## Propagation of an Airy beam through the atmosphere |

Optics Express, Vol. 21, Issue 2, pp. 2154-2164 (2013)

http://dx.doi.org/10.1364/OE.21.002154

Acrobat PDF (4195 KB)

### Abstract

In this paper, the effect of thermal blooming of an Airy beam propagating through the atmosphere is examined, and the effect of atmospheric turbulence is not considered. The changes of the intensity distribution, the centroid position and the mean-squared beam width of an Airy beam propagating through the atmosphere are studied by using the four-dimensional (4D) computer code of the time-dependent propagation of Airy beams through the atmosphere. It is shown that an Airy beam can’t retain its shape and the structure when the Airy beam propagates through the atmosphere due to thermal blooming except for the short propagation distance, or the short time, or the low beam power. The thermal blooming results in a central dip of the center lobe, and causes the center lobe to spread and decrease. In contrast with the center lobe, the side lobes are less affected by thermal blooming, such that the intensity maximum of the side lobe may be larger than that of the center lobe. However, the cross wind can reduce the effect of thermal blooming. When there exists the cross wind velocity *v _{x}* in x direction, the dependence of centroid position in x direction on

*v*is not monotonic, and there exists a minimum, but the centroid position in y direction is nearly independent of

_{x}*v*.

_{x}© 2013 OSA

## 1. Introduction

1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**(3), 264–267 (1979). [CrossRef]

2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. **99**(21), 213901 (2007). [CrossRef] [PubMed]

2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

5. I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. **32**(16), 2447–2449 (2007). [CrossRef] [PubMed]

6. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express **16**(17), 12880–12891 (2008). [CrossRef] [PubMed]

7. Y. Gu and G. Gbur, “Scintillation of Airy beam arrays in atmospheric turbulence,” Opt. Lett. **35**(20), 3456–3458 (2010). [CrossRef] [PubMed]

8. H. T. Eyyuboğlu, “Scintillation behavior of Airy beam,” Opt. Laser Technol. **47**, 232–236 (2013). [CrossRef]

5. I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. **32**(16), 2447–2449 (2007). [CrossRef] [PubMed]

9. M. A. Bandres and J. C. Gutiérrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express **15**(25), 16719–16728 (2007). [CrossRef] [PubMed]

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

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

11. D. C. Smith, “High-power laser propagation: Thermal blooming,” Proc. IEEE **65**(12), 1679–1714 (1977). [CrossRef]

12. J. A. Fleck Jr, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) **10**(2), 129–160 (1976). [CrossRef]

13. J. A. Fleck Jr, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere: II,” Appl. Phys. (Berl.) **14**(1), 99–115 (1977). [CrossRef]

14. B. V. Fortes and V. P. Lukin, “Estimation of turbulent and thermal blooming degradation and required characterization of adaptive system,” Proc. SPIE **3706**, 361–367 (1999). [CrossRef]

15. F. G. Gebhardt, “Twenty-five years of thermal blooming: an overview,” Proc. SPIE **1221**, 2–25 (1990). [CrossRef]

## 2. Theoretical model

*I*is given in terms of

*U*, i.e.,where

*α*is the absorption coefficient.

12. J. A. Fleck Jr, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) **10**(2), 129–160 (1976). [CrossRef]

13. J. A. Fleck Jr, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere: II,” Appl. Phys. (Berl.) **14**(1), 99–115 (1977). [CrossRef]

12. J. A. Fleck Jr, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) **10**(2), 129–160 (1976). [CrossRef]

*ρ*and

*v*are perturbations in density and velocity,

*c*is the sound speed, and

_{s}*γ*is the specific heat ratio.

*z*= 0) can be expressed as [2

2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

7. Y. Gu and G. Gbur, “Scintillation of Airy beam arrays in atmospheric turbulence,” Opt. Lett. **35**(20), 3456–3458 (2010). [CrossRef] [PubMed]

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

*w*

_{0}and

*a*are the arbitrary transverse scale and the exponential truncation factor, respectively.

16. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. **24**(9), S1027–S1049 (1992). [CrossRef]

**10**(2), 129–160 (1976). [CrossRef]

13. J. A. Fleck Jr, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere: II,” Appl. Phys. (Berl.) **14**(1), 99–115 (1977). [CrossRef]

## 3. Numerical calculation results and analysis

*γ*= 1.4,

*n*

_{0}= 1.00035,

*c*= 340m/s,

_{s}*α*= 1.252 × 10

^{−5}/m,

*w*

_{0}= 0.05m and

*a*= 0.2. The standard atmosphere density

*ρ*

_{0}= 1.302461kg/m

^{3}is taken as the initial value to solve Eq. (4).

*I*(

*x*,

*y*,

*z*) and its counter lines versus the propagation distance

*z*, the time

*t*, the power

*P*and the cross wind velocity

*v*in x direction are examined by using the 4D computer code of the time-dependent propagation of Airy beams through the atmosphere and are depicted in Figs. 1 -8, respectively. It is mentioned that in Figs. 1-8 the “12.5 mm” is referring to multiplication for the numerals on the transverse axes, and the middle value of the transverse axes is corresponding to the position of the propagation axis z. From Figs. 1 and 2 , it can be seen that an Airy beam can’t retain its shape and the structure when the Airy beam propagates through the atmosphere except for the short propagation distance (e.g.,

_{x}*z*= 0.1km in Fig. 1(b) and Fig. 2(b)). The physical reason is the effect of thermal blooming of atmosphere. When the propagation distance

*z*increases, the thermal blooming causes the center lobe to dip centrally, spread and decrease. In contrast with the center lobe, the side lobes are less affected by thermal blooming because the energy within the side lobe is lower than that within the center lobe, such that the intensity maximum of the side lobe may be larger than that of the center lobe (see Fig. 1(e) and 1(f)). When the

*z*is long enough, the side lobe may dip centrally due to thermal blooming (e.g.,

*z*= 1.2km in Fig. 1(f)). In this paper, we adopt the Strehl ratio

*S*to describe the effect of thermal blooming on the maximum intensity, which is defined as

_{R}*S*=

_{R}*I*

_{max}/

*I*

_{0max}, where

*I*

_{max}and

*I*

_{0max}are the maximum intensity in the atmosphere and in vacuum respectively. The smaller

*S*means the maximum intensity is more affected by thermal blooming. When

_{R}*z*= 0, 0.1km, 0.4km, 0.6km, 0.8km and 1.2km (see Figs. 1(a)-1(f)), we have

*S*= 1, 0.899, 0.411, 0.307, 0.273 and 0.204, respectively.

_{R}*t*and the power

*P*increase. However, when the time

*t*is short enough (e.g.,

*t*= 0.024s in Fig. 3(b) and Fig. 4(b) ) and the power

*P*is low enough (e.g.,

*P*= 10

^{4}W in Fig. 5(b) and Fig. 6(b) ), the effect of thermal blooming can be ignored, and the Airy beam can retain its shape and the structure, which is similar to the behavior in vacuum. On the other hand, for Figs. 3(a)-3(f) we have

*S*= 1, 0.762, 0.411, 0.285, 0.247 and 0.226, respectively. For Figs. 5(a)-5(f) we have

_{R}*S*= 1, 0.877, 0.397, 0.280, 0.247 and 0.201, respectively.

_{R}*I*(

*x*,

*y*,

*z*) and its counter lines versus the cross wind velocity

*v*in x direction. It can be seen that the center lobe takes as a crescent-like pattern (see Fig. 7(b) and Fig. 8(b) where

_{x}*v*= 0.15m/s), which is associated with thermal distortion. In addition, it is clear that the cross wind results in a decrease of the effect of thermal blooming. The physical reason is that the cross wind constantly brings cooler air onto the beam path, allowing more resistance to the effect of thermal blooming. For example, for the center lobe, the central dip disappears, the intensity maximum increases, and the spreading decreases due to the cross wind. When the cross wind velocity is large enough (e.g.,

_{x}*v*= 1m/s in Fig. 7(f) and Fig. 8(c)), the intensity distribution is some similar to that of the Airy beam propagating in vacuum. On the other hand, for Figs. 7(a)-7(f) we have

_{x}*S*= 0.223, 0.384, 0.563, 0.652, 0.698 and 0.744, respectively.

_{R}*w*(

*w*=

_{x}*w*=

_{y}*w*) versus the propagation distance

*z*, the time

*t*and the power

*P*are plotted in Figs. 9 , 10 , and 11 , respectively. It can be seen that,

*w*in the atmosphere is larger than that in vacuum due to the effect of thermal blooming. In the atmosphere

*w*increases with increasing

*z*,

*t*and

*P*, but in vacuum

*w*is unchanged versus

*t*and

*P*. The changes of centroid position versus the propagation distance

*z*, the time

*t*and the power

*P*are also examined by using the 4D computer code of the time-dependent propagation of Airy beams through the atmosphere. It can be shown that the changes of centroid position versus

*z*,

*t*and

*P*are very small, e.g., only within 1mm for the cases in Figs. 9-11, and the numerical results are all omitted here. It is mentioned that the centroid doesn’t change in vacuum or turbulence [10

**36**(14), 2701–2703 (2011). [CrossRef] [PubMed]

*v*in x direction are given in Figs. 12 and 13 , respectively. Figure 12 indicates that the change of centroid position

_{x}*v*, and there exists a minimum, where the centroid position is the furthest away from the propagation axis z. On the other hand, the centroid position

_{x}*v*. Figure 13 shows that

_{x}*w*and

_{x}*w*are all decrease with increasing

_{y}*v*, but

_{x}*w*is smaller than

_{x}*w*. It implies that, in the atmosphere the effect of the cross wind velocity

_{y}*v*on the intensity distribution of an Airy beam in x direction is larger than that in y direction.

_{x}## 4. Conclusions

*z*or the time

*t*is short enough, or the beam power

*P*is low enough, the effect of thermal blooming can be ignored, and the Airy beam can behave as that in vacuum. The thermal blooming causes the center lobe to dip centrally, spread and decrease. In contrast with the center lobe, the side lobes are less affected by thermal blooming because the energy within the side lobe is lower than that within the center lobe, such that the intensity maximum of the side lobe may be larger than that of the center lobe. However, the cross wind can reduce the effect of thermal blooming.

*v*is not monotonic, and there exists a minimum, but the centroid position

_{x}*v*. The mean-squared beam width

_{x}*w*and

_{x}*w*are all decrease with increasing

_{y}*v*, but

_{x}*w*is smaller than

_{x}*w*. In addition, changes of centroid position versus

_{y}*z*,

*t*and

*P*are very small. In the atmosphere

*w*increases with increasing

*z*,

*t*and

*P*, but in vacuum

*w*is independent of

*t*and

*P*. The results obtained in this paper are very useful for applications of Airy beams.

## Acknowledgments

## References and links

1. | M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. |

2. | G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. |

3. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. |

4. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. |

5. | I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. |

6. | J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express |

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

8. | H. T. Eyyuboğlu, “Scintillation behavior of Airy beam,” Opt. Laser Technol. |

9. | M. A. Bandres and J. C. Gutiérrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express |

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

11. | D. C. Smith, “High-power laser propagation: Thermal blooming,” Proc. IEEE |

12. | J. A. Fleck Jr, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) |

13. | J. A. Fleck Jr, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere: II,” Appl. Phys. (Berl.) |

14. | B. V. Fortes and V. P. Lukin, “Estimation of turbulent and thermal blooming degradation and required characterization of adaptive system,” Proc. SPIE |

15. | F. G. Gebhardt, “Twenty-five years of thermal blooming: an overview,” Proc. SPIE |

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

**OCIS Codes**

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(050.1940) Diffraction and gratings : Diffraction

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: September 14, 2012

Revised Manuscript: December 29, 2012

Manuscript Accepted: January 4, 2013

Published: January 22, 2013

**Citation**

Xiaoling Ji, Halil T. Eyyuboğlu, Guangming Ji, and Xinhong Jia, "Propagation of an Airy beam through the atmosphere," Opt. Express **21**, 2154-2164 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-2-2154

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### References

- M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 264–267 (1979). [CrossRef]
- G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32(8), 979–981 (2007). [CrossRef] [PubMed]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99(21), 213901 (2007). [CrossRef] [PubMed]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett.33(3), 207–209 (2008). [CrossRef] [PubMed]
- I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett.32(16), 2447–2449 (2007). [CrossRef] [PubMed]
- J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express16(17), 12880–12891 (2008). [CrossRef] [PubMed]
- Y. Gu and G. Gbur, “Scintillation of Airy beam arrays in atmospheric turbulence,” Opt. Lett.35(20), 3456–3458 (2010). [CrossRef] [PubMed]
- H. T. Eyyuboğlu, “Scintillation behavior of Airy beam,” Opt. Laser Technol.47, 232–236 (2013). [CrossRef]
- M. A. Bandres and J. C. Gutiérrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express15(25), 16719–16728 (2007). [CrossRef] [PubMed]
- X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett.36(14), 2701–2703 (2011). [CrossRef] [PubMed]
- D. C. Smith, “High-power laser propagation: Thermal blooming,” Proc. IEEE65(12), 1679–1714 (1977). [CrossRef]
- J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.)10(2), 129–160 (1976). [CrossRef]
- J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-Dependent propagation of high energy laser beams through the atmosphere: II,” Appl. Phys. (Berl.)14(1), 99–115 (1977). [CrossRef]
- B. V. Fortes and V. P. Lukin, “Estimation of turbulent and thermal blooming degradation and required characterization of adaptive system,” Proc. SPIE3706, 361–367 (1999). [CrossRef]
- F. G. Gebhardt, “Twenty-five years of thermal blooming: an overview,” Proc. SPIE1221, 2–25 (1990). [CrossRef]
- H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron.24(9), S1027–S1049 (1992). [CrossRef]

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