## Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere |

Optics Express, Vol. 18, Issue 24, pp. 24661-24672 (2010)

http://dx.doi.org/10.1364/OE.18.024661

Acrobat PDF (1285 KB)

### 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

## 1. Introduction

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]

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]

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]

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

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]

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]

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]

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]

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]

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]

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]

## 2. Second-order moments of the Wigner distribution function of a TGSM beam in turbulent atmosphere

*z*= 0) is expressed as [14

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

*λ*being the wavelength of light field.

**J**denotes an anti-symmetric matrix given by [14

**10**(1), 95–109 (1993). [CrossRef]

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

*z*is expressed as [11,30

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]

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]

*κ*is the magnitude of the spatial wave-number.

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, “M^{2}-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express **17**(20), 17344–17356 (2009). [CrossRef] [PubMed]

*z*-direction;

*x-*axis and

*y*-axis, respectively.

**16**(20), 15563–15575 (2008). [CrossRef] [PubMed]

## 3. Propagation factor of a TGSM beam in turbulent atmosphere

**16**(20), 15563–15575 (2008). [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]

*M*

^{2}-factor of the TGSM beam in free space or in the source plane given by

*M*

^{2}-factor of a TGSM beam in free space. Under the condition of

*M*

^{2}-factor of a GSM beam without twist phase in turbulent atmosphere. From Eq. (25), it is clear that the

*M*

^{2}-factor of a TGSM beam in free space is independent of the propagation distance, and increases with the increase of the absolute value of the twist factor. This phenomenon is caused by the fact that the twist factor cause more rapid spreading of a TGSM beam on propagation.

*M*

^{2}-factor of a TGSM beam in turbulent atmosphere. In the following numerical examples, we adopt the Tatarskii spectrum for the spectral density of the index-of-refraction fluctuations, which is expressed as [11]where

*M*

^{2}-factor of a TGSM beam defined as

*M*

^{2}-factor of a TGSM beam on propagation in turbulent atmosphere for different values of the structure constant

*M*

^{2}-factor of a TGSM beam in turbulent atmosphere increases on propagation, which is much different from its propagation-invariant properties in free space (

*M*

^{2}-factor increases more rapid on propagation. Figure 2 shows the normalized

*M*

^{2}-factor of a TGSM beam on propagation in turbulent atmosphere for different values of

*M*

^{2}-factor of a TGSM beam increases slower on propagation as its initial coherence width

**16**(20), 15563–15575 (2008). [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]

*M*

^{2}-factor of a TGSM beam increases slower than that of a GSM beam without twist phase (

*M*

^{2}-factor to show the difference between the normalized

*M*

^{2}-factor of a TGSM beam and that of a GSM beam. The deviation percentage of the normalized

*M*

^{2}-factor is defined as

*M*

^{2}-factor versus the propagation distance z for different values of

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]

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]

## 4. Effective radius of curvature of a TGSM beam in turbulent atmosphere

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]

*z*is defined in terms of the ratio of

*λ*and the twisted factor

*z*with

## 5. Rayleigh range of a TGSM beam in turbulent atmosphere

*T*= 0 (free space), Eqs. (35) and (36) reduce to the same quadratic equation. After some calculation, we obtain following analytical expression for

## 6. Conclusion

## Acknowledgments

## References and links

1. | L. Mandel, and E. Wolf, |

2. | Y. Baykal, “Average transmittance in turbulence for partially coherent sources,” Opt. Commun. |

3. | M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. |

4. | Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

5. | Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express |

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 |

7. | A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. |

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 |

9. | E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A |

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. |

11. | L. C. Andrews, and R. L. Phillips, |

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 |

13. | Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express |

14. | R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A |

15. | A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A |

16. | D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. |

17. | R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. |

18. | J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. |

19. | Q. Lin and Y. Cai, “Tensor |

20. | Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. |

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 |

22. | Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. |

23. | Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express |

24. | Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express |

25. | C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express |

26. | Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B |

27. | H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express |

28. | Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express |

29. | Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. |

30. | Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express |

31. | Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M |

32. | S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express |

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 |

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 |

35. | A. Erdelyi, W. Magnus, and F. Oberhettinger, |

36. | G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. |

**OCIS Codes**

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(030.1670) Coherence and statistical optics : Coherent optical effects

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: September 13, 2010

Revised Manuscript: October 13, 2010

Manuscript Accepted: October 28, 2010

Published: November 10, 2010

**Citation**

Fei Wang and Yangjian Cai, "Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere," Opt. Express **18**, 24661-24672 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-24661

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### References

- L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
- Y. Baykal, “Average transmittance in turbulence for partially coherent sources,” Opt. Commun. 231(1-6), 129–136 (2004). [CrossRef]
- M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008). [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]
- 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]
- A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982). [CrossRef]
- 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]
- 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]
- 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]
- L. C. Andrews, and R. L. Phillips, Laser beam propagation in the turbulent atmosphere, 2nd edition, (SPIE Press, Bellington, 2005).
- 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]
- 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]
- R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993). [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]
- 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]
- 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]
- J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26(7), 405–407 (2001). [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]
- 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]
- 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 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. 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 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 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. 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]
- 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]
- A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).
- G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2001).

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