Abstract

The propagation of a higher-order cosh-Gaussian beam through a paraxial and real ABCD optical system in turbulent atmosphere has been investigated. The analytical expressions for the average intensity, the effective beam size, and the kurtosis parameter of a higher-order cosh-Gaussian beam through a paraxial and real ABCD optical system are derived in turbulent atmosphere. The average intensity distribution and the spreading properties of a higher-order cosh-Gaussian in turbulent atmosphere are numerically demonstrated. The influences of the beam parameters and the structure constant of the atmospheric turbulence on the propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere are also examined in detail.

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References

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    [CrossRef]
  4. D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  10. Y. Zhang, Y. Song, Z. Chen, J. Ji, and Z. Shi, “Virtual sources for a cosh-Gaussian beam,” Opt. Lett. 32(3), 292–294 (2007).
    [CrossRef] [PubMed]
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    [CrossRef]
  12. K. Zhu, H. Tang, X. Wang, and T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Stuttg.) 113(5), 222–226 (2002).
    [CrossRef]
  13. Q. Tang, Y. Yu, and Q. Hu, “A new method to generate flattened Gaussian beam by incoherent combination of cosh Gaussian beams,” Chin. Opt. Lett. 5, S46–S48 (2007).
  14. G. Zhou and J. Zheng, “Beam propagation of a higher-order cosh-Gaussian beam,” Opt. Laser Technol. 41(2), 202–208 (2009).
    [CrossRef]
  15. G. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 56(7), 886–892 (2009).
    [CrossRef]
  16. J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 57(20), 2039–2047 (2010).
    [CrossRef]
  17. J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
    [CrossRef]

2010 (2)

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 57(20), 2039–2047 (2010).
[CrossRef]

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

2009 (2)

G. Zhou and J. Zheng, “Beam propagation of a higher-order cosh-Gaussian beam,” Opt. Laser Technol. 41(2), 202–208 (2009).
[CrossRef]

G. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 56(7), 886–892 (2009).
[CrossRef]

2008 (1)

G. Zhou and F. Liu, “Far field structural characteristics of cosh-Gaussian beam,” Opt. Laser Technol. 40(2), 302–308 (2008).
[CrossRef]

2007 (5)

2005 (1)

2004 (2)

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
[CrossRef]

H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12(20), 4659–4674 (2004).
[CrossRef] [PubMed]

2003 (1)

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224(1-3), 5–12 (2003).
[CrossRef]

2002 (1)

K. Zhu, H. Tang, X. Wang, and T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Stuttg.) 113(5), 222–226 (2002).
[CrossRef]

1998 (2)

Baykal, Y.

Casperson, L. W.

Chen, F.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 57(20), 2039–2047 (2010).
[CrossRef]

Chen, Y.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 57(20), 2039–2047 (2010).
[CrossRef]

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

Chen, Z.

Chu, X.

X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express 15(26), 17613–17618 (2007).
[CrossRef] [PubMed]

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87(3), 547–552 (2007).
[CrossRef]

Eyyuboglu, H. T.

Hu, Q.

Ji, J.

Jing, F.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
[CrossRef]

Li, J.

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 57(20), 2039–2047 (2010).
[CrossRef]

Liu, F.

G. Zhou and F. Liu, “Far field structural characteristics of cosh-Gaussian beam,” Opt. Laser Technol. 40(2), 302–308 (2008).
[CrossRef]

Liu, H.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
[CrossRef]

Liu, T.

K. Zhu, H. Tang, X. Wang, and T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Stuttg.) 113(5), 222–226 (2002).
[CrossRef]

Mao, H.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
[CrossRef]

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224(1-3), 5–12 (2003).
[CrossRef]

Ni, Y.

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87(3), 547–552 (2007).
[CrossRef]

Shi, Z.

Song, Y.

Tang, H.

K. Zhu, H. Tang, X. Wang, and T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Stuttg.) 113(5), 222–226 (2002).
[CrossRef]

Tang, Q.

Tovar, A. A.

Wang, S.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
[CrossRef]

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224(1-3), 5–12 (2003).
[CrossRef]

Wang, X.

K. Zhu, H. Tang, X. Wang, and T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Stuttg.) 113(5), 222–226 (2002).
[CrossRef]

Wang, Y.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 57(20), 2039–2047 (2010).
[CrossRef]

Wei, X.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
[CrossRef]

Xin, Y.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 57(20), 2039–2047 (2010).
[CrossRef]

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

Xu, S.

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 57(20), 2039–2047 (2010).
[CrossRef]

Yu, Y.

Zhang, W.

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224(1-3), 5–12 (2003).
[CrossRef]

Zhang, Y.

Zhao, D.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
[CrossRef]

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224(1-3), 5–12 (2003).
[CrossRef]

Zhao, Q.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 57(20), 2039–2047 (2010).
[CrossRef]

Zheng, J.

G. Zhou and J. Zheng, “Beam propagation of a higher-order cosh-Gaussian beam,” Opt. Laser Technol. 41(2), 202–208 (2009).
[CrossRef]

Zhou, G.

G. Zhou and J. Zheng, “Beam propagation of a higher-order cosh-Gaussian beam,” Opt. Laser Technol. 41(2), 202–208 (2009).
[CrossRef]

G. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 56(7), 886–892 (2009).
[CrossRef]

G. Zhou and F. Liu, “Far field structural characteristics of cosh-Gaussian beam,” Opt. Laser Technol. 40(2), 302–308 (2008).
[CrossRef]

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87(3), 547–552 (2007).
[CrossRef]

Zhou, M.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 57(20), 2039–2047 (2010).
[CrossRef]

Zhu, K.

K. Zhu, H. Tang, X. Wang, and T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Stuttg.) 113(5), 222–226 (2002).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87(3), 547–552 (2007).
[CrossRef]

Chin. Opt. Lett. (1)

Eur. Phys. J. D (1)

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

J. Mod. Opt. (2)

G. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 56(7), 886–892 (2009).
[CrossRef]

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 57(20), 2039–2047 (2010).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224(1-3), 5–12 (2003).
[CrossRef]

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004).
[CrossRef]

Opt. Express (2)

Opt. Laser Technol. (2)

G. Zhou and J. Zheng, “Beam propagation of a higher-order cosh-Gaussian beam,” Opt. Laser Technol. 41(2), 202–208 (2009).
[CrossRef]

G. Zhou and F. Liu, “Far field structural characteristics of cosh-Gaussian beam,” Opt. Laser Technol. 40(2), 302–308 (2008).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttg.) (1)

K. Zhu, H. Tang, X. Wang, and T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Stuttg.) 113(5), 222–226 (2002).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Normalized average intensity distribution in the x-direction of a higher-order cosh-Gaussian beam in the reference plane of z = 1km in turbulent atmosphere. (a) n = 2 and w 0 = 0.02m. (b) w 0 = 0.02m and Ω = 60m−1. (c) n = 2 and Ω = 60m−1.

Fig. 2
Fig. 2

Normalized average intensity distribution in the x-direction of a higher-order cosh-Gaussian beam in the reference plane of z = 2km in turbulent atmosphere. (a) n = 2 and w 0 = 0.02m. (b) w 0 = 0.02m and Ω = 60m−1. (c) n = 2 and Ω = 60m−1.

Fig. 3
Fig. 3

The effective beam size in the x-direction of a higher-order cosh-Gaussian beam versus the propagation distance z in turbulent atmosphere. (a) n = 2 and w 0 = 0.02m. (b) w 0 = 0.02m and Ω = 60m−1. (c) n = 2 and Ω = 60m−1.

Fig. 4
Fig. 4

The kurtosis parameter in the x-direction of a higher-order cosh-Gaussian beam versus the propagation distance z in turbulent atmosphere. (a) n = 2 and w 0 = 0.02m. (b) w 0 = 0.02m and Ω = 60m−1. (c) n = 2 and Ω = 60m−1.

Fig. 5
Fig. 5

(a) Normalized average intensity distribution in the x-direction of a higher-order cosh-Gaussian beam in the reference plane of z = 2km. (b) The effective beam size in the x-direction of a higher-order cosh-Gaussian beam versus the propagation distance z. (c) The kurtosis parameter in the x-direction of a higher-order cosh-Gaussian beam versus the propagation distance z. n = 2, Ω = 80m−1, and w 0 = 0.02m.

Equations (15)

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E ( x 0 , y 0 , 0 ) n = E ( x 0 , 0 ) n E ( y 0 , 0 ) n ,
E ( j 0 , 0 ) n = cosh n ( Ω j 0 ) exp ( j 0 2 w 0 2 ) , n = 1 , 2 , 3 , ,
E ( j 0 , 0 ) n = m = 0 n C m exp [ - ( j 0 b m ) 2 / w 0 2 ] ,
C m = a m exp [ ( m n 2 ) 2 δ ] , a m = n ! 2 n m ! ( n m ) ! , b m = ( m n 2 ) δ Ω ,
E ( x , y , z ) = 1 i λ B exp [ i k z + i k D ( x 2 + y 2 ) 2 B ] E n ( x 0 , y 0 , 0 ) × exp { i k 2 B [ A ( x 0 2 + y 0 2 ) 2 ( x x 0 + 2 y y 0 ) ] + ψ ( x 0 , y 0 , x , y ) } d x 0 d y 0 ,
< I ( x , y , z ) > = 1 λ 2 B 2 E n ( x 01 , y 01 , 0 ) E n ( x 02 , y 02 , 0 ) < exp [ ψ ( x 01 , y 01 , x , y ) + ψ ( x 02 , y 02 , x , y ) ] > × exp { i k 2 B [ A ( x 01 2 x 02 2 + y 01 2 y 02 2 ) 2 x ( x 01 x 02 ) 2 y ( y 01 y 02 ) ] } d x 01 d y 01 d x 02 d y 02 ,
< exp [ ψ ( x 01 , y 01 , x , y ) + ψ ( x 02 , y 02 , x , y ) ] > = exp [ ( x 01 x 02 ) 2 + ( y 01 y 02 ) 2 ρ 0 2 ] ,
ρ 0 = B [ 1.46 k 2 C n 2 0 L b 5 / 3 ( z ) d z ] 3 / 5 ,
< I ( x , y , z ) > = < I ( x , z ) > < I ( y , z ) > ,
< I ( j , z ) > = π λ B α 1 j α 2 j m = 0 n l = 0 n a m a l exp ( β 1 j 2 4 α 1 j + β 2 j 2 4 α 2 j ) ,
α 1 j = 1 w 0 2 + 1 ρ 0 2 + i k A 2 B , α 2 j = 1 w 0 2 + 1 ρ 0 2 1 α 1 j ρ 0 4 i k A 2 B , β 1 j = 2 b l w 0 2 + i k j B , β 2 j = 2 b m w 0 2 i k j B + β 1 j α 1 j ρ 0 2 .
W j = [ 2 j 2 < I ( x , y , z ) > d x d y < I ( x , y , z ) > d x d y ] 1 / 2 = 1 2 ξ [ m = 0 n l = 0 n a m a l ( 2 ξ j + η j 2 ) exp ( τ ) m = 0 n l = 0 n a m a l exp ( τ ) ] 1 / 2 ,
τ = η j 2 4 ξ j + b l 2 α 1 j w 0 4 + γ 1 j 2 4 α 2 j ,
γ 1 j = 2 b m w 0 2 + 2 b l α 1 j w 0 2 ρ 0 2 , γ 2 j = i k α 1 j B ρ 0 2 i k B , ξ j = k 2 4 α 1 j B 2 γ 2 j 2 4 α 2 j , η j = i k b l B α 1 j w 0 2 + γ 1 j γ 2 j 2 α 2 j .
K j = [ j 4 < I ( x , y , z ) > d x d y ] [ < I ( x , y , z ) > d x d y ] [ j 2 < I ( x , y , z ) > d x d y ] 2 = m = 0 n l = 0 n a m a l exp ( τ ) m = 0 n l = 0 n a m a l ( 12 ξ j 2 + 12 ξ j η j 2 + η j 4 ) exp ( τ ) [ m = 0 n l = 0 n a m a l ( 2 ξ j + η j 2 ) exp ( τ ) ] 2 .

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