Abstract

The propagation of a partially coherent cosine-Gaussian beam through a paraxial ABCD optical system in turbulent atmosphere is investigated. Analytical expression for the average intensity in the output plane is derived. The presented formula only covers optical systems without inherent apertures, where ABCD are all real-valued. As a special case of the general formula, the analytical formula for the average intensity of a partially coherent cosh-Gaussian beam through an ABCD optical system in turbulent atmosphere is also presented, respectively. The properties of the average intensity of the partially coherent cosine-Gaussian beam are investigated with a numerical example, and the dependence of the average intensity distribution on the spatial correlation length of a partially coherent cosine-Gaussian beam is mainly discussed.

© 2009 OSA

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  22. X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express 15(26), 17613–17618 (2007).
    [CrossRef] [PubMed]
  23. 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]
  24. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891–2901 (2007).
    [CrossRef]
  25. H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278(1), 17–22 (2007).
    [CrossRef]
  26. A. Yang, E. Zhang, X. Ji, and B. Lü, “Angular spread of partially coherent Hermite-cosh-Gaussian beams propagating through atmospheric turbulence,” Opt. Express 16(12), 8366–8380 (2008).
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    [CrossRef]

2008 (4)

G. Zhou, R. Qü, and L. Sun, “The structural properties of cosine-Gaussian beam in the far field,” J. Mod. Opt. 55(15), 2485–2495 (2008).
[CrossRef]

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

A. Yang, E. Zhang, X. Ji, and B. Lü, “Angular spread of partially coherent Hermite-cosh-Gaussian beams propagating through atmospheric turbulence,” Opt. Express 16(12), 8366–8380 (2008).
[CrossRef] [PubMed]

X. Chu, Z. Lu, and Y. Wu, “The propagation of a flattened circular Gaussian beam through an optical system in turbulent atmosphere,” Appl. Phys. B 92(1), 119–122 (2008).
[CrossRef]

2007 (7)

2006 (1)

X. Kang and B. Lü, “Characterization of nonparaxial truncated cosine-Gaussian beams and the beam quality in the far field,” Chin. Phys. Lett. 23(9), 2430–2433 (2006).
[CrossRef]

2005 (3)

2004 (3)

Z. Mei, D. Zhao, D. Sun, and J. Gu, “The M2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttg.) 115(2), 89–93 (2004).

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

H. T. Eyyuboğlu and Y. Baykal, “Cosine-Gaussian laser beam intensity in turbulent atmosphere,” Proc. SPIE 5743, 131–141 (2004).
[CrossRef]

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 (2)

S. Yu, H. Guo, X. Fu, and W. Hu, “Propagation perties of elegant Hermite-cosh-Gaussian laser beams,” Opt. Commun. 204, 59–66 (2002).

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).

2001 (1)

X. Wang and B. Lü, “The M2 factor of Hermite-cosh-Gaussian beams,” J. Mod. Opt. 48, 2097–2103 (2001).

2000 (1)

B. Lü and S. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178(4-6), 275–281 (2000).
[CrossRef]

1999 (2)

1998 (2)

1987 (1)

Baykal, Y.

Cai, Y.

Casperson, L. W.

Chen, Z.

Chu, X.

X. Chu, Z. Lu, and Y. Wu, “The propagation of a flattened circular Gaussian beam through an optical system in turbulent atmosphere,” Appl. Phys. B 92(1), 119–122 (2008).
[CrossRef]

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.

Fu, X.

S. Yu, H. Guo, X. Fu, and W. Hu, “Propagation perties of elegant Hermite-cosh-Gaussian laser beams,” Opt. Commun. 204, 59–66 (2002).

Gu, J.

Z. Mei, D. Zhao, D. Sun, and J. Gu, “The M2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttg.) 115(2), 89–93 (2004).

Guo, H.

S. Yu, H. Guo, X. Fu, and W. Hu, “Propagation perties of elegant Hermite-cosh-Gaussian laser beams,” Opt. Commun. 204, 59–66 (2002).

Hanson, S. G.

Hu, W.

S. Yu, H. Guo, X. Fu, and W. Hu, “Propagation perties of elegant Hermite-cosh-Gaussian laser beams,” Opt. Commun. 204, 59–66 (2002).

Ji, J.

Ji, X.

Kang, X.

X. Kang and B. Lü, “Characterization of nonparaxial truncated cosine-Gaussian beams and the beam quality in the far field,” Chin. Phys. Lett. 23(9), 2430–2433 (2006).
[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, 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).

Lu, Z.

X. Chu, Z. Lu, and Y. Wu, “The propagation of a flattened circular Gaussian beam through an optical system in turbulent atmosphere,” Appl. Phys. B 92(1), 119–122 (2008).
[CrossRef]

Lü, B.

A. Yang, E. Zhang, X. Ji, and B. Lü, “Angular spread of partially coherent Hermite-cosh-Gaussian beams propagating through atmospheric turbulence,” Opt. Express 16(12), 8366–8380 (2008).
[CrossRef] [PubMed]

X. Kang and B. Lü, “Characterization of nonparaxial truncated cosine-Gaussian beams and the beam quality in the far field,” Chin. Phys. Lett. 23(9), 2430–2433 (2006).
[CrossRef]

X. Wang and B. Lü, “The M2 factor of Hermite-cosh-Gaussian beams,” J. Mod. Opt. 48, 2097–2103 (2001).

B. Lü and S. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178(4-6), 275–281 (2000).
[CrossRef]

B. Lü, H. Ma, and B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164(4-6), 165–170 (1999).
[CrossRef]

B. Lü, B. Zhang, and H. Ma, “Beam-propagation factor and mode-coherence coefficients of hyperbolic-cosine Gaussian beams,” Opt. Lett. 24(10), 640–642 (1999).
[CrossRef]

Luo, S.

B. Lü and S. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178(4-6), 275–281 (2000).
[CrossRef]

Ma, H.

Mao, H.

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]

Mei, Z.

Z. Mei, D. Zhao, D. Sun, and J. Gu, “The M2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttg.) 115(2), 89–93 (2004).

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]

Qü, R.

G. Zhou, R. Qü, and L. Sun, “The structural properties of cosine-Gaussian beam in the far field,” J. Mod. Opt. 55(15), 2485–2495 (2008).
[CrossRef]

Shi, Z.

Song, Y.

Sun, D.

Z. Mei, D. Zhao, D. Sun, and J. Gu, “The M2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttg.) 115(2), 89–93 (2004).

Sun, L.

G. Zhou, R. Qü, and L. Sun, “The structural properties of cosine-Gaussian beam in the far field,” J. Mod. Opt. 55(15), 2485–2495 (2008).
[CrossRef]

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).

Tovar, A. A.

Wang, S.

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).

X. Wang and B. Lü, “The M2 factor of Hermite-cosh-Gaussian beams,” J. Mod. Opt. 48, 2097–2103 (2001).

Wu, Y.

X. Chu, Z. Lu, and Y. Wu, “The propagation of a flattened circular Gaussian beam through an optical system in turbulent atmosphere,” Appl. Phys. B 92(1), 119–122 (2008).
[CrossRef]

Yang, A.

Yu, S.

S. Yu, H. Guo, X. Fu, and W. Hu, “Propagation perties of elegant Hermite-cosh-Gaussian laser beams,” Opt. Commun. 204, 59–66 (2002).

Yura, H. T.

Zhang, B.

Zhang, E.

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.

Z. Mei, D. Zhao, D. Sun, and J. Gu, “The M2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttg.) 115(2), 89–93 (2004).

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]

Zhou, G.

G. Zhou, R. Qü, and L. Sun, “The structural properties of cosine-Gaussian beam in the far field,” J. Mod. Opt. 55(15), 2485–2495 (2008).
[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]

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).

Appl. Opt. (2)

Appl. Phys. B (2)

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]

X. Chu, Z. Lu, and Y. Wu, “The propagation of a flattened circular Gaussian beam through an optical system in turbulent atmosphere,” Appl. Phys. B 92(1), 119–122 (2008).
[CrossRef]

Chin. Phys. Lett. (1)

X. Kang and B. Lü, “Characterization of nonparaxial truncated cosine-Gaussian beams and the beam quality in the far field,” Chin. Phys. Lett. 23(9), 2430–2433 (2006).
[CrossRef]

J. Mod. Opt. (2)

G. Zhou, R. Qü, and L. Sun, “The structural properties of cosine-Gaussian beam in the far field,” J. Mod. Opt. 55(15), 2485–2495 (2008).
[CrossRef]

X. Wang and B. Lü, “The M2 factor of Hermite-cosh-Gaussian beams,” J. Mod. Opt. 48, 2097–2103 (2001).

J. Opt. Soc. Am. A (6)

Opt. Commun. (6)

H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278(1), 17–22 (2007).
[CrossRef]

H. T. Eyyuboğlu, “Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Commun. 245(1-6), 37–47 (2005).
[CrossRef]

B. Lü, H. Ma, and B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164(4-6), 165–170 (1999).
[CrossRef]

B. Lü and S. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178(4-6), 275–281 (2000).
[CrossRef]

S. Yu, H. Guo, X. Fu, and W. Hu, “Propagation perties of elegant Hermite-cosh-Gaussian laser beams,” Opt. Commun. 204, 59–66 (2002).

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]

Opt. Express (3)

Opt. Laser Technol. (1)

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

Opt. Lett. (2)

Optik (Stuttg.) (2)

Z. Mei, D. Zhao, D. Sun, and J. Gu, “The M2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttg.) 115(2), 89–93 (2004).

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).

Proc. SPIE (1)

H. T. Eyyuboğlu and Y. Baykal, “Cosine-Gaussian laser beam intensity in turbulent atmosphere,” Proc. SPIE 5743, 131–141 (2004).
[CrossRef]

Other (1)

S. Wang, and D. Zhao, Matrix Optics (CHEP and Springer, Beijing, 2000), p. 15.

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of a two-lens system in turbulent atmosphere.

Fig. 2
Fig. 2

The intensity distribution in the x direction of a partially coherent cosine-Gaussian beam in the source plane. (a) Ω = 30m−1. (b) Ω = 10m−1.

Fig. 3
Fig. 3

Average intensity distribution in the x direction of a partially coherent cosine-Gaussian beam at different propagation distance in turbulent atmosphere. Ω = 30m−1, and Cn 2 = 10−14m2/3. The solid and the dashed curves correspond to σ = 0.2m and σ = 0.007m, respectively. (a) L = 2km. (b) L = 3.5km. (c) L = 10km.

Fig. 5
Fig. 5

Average intensity distribution in the x direction of a partially coherent cosine-Gaussian beam at different propagation distance in turbulent atmosphere. Ω = 30m−1, Cn 2 = 10−16m-2/3, and the rest of parameters are same as those in Fig. 3.

Fig. 4
Fig. 4

Average intensity distribution in the x direction of a partially coherent cosine-Gaussian beam at different propagation distance in turbulent atmosphere. Ω = 10m−1, and the rest of parameters are same as those in Fig. 3.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

W(x01,x02,y01,y02,0)=W(x01,x02,0)W(y01,y02,0),
with W(x01,x02,0) and W(y01,y02,0) given by
W(j01,j02,0)=exp(j012+j022w02)cos(Ωj01)cos(Ωj02)exp[(j01j02)22σ2],
<I(x,y,z)>=W(x,x,y,y,z)=W(x01,x02,y01,y02,0)<exp[ψ(x01,y01,x,y)+ψ(x02,y02,x,y)]>×(k2πB)2exp{ik2B[A(x012x022+y012y022)2x(x01x02)2y(y01y02)]}dx01dy01dx02dy02,
<exp[ψ(x01,y01,x,y)+ψ(x02,y02,x,y)]>=exp[(x01x02)2+(y01y02)2ρ02],
ρ0=Bβ=B{1.46k2Cn20Lb5/3(z)dz}3/5,
<I(x,y,z)>=<I(x,z)><I(y,z)>,
<I(j,z)>=w02wexp(2j2w22Ω2B2k2w2){cosh(4ΩBkw2j)+exp[2Ω2B2k2w2(1τ12+1τ22)]cos(2Ωw02Aw2j)}
τ1=σ/w0,τ2=Bβ/(2w0),
w=[A2w02+4B2k2w02(1+1τ12+1τ22)]1/2.
<I(j,z)>=w02wexp(2j2w2)[1+exp(4Ω2w02k2w2β2)cos(2Ωw02Aw2j)],
w=[A2w02+8/(k2β2]1/2.
<I(j,z)>=w02wexp(2j2w22Ω2B2k2w2)[cosh(4ΩBkw2j)+exp(4Ω2w02k2w2β2)cos(2Ωw02Aw2j)],
w=[A2w02+4(B2β2+2w02)/(k2w02β2)]1/2.
<I(x,y,z)>=<I(x,z)><I(y,z)>,
<I(j,z)>=w02wexp(2j2w2+2Ω2B2k2w2){cos(4ΩBkw2j)+exp[2Ω2w02k2w2(1τ12+1τ22)]cosh(2Ωw02Aw2j)}.
A=[f1f2(z2+z3)f2+z3(z2f1)]/f1f2,
B=[Lf1f2z1(z2+z3)f2z3(z1+z2)f1+z1z2z3]/f1f2,
b(z)={L/3+z,0<zL/4,7L/12z/2,L/4<z3L/4,Lz,3L/4<zL.

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