## Average intensity and spreading of an elegant Hermite–Gaussian beam in turbulent atmosphere

Optics Express, Vol. 17, Issue 13, pp. 11130-11139 (2009)

http://dx.doi.org/10.1364/OE.17.011130

Acrobat PDF (370 KB)

### Abstract

The propagation of an elegant Hermite-Gaussian beam (EHGB) in turbulent atmosphere is investigated. Analytical propagation formulae for the average intensity and effective beam size of an EHGB in turbulent atmosphere are derived based on the extended Huygens-Fresnel integral. The corresponding results of a standard Hermite-Gaussian beam (SHGB) in turbulent atmosphere are also derived for the convenience of comparison. The intensity and spreading properties of EHGBs and SHGBs in turbulent atmosphere are studied numerically and comparatively. It is found that the propagation properties of EHGBs and SHGBs are much different from their properties in free space, and the EHGB and SHGB with higher orders are less affected by the turbulence. What’s more, the SHGB spreads more rapidly than the EHGB in turbulent atmosphere under the same conditions. Our results will be useful in long-distance free-space optical communications.

© 2009 OSA

## 1. Introduction

10. Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. **245**(1-6), 21–26 (2005). [CrossRef]

2. W. H. Carter, “Spot size and divergence for Hermite-Gaussian beams of any order,” Appl. Opt. **19**(7), 1027–1029 (1980). [CrossRef]

3. T. Kojima, “Diffraction of Hermite-Gaussian beams from a sinusoidal conducting grating,” J. Opt. Soc. Am. A **7**(9), 1740–1744 (1990). [CrossRef]

4. O. Mata-Mendez and F. Chavez-Rivas, “Diffraction of Gaussian and Hermite-Gaussia beams by finite gratings,” J. Opt. Soc. Am. A **18**(3), 537–545 (2001). [CrossRef]

5. K. M. Luk and P. K. Yu, “Generation of Hermite-Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. A **2**(11), 1818–1820 (1985). [CrossRef]

6. Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. **33**(6), 1025–1031 (1997). [CrossRef]

7. H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. **147**(1-3), 1–4 (1998). [CrossRef]

9. Y. Cai and Q. Lin, “Decentered elliptical Hermite–Gaussian beam,” J. Opt. Soc. Am. A **20**(6), 1111–1119 (2003). [CrossRef]

10. Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. **245**(1-6), 21–26 (2005). [CrossRef]

11. Y. Cai and C. Chen, “Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems,” J. Opt. Soc. Am. A **24**(8), 2394–2401 (2007). [CrossRef]

12. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. **63**(9), 1093–1094 (1973). [CrossRef]

13. 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]

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

16. S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. **153**(4-6), 207–210 (1998). [CrossRef]

18. B. Lu and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. **174**(1-4), 99–104 (2000). [CrossRef]

19. Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. **10**(1), 33–35 (1967). [CrossRef]

20. C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. **41**(5), 1097–1103 (2002). [CrossRef]

25. Y. Baykal, “Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere,” J. Opt. Soc. Am. A **21**(7), 1290–1299 (2004). [CrossRef]

26. H. T. Eyyuboğlu, “Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere,” J. Opt. Soc. Am. A **22**(8), 1527–1535 (2005). [CrossRef]

## 2. Formulation

12. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. **63**(9), 1093–1094 (1973). [CrossRef]

18. B. Lu and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. **174**(1-4), 99–104 (2000). [CrossRef]

*m*or

*n*in x and y directions, respectively,

*x*

_{1},

*y*

_{1},0)/I(

*x*

_{1},

*y*

_{1},0)

_{max}of an EHGB for different values of

*m*with

*m*=

*n*and

2. W. H. Carter, “Spot size and divergence for Hermite-Gaussian beams of any order,” Appl. Opt. **19**(7), 1027–1029 (1980). [CrossRef]

*m*with

*m*=

*n*and

20. C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. **41**(5), 1097–1103 (2002). [CrossRef]

35. X. Ji, X. Chen, and B. Lu, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A **25**(1), 21–28 (2008). [CrossRef]

20. C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. **41**(5), 1097–1103 (2002). [CrossRef]

35. X. Ji, X. Chen, and B. Lu, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A **25**(1), 21–28 (2008). [CrossRef]

**41**(5), 1097–1103 (2002). [CrossRef]

35. X. Ji, X. Chen, and B. Lu, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A **25**(1), 21–28 (2008). [CrossRef]

**41**(5), 1097–1103 (2002). [CrossRef]

**25**(1), 21–28 (2008). [CrossRef]

*x*] gives the greatest integer less than or equal to

*x,*and

2. W. H. Carter, “Spot size and divergence for Hermite-Gaussian beams of any order,” Appl. Opt. **19**(7), 1027–1029 (1980). [CrossRef]

*x*or

*y*, the effective beam size of an EHGB in

*x*direction at plane z is defined asSubstituting Eq. (4) into Eq. (9), after tedious integration, we obtain the expression for the effective beam size

## 3. Numerical examples

*m*=

*n =*2,

*m*and

*n*. For a SHGB however, its beam profile remains invariant on propagation in free space, although its beam spot spreads on propagation. Our results agree well with those reported in [1,2

**19**(7), 1027–1029 (1980). [CrossRef]

17. S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. **191**(3-6), 173–179 (2001). [CrossRef]

18. B. Lu and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. **174**(1-4), 99–104 (2000). [CrossRef]

*m*=

*n =*2,

*m*with

*m*=

*n*at z = 3

*km*in turbulent atmosphere. One finds from Fig. 5 that the intensity distribution of the EHGB with larger

*m*and

*n*in turbulent atmosphere is more similar to its far field intensity distribution in free space, which means the EHGB with larger

*m*and

*n*is less affected by the turbulence. Our numerical results (not shown here to save space) also show that the EHGB with larger values of

*m*,

*n*,

**41**(5), 1097–1103 (2002). [CrossRef]

*m*=

*n*= 2 and

## 4. Conclusion

## References and links

1. | A. E. Siegman, LASERS, University Science Books, Mill Valley, 1986. |

2. | W. H. Carter, “Spot size and divergence for Hermite-Gaussian beams of any order,” Appl. Opt. |

3. | T. Kojima, “Diffraction of Hermite-Gaussian beams from a sinusoidal conducting grating,” J. Opt. Soc. Am. A |

4. | O. Mata-Mendez and F. Chavez-Rivas, “Diffraction of Gaussian and Hermite-Gaussia beams by finite gratings,” J. Opt. Soc. Am. A |

5. | K. M. Luk and P. K. Yu, “Generation of Hermite-Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. A |

6. | Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. |

7. | H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. |

8. | Y. Cai and Q. Lin, “The elliptical Hermite-Gaussian beam and its propagation through paraxial systems,” Opt. Commun. |

9. | Y. Cai and Q. Lin, “Decentered elliptical Hermite–Gaussian beam,” J. Opt. Soc. Am. A |

10. | Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. |

11. | Y. Cai and C. Chen, “Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems,” J. Opt. Soc. Am. A |

12. | A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. |

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

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

15. | H. Laabs, C. Gao, and H. Weber, “Twisting of three-dimensional Hermite-Gaussian beams,” J. Mod. Opt. |

16. | S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. |

17. | S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. |

18. | B. Lu and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. |

19. | Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. |

20. | C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. |

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

22. | X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express |

23. | O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media |

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

25. | Y. Baykal, “Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere,” J. Opt. Soc. Am. A |

26. | H. T. Eyyuboğlu, “Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere,” J. Opt. Soc. Am. A |

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

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

29. | M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. |

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

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

32. | Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. |

33. | K. Zhu, G. Zhou, X. Li, X. Zheng, and H. Tang, “Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere,” Opt. Express |

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

35. | X. Ji, X. Chen, and B. Lu, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A |

36. | A. Belmonte, A. Comerón, J. A. Rubio, J. Bará, and E. Fernández, “Atmospheric-turbulence-induced power-fade statistics for a multiaperture optical receiver,” Appl. Opt. |

37. | I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple apertures,” in |

38. | Y. E. Yenice, and B. G. Evans, “Adaptive beam-size control for ground-to-space laser communications,” in |

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

**OCIS Codes**

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.3310) Atmospheric and oceanic optics : Laser beam transmission

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: May 12, 2009

Revised Manuscript: June 5, 2009

Manuscript Accepted: June 11, 2009

Published: June 18, 2009

**Citation**

Yangsheng Yuan, Yangjian Cai, Jun Qu, Halil T. Eyyuboğlu, and Yahya Baykal, "Average intensity and spreading of an elegant Hermite–Gaussian beam in turbulent atmosphere," Opt. Express **17**, 11130-11139 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-13-11130

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### References

- A. E. Siegman, LASERS, University Science Books, Mill Valley, 1986.
- W. H. Carter, “Spot size and divergence for Hermite-Gaussian beams of any order,” Appl. Opt. 19(7), 1027–1029 (1980). [CrossRef]
- T. Kojima, “Diffraction of Hermite-Gaussian beams from a sinusoidal conducting grating,” J. Opt. Soc. Am. A 7(9), 1740–1744 (1990). [CrossRef]
- O. Mata-Mendez and F. Chavez-Rivas, “Diffraction of Gaussian and Hermite-Gaussia beams by finite gratings,” J. Opt. Soc. Am. A 18(3), 537–545 (2001). [CrossRef]
- K. M. Luk and P. K. Yu, “Generation of Hermite-Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. A 2(11), 1818–1820 (1985). [CrossRef]
- Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997). [CrossRef]
- H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998). [CrossRef]
- Y. Cai and Q. Lin, “The elliptical Hermite-Gaussian beam and its propagation through paraxial systems,” Opt. Commun. 207, 139–147 (2002).
- Y. Cai and Q. Lin, “Decentered elliptical Hermite–Gaussian beam,” J. Opt. Soc. Am. A 20(6), 1111–1119 (2003). [CrossRef]
- Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005). [CrossRef]
- Y. Cai and C. Chen, “Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems,” J. Opt. Soc. Am. A 24(8), 2394–2401 (2007). [CrossRef]
- A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63(9), 1093–1094 (1973). [CrossRef]
- 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]
- E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. 3(4), 465–469 (1986). [CrossRef]
- H. Laabs, C. Gao, and H. Weber, “Twisting of three-dimensional Hermite-Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).
- S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998). [CrossRef]
- S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001). [CrossRef]
- B. Lu and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. 174(1-4), 99–104 (2000). [CrossRef]
- Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10(1), 33–35 (1967). [CrossRef]
- C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41(5), 1097–1103 (2002). [CrossRef]
- 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]
- X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express 15(26), 17613–17618 (2007). [CrossRef]
- O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005). [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]
- Y. Baykal, “Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 21(7), 1290–1299 (2004). [CrossRef]
- H. T. Eyyuboğlu, “Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere,” J. Opt. Soc. Am. A 22(8), 1527–1535 (2005). [CrossRef]
- 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, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007). [CrossRef] [PubMed]
- M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008). [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]
- R. J. Noriega-Manez and J. C. Gutierrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007). [CrossRef]
- Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007). [CrossRef]
- K. Zhu, G. Zhou, X. Li, X. Zheng, and H. Tang, “Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere,” Opt. Express 16(26), 21315–21320 (2008). [CrossRef] [PubMed]
- Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007). [CrossRef] [PubMed]
- X. Ji, X. Chen, and B. Lu, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 25(1), 21–28 (2008). [CrossRef]
- A. Belmonte, A. Comerón, J. A. Rubio, J. Bará, and E. Fernández, “Atmospheric-turbulence-induced power-fade statistics for a multiaperture optical receiver,” Appl. Opt. 36(33), 8632–8638 (1997). [CrossRef] [PubMed]
- I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple apertures,” in Free-Space Laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE 2990, 102–113 (1997).
- Y. E. Yenice, and B. G. Evans, “Adaptive beam-size control for ground-to-space laser communications,” in Free-Space Laser Communication Technologies X, G. Mecherle, ed., Proc. SPIE 3266, 221–230 (1998).
- A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

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