## Power coupling of a two-Cassegrain-telescopes system in turbulent atmosphere in a slant path

Optics Express, Vol. 15, Issue 12, pp. 7697-7707 (2007)

http://dx.doi.org/10.1364/OE.15.007697

Acrobat PDF (862 KB)

### Abstract

The characteristics of dark hollow beams passing through a two-Cassegrain-telescopes system in turbulent atmosphere in a slant path have been investigated. The distribution of the average intensity at the receiver telescope and the efficiency of power coupling with respect to propagation distance with different parameters are derived and numerically calculated. These studies illuminate that the power of the dark hollow beams is concentrated on a narrow annular aperture at the source plane and its power coupling with a transmitter Cassegrain telescope can remain quite high. For short distance between the two Cassegrain telescopes, the normalized average intensity distribution at receiver plane holds shape similar to that at the source plane, and the two Cassegrain telescopes keep high efficiency of the power coupling. But with the increment in the propagation distance, the power of the dark hollow beams gradually converges to the central and the spot spreads. The central obscuration of the receiver telescope blocks more of the power; meanwhile more of the power moves out beyond the edge of the receiving aperture. Therefore, the efficiency of the power coupling decreases with the increment in the propagation distance. In addition, the relations between the efficiency of power coupling and wavelength of laser beams are also numerically calculated and discussed.

© 2007 Optical Society of America

## 1. Introduction

1. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A **20**, 1094–1102 (2003). [CrossRef]

9. C. Arpali, C. Yazicioglu, H. T. Eyyuboglu, S. A. Arpali, and Y. Baykal, “Simulator for general-type beam propagation in turbulent atmosphere,” Opt. Express **14**, 8918–8928 (2006). [CrossRef] [PubMed]

## 2. Analysis of theory

*H*is the altitude between the transmitter and the receiver,

*ζ*is the zenith angle,

*L*is the propagation distance of the laser beam along

*z*. In the Cartesian coordinate system, the source plane is located at the transmitter aperture (

*z*=0), the receiver plane is located at the receiver aperture (

*z*=

*L*). The (

*x*,

*y*) and (

*p*,

*q*) denote the transverse coordinates of the source plane and receiver plane, respectively. When laser beams enter into the transmitter telescope, the power will be lost due to the limitation of the telescope. Laser beams at the source plane can be regarded as a collimated laser beam limited by aperture of the primary mirror and the obscuration of the second mirror of the Cassegrain telescope. If

*E*

_{0}(

*x*,

*y*,0) denotes the optical field of the reflected collimated laser beam from the primary mirror without any loss of energy, the optical field at the source plane can been expressed as

*a*and

*b*are the radius of the secondary and primary mirror, respectively. The loss due to the aperture on the primary mirror is neglected. In order to reduce the loss due to the transmitter telescope, dark hollow beams are adapted in this paper. The optical field of a circular dark hollow beam (with circular symmetry) at

*z*=0 can be expressed as the following finite sum of Gaussian beams [15

15. Z. Mei and D. Zhao, “Controllable dark-hollow beams and their propagation characteristics,” J. Opt. Soc. Am. A **22**, 1898–1902 (2005). [CrossRef]

*N*is the order of a circular dark hollow beam,

*σ*→ 0 , Eq. (3) reduces to a flattened Gaussian beam [16

16. Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. **27**, 1007–1009 (2002). [CrossRef]

*N*=1 and

*σ*→ 0 , Eq. (3) reduces to a Gaussian beam. Contour plots (see Fig. 1 in Ref [11

11. Y. Cai and S. He, “Propagation of various dark hollow beams in turbulent atmosphere,” Opt. Express **14**, 1353–1367 (2006). [CrossRef] [PubMed]

*N*or σ increases. By using the extended Huygens-Fresnel principle, the average intensity distribution at the receiver plane can been expressed as [2

2. H. T. Eyyuboglu and Y. Baykal, “Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Express **12**, 4659–4674 (2004). [CrossRef] [PubMed]

*k*is the wave number, the asterisk denotes the complex conjugation, and the <> indicates the ensemble average over the medium statistics covering the log-amplitude and phase fluctuations due to the turbulent atmosphere.

*Φ*(

*x*,

*y*,

*p*,

*q*) represents the random part of the complex phase of a spherical wave that propagates from point (

*x*,

*y*, 0) at the source plane to the point (

*p*,

*q*,

*L*) at the receiver plane and can be expressed as [17

17. 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**, 1297–1304 (1979). [CrossRef]

*D*(

_{Φ}*x*-

*ξ*,

*y*-

*η*) is the wave structure function,

*ρ*

_{0}is the coherence length. In longdistance transmission systems, the coherence length may be reduced by propagation factors such as turbulence, scattering, and diffraction. In turbulent atmosphere the coherence length of spherical wave is

*C*

^{2}

*(*

_{n}*h*) is altitude dependent. In this paper the ITU-R model presented in 2001 [18] is selected to describe the model of the structure constant and can be expressed as

*h*is the altitude from the ground,

*V*= [

*v*

_{g}^{2}+ 30.69

*v*+ 348.91)

_{g}^{1/2}is the wind speed along the vertical path,

*v*is the ground wind speed (in this paper we set

_{g}*v*=0),

_{g}*C*

_{0}is the nominal value of at ground level (typically value is 1.7×10

^{-14}/

*m*

^{-2/3}). Generally, the hard-edge aperture function can be expanded as the sum of complex Gaussian functions with finite numbers [19

19. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. **83**, 1752 (1988). [CrossRef]

20. Y. Zhang, “Nonparaxial propagation analysis of elliptical Gaussian beams diffracted by a circular aperture,” Opt Commun. **248**, 317–326 (2005). [CrossRef]

*B*and

_{j}*C*, are the expansion coefficients.

_{j}*M*is the number of the expansion coefficients. In this paper we adopt the expansion coefficients in Ref [19

19. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. **83**, 1752 (1988). [CrossRef]

*M*=16).

*R*= [

*p*

^{2}+

*q*

^{2})

^{1/2}, the average intensity at receiver plane of the laser beams limited by the Cassegrain telescope can be written as

*σ*→ 0 and

*a*=0, the average intensity of a circular flattened Gaussian beam with aperture in turbulent atmosphere can be obtained as

14. X. Chu, Y. Ni, and G. Zhou, “Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere,” Opt. Commun. (2007), doi:10.1016/ j.optcom.2007.02.035. [CrossRef]

*b*→ ∞ and

*a*=0, the average intensity of a circular dark hollow beam without aperture in turbulent atmosphere can be obtained. It agrees with the results in Ref. [11

11. Y. Cai and S. He, “Propagation of various dark hollow beams in turbulent atmosphere,” Opt. Express **14**, 1353–1367 (2006). [CrossRef] [PubMed]

19. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. **83**, 1752 (1988). [CrossRef]

21. Z. Mei, D. Zhao, and J. Gu, “Comparison of two approximate methods for hard-edged diffracted flat-topped light beams,” Opt. Commun. **267**, 58–64 (2006). [CrossRef]

## 3. Numerical calculation and analysis

### 3.1. Power distribution of dark hollow beam

*I*(

*r*,0) =

*E*

^{2}(

*r*,0) (power on unit area). Another one is the power at annular aperture with unit radius and can be expressed as

*I*(

*r*, 0) and

*S*(

*r*, 0) with different parameters. The vertical axis denotes the coordinates of the position where the value of

*I*(

*r*, 0) or

*S*(

*r*, 0) is the maximum.

*I*(

*r*, 0) and

*S*(

*r*, 0) are different when the parameters of the dark hollow beams are the same. The value of coordinates of the peak value position of

*S*(

*r*, 0) is bigger than that of

*I*(

*r*, 0). Meanwhile, the coordinates of peak value positions are changed with the change of the parameters. Figure 2 shows that the value of the coordinates of the peak value position of

*S*(

*r*, 0) is increasing with σ,

*N*, and

*w*

_{0}. The principles for

*I*(

*r*, 0) are the same, but the speed of the change of the coordinates versus

*w*

_{0}is faster than those with σ and

*N*. In this paper the parameters of the dark hollow beams are selected as σ = 0.7,

*w*

_{0}= 0.2

*m*, and

*N*= 3. Numerical calculation shows that the coordinates of the peak value position of

*I*(

*r*, 0) and σ (

*r*, 0) are 0.24

*m*and 0.26

*m*, respectively. In order to optimize the parameters of the receiver Cassegrain telescope, the profile of the power in bucket (

*P*(

*r*, 0)/

*P*

_{0}) of the dark hollow beam without the limitation of the transmitter telescope is plotted in Fig. 3, where

*P*(

*r*,0) = ∫

^{r}_{0}

*S*(

*r*,0)

*dr*.

*a*=0.15

*m*and

*b*=0.4

*m*, 98% power of the dark hollow beams can pass through the transmitter telescope.

### 3.2. Average intensity distribution at the receiver plane

*w*

_{0}0=0.2

*m*, and

*N*=3. The normalized intensity of the dark hollow beams before entering the transmitter is defined as

*I*(

_{Max}*x*,

*y*,0) is the peak value of

*I*(

*x*,

*y*,0) (

*I*(

*x*,

*y*,0) =

*E*

^{2}(

*x*,

*y*,0)). The normalized average intensity at the receiver plane is defined as

*b*-

*a*) is larger, namely the truncation by the annular aperture is small, the profiles of the normalized average intensity with annular aperture is similar to those without the limitation of the annular aperture [see the solid curves and the dashed curves in Figs. 4(b) and 4(c)]. But with more truncation, the profile of the normalized average intensity is complex [see the dotted curve in Fig. 4(b)]. With further increases in the propagation distance, the peak of the profile of the normalized average intensity around the central point gradually disappears and concentrates on the centre. The speed of the concentration with more truncation is faster than that with less truncation or without truncation. If we chose the same parameters as those in Ref. 11, numerical calculation shows that the effects of turbulence on the average intensity distribution are smaller in a slant path than those in a horizontal path.

### 3.3. Efficiency of the power coupling

*η*=

*P*(

*R*,

*L*)/

*P*

_{0}, if the radius of the receiver telescope (without obscuration on the center) is

*R*. Figure 5 illuminates the efficiency of the power coupling with different parameters.

*R*, the speed of the increase of the efficiency becomes faster. However, with further increases in

*R*, the speed of the increase of the efficiency becomes slower and reaches to zero. Namely 100% energy transmitted from the transmitter enters into the receiver aperture. From Figs. 5(b) and 5(c) we can see that the speed of the increase of the efficiency gradually reaches to a constant within a magnitude of

*R*with the increase of the propagation distance. Namely the power is more diffused with longer propagation distance. The principle for a truncated beam is the same way, but the speed of the variations is faster than this without truncation. From the discussion above we can see that the size of the central obscuration and receiver aperture at the receiver plane is an important factor to influence the efficiency of the power coupling. To show the effect of the receiver telescope and propagation distance on the efficiency of the power coupling, some values of the efficiency with different parameters are listed in Table.1.

*μm*than those with λ=3.8

*μm*and λ=10.6

*μm*. The density of the power on the center of receiver plane is larger with λ=3.8

*μm*than that with λ=1.06

*μm*and λ=10.6

*μm*; namely, the loss due to obscuration of the receiver telescope is bigger with λ=3.8

*μm*than that with λ=1.06

*μm*and λ=10.6

*μm*. The transmitted power moving out beyond the edge of the receiving aperture is bigger with λ=10.6

*μm*than that with λ=1.06

*μm*and λ=3.8

*μm*.

## 4. Conclusion

## Acknowledgments

## References and links

1. | T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A |

2. | H. T. Eyyuboglu and Y. Baykal, “Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Express |

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

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

5. | H. T. Eyyuboglu, “Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Commun. |

6. | Y. Baykal, “Log-amplitude and phase fluctuations of higher-order annular laser beams in a turbulent medium,” J. Opt. Soc. Am. A |

7. | H. T. Eyyuboglu, C. Arpali, and Y. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express. |

8. | Y. Baykal and H. T. Eyyuboglu, “Scintillation index of flat-topped Gaussian beams,” Appl. Opt. |

9. | C. Arpali, C. Yazicioglu, H. T. Eyyuboglu, S. A. Arpali, and Y. Baykal, “Simulator for general-type beam propagation in turbulent atmosphere,” Opt. Express |

10. | H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere,” Opt. Commun. |

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

12. | Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A: Pure Appl. Opt. |

13. | Y. Cai and D. Ge, “Analytical formula for a decentered elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Commun. |

14. | X. Chu, Y. Ni, and G. Zhou, “Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere,” Opt. Commun. (2007), doi:10.1016/ j.optcom.2007.02.035. [CrossRef] |

15. | Z. Mei and D. Zhao, “Controllable dark-hollow beams and their propagation characteristics,” J. Opt. Soc. Am. A |

16. | Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. |

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

18. | ITU-R Document 3J/31-E, “On propagation data and prediction methods required for the design of space-to-earth and earth to-space optical communication systems,” Radio-communication Study Group Meeting, Budapest (2001), p. 7. |

19. | J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. |

20. | Y. Zhang, “Nonparaxial propagation analysis of elliptical Gaussian beams diffracted by a circular aperture,” Opt Commun. |

21. | Z. Mei, D. Zhao, and J. Gu, “Comparison of two approximate methods for hard-edged diffracted flat-topped light beams,” Opt. Commun. |

**OCIS Codes**

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(010.3310) Atmospheric and oceanic optics : Laser beam transmission

**ToC Category:**

Atmospheric and ocean optics

**History**

Original Manuscript: March 23, 2007

Revised Manuscript: April 17, 2007

Manuscript Accepted: May 8, 2007

Published: June 7, 2007

**Citation**

Xiuxiang Chu and Guoquan Zhou, "Power coupling of a two-Cassegrain-telescopes system in turbulent atmosphere in a slant path," Opt. Express **15**, 7697-7707 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-12-7697

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### References

- T. Shirai, A. Dogariu, and E. Wolf, "Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. A 20, 1094-1102 (2003). [CrossRef]
- H. T. Eyyuboðlu and Y. Baykal, "Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere," Opt. Express 12, 4659-4674 (2004). [CrossRef] [PubMed]
- H. T. Eyyuboðlu and Y. Baykal, "Average intensity and spreading of cosh-Gaussian beams in the turbulent atmosphere," Appl. Opt. 44, 976-983 (2005). [CrossRef] [PubMed]
- H. T. Eyyuboðlu, "Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere," J. Opt. Soc. Am. A 22, 1527-1535 (2005). [CrossRef]
- H. T. Eyyuboðlu, "Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere," Opt. Commun. 245, 37-47 (2005). [CrossRef]
- Y. Baykal, "Log-amplitude and phase fluctuations of higher-order annular laser beams in a turbulent medium," J. Opt. Soc. Am. A 22, 672-679 (2005). [CrossRef]
- H. T. Eyyuboðlu, C. Arpali, and Y. Baykal, "Flat topped beams and their characteristics in turbulent media," Opt. Express. 14, 4196-4207 (2006). [CrossRef] [PubMed]
- Y. Baykal and H. T. Eyyuboðlu, "Scintillation index of flat-topped Gaussian beams," Appl. Opt. 45, 3793-3797 (2006). [CrossRef] [PubMed]
- C. Arpali, C. Yazýcýoðlu, H. T. Eyyuboðlu, S. A. Arpali, and Y. Baykal, "Simulator for general-type beam propagation in turbulent atmosphere," Opt. Express 14, 8918-8928 (2006). [CrossRef] [PubMed]
- H. T. Eyyuboðlu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405 (2006). [CrossRef]
- Y. Cai and S. He, "Propagation of various dark hollow beams in turbulent atmosphere," Opt. Express 14,1353-1367 (2006). [CrossRef] [PubMed]
- Y. Cai, "Propagation of various flat-topped beams in a turbulent atmosphere," J. Opt. A: Pure Appl. Opt. 8, 537-545 (2006). [CrossRef]
- Y. Cai and D. Ge, "Analytical formula for a decentered elliptical Gaussian beam propagating in a turbulent atmosphere," Opt. Commun. 271, 509-516 (2007). [CrossRef]
- X. Chu, Y. Ni, and G. Zhou, "Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere," Opt. Commun. (2007), doi:10.1016/ j.optcom.2007.02.035. [CrossRef]
- Z. Mei and D. Zhao, "Controllable dark-hollow beams and their propagation characteristics," J. Opt. Soc. Am. A 22, 1898-1902 (2005). [CrossRef]
- Y. Li, "Light beams with flat-topped profiles," Opt. Lett. 27, 1007-1009 (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, 1297-1304 (1979). [CrossRef]
- ITU-R Document 3J/31-E, "On propagation data and prediction methods required for the design of space-to-earth and earth to-space optical communication systems," Radio-communication Study Group Meeting, Budapest (2001), p. 7.
- J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83,1752 (1988). [CrossRef]
- Y. Zhang, "Nonparaxial propagation analysis of elliptical Gaussian beams diffracted by a circular aperture," Opt Commun. 248, 317-326 (2005). [CrossRef]
- Z. Mei, D. Zhao, and J. Gu, "Comparison of two approximate methods for hard-edged diffracted flat-topped light beams," Opt. Commun. 267, 58-64 (2006). [CrossRef]

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