## Propagation of elegant Laguerre–Gaussian beam in non-Kolmogorov turbulence |

Optics Express, Vol. 19, Issue 22, pp. 21163-21173 (2011)

http://dx.doi.org/10.1364/OE.19.021163

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### Abstract

The analytical formulas for the average intensity and the beam width of the elegant Laguerre–Gaussian beam (ELGB) in non-Kolmogorov turbulence have been derived based on the extended Huygens–Fresnel principle. Numerical results reveal that the ELGB converts to Gaussian form quicker for smaller values of beam order and for smaller wavelengths. The root-mean-square (rms) beam width of ELGB increases markedly with the propagation distance for higher beam order, smaller waist width, and larger wavelength. Furthermore, discussions of the influence of ELGB by the non-Kolmogorov turbulence reveal that the normalized intensity distribution of ELGB converts into Gaussian form more quickly and that the rms beam width of ELGB increases more rapidly in non-Kolmogorov turbulence with smaller parameter

© 2011 OSA

## 1. Introduction

1. X. Chu, C. Qiao, X. Feng, and R. Chen, “Propagation of Gaussian-Schell beam in turbulent atmosphere of three-layer altitude model,” Appl. Opt. **50**(21), 3871–3878 (2011). [CrossRef] [PubMed]

2. X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. **36**(14), 2701–2703 (2011). [CrossRef] [PubMed]

7. M. S. Belen’kii, S. J. Kars, and C. L. Osmon, “Experimental evidence of the effects of non-Kolmogorov turbulence,” Proc. SPIE **3749**, 50–51 (1999). [CrossRef]

8. D. Dayton, B. Pierson, B. Spielbusch, and J. Gonglewski, “Atmospheric structure function measurements with a Shack-Hartmann wave-front sensor,” Opt. Lett. **17**(24), 1737–1739 (1992). [CrossRef] [PubMed]

*et al.*[9

9. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE **6551**, 65510E–1 65510E-12 (2007). [CrossRef]

10. 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**(2), 026003 (2008). [CrossRef]

11. 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**(5), 715–717 (2010). [CrossRef] [PubMed]

12. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. **35**(7), 1043–1045 (2010). [CrossRef] [PubMed]

13. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express **18**(10), 10650–10658 (2010). [CrossRef] [PubMed]

14. Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre–Gaussian beams in turbulent atmosphere,” Appl. Phys. B **102**(4), 937–944 (2011). [CrossRef]

18. F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and Elegant Laguerre-Gaussian beam in turbulent atmosphere,” Prog. Electromagn. Res. **103**, 33–56 (2010). [CrossRef]

19. T. Shirai, “Polarization properties of a class of electromagnetic Gaussian Schell-model beams which have the same far-zone intensity distribution as a fully coherent laser beam,” Opt. Commun. **256**(4-6), 197–209 (2005). [CrossRef]

14. Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre–Gaussian beams in turbulent atmosphere,” Appl. Phys. B **102**(4), 937–944 (2011). [CrossRef]

17. J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. **283**(14), 2772–2781 (2010). [CrossRef]

18. F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and Elegant Laguerre-Gaussian beam in turbulent atmosphere,” Prog. Electromagn. Res. **103**, 33–56 (2010). [CrossRef]

## 2. Theoretical model

### 2.1 **Average intensity of ELGB in non-Kolmogorov turbulence**

20. E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A **3**(4), 465–469 (1986). [CrossRef]

23. D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite–Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt. **5**(5), 489–494 (2003). [CrossRef]

31. H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. **44**(6), 976–983 (2005). [CrossRef] [PubMed]

11. 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**(5), 715–717 (2010). [CrossRef] [PubMed]

12. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. **35**(7), 1043–1045 (2010). [CrossRef] [PubMed]

32. T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. **28**(8), 610–612 (2003). [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]

9. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE **6551**, 65510E–1 65510E-12 (2007). [CrossRef]

12. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. **35**(7), 1043–1045 (2010). [CrossRef] [PubMed]

34. L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express **18**(20), 21269–21283 (2010). [CrossRef] [PubMed]

35. G. Wu, T. Zhao, J. Ren, J. Zhang, X. Zhang, and W. Li, “Beam propagation factor of partially coherent Hermite-Gassian beams through non-Kolmogorov turbulence,” Opt. Laser Technol. **43**(7), 1225–1228 (2011). [CrossRef]

11. 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**(5), 715–717 (2010). [CrossRef] [PubMed]

**35**(7), 1043–1045 (2010). [CrossRef] [PubMed]

**35**(5), 715–717 (2010). [CrossRef] [PubMed]

**35**(7), 1043–1045 (2010). [CrossRef] [PubMed]

### 2.2 Rms beam width of ELGB in non-Kolmogorov turbulence

**35**(5), 715–717 (2010). [CrossRef] [PubMed]

**35**(7), 1043–1045 (2010). [CrossRef] [PubMed]

31. H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. **44**(6), 976–983 (2005). [CrossRef] [PubMed]

32. T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. **28**(8), 610–612 (2003). [CrossRef] [PubMed]

## 3. Numerical results

## 4. Conclusion

## Acknowledgments

## References and links

1. | X. Chu, C. Qiao, X. Feng, and R. Chen, “Propagation of Gaussian-Schell beam in turbulent atmosphere of three-layer altitude model,” Appl. Opt. |

2. | X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. |

3. | R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE |

4. | C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. |

5. | A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE |

6. | B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE |

7. | M. S. Belen’kii, S. J. Kars, and C. L. Osmon, “Experimental evidence of the effects of non-Kolmogorov turbulence,” Proc. SPIE |

8. | D. Dayton, B. Pierson, B. Spielbusch, and J. Gonglewski, “Atmospheric structure function measurements with a Shack-Hartmann wave-front sensor,” Opt. Lett. |

9. | I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE |

10. | 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. |

11. | G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. |

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

13. | E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express |

14. | Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre–Gaussian beams in turbulent atmosphere,” Appl. Phys. B |

15. | F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express |

16. | H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B |

17. | J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. |

18. | F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and Elegant Laguerre-Gaussian beam in turbulent atmosphere,” Prog. Electromagn. Res. |

19. | T. Shirai, “Polarization properties of a class of electromagnetic Gaussian Schell-model beams which have the same far-zone intensity distribution as a fully coherent laser beam,” Opt. Commun. |

20. | E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A |

21. | Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express |

22. | S. Y. Shin and L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. |

23. | D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite–Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt. |

24. | K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. |

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

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

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

28. | 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 |

29. | Y. Cai, H.T. Eyyuboğlu, and Y. Baykal, “Scintillation of astigmatic dark hollow beams in weak atmospheric turbulence,” J. Opt. Soc. Am. A |

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

31. | H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. |

32. | T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. |

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. | L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express |

35. | G. Wu, T. Zhao, J. Ren, J. Zhang, X. Zhang, and W. Li, “Beam propagation factor of partially coherent Hermite-Gassian beams through non-Kolmogorov turbulence,” Opt. Laser Technol. |

**OCIS Codes**

(010.1290) Atmospheric and oceanic optics : Atmospheric optics

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: July 12, 2011

Revised Manuscript: September 8, 2011

Manuscript Accepted: September 13, 2011

Published: October 10, 2011

**Citation**

Huafeng Xu, Zhifeng Cui, and Jun Qu, "Propagation of elegant Laguerre–Gaussian beam in non-Kolmogorov turbulence," Opt. Express **19**, 21163-21173 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21163

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### References

- X. Chu, C. Qiao, X. Feng, and R. Chen, “Propagation of Gaussian-Schell beam in turbulent atmosphere of three-layer altitude model,” Appl. Opt.50(21), 3871–3878 (2011). [CrossRef] [PubMed]
- X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett.36(14), 2701–2703 (2011). [CrossRef] [PubMed]
- R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE2375A, 1111–1126 (1995).
- C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt.47(6), 1111–1126 (2000). [CrossRef]
- A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE5987, 598702, 598702-12 (2005). [CrossRef]
- B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE2471, 2471–22 (1995).
- M. S. Belen’kii, S. J. Kars, and C. L. Osmon, “Experimental evidence of the effects of non-Kolmogorov turbulence,” Proc. SPIE3749, 50–51 (1999). [CrossRef]
- D. Dayton, B. Pierson, B. Spielbusch, and J. Gonglewski, “Atmospheric structure function measurements with a Shack-Hartmann wave-front sensor,” Opt. Lett.17(24), 1737–1739 (1992). [CrossRef] [PubMed]
- I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE6551, 65510E–1 65510E-12 (2007). [CrossRef]
- 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(2), 026003 (2008). [CrossRef]
- 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(5), 715–717 (2010). [CrossRef] [PubMed]
- P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett.35(7), 1043–1045 (2010). [CrossRef] [PubMed]
- E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express18(10), 10650–10658 (2010). [CrossRef] [PubMed]
- Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre–Gaussian beams in turbulent atmosphere,” Appl. Phys. B102(4), 937–944 (2011). [CrossRef]
- F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express17(25), 22366–22379 (2009). [CrossRef] [PubMed]
- H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B98(4), 857–863 (2010). [CrossRef]
- J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun.283(14), 2772–2781 (2010). [CrossRef]
- F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and Elegant Laguerre-Gaussian beam in turbulent atmosphere,” Prog. Electromagn. Res.103, 33–56 (2010). [CrossRef]
- T. Shirai, “Polarization properties of a class of electromagnetic Gaussian Schell-model beams which have the same far-zone intensity distribution as a fully coherent laser beam,” Opt. Commun.256(4-6), 197–209 (2005). [CrossRef]
- E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A3(4), 465–469 (1986). [CrossRef]
- Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express17(13), 11130–11139 (2009). [CrossRef] [PubMed]
- S. Y. Shin and L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am.67(5), 699–700 (1977). [CrossRef]
- D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite–Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt.5(5), 489–494 (2003). [CrossRef]
- K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron.29, 2563–2567 (1993).
- L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications, SPIE Press, Washington, 2001.
- L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd edition SPIE Press, Bellington, (2005)
- 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. Express17(20), 17344–17356 (2009). [CrossRef] [PubMed]
- 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. Express16(11), 7665–7673 (2008). [CrossRef] [PubMed]
- Y. Cai, H.T. Eyyuboğlu, and Y. Baykal, “Scintillation of astigmatic dark hollow beams in weak atmospheric turbulence,” J. Opt. Soc. Am. A25(7), 1497–1503 (2008). [CrossRef]
- H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express12(20), 4659–4674 (2004). [CrossRef] [PubMed]
- H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt.44(6), 976–983 (2005). [CrossRef] [PubMed]
- T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett.28(8), 610–612 (2003). [CrossRef] [PubMed]
- T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A20(6), 1094–1102 (2003). [CrossRef] [PubMed]
- L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express18(20), 21269–21283 (2010). [CrossRef] [PubMed]
- G. Wu, T. Zhao, J. Ren, J. Zhang, X. Zhang, and W. Li, “Beam propagation factor of partially coherent Hermite-Gassian beams through non-Kolmogorov turbulence,” Opt. Laser Technol.43(7), 1225–1228 (2011). [CrossRef]

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