 2 c 4 w 4 ) N 3 S M 3 ] N 4 2 M 4 + [ i Δ S 2 M 2 c 2 w 2 + ( Δ Δ 4 M 1 M 2 c 4 w 4 ) S 2 2 M 3 ] 1 2 M 4 ( 1 + N 4 2 2 M 4 ) } ,
W r x y ( ρ 1 , ρ 2 , z ) = Q M 1 M 4 { [ N 1 i N 2 2 M 2 c 2 w 2 + ( Δ Δ 4 M 1 M 2 c 4 w 4 ) N 3 2 M 3 ] N 4   + [ i Δ 2 M 2 c 2 w 2 + ( Δ Δ 4 M 1 M 2 c 4 w 4 ) S 2 M 3 ] ( 1 + N 4 2 2 M 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 ) = Q M 2 M 4 { ( N 2 i Δ N 3 4 M 1 M 3 c 2 w 2 ) N 4 + ( Δ i Δ S 4 M 1 M 3 c 2 w 2 ) ( 1 + n 4 2 2 M 4 ) } ,
Q = k 2 16 w 2 z 2 M 1 M 2 M 3 M 4 exp { i k 2 z ( ρ 1   2 ρ 2   2 ) ρ o c ξ 2     ( ρ 1 ρ 2 ) 2 } exp { N 1 2 4 M 1 + N 2 2 4 M 2 + N 3 2 4 M 3 + N 4 2 4 M 4 } , M 1 = 1 a 2 w 0 2 + 1 2 σ 0 2 + ρ o c ξ 2 + i k 2 z , N 1 = ρ o c ξ 2 ( u 1 u 2 ) + i k u 1 z , Δ = 1 σ 0 2 + 2 ρ o c ξ 2 , M 2 = 1 4 M 1 c 4 w 0 4 + 1 b 2 w 0 2 + 1 2 σ 0 2 + ρ o c ξ 2 + i k 2 z , N 2 = i N 1 2 M 1 c 2 w 0 2 ρ o c ξ 2 ( v 1 v 2 ) + i k v 1 z , M 3 = Δ 2 16 M 1 2 M 1 c 4 w 0 4 Δ 2 4 M 1 + 1 a 2 w 0 2 + 1 2 σ 0 2 + ρ o c ξ 2 i k 2 z , S = i Δ 2 4 M 1 M 2 c 2 w 0 2 + i 1 c 2 w 0 2 N 3 = i N 2 Δ 4 M 1 M 2 c 2 w 0 2 + N 1 Δ 2 M 1 + ρ o c ξ 2 ( u 1 u 2 ) i k u 2 z , M 4 = S 2 4 M 3 Δ 2 4 M 2 + 1 b 2 w 0 2 + 1 2 σ 0 2 + ρ o c ξ 2 i k 2 z , N 4 = N 3 S 2 M 3 + N 2 Δ 2 M 2 + ρ o c ξ 2 ( v 1 v 2 ) i k v 2 z .
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 ) = T r W ( ρ , ρ ; z ) = W r x x ( ρ , ρ , z ) + W r y y ( ρ , ρ , z ) ,
μ ( ρ 1 , ρ 2 , z ) = T r W ( ρ 1 , ρ 2 ; z ) I ( ρ 1 , z ) I ( ρ 2 , z ) ,
P ( ρ , z ) = 1 4 D e t W ( ρ , ρ ; z ) [ T r 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 ,
θ ( ρ ; z ) = 1 2 arctan  [ R e [ W r x y ( ρ , ρ , z ) ] + R e [ W r y x ( ρ , ρ , z ) ] W r x x ( ρ , ρ , z ) W r y y ( ρ , ρ , z ) ] .

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