## Propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere |

Optics Express, Vol. 19, Issue 5, pp. 3945-3951 (2011)

http://dx.doi.org/10.1364/OE.19.003945

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### Abstract

The propagation of a higher-order cosh-Gaussian beam through a paraxial and real *ABCD* optical system in turbulent atmosphere has been investigated. The analytical expressions for the average intensity, the effective beam size, and the kurtosis parameter of a higher-order cosh-Gaussian beam through a paraxial and real *ABCD* optical system are derived in turbulent atmosphere. The average intensity distribution and the spreading properties of a higher-order cosh-Gaussian in turbulent atmosphere are numerically demonstrated. The influences of the beam parameters and the structure constant of the atmospheric turbulence on the propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere are also examined in detail.

© 2011 OSA

## 1. Introduction

1. L. W. Casperson and A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A **15**(4), 954–961 (1998). [CrossRef]

2. A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite-sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A **15**(9), 2425–2432 (1998). [CrossRef]

1. L. W. Casperson and A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A **15**(4), 954–961 (1998). [CrossRef]

4. D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. **236**(4-6), 225–235 (2004). [CrossRef]

5. H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express **12**(20), 4659–4674 (2004). [CrossRef] [PubMed]

9. X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express **15**(26), 17613–17618 (2007). [CrossRef] [PubMed]

10. Y. Zhang, Y. Song, Z. Chen, J. Ji, and Z. Shi, “Virtual sources for a cosh-Gaussian beam,” Opt. Lett. **32**(3), 292–294 (2007). [CrossRef] [PubMed]

11. G. Zhou and F. Liu, “Far field structural characteristics of cosh-Gaussian beam,” Opt. Laser Technol. **40**(2), 302–308 (2008). [CrossRef]

12. K. Zhu, H. Tang, X. Wang, and T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Stuttg.) **113**(5), 222–226 (2002). [CrossRef]

14. G. Zhou and J. Zheng, “Beam propagation of a higher-order cosh-Gaussian beam,” Opt. Laser Technol. **41**(2), 202–208 (2009). [CrossRef]

15. G. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. **56**(7), 886–892 (2009). [CrossRef]

16. J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. **57**(20), 2039–2047 (2010). [CrossRef]

17. J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D **57**(3), 419–425 (2010). [CrossRef]

*ABCD*optical system in turbulent atmosphere is investigated. Analytical propagation formulae of the average intensity, the effective beam size, and the kurtosis parameter are derived and illustrated by a numerical example.

## 2. Propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere

*z*-axis is taken to be the propagation axis. The higher-order cosh-Gaussian beam in the source plane

*z*= 0 takes the form aswith

*E*(

_{n}*x*

_{0}, 0) and

*E*(

_{n}*y*

_{0}, 0) given bywhere

*j*

_{0}=

*x*

_{0}or

*y*

_{0}.

*n*is the beam order. Ω is the cosh parameter, and

*w*

_{0}is the waist width of Gaussian part. If

*n*= 0, Eq. (1) reduces to be the well-known Gaussian beam. If

*n*= 1, Eq. (1) reduces to be a cosh-Gaussian beam.

*E*(

_{n}*j*

_{0}, 0) can also be written in the form aswith the coefficients

*C*and

_{m}*b*given bywhere

_{m}*δ*=

*w*

_{0}

^{2}Ω

^{2}. Therefore,

*E*(

_{n}*j*

_{0}, 0) can be produced by superposition of

*n*+ 1 decentered Gaussian beams with the same waist width. Based on the extended Huygens-Fresnel diffraction integral, the higher-order cosh-Gaussian beam propagating through a paraxial and real

*ABCD*optical system in turbulent atmosphere can be obtained by

*k*= 2

*π*/

*λ*with

*λ*being the optical wavelength.

*ψ*(

*x*

_{0},

*y*

_{0},

*x*,

*y*) is the solution to the Rytov method that represents the random part of the complex phase. The optical system of concern is orthogonal and is described by the same

*ABCD*matrix in each of the mutually orthogonal planes,

*x*-

*z*and

*y*-

*z*.

*A*,

*B*,

*C*, and

*D*are the transfer matrix elements of the paraxial optical system. Moreover, there is no inherent aperture between the source and the output planes, which denotes that

*A*,

*B*,

*C*, and

*D*are all realvalued. Therefore, the average intensity of the higher-order cosh-Gaussian beam passing through a paraxial and real

*ABCD*optical system in turbulent atmosphere is found to be

*ρ*

_{0}is the spherical-wave lateral coherence radius due to the turbulence of the entire optical system and is defined aswhere

*C*

_{n}^{2}is the constant of refraction index structure and describes the turbulence level.

*b*(

*z*) corresponds to the approximate matrix element for a ray propagating backwards through the system.

*L*is the axial distance between the source and the output planes. Substituting Eqs. (1), (3), and (7) into Eq. (6), the average intensity of the higher-order cosh-Gaussian beam in the output plane reads aswith <

*I*(

*x*,

*z*) and <

*I*(

*y*,

*z*) given bywhere

*j*=

*x*or

*y*(hereafter). The parameters

*α*

_{1}

*,*

_{j}*α*

_{2}

*,*

_{j}*β*

_{1}

*, and*

_{j}*β*

_{2}

*are defined byThe effective beam size of the higher-order cosh-Gaussian beam in the*

_{j}*j*-direction of the output plane yields

*τ*given bywhere

*γ*

_{1}

*,*

_{j}*γ*

_{2}

*,*

_{j}*ξ*, and

_{j}*η*are defined byThe kurtosis parameter, which is defined by the fourth-order moment, is employed to describe the flatness degree of the beams and is an important parameter to valuate the beam propagation. The kurtosis parameter of the higher-order cosh-Gaussian beam in the

_{j}*j-*direction turns out to be

## 3. The numerical results and analyses

*A*= 1,

*B*=

*z*,

*C*= 0, and

*D*= 1. As the

*x*- and

*y*-directions are separable in the formulae derived above, only the

*x*-direction is considered hereafter. Moreover,

*λ*is set to be 0.8μm. Figures 1 and 2 represent the normalized intensity distributions of higher-order cosh-Gaussian beams with different beam parameters in the reference plane

*z*= 1km and

*z*= 2km in turbulent atmosphere. In Figs. 1 and 2,

*C*

_{n}^{2}is set to be 10

^{−14}m

^{-2/3}. Under different conditions of the beam parameters, the higher-order cosh-Gaussian beam in the source plane can be a Gaussian-like beam, a flattened beam, and a dark hollow beam. The intensity distributions of a flattened beam and a dark hollow beam will change upon propagation. When the propagation distance is large enough, the intensity distributions of a flattened beam and a dark hollow beam will tend to a Gaussian-like distribution. With varying one of the cosh parameter Ω, the beam order

*n*, and the Gaussian waist

*w*

_{0}, the normalized intensity distribution of a higher-order cosh-Gaussian beam propagating in turbulent atmosphere also takes on different distribution, which is shown in Fig. 1. From Figs. 1 and 2, we can conclude the following conclusion. When one of the cosh parameter Ω, the beam order

*n*, and the Gaussian waist

*w*

_{0}is large, the normalized intensity distribution of a higher-order cosh-Gaussian beam will undergo first the dark hollow distribution, then the fattened distribution, and finally the Gaussian-like distribution with increasing the propagation distance in turbulent atmosphere. To further reveal the spreading properties of a higher-order cosh-Gaussian beam in turbulent atmosphere, the effective beam sizes of higher-order cosh-Gaussian beams versus the propagation distance

*z*in turbulent atmosphere are depicted in Fig. 3 , where

*C*

_{n}^{2}= 10

^{−14}m

^{-2/3}. When the propagation distance in turbulent atmosphere is small, the higher-order cosh-Gaussian beam with the large cosh parameter has the large effective beam size. When the propagation distance in turbulent atmosphere is large enough, the effective beam sizes of higher-order cosh-Gaussian beams with the different cosh parameters are approximately equivalent. The higher-order cosh-Gaussian beam with the large beam order has the large effective beam size. However, the difference between the effective beam sizes of higher-order cosh-Gaussian beams with different beam orders will shrink upon propagation in turbulent atmosphere. The higher-order cosh-Gaussian beam with the smaller Gaussian waist first has the smaller effective beam size and finally has the larger effective beam size with increasing the propagation distance in turbulent atmosphere. The gradients of the curves in Fig. 3 denote that the higher-order cosh-Gaussian beam spreads more rapidly in turbulent atmosphere for the smaller beam parameters. Figure 4 shows the kurtosis parameters of higher-order cosh-Gaussian beams versus the propagation distance

*z*in turbulent atmosphere. When first propagating in turbulent atmosphere, the higher-order cosh-Gaussian beam with the small cosh parameter, or the small beam order, or the small Gaussian waist has the large kurtosis parameter. When the propagation distance in turbulent atmosphere is large enough, the kurtosis parameters of the higher-order cosh-Gaussian beams all tend to 3, which is the value of the kurtosis parameter of the Gaussian distribution. The influence of the structure constant of the atmospheric turbulence on the propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere is shown in Fig. 5 . We find that the higher-order cosh-Gaussian beam spreads more rapidly in turbulent atmosphere for a larger structure constant.

## 4. Conclusions

*ABCD*optical system in turbulent atmosphere is investigated. Based on the extended Huygens-Fresnel integral, the analytical expressions of the average intensity, the effective beam size, and the kurtosis parameter of a higher-order cosh-Gaussian beam are derived in turbulent atmosphere. The average intensity distribution and spreading properties of a higher-order cosh-Gaussian are numerically examined. The higher-order cosh-Gaussian beam spreads more rapidly in turbulent atmosphere for the smaller beam parameters and a larger structure constant. This research is useful to the practical applications in free-space optical communications and remote sensing involving the higher-order cosh-Gaussian beam.

## Acknowledgements

## References and links

1. | L. W. Casperson and A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A |

2. | A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite-sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A |

3. | D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned |

4. | D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. |

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

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

7. | H. T. Eyyuboğlu and Y. Baykal, “Scintillation characteristics of cosh-Gaussian beams,” Appl. Opt. |

8. | X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B |

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

10. | Y. Zhang, Y. Song, Z. Chen, J. Ji, and Z. Shi, “Virtual sources for a cosh-Gaussian beam,” Opt. Lett. |

11. | G. Zhou and F. Liu, “Far field structural characteristics of cosh-Gaussian beam,” Opt. Laser Technol. |

12. | K. Zhu, H. Tang, X. Wang, and T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Stuttg.) |

13. | Q. Tang, Y. Yu, and Q. Hu, “A new method to generate flattened Gaussian beam by incoherent combination of cosh Gaussian beams,” Chin. Opt. Lett. |

14. | G. Zhou and J. Zheng, “Beam propagation of a higher-order cosh-Gaussian beam,” Opt. Laser Technol. |

15. | G. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. |

16. | J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. |

17. | J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D |

**OCIS Codes**

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(080.2730) Geometric optics : Matrix methods in paraxial optics

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: January 4, 2011

Revised Manuscript: February 6, 2011

Manuscript Accepted: February 7, 2011

Published: February 14, 2011

**Citation**

Guoquan Zhou, "Propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere," Opt. Express **19**, 3945-3951 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-5-3945

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### References

- L. W. Casperson and A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15(4), 954–961 (1998). [CrossRef]
- A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite-sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A 15(9), 2425–2432 (1998). [CrossRef]
- D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224(1-3), 5–12 (2003). [CrossRef]
- D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004). [CrossRef]
- H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12(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]
- H. T. Eyyuboğlu and Y. Baykal, “Scintillation characteristics of cosh-Gaussian beams,” Appl. Opt. 46(7), 1099–1106 (2007). [CrossRef] [PubMed]
- X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87(3), 547–552 (2007). [CrossRef]
- X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express 15(26), 17613–17618 (2007). [CrossRef] [PubMed]
- Y. Zhang, Y. Song, Z. Chen, J. Ji, and Z. Shi, “Virtual sources for a cosh-Gaussian beam,” Opt. Lett. 32(3), 292–294 (2007). [CrossRef] [PubMed]
- G. Zhou and F. Liu, “Far field structural characteristics of cosh-Gaussian beam,” Opt. Laser Technol. 40(2), 302–308 (2008). [CrossRef]
- K. Zhu, H. Tang, X. Wang, and T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Stuttg.) 113(5), 222–226 (2002). [CrossRef]
- Q. Tang, Y. Yu, and Q. Hu, “A new method to generate flattened Gaussian beam by incoherent combination of cosh Gaussian beams,” Chin. Opt. Lett. 5, S46–S48 (2007).
- G. Zhou and J. Zheng, “Beam propagation of a higher-order cosh-Gaussian beam,” Opt. Laser Technol. 41(2), 202–208 (2009). [CrossRef]
- G. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 56(7), 886–892 (2009). [CrossRef]
- J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 57(20), 2039–2047 (2010). [CrossRef]
- J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010). [CrossRef]

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