## Influence of temperature on divergence angle of a focal telescope used in laser optical communication |

Optics Express, Vol. 20, Issue 12, pp. 13208-13214 (2012)

http://dx.doi.org/10.1364/OE.20.013208

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### Abstract

Divergence angle of antenna is an important parameter in laser optical communication. It determines the power of the receiver terminal. In this paper, the influence of temperature on the divergence angle is discussed. Theoretical analysis and experiment results demonstrate that the relationship between the variance of temperature and of divergence angle is linear.

© 2012 OSA

## 1. Introduction

1. L. Y. Tan, Y. Q. Yang, J. Ma, and J. J. Yu, “Pointing and tracking errors due to localized deformation in inter-satellite laser communication links,” Opt. Express **16**(17), 13372–13380 (2008). [CrossRef] [PubMed]

7. A. Shlomi and N. S. Kopeika, “Free-space satellite optical communication: adaptive information bandwidth to maintain constant bit error rate during periods of high satellite vibration amplitudes,” Proc. SPIE **2699**, 327–338 (1996). [CrossRef]

8. E. Fischer, P. Adolph, T. Weigel, C. Haupt, and G. Baister, “Advanced optical solutions for inter-satellite communications,” Optik (Stuttg.) **112**(9), 442–448 (2001). [CrossRef]

9. M. Toyoshima, N. Takahashi, T. Jono, T. Yamawaki, K. Nakagawa, and A. Yamamoto, “Mutual alignment errors due to the variation of wave-front aberrations in a free-space laser communication link,” Opt. Express **9**(11), 592–602 (2001). [CrossRef] [PubMed]

*et al*. have studied the influence of temperature on the link length between mirrors and the resulting defocus of camera [10

10. T. P. O’Brien and B. Atwood, “Adjustable Truss for support, optical alignment, and athermalization of a schmidt camera,” Proc. SPIE **4841**, 403–410 (2003). [CrossRef]

## 2. Theoretical analysis

_{1}and M

_{2}denotes secondary and primary mirrors, respectively. M

_{1}is spherical and its radius is

*r*

_{1}, while M

_{2}is a high-order aspheric surface with vertex radius

*r*

_{2}, which can be expressed using Eq. (1):where

*K*is conic constant, and

*A*,

*B*are high-order coefficients of aspheric surface with

^{′}is the projection P(

*z*

_{2},

*h*

_{2}) on

*xoy*plane and its coordinate is (

*x*

_{2},

*y*

_{2}).

_{1}and M

_{2}is

*d*. The height of marginal ray on M

_{1}and M

_{2}are

*h*

_{1}and

*h*

_{2}respectively. The incident angle of marginal ray on M

_{1}and M

_{2}are

*i*

_{1}and

*i*

_{2}respectively. Suppose

*α*

_{1}and

*α*

_{2}are thermal coefficient of the substrate of M

_{1}and M

_{2}, and

*α*

_{3}is thermal coefficient of supporting frame material between M

_{1}and M

_{2}. Thus, when temperature changes Δ

*T*,

*r*

_{1},

*r*

_{2}and

*d*will change to

*r*

_{1}′,

*r*

_{2}′ and

*d*′ expressed by Eqs. (2), (3) and (4) respectively (high-order terms of aspherical polynomial of M

_{2}are omitted, because they are very small compared to the change of radius):

_{2}is

*h*

_{2}′. Then, the relationship between

*h*

_{2}′ and

*d*′ is:

*z*

_{2}′,

*h*

_{2}′), the point of intersection between marginal ray and M

_{2}.

*z*can be expressed by Eq. (7):

*z*′ is the normal line of the surface on point P(

*z*

_{2}′,

*h*

_{2}′). So the angle between

*z*axis and the normal line of marginal point P(

*z*

_{2}′,

*h*

_{2}′) on M

_{2}can be expressed using Eq. (8):

*r*

_{1}and

*r*

_{2}, the incident angle of marginal ray on M

_{2}changes to

*i*

_{2}′:

*i*

_{1}is equal to

*i*

_{2}, and the angle of divergence

*θ*is zero. Hence, the change of divergence angle Δ

*θ*induced by the temperature change is:

*θ*<0 means that the beam emitted from the antenna is convergent, while the beam is divergent in case of Δ

*θ*>0. The result is based on condition that the design effective aperture is smaller than actual aperture. Otherwise in the former case, the marginal ray on M

_{1}will not hit M

_{2}.

*r*

_{1}= 25mm,

*r*

_{2}= 300mm,

*h*

_{1}= 3.6mm,

*d*= 137.5mm,

*K*= −0.9713, and aperture

*D*= 87.2346mm. The materials and CTE of the substrate of M

_{1}, M

_{2}and the link between M

_{1}and M

_{2}, are listed in Table 1 .

## 3. Simulation result

_{2}.

*d*,

*r*

_{2}and

*r*

_{1}respectively. The contribution of each factor to the overall variance of divergence angle with 1 degree temperature change is shown in Table 4 . The result shows that the contribution of factor a and c is negative and the contribution of b is positive. Factor

*d*and

*r*

_{2}are major influences on afocal telescope divergence angle due to temperature change. The overall variance of divergence angle of all-aluminum system is even smaller than that of our elaborate system.

## 4. Experiment result

*f*,

*D*and sag which is expressed by “Power” in MetrPro software. The measurements were done using Zygo interferometer and MetrPro. The experiment results are showed in Fig. 5 .

*θ*value at different temperatures from 22°C to 24°C are calculated in Table 5 and their relationship is shown in Fig. 6 . The results also illustrate that the relationship between Δ

*θ*and Δ

*T*is linear. And Δ

*θ*/Δ

*T*equals to −0.611μard/°C. The experiment result exactly matches the theory. The experiment result of Δ

*θ*/Δ

*T*is a little larger than that of the theory because of the M

_{2}mount is made of titanium alloy which has a lager CTE and few of it is between M

_{1}and M

_{2}.

## 5. Conclusion

## Acknowledgments

## References and links

1. | L. Y. Tan, Y. Q. Yang, J. Ma, and J. J. Yu, “Pointing and tracking errors due to localized deformation in inter-satellite laser communication links,” Opt. Express |

2. | R. A. Conrad, W. E. Wilcox, T. H. Williams, S. Michael, and J. M. Roth, “Emulation of dynamic wavefront disturbances using a deformable mirror,” Opt. Express |

3. | M. Jeganathan, A. Portillo, C. Racho, S. Lee, D. Erickson, J. DePew, S. Monacos, and A. Biswas, “Lessons learnt from the Optical Communications Demonstrator (OCD),” Proc. SPIE |

4. | M. R. García-Talavera, Á. Alonso, S. Chueca, J. J. Fuensalida, Z. Sodnik, V. Cessa, A. Bird, A. Comerón, A. Rodríguez, V. F. Dios, and J. A. Rubio, “Ground to space optical communication characterization,” Proc. SPIE |

5. | C. Chen and J. R. Lesh, “Overview of the Optical Communications Demonstrator,” Proc. SPIE |

6. | S. Arnon, S. Rotman, and N. S. Kopeika, “Optimum transmitter optics aperture for satellite optical communication,” IEEE Trans. Aerosp. Electron. Syst. |

7. | A. Shlomi and N. S. Kopeika, “Free-space satellite optical communication: adaptive information bandwidth to maintain constant bit error rate during periods of high satellite vibration amplitudes,” Proc. SPIE |

8. | E. Fischer, P. Adolph, T. Weigel, C. Haupt, and G. Baister, “Advanced optical solutions for inter-satellite communications,” Optik (Stuttg.) |

9. | M. Toyoshima, N. Takahashi, T. Jono, T. Yamawaki, K. Nakagawa, and A. Yamamoto, “Mutual alignment errors due to the variation of wave-front aberrations in a free-space laser communication link,” Opt. Express |

10. | T. P. O’Brien and B. Atwood, “Adjustable Truss for support, optical alignment, and athermalization of a schmidt camera,” Proc. SPIE |

**OCIS Codes**

(060.4510) Fiber optics and optical communications : Optical communications

(080.2740) Geometric optics : Geometric optical design

(110.6770) Imaging systems : Telescopes

(120.6810) Instrumentation, measurement, and metrology : Thermal effects

**History**

Original Manuscript: January 17, 2012

Revised Manuscript: March 6, 2012

Manuscript Accepted: March 7, 2012

Published: May 29, 2012

**Citation**

Guoxian Zheng, Feng Zhou, Jianfeng Liu, Tuotuo Li, Ning An, and Binglong Zhang, "Influence of temperature on divergence angle of a focal telescope used in laser optical communication," Opt. Express **20**, 13208-13214 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13208

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### References

- L. Y. Tan, Y. Q. Yang, J. Ma, and J. J. Yu, “Pointing and tracking errors due to localized deformation in inter-satellite laser communication links,” Opt. Express16(17), 13372–13380 (2008). [CrossRef] [PubMed]
- R. A. Conrad, W. E. Wilcox, T. H. Williams, S. Michael, and J. M. Roth, “Emulation of dynamic wavefront disturbances using a deformable mirror,” Opt. Express17(5), 3447–3460 (2009). [CrossRef] [PubMed]
- M. Jeganathan, A. Portillo, C. Racho, S. Lee, D. Erickson, J. DePew, S. Monacos, and A. Biswas, “Lessons learnt from the Optical Communications Demonstrator (OCD),” Proc. SPIE3615, 23–30 (1999). [CrossRef]
- M. R. García-Talavera, Á. Alonso, S. Chueca, J. J. Fuensalida, Z. Sodnik, V. Cessa, A. Bird, A. Comerón, A. Rodríguez, V. F. Dios, and J. A. Rubio, “Ground to space optical communication characterization,” Proc. SPIE5892, 201–216 (2005).
- C. Chen and J. R. Lesh, “Overview of the Optical Communications Demonstrator,” Proc. SPIE2123, 85–94 (1994). [CrossRef]
- S. Arnon, S. Rotman, and N. S. Kopeika, “Optimum transmitter optics aperture for satellite optical communication,” IEEE Trans. Aerosp. Electron. Syst.34(2), 590–596 (1998). [CrossRef]
- A. Shlomi and N. S. Kopeika, “Free-space satellite optical communication: adaptive information bandwidth to maintain constant bit error rate during periods of high satellite vibration amplitudes,” Proc. SPIE2699, 327–338 (1996). [CrossRef]
- E. Fischer, P. Adolph, T. Weigel, C. Haupt, and G. Baister, “Advanced optical solutions for inter-satellite communications,” Optik (Stuttg.)112(9), 442–448 (2001). [CrossRef]
- M. Toyoshima, N. Takahashi, T. Jono, T. Yamawaki, K. Nakagawa, and A. Yamamoto, “Mutual alignment errors due to the variation of wave-front aberrations in a free-space laser communication link,” Opt. Express9(11), 592–602 (2001). [CrossRef] [PubMed]
- T. P. O’Brien and B. Atwood, “Adjustable Truss for support, optical alignment, and athermalization of a schmidt camera,” Proc. SPIE4841, 403–410 (2003). [CrossRef]

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