Abstract

The analytical expressions for the cross-spectral density and average intensity of Gaussian Schell-model (GSM) vortex beams propagating through oceanic turbulence are obtained by using the extended Huygens–Fresnel principle and the spatial power spectrum of the refractive index of ocean water. The evolution behavior of GSM vortex beams through oceanic turbulence is studied in detail by numerical simulation. It is shown that the evolution behavior of coherent vortices and average intensity depends on the oceanic turbulence including the rate of dissipation of turbulent kinetic energy per unit mass of fluid, rate of dissipation of mean-square temperature, relative strength of temperature salinity fluctuations, and beam parameters including the spatial correlation length and topological charge of the beams, as well as the propagation distance.

© 2014 Optical Society of America

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References

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  1. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25(1), 225–230 (2008).
    [CrossRef] [PubMed]
  2. T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
    [CrossRef]
  3. J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
    [CrossRef]
  4. J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11(4), 045710 (2009).
    [CrossRef]
  5. 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]
  6. Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel–Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
    [CrossRef]
  7. X. Sheng, Y. Zhu, Y. Zhu, and Y. Zhang, “Orbital angular momentum entangled states of vortex beam pump in non-Kolmogorov turbulence channel,” Optik (Stuttg.) 124(17), 2635–2638 (2013).
    [CrossRef]
  8. R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. A 68(8), 1067–1072 (1978).
    [CrossRef]
  9. V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuation of the sea-water refractive index,” Int. J. Fluid Mech. Res. 27, 82–98 (2000).
  10. S. A. Thorpe, The Turbulent Ocean (Cambridge University, 2007).
  11. O. Korotkova and N. Farwell, “Polarization changes in stochastic electromagnetic beams propagating in the oceanic turbulence,” Proc. SPIE 7588, 75880S (2010).
    [CrossRef]
  12. O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284(7), 1740–1746 (2011).
    [CrossRef]
  13. E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105(2), 415–420 (2011).
    [CrossRef]
  14. O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Wave Random Complex 22(2), 260–266 (2012).
    [CrossRef]
  15. N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
    [CrossRef]
  16. 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(12), 1052–1058 (2006).
    [CrossRef]
  17. J. Xu and D. Zhao, “Propagation of a stochastic electromagnetic vortex beam in the oceanic turbulence,” Opt. Laser Technol. 57, 189–193 (2014).
    [CrossRef]
  18. Y. Zhou, K. Huang, and D. Zhao, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams propagating through the oceanic turbulence,” Appl. Phys. B 109(2), 289–294 (2012).
    [CrossRef]
  19. M. Tang and D. Zhao, “Spectral changes in stochastic anisotropic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312, 89–93 (2014).
    [CrossRef]
  20. 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]
  21. I. S. Gradysteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).
  22. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  23. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1–6), 117–125 (2003).
    [CrossRef]
  24. I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
    [CrossRef] [PubMed]

2014 (3)

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel–Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[CrossRef]

J. Xu and D. Zhao, “Propagation of a stochastic electromagnetic vortex beam in the oceanic turbulence,” Opt. Laser Technol. 57, 189–193 (2014).
[CrossRef]

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

2013 (1)

X. Sheng, Y. Zhu, Y. Zhu, and Y. Zhang, “Orbital angular momentum entangled states of vortex beam pump in non-Kolmogorov turbulence channel,” Optik (Stuttg.) 124(17), 2635–2638 (2013).
[CrossRef]

2012 (3)

Y. Zhou, K. Huang, and D. Zhao, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams propagating through the oceanic turbulence,” Appl. Phys. B 109(2), 289–294 (2012).
[CrossRef]

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Wave Random Complex 22(2), 260–266 (2012).
[CrossRef]

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
[CrossRef]

2011 (4)

O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284(7), 1740–1746 (2011).
[CrossRef]

E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105(2), 415–420 (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]

J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
[CrossRef]

2010 (1)

O. Korotkova and N. Farwell, “Polarization changes in stochastic electromagnetic beams propagating in the oceanic turbulence,” Proc. SPIE 7588, 75880S (2010).
[CrossRef]

2009 (1)

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11(4), 045710 (2009).
[CrossRef]

2008 (2)

G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25(1), 225–230 (2008).
[CrossRef] [PubMed]

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

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(12), 1052–1058 (2006).
[CrossRef]

2003 (1)

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

2002 (1)

2000 (1)

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuation of the sea-water refractive index,” Int. J. Fluid Mech. Res. 27, 82–98 (2000).

1994 (1)

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
[CrossRef] [PubMed]

1978 (1)

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

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]

Farwell, N.

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Wave Random Complex 22(2), 260–266 (2012).
[CrossRef]

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
[CrossRef]

E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105(2), 415–420 (2011).
[CrossRef]

O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284(7), 1740–1746 (2011).
[CrossRef]

O. Korotkova and N. Farwell, “Polarization changes in stochastic electromagnetic beams propagating in the oceanic turbulence,” Proc. SPIE 7588, 75880S (2010).
[CrossRef]

Freund, I.

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
[CrossRef] [PubMed]

Gbur, G.

He, X.

Hill, R. J.

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

Huang, K.

Y. Zhou, K. Huang, and D. Zhao, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams propagating through the oceanic turbulence,” Appl. Phys. B 109(2), 289–294 (2012).
[CrossRef]

Korotkova, O.

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
[CrossRef]

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Wave Random Complex 22(2), 260–266 (2012).
[CrossRef]

E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105(2), 415–420 (2011).
[CrossRef]

O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284(7), 1740–1746 (2011).
[CrossRef]

O. Korotkova and N. Farwell, “Polarization changes in stochastic electromagnetic beams propagating in the oceanic turbulence,” Proc. SPIE 7588, 75880S (2010).
[CrossRef]

Li, J.

J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
[CrossRef]

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11(4), 045710 (2009).
[CrossRef]

Liu, L.

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(12), 1052–1058 (2006).
[CrossRef]

Liu, Z.

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel–Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[CrossRef]

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(12), 1052–1058 (2006).
[CrossRef]

Lü, B.

J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (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]

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11(4), 045710 (2009).
[CrossRef]

Nikishov, V. I.

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuation of the sea-water refractive index,” Int. J. Fluid Mech. Res. 27, 82–98 (2000).

Nikishov, V. V.

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuation of the sea-water refractive index,” Int. J. Fluid Mech. Res. 27, 82–98 (2000).

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]

Qin, Z.

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel–Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[CrossRef]

Shchepakina, E.

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Wave Random Complex 22(2), 260–266 (2012).
[CrossRef]

E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105(2), 415–420 (2011).
[CrossRef]

Sheng, X.

X. Sheng, Y. Zhu, Y. Zhu, and Y. Zhang, “Orbital angular momentum entangled states of vortex beam pump in non-Kolmogorov turbulence channel,” Optik (Stuttg.) 124(17), 2635–2638 (2013).
[CrossRef]

Shvartsman, N.

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
[CrossRef] [PubMed]

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(12), 1052–1058 (2006).
[CrossRef]

Tang, M.

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

Tao, R.

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel–Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[CrossRef]

Tyson, R. K.

Visser, T. D.

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

Wolf, E.

Xu, J.

J. Xu and D. Zhao, “Propagation of a stochastic electromagnetic vortex beam in the oceanic turbulence,” Opt. Laser Technol. 57, 189–193 (2014).
[CrossRef]

Xu, X.

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel–Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[CrossRef]

Zhang, Y.

X. Sheng, Y. Zhu, Y. Zhu, and Y. Zhang, “Orbital angular momentum entangled states of vortex beam pump in non-Kolmogorov turbulence channel,” Optik (Stuttg.) 124(17), 2635–2638 (2013).
[CrossRef]

Zhao, D.

J. Xu and D. Zhao, “Propagation of a stochastic electromagnetic vortex beam in the oceanic turbulence,” Opt. Laser Technol. 57, 189–193 (2014).
[CrossRef]

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

Y. Zhou, K. Huang, and D. Zhao, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams propagating through the oceanic turbulence,” Appl. Phys. B 109(2), 289–294 (2012).
[CrossRef]

Zhou, P.

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel–Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[CrossRef]

Zhou, Y.

Y. Zhou, K. Huang, and D. Zhao, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams propagating through the oceanic turbulence,” Appl. Phys. B 109(2), 289–294 (2012).
[CrossRef]

Zhu, Y.

X. Sheng, Y. Zhu, Y. Zhu, and Y. Zhang, “Orbital angular momentum entangled states of vortex beam pump in non-Kolmogorov turbulence channel,” Optik (Stuttg.) 124(17), 2635–2638 (2013).
[CrossRef]

X. Sheng, Y. Zhu, Y. Zhu, and Y. Zhang, “Orbital angular momentum entangled states of vortex beam pump in non-Kolmogorov turbulence channel,” Optik (Stuttg.) 124(17), 2635–2638 (2013).
[CrossRef]

Appl. Phys. B (2)

E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105(2), 415–420 (2011).
[CrossRef]

Y. Zhou, K. Huang, and D. Zhao, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams propagating through the oceanic turbulence,” Appl. Phys. B 109(2), 289–294 (2012).
[CrossRef]

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

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuation of the sea-water refractive index,” Int. J. Fluid Mech. Res. 27, 82–98 (2000).

J. Opt. A, Pure Appl. Opt. (2)

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(12), 1052–1058 (2006).
[CrossRef]

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11(4), 045710 (2009).
[CrossRef]

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

Opt. Commun. (5)

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

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

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
[CrossRef]

J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
[CrossRef]

O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284(7), 1740–1746 (2011).
[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).
[CrossRef]

Opt. Laser Technol. (2)

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel–Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[CrossRef]

J. Xu and D. Zhao, “Propagation of a stochastic electromagnetic vortex beam in the oceanic turbulence,” Opt. Laser Technol. 57, 189–193 (2014).
[CrossRef]

Optik (Stuttg.) (1)

X. Sheng, Y. Zhu, Y. Zhu, and Y. Zhang, “Orbital angular momentum entangled states of vortex beam pump in non-Kolmogorov turbulence channel,” Optik (Stuttg.) 124(17), 2635–2638 (2013).
[CrossRef]

Phys. Rev. A (1)

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
[CrossRef] [PubMed]

Proc. SPIE (1)

O. Korotkova and N. Farwell, “Polarization changes in stochastic electromagnetic beams propagating in the oceanic turbulence,” Proc. SPIE 7588, 75880S (2010).
[CrossRef]

Wave Random Complex (1)

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Wave Random Complex 22(2), 260–266 (2012).
[CrossRef]

Other (3)

S. A. Thorpe, The Turbulent Ocean (Cambridge University, 2007).

I. S. Gradysteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

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

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Figures (5)

Fig. 1
Fig. 1

Evolution of normalized intensity profiles of GSM vortex beams propagating in oceanic turbulence.

Fig. 2
Fig. 2

T(η,ω,ε,χT) versus the rate of dissipation of turbulent kinetic energy per unit mass of fluid ε, (a) for different values of the rate of dissipation of mean-square temperatureχT and (b) for different values of the relative strength of temperature and salinity fluctuation ω.

Fig. 3
Fig. 3

Curves of Reμ = 0 and Imμ = 0 of a GSM vortex beam with m = + 1 in oceanic turbulence at the propagation distance (a) z = 50m and (b) z = 220m.

Fig. 4
Fig. 4

Contour lines of phase of a GSM vortex beam with m = + 1 in oceanic turbulence at the plane (a) z = 50m and (b) z = 220m.

Fig. 5
Fig. 5

Position and number of coherent vortices of a GSM with m = + 1 propagation through oceanic turbulence for different values of ω, ε, χT.

Equations (33)

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U(ρ,z=0)=u(ρ) [ ρ x +isgn(m) ρ y ] | m | ,
W ( 0 ) ( ρ 1 , ρ 2 ,0)= [( ρ 1x ρ 2x + ρ 1y ρ 2y )+isgn( m )( ρ 1x ρ 2y ρ 2x ρ 1y )] m ×exp( ρ 1 2 + ρ 2 2 w 0 2 )exp( | ρ 1 ρ 2 | 2 2 σ 0 2 ).
W( ρ 1 , ρ 2 ,z)= ( k 2πz ) 2 d 2 ρ 1 d 2 ρ 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 )] ,
exp[ ψ ( ρ 1 , ρ 1 )+ψ( ρ 2 , ρ 2 )] =exp{ 4 π 2 k 2 z 0 1 0 dκdξ κ Φ n (κ) [ 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 ) dκ[ ( ρ 1 ρ 2 ) 2 +( ρ 1 ρ 2 )( ρ 1 ρ 2 )+ ( ρ 1 ρ 2 ) 2 ] } =exp{ k 2 zT( η,ε, χ T ,ω )[ ( ρ 1 ρ 2 ) 2 +( ρ 1 ρ 2 )( ρ 1 ρ 2 )+ ( ρ 1 ρ 2 ) 2 ] } ,
T(η,ε, χ T ,ω)= π 2 3 0 κ 3 Φ n ( κ )dκ ,
Φ n (κ,η,ε, χ T ,ω)=0.388× 10 8 ε 1/3 κ 11/3 χ T ω 2 [1+2.35 (κη) 2/3 ] ×( ω 2 e A T δ + e A S δ 2ω e A TS δ ),
δ=8.284 (κη) 4/3 +12.978 (κη) 2 ,
T(η,ε, χ T ,ω)=1.2765× 10 8 ω 2 ε 1/3 η 1/3 χ T (47.570817.6701ω+6.78335 ω 2 ).
W( ρ 1 , ρ 2 ,z)= ( k 2πz ) 2 exp[ ik 2z ( ρ 1 2 ρ 2 2 )]exp[T( η,ε, χ T ,ω ) k 2 z ( ρ 1 ρ 2 ) 2 ] × d 2 s d 2 t[ ( s 2 t 2 4 )i( s x t y s y t x ) ] ×exp( a t 2 ik z st )exp( 2 w 0 2 s 2 )exp[ ik z ( ρ 1 ρ 2 )s ] ×exp{ t[ ik 2z ( ρ 1 + ρ 2 )T( η,ε, χ T ,ω ) k 2 z( ρ 1 ρ 2 ) ] } ,
a= 1 2 ( 1 w 0 2 + 1 σ 0 2 )+ k 2 zT( η,ε, χ T ,ω ),
exp(p x 2 +2qx) dx=exp( q 2 p ) π p ,
xexp(p x 2 +2qx) dx=exp( q 2 p ) π p ( q p ),
x 2 exp(p x 2 +2qx) dx= 1 2p exp( q 2 p ) π p (1+ 2 q 2 p ),
W( ρ 1 , ρ 2 ,z)= ( k 2πz ) 2 exp[ ik 2z ( ρ 1 2 ρ 2 2 )] ×exp[ k 2 zT( η,ε, χ T ,ω ) | ρ 1 ρ 2 | 2 ]×[ ( N 1 N 2 )( N 3 N 4 ) ],
N 1 = π 2 aC E x E y exp( F x 2 + F y 2 C )( F x 2 + F y 2 C 2 + 1 C ),
N 2 = π 2 w 0 2 8D exp( k 2 w 0 2 8 z 2 | ρ 1 ρ 2 | 2 )×exp( G x 2 + G y 2 D )( G x 2 + G y 2 D 2 + 1 D ),
N 3 = i π 2 w 0 2aCD E x F x G y CD exp[ k 2 w 0 2 8 z 2 ( ρ 1y ρ 2y ) 2 ]exp( F x 2 C + G y 2 D ),
N 4 = i π 2 w 0 2aCD E y F y G x CD exp[ k 2 w 0 2 8 z 2 ( ρ 1x ρ 2x ) 2 ]exp( F y 2 C + G x 2 D ),
b= ik z ,
C= 2 w 0 2 b 2 4a ,
D=a b 2 w 0 2 8 ,
E x =exp{ 1 4a [ ik 2z ( ρ 1x + ρ 2x )T( η,ε, χ T ,ω ) k 2 z( ρ 1x ρ 2x ) ] 2 },
F x = 1 2 [ ik z ( ρ 1x ρ 2x ) ikb 4az ( ρ 1x + ρ 2x )+ b k 2 2a T( η,ε, χ T ,ω )( ρ 1x ρ 2x )],
G x = 1 2 [ ik 2z ( ρ 1x + ρ 2x )T( η,ε, χ T ,ω ) k 2 z( ρ 1x ρ 2x ) b 2 w 0 2 4 ( ρ 1x ρ 2x )].
I( ρ ,z)=W( ρ , ρ ,z) = ( k 2πz ) 2 d 2 s d 2 t [( s 2 t 2 4 )i( s x t y s y t x )] ×exp( a t 2 ik z st )exp( 2 w 0 2 s 2 )exp( ik z ρ t),
I( ρ ,z)= ( k 2πz ) 2 { π 2 aC ( k 2 ρ 2 4a z 2 C 2 + 1 C ) ×exp[ k 2 4a z 2 ρ 2 ( 1 C 1)] π 2 w 0 2 8D ( ik ρ 2 2z D 2 + 1 D )exp( ik ρ 2 2zD )}.
a = 1 2 ( 1 w 0 2 + 1 σ 0 2 ),
C = 2 w 0 2 b 2 4 a ,
D = a b 2 w 0 2 8 ,
I ( ρ ,z)= ( k 2πz ) 2 { π 2 a C ( k 2 ρ 2 4 a z 2 C 2 + 1 C ) ×exp[ k 2 4 a z 2 ρ 2 ( 1 C 1)] π 2 w 0 2 8 D ( ik ρ 2 2z D 2 + 1 D )exp( ik ρ 2 2z D )},
μ( ρ 1 , ρ 2 ,z)= W( ρ 1 , ρ 2 ,z) [I( ρ 1 ,z)I( ρ 2 ,z)] 1/2 ,
Re[μ( ρ 1 , ρ 2 ,z)]=0,
Im[μ( ρ 1 , ρ 2 ,z)]=0.

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