Abstract

The analytical formula for the effective radius of curvature of radial Gaussian array beams propagating through atmospheric turbulence is derived, where coherent and incoherent beam combinations are considered. The influence of turbulence on the effective radius of curvature of radial Gaussian array beams is studied by using numerical calculation examples.

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References

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  1. R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics 22, E. Wolf, ed., (Elsevier, Amsterdam, 1985), Chap. 6.
  2. L. C. Andrews, and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).
  3. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28(1), 10–12 (2003).
    [CrossRef] [PubMed]
  4. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
    [CrossRef]
  5. J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. 282(9), 1691–1698 (2009).
    [CrossRef]
  6. X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24(11), 3554–3563 (2007).
    [CrossRef]
  7. H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10(1), 015005 (2008).
    [CrossRef]
  8. 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]
  9. M. A. Porras, J. Alda and E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams,” Appl. Opt. 31(30), 6389–6402 (1992).
    [CrossRef] [PubMed]
  10. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002).
    [CrossRef]
  11. H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
    [CrossRef]
  12. W. D. Bilida, J. D. Strohschein, and H. J. J. Seguin, “High-power 24 channel radial array slab RF-excited carbon dioxide laser,” in Gas and Chemical Lasers and Applications, H. R. C. Sze and E. A. Dorko, eds., Proc. SPIE 2987, 13–21 (1997).
  13. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
    [CrossRef] [PubMed]
  14. Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
    [CrossRef]
  15. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91(2), 265–271 (2008).
    [CrossRef]
  16. X. Ji and X. Li, “Directionality of Gaussian array beams propagating in atmospheric turbulence,” J. Opt. Soc. Am. A 26(2), 236–243 (2009).
    [CrossRef]
  17. X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
    [CrossRef]
  18. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–1049 (1992).
    [CrossRef]
  19. Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009).
    [CrossRef] [PubMed]
  20. To derive Eq. (4), we first introduce the new variables of integration by setting u=(r′2+r′1)/2 and v=r′2−r′1. Then we use the formulae∫exp(−i2πxs)dx=δ(s), ∫x2exp(−i2πxs)dx=−δ″(s)/(2π)2, ∫f(x)δ″(x)dx=f″(0) and ∫f(x)δ(x)dx=f(0), where δ denotes the Dirac delta function and δ″ is its second derivative, and f is an arbitrary function and f″ is its second derivative.
  21. J. D. Strohschein, H. J. Seguin, and C. E. Capjack, “Beam propagation constants for a radial laser array,” Appl. Opt. 37(6), 1045–1048 (1998).
    [CrossRef]

2010 (1)

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[CrossRef]

2009 (3)

2008 (3)

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

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91(2), 265–271 (2008).
[CrossRef]

H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10(1), 015005 (2008).
[CrossRef]

2007 (3)

2006 (1)

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

2003 (1)

2002 (1)

1998 (1)

1996 (1)

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[CrossRef]

1992 (2)

Alda, .

Amarande, S.

Baker, H. J.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[CrossRef]

Baykal, Y.

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[CrossRef]

H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10(1), 015005 (2008).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91(2), 265–271 (2008).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[CrossRef]

Bernabeu, E.

Cai, Y.

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91(2), 265–271 (2008).
[CrossRef]

H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10(1), 015005 (2008).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[CrossRef]

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

Capjack, C. E.

Chen, S.

Chen, X.

Chen, Y.

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[CrossRef]

Dan, Y.

Davidson, F. M.

Dogariu, A.

Du, X.

Eyyuboglu, H. T.

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91(2), 265–271 (2008).
[CrossRef]

H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10(1), 015005 (2008).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[CrossRef]

Hall, D. R.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[CrossRef]

He, S.

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

Hornby, A. M.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[CrossRef]

Ji, X.

Korotkova, O.

J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. 282(9), 1691–1698 (2009).
[CrossRef]

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

Li, X.

Lü, B.

Morley, R. J.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[CrossRef]

Porras, M. A.

Pu, J.

J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. 282(9), 1691–1698 (2009).
[CrossRef]

Ricklin, J. C.

Seguin, H. J.

Strohschein, J. D.

Taghizadeh, M. R.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[CrossRef]

Weber, H.

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–1049 (1992).
[CrossRef]

Yelden, E. F.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[CrossRef]

Zhang, B.

Zhao, D.

Zhu, Y.

Appl. Opt. (2)

Appl. Phys. B (3)

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91(2), 265–271 (2008).
[CrossRef]

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[CrossRef]

Appl. Phys. Lett. (1)

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

IEEE J. Quantum Electron. (1)

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[CrossRef]

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

H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10(1), 015005 (2008).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. 282(9), 1691–1698 (2009).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–1049 (1992).
[CrossRef]

Other (4)

To derive Eq. (4), we first introduce the new variables of integration by setting u=(r′2+r′1)/2 and v=r′2−r′1. Then we use the formulae∫exp(−i2πxs)dx=δ(s), ∫x2exp(−i2πxs)dx=−δ″(s)/(2π)2, ∫f(x)δ″(x)dx=f″(0) and ∫f(x)δ(x)dx=f(0), where δ denotes the Dirac delta function and δ″ is its second derivative, and f is an arbitrary function and f″ is its second derivative.

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics 22, E. Wolf, ed., (Elsevier, Amsterdam, 1985), Chap. 6.

L. C. Andrews, and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

W. D. Bilida, J. D. Strohschein, and H. J. J. Seguin, “High-power 24 channel radial array slab RF-excited carbon dioxide laser,” in Gas and Chemical Lasers and Applications, H. R. C. Sze and E. A. Dorko, eds., Proc. SPIE 2987, 13–21 (1997).

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Figures (3)

Fig. 1
Fig. 1

Effective radius of curvature R versus the propagation distance z, r 0 = 2 , N = 15

Fig. 3
Fig. 3

Effective radius of curvature R versus the beam number N, z = 4km, r 0 = 2 .

Fig. 2
Fig. 2

Effective radius of curvature R versus the inverse radial fill-factor r 0 , z = 4km.

Equations (16)

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W ( x 1 , x 2 , y 1 , y 2 , z = 0 ) = p = 0 N 1 q = 0 N 1 exp [ ( x 1 cos α p + y 1 sin α p r 0 ) 2 + ( y 1 cos α p x 1 sin α p ) 2 w 0 2 ]
× exp [ ( x 2 cos α q + y 2 sin α q r 0 ) 2 + ( y 2 cos α q x 2 sin α q ) 2 w 0 2 ] ,
I ( r , z ) = ( k 2 π z ) 2 d 2 r 1 d 2 r 2 W ( r 1  , r 2 , z = 0 ) exp { i k 2 z [ ( r 1 ′2 r 2 ′2 ) 2 r ( r 1 r 2 ) ] } ,
< r 2 > = r 2 I ( r , z ) d 2 r / I ( r , z ) d 2 r .
< r 2 > = G + ( Q / k 2 ) z 2 ,
G = p = 0 N 1 q = 0 N 1 w 0 2 { r 0 ′2 [ 1 + cos ( α p α q ) ] + 1 } S / 2 p = 0 N 1 q = 0 N 1 S ,
Q = 2 w 0 2 p = 0 N 1 q = 0 N 1 { 1 r 0 ′2 [ 1 cos ( α p α q ) ] } S / p = 0 N 1 q = 0 N 1 S ,
S = exp { r 0 ′2 [ 1 + cos ( α p α q ) ] } ,
< r 2 2 > = < ( A r 1 + B θ 1 ) 2 > ,
R = < r 2 > / < r θ > .
R = z + z R 2 z ,
< r 2 > = < r 2 > 0 + 2 < r θ > 0 z + < θ 2 > 0 z 2 + ( 4 / 3 ) T z 3 ,
< r θ > = < r θ > 0 + < θ 2 > 0 z + 2 T z 2 ,
T = π 2 0 κ 3 Φ n ( κ ) d κ .
R = A + ( B / k 2 ) z 2 + ( 4 / 3 ) T z 3 ( B / k 2 ) z + 2 T z 2 .
R = w 0 2 ( 1 / 2 + r 0 ′2 ) + ( 2 / k 2 w 0 2 ) z 2 + ( 4 / 3 ) T z 3 ( 2 / k 2 w 0 2 ) z + 2 T z 2 .

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