Abstract

A stochastic beam generated by a recently introduced source of Schell type with cosine-Gaussian spectral degree of coherence is shown to possess interesting novel features on propagation in isotropic and homogeneous atmospheric turbulence with general non-Kolmogorov power spectrum. It is shown that while at small distances from the source the beam’s intensity exhibits annular profile with adjustable area of the dark region, the center disappears at sufficiently large distances and the beam’s intensity tends to Gaussian form. Hence the 3D bottle beam is produced by the cumulative effect of the random source and the atmosphere. The distances at which the on-axis beam intensity has local minima and maxima are shown to have analytic dependence on the source and the atmospheric parameters. And the influence of the fractal constant of the atmospheric power spectrum and refractive-index structure constant on beam characteristics is analyzed in depth. The novel double-cycle qualitative change in the degree of coherence is shown to occur on atmospheric propagation which was not previously known for any other beams.

© 2013 OSA

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics(Cambridge University, Cambridge, 1995).
    [CrossRef]
  2. F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun.64, 311–316 (1987).
    [CrossRef]
  3. F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett.33, 1857–1859 (2008).
    [CrossRef] [PubMed]
  4. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett.36, 4104–4106 (2011).
    [CrossRef] [PubMed]
  5. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett.37, 2970–2972 (2012).
    [CrossRef] [PubMed]
  6. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A29, 2159–2164 (2012).
    [CrossRef]
  7. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett.38, 91–93 (2013).
    [CrossRef] [PubMed]
  8. Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” (Opt. Lett., in press).
  9. Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A372, 4654–4660 (2008).
    [CrossRef]
  10. M. Alavinejad, G. Taherabadi, N. Hadilou, and B. Ghafary, “Changes in the coherence properties of partially coherent dark hollow beam propagating through atmospheric turbulence,” Opt. Commun.288, 1–6 (2013).
    [CrossRef]
  11. H. T. Eyyuboğlu and Y. Baykal, “Cosine-Gaussian laser beam intensity in turbulent atmosphere,”Proc. SPIE5743, 131–141 (2004).
    [CrossRef]
  12. 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]
  13. J. Pu, X. Liu, and S. Nemoto, “Partially coherent bottle beams,” Opt. Commun.252, 7–11 (2005).
    [CrossRef]
  14. L.C. Andrews and R. L. Phillips, Laser Beam Propagation in Random Media(SPIE Press, Bellington, 2005).
    [CrossRef]
  15. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng.47, 026003 (2008).
    [CrossRef]
  16. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett.35, 715–718 (2010).
    [CrossRef] [PubMed]
  17. W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun.271(1), 1–8 (2007).
    [CrossRef]
  18. O. Korotkova and E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett.35, 3772–3774 (2010).
    [CrossRef] [PubMed]
  19. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express18, 10650–10658 (2010).
    [CrossRef] [PubMed]

2013 (2)

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett.38, 91–93 (2013).
[CrossRef] [PubMed]

M. Alavinejad, G. Taherabadi, N. Hadilou, and B. Ghafary, “Changes in the coherence properties of partially coherent dark hollow beam propagating through atmospheric turbulence,” Opt. Commun.288, 1–6 (2013).
[CrossRef]

2012 (2)

2011 (1)

2010 (3)

2008 (3)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng.47, 026003 (2008).
[CrossRef]

F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett.33, 1857–1859 (2008).
[CrossRef] [PubMed]

Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A372, 4654–4660 (2008).
[CrossRef]

2007 (2)

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]

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun.271(1), 1–8 (2007).
[CrossRef]

2005 (1)

J. Pu, X. Liu, and S. Nemoto, “Partially coherent bottle beams,” Opt. Commun.252, 7–11 (2005).
[CrossRef]

2004 (1)

H. T. Eyyuboğlu and Y. Baykal, “Cosine-Gaussian laser beam intensity in turbulent atmosphere,”Proc. SPIE5743, 131–141 (2004).
[CrossRef]

1987 (1)

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun.64, 311–316 (1987).
[CrossRef]

Alavinejad, M.

M. Alavinejad, G. Taherabadi, N. Hadilou, and B. Ghafary, “Changes in the coherence properties of partially coherent dark hollow beam propagating through atmospheric turbulence,” Opt. Commun.288, 1–6 (2013).
[CrossRef]

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng.47, 026003 (2008).
[CrossRef]

Andrews, L.C.

L.C. Andrews and R. L. Phillips, Laser Beam Propagation in Random Media(SPIE Press, Bellington, 2005).
[CrossRef]

Baykal, Y.

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 and Y. Baykal, “Cosine-Gaussian laser beam intensity in turbulent atmosphere,”Proc. SPIE5743, 131–141 (2004).
[CrossRef]

Borghi, R.

Cai, Y.

Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A372, 4654–4660 (2008).
[CrossRef]

Eyyuboglu, H. T.

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 and Y. Baykal, “Cosine-Gaussian laser beam intensity in turbulent atmosphere,”Proc. SPIE5743, 131–141 (2004).
[CrossRef]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng.47, 026003 (2008).
[CrossRef]

Ghafary, B.

M. Alavinejad, G. Taherabadi, N. Hadilou, and B. Ghafary, “Changes in the coherence properties of partially coherent dark hollow beam propagating through atmospheric turbulence,” Opt. Commun.288, 1–6 (2013).
[CrossRef]

Gori, F.

F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett.33, 1857–1859 (2008).
[CrossRef] [PubMed]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun.64, 311–316 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun.64, 311–316 (1987).
[CrossRef]

Guo, H.

Hadilou, N.

M. Alavinejad, G. Taherabadi, N. Hadilou, and B. Ghafary, “Changes in the coherence properties of partially coherent dark hollow beam propagating through atmospheric turbulence,” Opt. Commun.288, 1–6 (2013).
[CrossRef]

Korotkova, O.

Lajunen, H.

Liu, L.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun.271(1), 1–8 (2007).
[CrossRef]

Liu, X.

J. Pu, X. Liu, and S. Nemoto, “Partially coherent bottle beams,” Opt. Commun.252, 7–11 (2005).
[CrossRef]

Lu, W.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun.271(1), 1–8 (2007).
[CrossRef]

Luo, B.

Mandel, L.

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

Mei, Z.

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett.38, 91–93 (2013).
[CrossRef] [PubMed]

Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” (Opt. Lett., in press).

Nemoto, S.

J. Pu, X. Liu, and S. Nemoto, “Partially coherent bottle beams,” Opt. Commun.252, 7–11 (2005).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun.64, 311–316 (1987).
[CrossRef]

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng.47, 026003 (2008).
[CrossRef]

L.C. Andrews and R. L. Phillips, Laser Beam Propagation in Random Media(SPIE Press, Bellington, 2005).
[CrossRef]

Pu, J.

J. Pu, X. Liu, and S. Nemoto, “Partially coherent bottle beams,” Opt. Commun.252, 7–11 (2005).
[CrossRef]

Saastamoinen, T.

Sahin, S.

Santarsiero, M.

Shchepakina, E.

Sun, J.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun.271(1), 1–8 (2007).
[CrossRef]

Taherabadi, G.

M. Alavinejad, G. Taherabadi, N. Hadilou, and B. Ghafary, “Changes in the coherence properties of partially coherent dark hollow beam propagating through atmospheric turbulence,” Opt. Commun.288, 1–6 (2013).
[CrossRef]

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng.47, 026003 (2008).
[CrossRef]

Wang, F.

Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A372, 4654–4660 (2008).
[CrossRef]

Wolf, E.

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

Wu, G.

Yang, Q.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun.271(1), 1–8 (2007).
[CrossRef]

Yu, S.

Zhu, Y.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun.271(1), 1–8 (2007).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (5)

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun.64, 311–316 (1987).
[CrossRef]

M. Alavinejad, G. Taherabadi, N. Hadilou, and B. Ghafary, “Changes in the coherence properties of partially coherent dark hollow beam propagating through atmospheric turbulence,” Opt. Commun.288, 1–6 (2013).
[CrossRef]

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]

J. Pu, X. Liu, and S. Nemoto, “Partially coherent bottle beams,” Opt. Commun.252, 7–11 (2005).
[CrossRef]

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun.271(1), 1–8 (2007).
[CrossRef]

Opt. Eng. (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng.47, 026003 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett (1)

Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” (Opt. Lett., in press).

Opt. Lett. (6)

Phys. Lett. A (1)

Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A372, 4654–4660 (2008).
[CrossRef]

Proc. SPIE (1)

H. T. Eyyuboğlu and Y. Baykal, “Cosine-Gaussian laser beam intensity in turbulent atmosphere,”Proc. SPIE5743, 131–141 (2004).
[CrossRef]

Other (2)

L.C. Andrews and R. L. Phillips, Laser Beam Propagation in Random Media(SPIE Press, Bellington, 2005).
[CrossRef]

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

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Figures (5)

Fig. 1
Fig. 1

The evolution of spectral density S [Eq. (9)] of the CGSM beam on propagation (a) in free space; (b) in the atmosphere: n = 0 (thick solid brown curve), n = 1 (dotted black curve), n = 2 (dashed green curve), n = 3 (dash-dotted red curve) and n = 4 (thin solid blue curve).

Fig. 2
Fig. 2

The evolution of transverse cross-sections of the spectral density S for the same five CGSM beams as in Fig. 1, for selected distances from the source: (a) z = 0 m; (b) z = 30 m; (c) z = 103 m and (d) z = 105 m.

Fig. 3
Fig. 3

The behavior of the transverse cross-sections of the spectral density S for n = 2, z = 10000 m for free space (black curve) and (a) C ˜ n 2 = 10 13 and α = 3.1 (blue dash-dotted curve), α = 3.67 (red dashed curve) ; (b) α = 3.1 and C ˜ n 2 = 10 13 (blue dash-dotted curve), C ˜ n 2 = 10 14 (red dashed curve).

Fig. 4
Fig. 4

Contour plots of the spectral density S as a function of propagation distance z [m] from the source (horizontal axis) and transverse position ρ [m] from the beam axis (vertical axis) for the CGSM beam with n = 4 and α = 3.1.

Fig. 5
Fig. 5

The transverse spectral degree of coherence μ [Eq. (10)] for the same beams as in Fig. 1 as a function of separation between two points at distances: (a) z = 0 m; (b) z = 300 m; (c) z = 600 m; (d) z = 1500 m; (e) z = 104 m and (f) z = 106 m.

Equations (11)

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W ( 0 ) ( ρ 1 ' , ρ 2 ' ) = exp ( | ρ 1 ' | 2 + | ρ 2 ' | 2 4 σ 2 ) cos [ n 2 π ( ρ 2 ' ρ 1 ' ) δ ] exp [ | ρ 2 ' ρ 1 ' | 2 2 δ 2 ] ,
W ( ρ 1 , ρ 2 , z ) = ( k / 2 π z ) 2 d 2 ρ 1 ' d 2 ρ 2 ' W ( 0 ) ( ρ 1 ' , ρ 2 ' ) × exp { ( i k z / 2 z ) [ ( ρ 1 ρ 1 ' ) 2 ( ρ 2 ρ 2 ' ) 2 ] } exp [ ϕ * ( ρ 1 , ρ 1 ' , z ) + ϕ ( ρ 2 , ρ 2 ' , z ) ] M ,
exp [ ϕ * ( ρ 1 , ρ 1 ' , z ) + ϕ ( ρ 2 , ρ 2 ' , z ) ] M = exp { ( π 2 k 2 z / 3 ) × [ ( ρ 1 ρ 2 ) 2 + ( ρ 1 ρ 2 ) ( ρ 1 ' ρ 2 ' ) + ( ρ 1 ' ρ 2 ' ) 2 ] 0 κ 3 Φ n ( κ ) d κ } ,
W ( ρ 1 , ρ 2 , z ) = k 2 σ 2 4 z 2 Δ ( z ) exp [ ( ρ 1 ρ 2 ) 2 R ( z ) i k 2 z ( ρ 1 2 ρ 2 2 ) ] [ exp ( γ + 2 Δ ( z ) ) + exp ( γ 2 Δ ( z ) ) ] ,
1 R ( z ) = k 2 σ 2 2 z 2 + k 2 π 2 z 3 I , I = 0 κ 3 Φ n ( κ ) d κ , Δ ( z ) = 1 R ( z ) + 1 2 ζ 2 , 1 ζ 2 = 1 4 σ 2 + 1 δ 2 , γ ± = ( 3 k 2 σ 2 4 z 2 1 2 R ( z ) ) ( ρ 1 ρ 2 ) + i k 4 z ( ρ 1 + ρ 2 ) ± in 2 π 2 δ .
Φ n ( κ ) = A ( α ) C ˜ n 2 exp [ κ 2 / κ m 2 ] / ( κ 2 + κ 0 2 ) α / 2 , 0 κ < , 3 < α < 4 ,
c ( α ) = { Γ [ ( 5 α ) / 2 ] A ( α ) ( 2 π / 3 ) } 1 / ( α 5 ) , A ( α ) = Γ ( α 1 ) cos ( α π / 2 ) / ( 4 π 2 ) ,
I = 0 κ 3 Φ n ( κ ) d κ = A ( α ) C ˜ n 2 2 ( α 2 ) [ κ m 2 α β exp ( κ 0 2 κ m 2 ) Γ ( 2 α 2 , κ 0 2 κ m 2 ) 2 κ 0 4 α ] ,
S ( ρ , z ) = W ( ρ , ρ , z ) ,
μ ( ρ 1 , ρ 2 , z ) = W ( ρ 1 , ρ 2 , z ) / W ( ρ 1 , ρ 1 , z ) W ( ρ 2 , ρ 2 , z ) .
z 4 + z 3 ( 5 ζ 2 π n 2 δ 2 ) 1 2 π 2 k 2 I + z 2 3 2 π 4 k 4 I 2 ζ 4 + z 2 σ 2 π 2 I + 3 σ 2 2 π 4 k 2 I 2 ( 1 ζ 2 π n 2 δ 2 ) = 0.

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