## Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence |

Optics Express, Vol. 19, Issue 18, pp. 16872-16884 (2011)

http://dx.doi.org/10.1364/OE.19.016872

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### Abstract

Kolmogorov turbulence theory based models cannot be directly applied in non-Kolmogorov turbulence case, which has been reported recently by increasing experimental evidence and theoretical investigation. In this study, based on the generalized von Karman spectral model, the theoretical expression of the irradiance scintillation index is derived for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence with horizontal path. In the derivation, the expression is divided into two parts for physical analysis purpose and mathematical analysis convenience. This expression considers the influences of finite turbulence inner and outer scales and has a general spectral power law value in the range 3 to 4 instead of standard power law value of 11/3 (for Kolmogorov turbulence). Numerical simulations are conducted to investigate the influences.

© 2011 OSA

## 1. Introduction

1. L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE **4489**, 23–34 (2002). [CrossRef]

4. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express **18**(6), 5356–5366 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-6-5356. [CrossRef] [PubMed]

11. L. C. Andrews, “An analytical model for the refractive index power dpectrum and its spplication to optical scintillations in the atmosphere,” J. Mod. Opt. **39**(9), 1849–1853 (1992), http://dx.doi.org/10.1080/09500349214551931. [CrossRef]

13. D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE **2120**, 43–55 (1994). [CrossRef]

18. M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. **20**(12), 1359–1361 (1995). [CrossRef] [PubMed]

19. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. **57**(6), 1783–1788 (2009). [CrossRef]

20. L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express **18**(2), 451–462 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-2-451. [CrossRef] [PubMed]

21. 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, 65510E-12 (2007). [CrossRef]

22. 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), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-20-21269. [CrossRef] [PubMed]

23. B. Xue, L. Cui, W. Xue, X. Bai, and F. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A **28**(5), 912–916 (2011). [CrossRef] [PubMed]

## 3. Irradiance scintillation index for Gaussian-beam wave

*ρ*and becomes zero at the beam centerline (

### 3.1 The radial component of the irradiance scintillation index

*ξ*, Eq. (11) becomes

19. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. **57**(6), 1783–1788 (2009). [CrossRef]

### 3.2 The longitudinal component of the irradiance scintillation index

7. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A **11**(10), 2719–2726 (1994). [CrossRef]

### 3.3 Irradiance scintillation index for Gaussian-beam wave

## 4. Numerical results

*α*on the irradiance scintillation index

7. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A **11**(10), 2719–2726 (1994). [CrossRef]

*L*and optical wavelength

*λ*are fixed (

### 4.1 Effect of inner scale’s variation on the irradiance scintillation index

7. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A **11**(10), 2719–2726 (1994). [CrossRef]

### 4.2 Effect of outer scale’s variation on the irradiance scintillation index

**11**(10), 2719–2726 (1994). [CrossRef]

*m*can lead to variation in the irradiance scintillation away from the center of the beam near the diffractive beam edge (

**11**(10), 2719–2726 (1994). [CrossRef]

### 4.3 Effect of *α*’s variation on the irradiance scintillation index

*α*, and then decreases with increasing

*α*lead to more obvious variation in the irradiance scintillation index away from the center of the beam near the diffractive beam edge (

## 5. Conclusions

**11**(10), 2719–2726 (1994). [CrossRef]

*α*produces obvious effects on the irradiance scintillation index especially for the case near the diffractive beam edge. The results in this study will help to better investigate the effects of turbulence on the Gaussian-beam wave propagating through weak non-Kolmogorov atmospheric turbulence with horizontal path.

## Acknowledgments

## References and links

1. | L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE |

2. | X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE |

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

4. | A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express |

5. | R. J. Sasiela, |

6. | W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A |

7. | W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A |

8. | J. D. Shelton, “Turbulence-induced scintillation on Gaussian-beam waves: theoretical predictions and observations from a laser-illuminated satellite,” J. Opt. Soc. Am. A |

9. | L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Complex Media |

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

11. | L. C. Andrews, “An analytical model for the refractive index power dpectrum and its spplication to optical scintillations in the atmosphere,” J. Mod. Opt. |

12. | V. I. Tatarskii, |

13. | D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE |

14. | M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE |

15. | M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE |

16. | A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. |

17. | A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A |

18. | M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. |

19. | I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. |

20. | L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express |

21. | 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 |

22. | 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 |

23. | B. Xue, L. Cui, W. Xue, X. Bai, and F. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A |

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

25. | L. C. Andrews, |

**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: June 14, 2011

Revised Manuscript: July 20, 2011

Manuscript Accepted: August 2, 2011

Published: August 15, 2011

**Citation**

Linyan Cui, Bindang Xue, Lei Cao, Shiling Zheng, Wenfang Xue, Xiangzhi Bai, Xiaoguang Cao, and Fugen Zhou, "Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence," Opt. Express **19**, 16872-16884 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-16872

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### References

- L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE 4489, 23–34 (2002). [CrossRef]
- X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE 7382, 73824O (2009). [CrossRef]
- W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-20-17829 . [CrossRef] [PubMed]
- A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express 18(6), 5356–5366 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-6-5356 . [CrossRef] [PubMed]
- R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer, 1994).
- W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10(4), 661–672 (1993). [CrossRef]
- W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994). [CrossRef]
- J. D. Shelton, “Turbulence-induced scintillation on Gaussian-beam waves: theoretical predictions and observations from a laser-illuminated satellite,” J. Opt. Soc. Am. A 12(10), 2172–2181 (1995). [CrossRef]
- L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Complex Media 11, 271–291 (2001). http://dx.doi.org/10.1080/13616670109409785 . [CrossRef]
- L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005), Chap.8.
- L. C. Andrews, “An analytical model for the refractive index power dpectrum and its spplication to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992), http://dx.doi.org/10.1080/09500349214551931 . [CrossRef]
- V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (trans.for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971).
- D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994). [CrossRef]
- M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997). [CrossRef]
- M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U, 63040U-12 (2006). [CrossRef]
- A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008). [CrossRef]
- A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12(11), 2517–2522 (1995). [CrossRef]
- M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 20(12), 1359–1361 (1995). [CrossRef] [PubMed]
- I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009). [CrossRef]
- L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(2), 451–462 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-2-451 . [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. SPIE 6551, 65510E, 65510E-12 (2007). [CrossRef]
- 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), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-20-21269 . [CrossRef] [PubMed]
- B. Xue, L. Cui, W. Xue, X. Bai, and F. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28(5), 912–916 (2011). [CrossRef] [PubMed]
- L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 1998).
- L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, 1998).

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