## Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence |

Optics Express, Vol. 22, Issue 15, pp. 17723-17734 (2014)

http://dx.doi.org/10.1364/OE.22.017723

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### 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

## 1. Introduction

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]

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]

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]

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]

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]

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

## 2. The cross-spectral density of Gaussian Schell-model vortex beams through oceanic turbulence

*z*= 0 source plane is expressed aswhere

*u*(

**) represents the profile of the background beam envelope,**

*ρ***(**

*ρ*≡*ρ*,

_{x}*ρ*) is two-dimensional coordinate vectors at the plane

_{y}*z*= 0, sgn(.) is the sign function,

*m*is the topological charge.

*z*= 0 is written as

*ρ*_{1}= (

*ρ*

_{1}

*,*

_{x}*ρ*

_{1}

*) and*

_{y}

*ρ*_{2}= (

*ρ*

_{2}

*,*

_{x}*ρ*

_{2}

*) are positions of two points at the plane*

_{y}*z*= 0, respectively. In this paper the

*m*is assumed to be

*m*= 0, Eq. (2) degenerates into the cross spectral density of GSM non-vortex beam,

*w*

_{0}is the waist width for the Gaussian part,

*σ*

_{0}is spatial correlation length.

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

*η*is the Kolmogorov micro scale,

*ε*is the rate of dissipation of turbulent kinetic energy per unit mass of fluid which may vary in range from 10

^{−4}m

^{2}/s

^{3}to10

^{−10}m

^{2}/s

^{3},

*χ*being the rate of dissipation of mean-square temperature, taking values in the range from 10

_{T}^{−4}K

^{2}/s to 10

^{−10}K

^{2}/s [10].

*ω*is the relative strength of temperature and salinity fluctuation, which in the ocean water the value can range from 0 to −5, the minus sign of the parameter

*ω*denotes that there is a reduction in temperature and an increase in salinity with depth. 0 corresponding to the case when temperature-driven turbulence dominates, −5 corresponding to the situation when salinity-driven turbulence prevails [10].

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

*χ*and decreasing

_{T}*ω*,

*ε*and

*η*.

*s*= (

*ρ*

_{1}+

*ρ*

_{2})/2,

*t*=

*ρ*

_{1}-

*ρ*

_{2}and with Eqs. (4)–(8) substituting into Eq. (3), we can obtain

*s*= (

_{x}*ρ*

_{1}

*+*

_{x}*ρ*

_{2}

*)/2,*

_{x}*t*=

_{x}*ρ*

_{1}

*-*

_{x}*ρ*

_{2}

*,*

_{x}*s*= (

_{y}*ρ*

_{1}

*+*

_{y}*ρ*

_{2}

*)/2,*

_{y}*t*=

_{y}*ρ*

_{1}

*-*

_{y}*ρ*

_{2y}, the symbol ‘

*m*= ± 1 at the source plane

*z*= 0.

*E*,

_{y}*F*and

_{y}*G*are obtained from

_{y}*E*,

_{x}*F*and

_{x}*G*by replacing

_{x}*ρ′*

_{1}

*and*

_{x}*ρ′*

_{2}

*with*

_{x}*ρ′*

_{1}

*and*

_{y}*ρ′*

_{2}

*, respectively.*

_{y}*m*= ± 1 propagating through oceanic turbulence. It follows from Eq. (14) that the cross–spectral density of GSM vortex beams depends on the oceanic turbulence and beam parameters such as the Kolmogorov micro scale

*η*, rate of dissipation of turbulent kinetic energy per unit mass of fluid

*ε*, rate of dissipation of mean-square temperature

*χ*, relative strength of temperature salinity fluctuation

_{T}*ω*, spatial correlation length

*σ*

_{0}, waist width

*w*

_{0}, sign and values of topological charge

*m*, as well as the propagation distance

*z*, and the positions of two points at the

*z*plane. In addition, for the case of

*T*(

*η*,

*ε*,

*χ*,

_{T}*ω*) = 0 in Eqs. (14)–(24), which can simplify to the analytical expression for the cross–spectral density of GSM vortex beams with

*m*= ± 1 in free space.

## 3. The evolution behavior of average intensity of Gaussian Schell-model vortex beams through oceanic turbulence

*z*plane is expressed asafter straightforward integral calculations deliver the expression

*σ*

_{0}, waist width

*w*

_{0}, topological charge |

*m*|, and parameter

*T*(

*η*,

*ε*,

*χ*,

_{T}*ω*) (including the Kolmogorov micro scale

*η*, the rate of dissipation of turbulent kinetic energy per unit mass of fluid

*ε*, the rate of dissipation of mean-square temperature

*χ*and the relative strength of temperature and salinity fluctuation

_{T}*ω*), but is independent of the sign of

*m*. It’s worth noting that Eq. (26) reduces to the analytical expression of a GSM vortex in free space for the case of

*T*(

*η*,

*ε*,

*χ*,

_{T}*ω*) = 0 in Eq. (10), namely,

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]

*C*

_{n}

^{2}= 0.

*I*/

*I*(

_{max}*I*—maximum intensity) of a GSM vortex beam versus the slanted axis

_{max}*ρ*′

*(*

_{r}*z*, (b) relative strength of temperature and salinity fluctuation

*ω*, (c) rate of dissipation of turbulent kinetic energy per unit mass of fluid

*ε*, (d) rate of dissipation of mean-square temperature

*χ*, (e) spatial correlation length

_{T}*σ*

_{0}, where the calculation parameters

*w*

_{0}= 3cm,

*λ*= 1060nm and

*η*= 10

^{−3}m are fixed in every figure, while the rest parameters including

*σ*

_{0}= 2.5cm,

*ω*= −2.5,

*ε*= 10

^{−7}m

^{2}/s

^{3},

*χ*= 10

_{T}^{−9}K

^{2}/s and

*z*= 300m are allowed to vary in different figures. As can be seen from Fig. 1(a), the average intensity of GSM vortex beams propagating through oceanic turbulence undergoes several stages of evolution like as GSM vortex beams propagating through ideal Kolmogorov atmospheric turbulence [4

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]

*z*= 0 plane there exists a zero intensity at the center

*ρ*′

*= 0, where the phase becomes singular. With increasing propagation distance to*

_{r}*z*= 560m, a flat-topped intensity profile occurs, and finally evolves into a Gaussian one at

_{flat}*z*= 700m. In Figs. 1(b)–1(e), we can see that at the given propagation distance (e.g.,

_{Gau}*z*= 300m), the evolution behavior of average intensity of GSM vortex beams in oceanic turbulence is significantly affected by the oceanic turbulence parameters including

*ω*,

*ε*and

*χ*

_{T}_{,}and beam parameter

*σ*

_{0}. From Figs. 1(a)–1(e), it is interesting to find that the beam profile will first take on a hollow shape, and then approach a Gaussian distribution, for example, when ω = −0.5 in Fig. 1(b),

*ε*= 10

^{−10}m

^{2}/s

^{3}in Fig. 1(c),

*χ*= 10

_{T}^{−7}K

^{2}/s in Fig. 1(d) and

*σ*

_{0}= 1cm, the

*I*/

*I*of a GSM vortex beam in oceanic turbulence extend to a Gaussian profile. The larger

_{max}*χ*, the smaller

_{T}*ω*,

*ε*and spatial correlation length and the longer propagation distance

*z*are, the faster the change of the beam profile is. We can see from Eq. (8) and Figs. 2(a) and 2(b) (the calculation parameters are the same as those of Fig. 1) that the larger

*χ*and the smaller

_{T}*ω*and

*ε*are, the stronger the oceanic turbulence is. Therefore, Figs. 1(a)–1(e) indicate that the evolution behavior of intensity depends on the oceanic turbulence strength, the beam coherent degree and the propagation distance. At the same time, it is shown that the larger

*χ*and

_{T}*z*or the smaller

*ω*and

*ε*are, the larger the spreading of the beam width is, but the effect on the beam width by the spatial correlation length is not significant for the GSM vortex beams through oceanic turbulence.

## 4. The evolution behavior of coherent vortices of Gaussian Schell-model vortex beams through oceanic turbulence

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]

*μ*= 0 and Im

*μ*= 0 of a GSM vortex beam with m = + 1 through oceanic turbulence at the propagation distance (a)

*z*= 50m and (b)

*z*= 220m, where

*ρ′*

_{1}= (6cm, 9cm),

*m*= + 1,

*σ*

_{0}= 2.5cm are kept fixed, and the other calculation parameters are the same as in Fig. 1. Figure 4 plots 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, where the calculation parameters are the same as in Fig. 3. Figures 3(a), 3(b), 4(a), and 4(b) indicate that there exists a coherent vortex A with

*m*= + 1 at

*z*= 50m whose position is (0.3783cm,-0.1178cm) and two coherent vortices A and B with

*m*= + 1, −1 appear at

*z*= 220m. The singularity A moves to the position (3.27cm, 0.2321cm), while a new singularity B with

*m*= −1 appears at the position (−7.351cm, 11.88cm). Therefore, the position and number of coherent vortices change with increasing propagation distance through oceanic turbulence.

*m*= + 1 through oceanic turbulence for different values of

*ω*,

*ε*,

*χ*, where

_{T}*z*= 300m is kept fixed, the other calculation parameters are same as those of Fig. 1, the white and black dots represent

*m*= −1 and

*m*= + 1, respectively. It can be seen that there appears only a coherent vortex with

*m*= + 1 and the topological charge is conserved within a certain propagation distance. But as the propagation distance increases, two coherent vortices with

*m*= + 1 and

*m*= −1 appear, whose positions change and gradually move closer. From Figs. 5(a)–5(c), we can see that the larger

*ω*is, the larger the distance

*z*for the conservation of the topological charge. It can be seen from Figs. 5(b), 5(d) and 5(e) that the larger

_{c}*ε*is, the larger

*z*is, for the case of

_{c}*ε*= 10

^{-10,}10

^{−7}and 10

^{−5},

*z*corresponding to 70m, 150m and 250m, respectively. The position and number of coherent vortices of GSM vortex beams with

_{c}*m*= + 1 propagation through oceanic turbulence for different values of

*χ*are plotted in Figs. 5(b), 5(f) and 5(g). We can see that the position and number of coherent vortices change with propagation distance, and two coherent vortices with

_{T}*m*= + 1 and

*m*= −1 appear at

*z*= 150m in Fig. 5(b), 70m in Fig. 5(f) and 40m in Fig. 5(g) respectively. Obviously, the smaller the

*χ*is, the larger the distance

_{T}*z*for the conservation of the topological charge, which indicates that the distance for the conservation of the topological charge increases with increasing

_{c}*ω*,

*ε*and decreasing

*χ*The physical reason can be seen from Fig. 2 that the larger

_{T}.*ω*,

*ε*and smaller

*χ*are, the smaller the value of

_{T}*T*(

*η*,

*ε*,

*χ*,

_{T}*ω*) is, which indicates that the weaker the oceanic turbulence is, the less effect on the conservation of the topological charge by the weaker oceanic turbulence. And then the positions of two singularities constantly change and the separation distance between them gradually becomes smaller, and this condition is more significant for the larger the rate of dissipation of mean-square temperature

*χ*in Fig. 5(g).

_{T}## 5. Concluding remarks

*ε*, rate of dissipation of mean-square temperature

*χ*, relative strength of temperature salinity fluctuations

_{T}*ω*, spatial correlation length and topological charge of the beams, as well as the propagation distance. The larger

*χ*, the smaller

_{T}*ω*,

*ε*and the shorter spatial correlation length and the longer propagation distance

*z*are, the faster the evolution of the beam intensity profile is. The smaller

*χ*and the larger

_{T}*ω*and

*ε*are, the longer the distance appearing two coherent vortices with

*m*= + 1 and

*m*= −1, namely, the larger the distance

*z*for the conservation of the topological charge. The results obtained in this paper would be useful for potential applications of vortex beams in oceanic optical communications.

_{c}## Acknowledgments

## References and links

1. | G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A |

2. | T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. |

3. | J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. |

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. |

5. | X. He and B. Lü, “Propagation of partially coherent flat-topped vortex beams through non-Kolmogorov atmospheric turbulence,” J. Opt. Soc. Am. A |

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. |

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.) |

8. | R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. A |

9. | V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuation of the sea-water refractive index,” Int. J. Fluid Mech. Res. |

10. | S. A. Thorpe, |

11. | O. Korotkova and N. Farwell, “Polarization changes in stochastic electromagnetic beams propagating in the oceanic turbulence,” Proc. SPIE |

12. | O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. |

13. | E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B |

14. | O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Wave Random Complex |

15. | N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. |

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. |

17. | J. Xu and D. Zhao, “Propagation of a stochastic electromagnetic vortex beam in the oceanic turbulence,” Opt. Laser Technol. |

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 |

19. | M. Tang and D. Zhao, “Spectral changes in stochastic anisotropic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. |

20. | G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A |

21. | I. S. Gradysteyn and I. M. Ryzhik, |

22. | L. Mandel and E. Wolf, |

23. | G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. |

24. | I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A |

**OCIS Codes**

(010.3310) Atmospheric and oceanic optics : Laser beam transmission

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(010.4455) Atmospheric and oceanic optics : Oceanic propagation

(260.6042) Physical optics : Singular optics

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: May 6, 2014

Revised Manuscript: June 7, 2014

Manuscript Accepted: June 22, 2014

Published: July 14, 2014

**Virtual Issues**

Vol. 9, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Yongping Huang, Bin Zhang, Zenghui Gao, Guangpu Zhao, and Zhichun Duan, "Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence," Opt. Express **22**, 17723-17734 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-17723

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### References

- G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A25(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]
- 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]
- X. He and B. Lü, “Propagation of partially coherent flat-topped vortex beams through non-Kolmogorov atmospheric turbulence,” J. Opt. Soc. Am. A28(9), 1941–1948 (2011). [CrossRef]
- 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]
- 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]
- R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. A68(8), 1067–1072 (1978). [CrossRef]
- 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).
- S. A. Thorpe, The Turbulent Ocean (Cambridge University, 2007).
- O. Korotkova and N. Farwell, “Polarization changes in stochastic electromagnetic beams propagating in the oceanic turbulence,” Proc. SPIE7588, 75880S (2010). [CrossRef]
- 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. B105(2), 415–420 (2011). [CrossRef]
- O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Wave Random Complex22(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]
- 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. Xu and D. Zhao, “Propagation of a stochastic electromagnetic vortex beam in the oceanic turbulence,” Opt. Laser Technol.57, 189–193 (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. B109(2), 289–294 (2012). [CrossRef]
- 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 E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A19(8), 1592–1598 (2002). [CrossRef] [PubMed]
- 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).
- G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun.222(1–6), 117–125 (2003). [CrossRef]
- I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A50(6), 5164–5172 (1994). [CrossRef] [PubMed]

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