## Influence of turbulence on the effective radius of curvature of radial Gaussian array beams

Optics Express, Vol. 18, Issue 7, pp. 6922-6928 (2010)

http://dx.doi.org/10.1364/OE.18.006922

Acrobat PDF (163 KB)

### Abstract

The analytical formula for the effective radius of curvature of radial Gaussian array beams propagating through atmospheric turbulence is derived, where coherent and incoherent beam combinations are considered. The influence of turbulence on the effective radius of curvature of radial Gaussian array beams is studied by using numerical calculation examples.

© 2010 OSA

## 1. **Introduction**

3. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. **28**(1), 10–12 (2003). [CrossRef] [PubMed]

8. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express **15**(25), 16909–16915 (2007). [CrossRef] [PubMed]

9. M. A. Porras, J. Alda and E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams,” Appl. Opt. **31**(30), 6389–6402 (1992). [CrossRef] [PubMed]

9. M. A. Porras, J. Alda and E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams,” Appl. Opt. **31**(30), 6389–6402 (1992). [CrossRef] [PubMed]

10. 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]

10. 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]

11. H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. **32**(3), 400–407 (1996). [CrossRef]

13. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express **16**(22), 18437–18442 (2008). [CrossRef] [PubMed]

17. X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B **98**(2-3), 557–565 (2010). [CrossRef]

18. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. **24**(9), S1027–1049 (1992). [CrossRef]

19. Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. **34**(5), 563–565 (2009). [CrossRef] [PubMed]

## 2. Analytical Formulae

### 2.1 Mean squared beam width in free space

*N*equal elements, which are Gaussian beams and located symmetrically on a ring with radius

*z*= 0 is expressed as

*j*=

*p*,

*q*= 0, 1, 2, …

*N*-1),

20. To derive Eq. (4), we first introduce the new variables of integration by setting *δ* denotes the Dirac delta function and *f* is an arbitrary function and

21. J. D. Strohschein, H. J. Seguin, and C. E. Capjack, “Beam propagation constants for a radial laser array,” Appl. Opt. **37**(6), 1045–1048 (1998). [CrossRef]

### 2.2 Effective radius of curvature in turbulence

18. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. **24**(9), S1027–1049 (1992). [CrossRef]

*A*,

*B*and

*D*are elements of the transfer matrix

*x*-axis and

*y*-axis respectively. The parameters with the subscripts “1” and “2” denote those before and after the optical ABCD system respectively.

18. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. **24**(9), S1027–1049 (1992). [CrossRef]

*z*= 0, from Eq. (9) we obtain the free-space propagation equation of the effective radius of curvature of an arbitrary field, i.e.,where

**24**(9), S1027–1049 (1992). [CrossRef]

*z*= 0. The Rayleigh range is used in the theory of lasers to characterize the distance over which a beam may be considered effectively non-spreading. It is clear that the effective radius of curvature of an arbitrary field defined by Eq. (9) obeys the same free-space propagation equation as does the wavefront curvature of an ideal Gaussian beam.

19. Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. **34**(5), 563–565 (2009). [CrossRef] [PubMed]

*T*= 0, we obtain the effective radius of curvature of radial Gaussian array beams propagating through atmospheric turbulence for the coherent combination case, i.e.

*R*increases with increasing

*R*is independent of

*N*.

## 3. Numerical calculation results and analysis

*R*, where

*R*decreases due to turbulence. But the decrease of

*R*is generally larger for the coherent combination than that for the incoherent combination, i.e., for the incoherent combination

*R*is less sensitive to turbulence than that for the coherent combination. Figure 1 shows that in free space for the coherent combination

*R*is always larger than that for the incoherent combination, while for the coherent combination

*R*may be smaller than that for the incoherent combination as the propagation distance

*z*increases in turbulence. Figure 1 also indicates that, there exists the minimum

*R*

_{min}as

*z*increases, and position

*z*

_{max}of

*R*

_{min}is further away from the source plane for the coherent combination than that for the incoherent combination. From Fig. 2 it can be seen that there may exist a minimum of

*R*as

*R*approaches the same value when

*R*increases with increasing

*N*in free space, but

*R*decreases with increasing

*N*in turbulence. Furthermore,

*R*tends to its asymptotical value when

*N*is large enough, and for the two types of beam combination

*R*approaches the same value when

*N*is small enough.

*N*is large enough since it is composed of multi-beams that are combined coherently or incoherently. Thus, propagation of radial array and annulus beams are different.

## 4. Conclusions

*R*of radial Gaussian array beams in turbulence has been derived in this paper. It has been shown that for the two types of beam combination

*R*decreases due to turbulence. However, for the incoherent combination

*R*is less sensitive to turbulence than that for the coherent combination. In free space

*R*for the coherent combination is always larger than that for the incoherent combination, but this situation may reverse as the propagation distance increases in turbulence. For the coherent combination,

*R*increases with increasing the beam number

*N*in free space, but

*R*decreases with increasing

*N*in turbulence. In addition, there may exist a minimum of

*R*as the inverse radial fill-factor

## Acknowledgment

## References and links

1. | R. L. Fante, “Wave propagation in random media: a systems approach,” in |

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

3. | A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. |

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

5. | J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. |

6. | X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A |

7. | H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. |

8. | X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express |

9. | M. A. Porras, J. Alda and E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams,” Appl. Opt. |

10. | 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 |

11. | H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. |

12. | W. D. Bilida, J. D. Strohschein, and H. J. J. Seguin, “High-power 24 channel radial array slab RF-excited carbon dioxide laser,” in |

13. | Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express |

14. | Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B |

15. | H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B |

16. | X. Ji and X. Li, “Directionality of Gaussian array beams propagating in atmospheric turbulence,” J. Opt. Soc. Am. A |

17. | X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B |

18. | H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. |

19. | Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. |

20. | To derive Eq. (4), we first introduce the new variables of integration by setting |

21. | J. D. Strohschein, H. J. Seguin, and C. E. Capjack, “Beam propagation constants for a radial laser array,” Appl. Opt. |

**OCIS Codes**

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(140.3295) Lasers and laser optics : Laser beam characterization

(140.3298) Lasers and laser optics : Laser beam combining

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: January 5, 2010

Revised Manuscript: February 28, 2010

Manuscript Accepted: March 11, 2010

Published: March 19, 2010

**Citation**

Xiaoling Ji, Halil T. Eyyuboğlu, and Yahya Baykal, "Influence of turbulence on the effective radius of curvature of radial Gaussian array beams," Opt. Express **18**, 6922-6928 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-6922

Sort: Year | Journal | Reset

### References

- R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics 22, E. Wolf, ed., (Elsevier, Amsterdam, 1985), Chap. 6.
- L. C. Andrews, and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).
- A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28(1), 10–12 (2003). [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]
- J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. 282(9), 1691–1698 (2009). [CrossRef]
- X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24(11), 3554–3563 (2007). [CrossRef]
- H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10(1), 015005 (2008). [CrossRef]
- X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007). [CrossRef] [PubMed]
- M. A. Porras, J. Alda and E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams,” Appl. Opt. 31(30), 6389–6402 (1992). [CrossRef] [PubMed]
- 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]
- H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996). [CrossRef]
- W. D. Bilida, J. D. Strohschein, and H. J. J. Seguin, “High-power 24 channel radial array slab RF-excited carbon dioxide laser,” in Gas and Chemical Lasers and Applications, H. R. C. Sze and E. A. Dorko, eds., Proc. SPIE 2987, 13–21 (1997).
- Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008). [CrossRef] [PubMed]
- Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007). [CrossRef]
- H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91(2), 265–271 (2008). [CrossRef]
- X. Ji and X. Li, “Directionality of Gaussian array beams propagating in atmospheric turbulence,” J. Opt. Soc. Am. A 26(2), 236–243 (2009). [CrossRef]
- X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010). [CrossRef]
- H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–1049 (1992). [CrossRef]
- Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009). [CrossRef] [PubMed]
- To derive Eq. (4), we first introduce the new variables of integration by setting u=(r′2+r′1)/2 and v=r′2−r′1. Then we use the formulae∫exp(−i2πxs)dx=δ(s), ∫x2exp(−i2πxs)dx=−δ″(s)/(2π)2, ∫f(x)δ″(x)dx=f″(0) and ∫f(x)δ(x)dx=f(0), where δ denotes the Dirac delta function and δ″ is its second derivative, and f is an arbitrary function and f″ is its second derivative.
- J. D. Strohschein, H. J. Seguin, and C. E. Capjack, “Beam propagation constants for a radial laser array,” Appl. Opt. 37(6), 1045–1048 (1998). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.