Abstract

We present a detailed investigation of the second-order statistics of a twisted Gaussian Schell-model (TGSM) beam propagating in turbulent atmosphere. Based on the extended Huygens-Fresnel integral, analytical expressions for the second-order moments of the Wigner distribution function of a TGSM beam in turbulent atmosphere are derived. Evolution properties of the second-order statistics, such as the propagation factor, the effective radius of curvature (ERC) and the Rayleigh range, of a TGSM beam in turbulent atmosphere are explored in detail. Our results show that a TGSM beam is less affected by the turbulence than a GSM beam without twist phase. In turbulent atmosphere the Rayleigh range doesn’t equal to the distance where the ERC takes a minimum value, which is much different from the result in free space. The second-order statistics are closely determined by the parameters of the turbulent atmosphere and the initial beam parameters. Our results will be useful in long-distance free-space optical communications.

© 2010 OSA

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References

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  1. L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  2. Y. Baykal, “Average transmittance in turbulence for partially coherent sources,” Opt. Commun. 231(1-6), 129–136 (2004).
    [CrossRef]
  3. M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008).
    [CrossRef]
  4. Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
    [CrossRef] [PubMed]
  5. Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007).
    [CrossRef] [PubMed]
  6. C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009).
    [CrossRef] [PubMed]
  7. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
    [CrossRef]
  8. F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007).
    [CrossRef]
  9. E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992).
    [CrossRef]
  10. M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008).
    [CrossRef] [PubMed]
  11. L. C. Andrews, and R. L. Phillips, Laser beam propagation in the turbulent atmosphere, 2nd edition, (SPIE Press, Bellington, 2005).
  12. 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]
  13. Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
    [CrossRef] [PubMed]
  14. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993).
    [CrossRef]
  15. A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994).
    [CrossRef]
  16. D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
    [CrossRef]
  17. R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5(3), 331–343 (1996).
    [CrossRef]
  18. J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26(7), 405–407 (2001).
    [CrossRef]
  19. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
    [CrossRef]
  20. Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27(19), 1672–1674 (2002).
    [CrossRef]
  21. Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19(10), 2036–2042 (2002).
    [CrossRef]
  22. Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31(6), 685–687 (2006).
    [CrossRef] [PubMed]
  23. Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007).
    [CrossRef] [PubMed]
  24. Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 (2009).
    [CrossRef] [PubMed]
  25. C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009).
    [CrossRef] [PubMed]
  26. Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009).
    [CrossRef]
  27. 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]
  28. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
    [CrossRef] [PubMed]
  29. 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]
  30. Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008).
    [CrossRef] [PubMed]
  31. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
    [CrossRef] [PubMed]
  32. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010).
    [CrossRef] [PubMed]
  33. X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18(7), 6922–6928 (2010).
    [CrossRef] [PubMed]
  34. H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010).
    [CrossRef]
  35. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).
  36. G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2001).

2010 (3)

2009 (5)

2008 (4)

2007 (3)

2006 (3)

2005 (1)

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
[CrossRef] [PubMed]

2004 (2)

2002 (4)

2001 (2)

J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26(7), 405–407 (2001).
[CrossRef]

G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2001).

1996 (1)

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5(3), 331–343 (1996).
[CrossRef]

1994 (2)

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994).
[CrossRef]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[CrossRef]

1993 (1)

1992 (1)

1982 (1)

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[CrossRef]

Alavinejad, M.

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008).
[CrossRef]

Ambrosini, D.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[CrossRef]

Bagini, V.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[CrossRef]

Baykal, Y.

Cai, Y.

S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010).
[CrossRef] [PubMed]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 (2009).
[CrossRef] [PubMed]

C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009).
[CrossRef] [PubMed]

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009).
[CrossRef]

C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009).
[CrossRef] [PubMed]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[CrossRef] [PubMed]

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008).
[CrossRef] [PubMed]

Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007).
[CrossRef] [PubMed]

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007).
[CrossRef]

Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007).
[CrossRef] [PubMed]

Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31(6), 685–687 (2006).
[CrossRef] [PubMed]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
[CrossRef] [PubMed]

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]

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
[CrossRef] [PubMed]

Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27(19), 1672–1674 (2002).
[CrossRef]

Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19(10), 2036–2042 (2002).
[CrossRef]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[CrossRef]

Dan, Y.

Davidson, F. M.

Eyyuboglu, H. T.

Friberg, A. T.

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5(3), 331–343 (1996).
[CrossRef]

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994).
[CrossRef]

E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992).
[CrossRef]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[CrossRef]

Gbur, G.

G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2001).

Ge, D.

Ghafary, B.

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008).
[CrossRef]

Gori, F.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[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]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
[CrossRef] [PubMed]

Hu, L.

Ji, X.

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010).
[CrossRef]

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18(7), 6922–6928 (2010).
[CrossRef] [PubMed]

Korotkova, O.

Lin, Q.

Lu, X.

Movilla, J. M.

Mukunda, N.

Peschel, U.

Qu, J.

Ricklin, J. C.

Santarsiero, M.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[CrossRef]

Serna, J.

Simon, R.

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5(3), 331–343 (1996).
[CrossRef]

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993).
[CrossRef]

Sudol, R. J.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[CrossRef]

Tervonen, E.

Turunen, J.

Wang, F.

Wolf, E.

G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2001).

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5(3), 331–343 (1996).
[CrossRef]

Yao, M.

Yuan, Y.

Zhang, B.

Zhao, C.

Zhu, S.

Zhu, S. Y.

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
[CrossRef] [PubMed]

Appl. Phys. B (2)

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (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]

J. Mod. Opt. (2)

G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2001).

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[CrossRef]

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

Opt. Commun. (2)

Y. Baykal, “Average transmittance in turbulence for partially coherent sources,” Opt. Commun. 231(1-6), 129–136 (2004).
[CrossRef]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[CrossRef]

Opt. Express (12)

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[CrossRef] [PubMed]

Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007).
[CrossRef] [PubMed]

C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009).
[CrossRef] [PubMed]

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]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
[CrossRef] [PubMed]

Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 (2009).
[CrossRef] [PubMed]

C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009).
[CrossRef] [PubMed]

Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008).
[CrossRef] [PubMed]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[CrossRef] [PubMed]

S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010).
[CrossRef] [PubMed]

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18(7), 6922–6928 (2010).
[CrossRef] [PubMed]

Opt. Lasers Eng. (1)

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5(3), 331–343 (1996).
[CrossRef]

Other (3)

L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

L. C. Andrews, and R. L. Phillips, Laser beam propagation in the turbulent atmosphere, 2nd edition, (SPIE Press, Bellington, 2005).

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

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Figures (7)

Fig. 1
Fig. 1

Normalized M 2-factor of a TGSM beam on propagation in turbulent atmosphere for different values of the structure constant C n 2 and the inner scale l 0 .

Fig. 2
Fig. 2

Normalized M 2-factor of a TGSM beam on propagation in turbulent atmosphere for different values of σ g 0 and μ 0 .

Fig. 3
Fig. 3

Deviation percentage of the normalized M 2-factor versus the propagation distance z for different values of μ 0 .

Fig. 4
Fig. 4

ERC of a TGSM beam in turbulent atmosphere versus the propagation distance z for different values of the structure constant C n 2 and the inner scale l 0 .

Fig. 5
Fig. 5

ERC of a TGSM beam in turbulent atmosphere versus the propagation distance for different values of μ 0 and σ g 0 .

Fig. 6
Fig. 6

Deviation percentage of the ERC of a TGSM or GSM beam versus the propagation distance z.

Fig. 7
Fig. 7

z R and z m of a TGSM beam in turbulent atmosphere different values of twist factor μ 0 and coherence width σ g 0

Equations (41)

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W 0 ( r 1 ' , r 2 ' ; 0 ) = exp [ r 1 ' 2 + r 2 ' 2 4 σ I 0 2 ( r 1 ' r 2 ' ) 2 2 σ g 0 2 i k μ 0 2 ( r 1 ' r 2 ' ) T J ( r 1 ' + r 2 ' ) ] ,
J = ( 0 1 1 0 ) .  
W ( r , r d ; z ) = 1 λ 2 z 2 W 0 ( r ' , r d ' ; 0 ) exp [ i k z ( r r ' ) ( r d r d ' ) H ( r d , r d ' ; z ) ] d 2 r ' d r d '
W 0 ' ( r ' , r d ' ; 0 ) = W 0 ( r 1 ' , r 2 ' ; 0 ) = W 0 ( r ' + r d ' / 2 , r ' r d ' / 2 ; 0 )
r = ( r 1 + r 2 ) / 2 , r d = r 1 r 2 , r ' = ( r 1 ' + r 2 ' ) / 2 , r d ' = r 1 ' r 2 ' .
exp [ H ( r d , r d ' ; z ) ] = 4 π 2 k 2 z 0 1 d ξ 0 [ 1 J 0 ( κ | r d ' ξ + ( 1 ξ ) r d | ) ] Φ n ( κ ) κ d κ ,
h ( r , θ ; z ) = ( 1 λ ) 2 W ( r , r d ; z ) exp [ i k θ r d ] d 2 r d ,
h ( r , θ ; z ) = 1 ( 2 π ) 2 ( 1 λ ) 2 W 0 ( r " , r d + z k κ d ; 0 ) exp ( i κ d r " i κ d r i k θ r d )                    × exp [ H ( r d , r d + z k κ d ; z ) ] d 2 κ d d 2 r " d 2 r d ,
W 0 ( r ' , r d ' ; 0 ) 1 ( 2 π ) 2 W 0 ( r " , r d ' ; 0 ) exp [ i κ d ( r " r ' ) ] d 2 κ d d 2 r " .
W 0 ( r " , r d + z k κ d ; 0 ) = exp [ r " 2 2 σ I 0 2 ( 1 8 σ I 0 2 + 1 2 σ g 0 2 ) ( r d + z k κ d ) 2 ] exp [ i k μ 0 ( r d + z k κ d ) T J r " ] .
h ( r , θ ; z ) = 1 ( 2 π ) 2 2 π σ I 0 2 ( 1 λ ) 2 exp [ σ I 0 2 2 ( κ d x + ( k μ 0 y d + μ 0 z κ d y ) ) 2 ]                    × exp [ σ I 0 2 2 ( κ d y ( k μ 0 x d + μ 0 z κ d x ) ) 2 ] exp [ ( 1 8 σ I 0 2 + 1 2 σ g 0 2 ) ( r d + z k κ d ) 2 ]                   × exp [ i r κ d i k θ r d ] exp ( H ( r d , r d + z k κ d ; z ) ) d 2 κ d d 2 r d .                    
x n 1 y n 2 θ x m 1 θ y m 2 = 1 P x n 1 y n 2 θ x m 1 θ y m 2 h ( r , θ , z ) d 2 r d 2 θ ,
P = h ( r , θ , z ) d 2 r d 2 θ .
r ( z ) 2 = x ( z ) 2 + y ( z ) 2 = 2 σ I 0 2 + 2 A z 2 + 4 π 2 T z 3 / 3 ,
r ( z ) θ ( z ) = x ( z ) θ x ( z ) + y ( z ) θ y ( z ) = 2 A z + 2 π 2 T z 2 ,
θ ( z ) 2 = θ x ( z ) 2 + θ y ( z ) 2 = 2 A + 4 π 2 T z ,
A = 1 / ( 4 k 2 σ I 0 2 ) + 1 / ( k 2 σ g 0 2 ) + μ 0 2 σ I 0 2 ,
T = 0 Φ n ( κ ) κ 3 d κ ,
P = 2 π σ I 0 2 .
δ ( s ) = 1 2 π exp ( i s x ) d x ,      
δ n ( s ) = 1 2 π ( i x ) n exp ( i s x ) d x ,   ( n = 0 ,   1 ,   2 ) ,
f ( x ) δ n ( x ) d x = ( 1 ) n f ( n ) ( 0 ) ,   ( n = 1 ,   2 ) .
M 2 ( z ) = k [ r ( z ) 2 θ ( z ) 2 r ( z ) θ ( z ) 2 ] 1 / 2 .
M 2 ( z ) = [ ( M 2 ( 0 ) ) 2 + ( 8 σ I 0 2 + 8 A z 2 / 3 + 4 π 2 T z 3 / 3 ) k 2 π 2 T z ] 1 / 2 ,
M 2 ( 0 ) = 1 + 4 μ 0 2 k 2 σ I 0 4 + 4 σ I 0 2 / σ g 0 2 .
Φ n ( κ ) = 0.033 C n 2 κ 11 / 3 exp ( κ 2 / κ m 2 ) ,
T = 0 Φ n ( κ ) κ 3 d κ = 0.1661 C n 2 l 0 1 / 3
Δ M 2 ( z ) = | M 2 ( z ) / M 2 ( 0 ) | μ 0 M 2 ( z ) / M 2 ( 0 ) | μ 0 = 0 | M 2 ( z ) / M 2 ( 0 ) | μ 0 = 0 .
R ( z ) = r ( z ) 2 / r ( z ) θ ( z ) .
R ( z ) = z + σ I 0 2 π 2 T z 3 / 3 A z + π 2 T z 2 .
Δ R ( z ) = | R ( z ) | t u r R ( z ) | f r e e | R ( z ) | f r e e .
z c = ( 3 σ I 0 2 π 2 T ) 1 / 3 .
r ( z R ) 2 2 r ( 0 ) 2 = 0 ,
d R ( z ) / d z | z = z m = 0.
4 π 2 T z R 3 / 3 + 2 A z 2 2 σ I 0 2 = 0 ,
2 π 2 T 2 z m 4 + 4 π 2 A T z m 3 + 3 A 2 z m 2 6 π 2 T z m σ I 0 2 3 A σ I 0 2 = 0.
z R = z m = σ I 0 ( 1 / ( 4 k 2 σ I 0 2 ) + 1 / ( k 2 σ g 0 2 ) + μ 0 2 σ I 0 2 ) 1 / 2 .
z R = ( A 2 + M 1 2 A M 1 ) / ( 2 π 2 T M 1 ) ,
z m = ( ( A 3 + 6 π 4 T 2 σ I 0 2 ) / N 3 N 3 2 + N 3 A ) / ( 2 π 2 T ) ,
M 1 = ( A 3 + 6 π 4 T 2 σ I 0 2 + 2 9 π 8 T 4 σ I 0 4 3 π 4 A 3 T 2 σ I 0 2 ) 1 / 3 , N 2 = 4 N 1 + 4 N 1 2 ( 54 A 6 ) 2
N 3 = 3 A 4 / N 2 1 / 3 + N 2 1 / 3 / 12 , N 1 = 54 ( A 3 + 6 π 4 T 2 σ I 0 2 ) 2 .

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