## Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere |

Optics Express, Vol. 22, Issue 2, pp. 1871-1883 (2014)

http://dx.doi.org/10.1364/OE.22.001871

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### Abstract

Laguerre-Gaussian Schell-model (LGSM) beam was proposed in theory [Opt. Lett. 38, 91 (2013
Opt. Lett. 38, 1814 (2013)] just recently. In this paper, we study the propagation of a LGSM beam in turbulent atmosphere. Analytical expressions for the cross-spectral density and the second-order moments of the Wigner distribution function of a LGSM beam in turbulent atmosphere are derived. The statistical properties, such as the degree of coherence and the propagation factor, of a LGSM beam in turbulent atmosphere are studied in detail. It is found that a LGSM beam with larger mode order *n* is less affected by turbulence than a LGSM beam with smaller mode order *n* or a GSM beam under certain condition, which will be useful in free-space optical communications.

© 2014 Optical Society of America

## 1. Introduction

7. 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] [PubMed]

11. G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. **36**(10), 1939–1941 (2011). [CrossRef] [PubMed]

16. Z. Tong and O. Korotkova, “Non-uniformly correlated beams in uniformly correlated media,” Opt. Lett. **37**(15), 3240–3242 (2012). [CrossRef] [PubMed]

18. Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. **38**(9), 1395–1397 (2013). [CrossRef] [PubMed]

20. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A **29**(10), 2159–2164 (2012). [CrossRef]

22. Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. **305**, 57–65 (2013). [CrossRef]

24. Z. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express **21**(15), 17512–17519 (2013). [CrossRef] [PubMed]

28. J. Cang, P. Xiu, and X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. **54**, 35–41 (2013). [CrossRef]

52. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. **69**(9), 1297–1304 (1979). [CrossRef]

16. Z. Tong and O. Korotkova, “Non-uniformly correlated beams in uniformly correlated media,” Opt. Lett. **37**(15), 3240–3242 (2012). [CrossRef] [PubMed]

18. Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. **38**(9), 1395–1397 (2013). [CrossRef] [PubMed]

20. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A **29**(10), 2159–2164 (2012). [CrossRef]

22. Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. **305**, 57–65 (2013). [CrossRef]

24. Z. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express **21**(15), 17512–17519 (2013). [CrossRef] [PubMed]

28. J. Cang, P. Xiu, and X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. **54**, 35–41 (2013). [CrossRef]

28. J. Cang, P. Xiu, and X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. **54**, 35–41 (2013). [CrossRef]

## 2. Cross-spectral density of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere

26. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. **38**(2), 91–93 (2013). [CrossRef] [PubMed]

*n*and 0. The degree of coherence of the LGSM beam at z = 0 is given asOne finds from Eq. (2) that the degree of coherence of a LGSM beam has a non-Gaussian distribution, which induces unique propagation properties of such beam [26

26. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. **38**(2), 91–93 (2013). [CrossRef] [PubMed]

**54**, 35–41 (2013). [CrossRef]

*n*= 0, Eq. (1) reduces to the expression for the CSD of a GSM beam.

7. 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] [PubMed]

33. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A **20**(6), 1094–1102 (2003). [CrossRef] [PubMed]

52. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. **69**(9), 1297–1304 (1979). [CrossRef]

7. 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] [PubMed]

33. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A **20**(6), 1094–1102 (2003). [CrossRef] [PubMed]

52. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. **69**(9), 1297–1304 (1979). [CrossRef]

**19**(9), 1794–1802 (2002). [CrossRef] [PubMed]

33. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A **20**(6), 1094–1102 (2003). [CrossRef] [PubMed]

**69**(9), 1297–1304 (1979). [CrossRef]

**r**, Eq. (10) reduces to

*n*, after integration over

## 3. Second-order moments of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere

39. 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]

39. 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]

*x-*axis and

*y*-axis, respectively.

39. 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]

*n*= 0, Eq. (37) reduces to the following expression for the propagation factor of a GSM beam in free space [55

55. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A **16**(1), 106–112 (1999). [CrossRef]

## 4. Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere

*n*= 2. One finds from Fig. 1 that the degree of coherence of the LGSM beam in the source plane has non-Gaussian distribution, and there are side robes around the main peak. The side robes in the degree of coherence disappear gradually on propagation in free space. In the far field, only the main peak exists which also has non-Gaussian distribution. Figure 2 shows the modulus of the degree of coherence of a LGSM beam at several propagation distances in turbulent atmosphere for different values of the structure constant

*n*= 1. From Fig. 2, one finds that the evolution properties of the degree of coherence at short propagation distance in turbulent atmosphere are similar to the corresponding evolution properties in free space, i.e., the side robes disappear gradually on propagation. While at long propagation distance, the distribution of the degree of coherence in turbulent atmosphere is much different from that in free space. In turbulent atmosphere, the degree of coherence becomes of Gaussian distribution in the far field. We may explain this phenomenon by the fact that at the short propagation distance, the influence of the turbulence can be neglected and the role of free-space diffraction plays a dominant role. At long propagation distance, the influence turbulence plays a dominant role, and the degree of coherence takes a Gaussian distribution due to the isotropic influence of the turbulence. Figure 3 shows modulus of the degree of coherence of a LGSM beam at several propagation distances in turbulent atmosphere for different values of the mode order

*n*. One finds from Fig. 3 that the evolution properties of the degree of coherence of the LGSM beam in turbulent atmosphere are also affected by the mode order

*n*. The conversion from the non-Gaussian distribution to Gaussian distribution becomes slower as the mode order

*n*increases, which means that a LGSM beam with larger

*n*is less affected by turbulence.

*n*and the structure constant

*n*and the coherence width

*n*and the wavelength

*n*increases slower than a LGSM beam with smaller

*n*or a GSM beam (

*n*= 0) on propagation, which means that the LGSM beam with larger

*n*is less affected by turbulence. Furthermore, we note that the advantage of a LGSM beam with larger

*n*over a LGSM beam with smaller

*n*or a GSM beam is enhanced for larger structure constant

## 5. Summary

*n*is less affected by turbulence than a LGSM beam with smaller mode order

*n*or a GSM beam by choosing suitable beam parameters. In [7

**19**(9), 1794–1802 (2002). [CrossRef] [PubMed]

## Acknowledgments

## References and links

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

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

3. | F. Wang, Y. Cai, and S. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express |

4. | D. Kermisch, “Partially coherent image processing by laser scanning,” J. Opt. Soc. Am. |

5. | Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. |

6. | A. Belendez, L. Carretero, and A. Fimia, “The use of partially coherent light to reduce the efficiency of silve-halide noise gratings,” Opt. Commun. |

7. | 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 |

8. | T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. |

9. | Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A |

10. | C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. |

11. | G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. |

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

13. | F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. |

14. | F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. |

15. | H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. |

16. | Z. Tong and O. Korotkova, “Non-uniformly correlated beams in uniformly correlated media,” Opt. Lett. |

17. | Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A |

18. | Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. |

19. | S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. |

20. | O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A |

21. | S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. |

22. | Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. |

23. | Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. |

24. | Z. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express |

25. | C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. Doc. ID 202151 (2014). |

26. | Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. |

27. | F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. |

28. | J. Cang, P. Xiu, and X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. |

29. | Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A |

30. | E. Wolf, |

31. | L. C. Andrews, R. L. Phillips, and C. Y. Hopen, |

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

33. | T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A |

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

35. | Y. Baykal and H. T. Eyyuboğlu, “Scintillation index of flat-topped Gaussian beams,” Appl. Opt. |

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

37. | Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. |

38. | R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express |

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

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

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

42. | O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. |

43. | Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express |

44. | W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express |

45. | Y. Gu and G. Gbur, “Scintillation of airy beam arrays in atmospheric turbulence,” Opt. Lett. |

46. | P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. |

47. | F. Wang and Y. Cai, “Second-order statistics of a twisted gaussian Schell-model beam in turbulent atmosphere,” Opt. Express |

48. | F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. |

49. | Y. Gu and G. Gbur, “Reduction of turbulence-induced scintillation by nonuniformly polarized beam arrays,” Opt. Lett. |

50. | F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. |

51. | X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. |

52. | S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. |

53. | M. Abramowitz and I. Stegun, |

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

55. | M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A |

**OCIS Codes**

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(030.0030) Coherence and statistical optics : Coherence and statistical optics

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: November 25, 2013

Revised Manuscript: January 11, 2014

Manuscript Accepted: January 13, 2014

Published: January 21, 2014

**Citation**

Rong Chen, Lin Liu, Shijun Zhu, Gaofeng Wu, Fei Wang, and Yangjian Cai, "Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere," Opt. Express **22**, 1871-1883 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-2-1871

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### References

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- Y. Cai, 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]
- F. Wang, Y. Cai, S. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14(16), 6999–7004 (2006). [CrossRef] [PubMed]
- D. Kermisch, “Partially coherent image processing by laser scanning,” J. Opt. Soc. Am. 65(8), 887–891 (1975). [CrossRef]
- Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984). [CrossRef]
- A. Belendez, L. Carretero, A. Fimia, “The use of partially coherent light to reduce the efficiency of silve-halide noise gratings,” Opt. Commun. 98(4-6), 236–240 (1993). [CrossRef]
- J. C. Ricklin, 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] [PubMed]
- T. van Dijk, D. G. Fischer, T. D. Visser, E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010). [CrossRef] [PubMed]
- Y. Dong, F. Wang, C. Zhao, Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012). [CrossRef]
- C. Zhao, Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011). [CrossRef] [PubMed]
- G. Wu, Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. 36(10), 1939–1941 (2011). [CrossRef] [PubMed]
- Y. Cai, U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007). [CrossRef] [PubMed]
- F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef] [PubMed]
- F. Gori, V. R. Sanchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009). [CrossRef]
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- Z. Tong, O. Korotkova, “Non-uniformly correlated beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). [CrossRef] [PubMed]
- Z. Tong, O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012). [CrossRef] [PubMed]
- Y. Gu, G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013). [CrossRef] [PubMed]
- S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef] [PubMed]
- O. Korotkova, S. Sahin, E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). [CrossRef]
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- Z. Mei, E. Schchepakina, O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013). [CrossRef] [PubMed]
- C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. Doc. ID 202151 (2014).
- Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). [CrossRef] [PubMed]
- F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013). [CrossRef] [PubMed]
- J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013). [CrossRef]
- Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014). [CrossRef]
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- T. Shirai, A. Dogariu, E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20(6), 1094–1102 (2003). [CrossRef] [PubMed]
- Y. Cai, S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006). [CrossRef] [PubMed]
- Y. Baykal, H. T. Eyyuboğlu, “Scintillation index of flat-topped Gaussian beams,” Appl. Opt. 45(16), 3793–3797 (2006). [CrossRef] [PubMed]
- Y. Cai, 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, Y. Chen, H. T. Eyyuboğlu, Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007). [CrossRef] [PubMed]
- R. J. Noriega-Manez, J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007). [CrossRef] [PubMed]
- Y. Dan, 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, 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, O. Korotkova, H. T. Eyyuboğlu, Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008). [CrossRef] [PubMed]
- O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008). [CrossRef]
- Y. Cai, Q. Lin, H. T. Eyyuboğlu, Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008). [CrossRef] [PubMed]
- W. Cheng, J. W. Haus, Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009). [CrossRef] [PubMed]
- Y. Gu, G. Gbur, “Scintillation of airy beam arrays in atmospheric turbulence,” Opt. Lett. 35(20), 3456–3458 (2010). [CrossRef] [PubMed]
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