## Pointing and tracking errors due to localized deformation in inter-satellite laser communication links

Optics Express, Vol. 16, Issue 17, pp. 13372-13380 (2008)

http://dx.doi.org/10.1364/OE.16.013372

Acrobat PDF (1409 KB)

### Abstract

Instead of Zernike polynomials, ellipse Gaussian model is proposed to represent localized wave-front deformation in researching pointing and tracking errors in inter-satellite laser communication links, which can simplify the calculation. It is shown that both pointing and tracking errors depend on the center deepness *h*, the radiuses *a* and *b*, and the distance *d* of the Gaussian distortion and change regularly as they increase. The maximum peak values of pointing and tracking errors always appear around *h*=0.2*λ*. The influence of localized deformation is up to 0.7*µ*rad for pointing error, and 0.5*µ*rad for tracking error. To reduce the impact of localized deformation on pointing and tracking errors, the machining precision of optical devices, which should be more greater than 0.2*λ*, is proposed. The principle of choosing the optical devices with localized deformation is presented, and the method that adjusts the pointing direction to compensate pointing and tracking errors is given. We hope the results can be used in the design of inter-satellite lasercom systems.

© 2008 Optical Society of America

## 1. Introduction

1. F. Cosson, P. Doubrere, and E. Perez, “Simulation model and on-ground performances validation of the PAT system for SILEX program, in Free-Space Laser Communication Technologies III, D. L. Begley and B. D. Seery, eds.,” Proc. SPIE **1417**, 262–276 (1991). [CrossRef]

6. Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, “Generation of hollow Gaussian beams by spatial filtering,” Opt. Lett. **32**, 2076–2078 (2007). [CrossRef] [PubMed]

7. Rodrigo J. Noriega-Manez and Julio C. Gutirrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express **15**, 16328–16341 (2007). [CrossRef] [PubMed]

*λ*and remain unchanged for long time, consequently localized distortions is almost inevitable. Both of the two reasons are equivalent to the deformation of the optical devices. When the beam transmits the optical devices with deformation, its wave-front will change locally.

*et al.*have studied mutual alignment errors in circle region using Zernike polynomials expressing wave-front aberrations. [10

10. 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**, 592–602 (2001). [CrossRef] [PubMed]

*et al.*developed the research to annular region. [11

11. J. F. Sun, L. R. Liu, M. J. Yun, and L. Y. Wan, “Mutual alignment errors due to wave-front aberrations in intersatellite laser communications,” Appl. Opt. **44**, 4953–4958 (2005). [CrossRef] [PubMed]

12. J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. **19**, 1510–1518 (1980). [CrossRef] [PubMed]

## 2. Ellipse Gaussian model

*A*is the center value of the ellipse Gaussian function (the center deepness

*h*=

*A*(1-1/e)),

*a*and

*b*are the radiuses of the localized distortion, (

*x*

_{0},

*y*

_{0}) is the coordinate of the center,

*d*is the distance from (

*0*,

*0*) to (

*x*

_{0},

*y*

_{0}), which can be represented as

*ψ*denotes the center amplitude of ellipse Gaussian function, which is considered to be 4

*Aπ*/

*λ*. Equation (3) is composed of two parts, Φ

_{1}and Φ

_{2}. Φ

_{1}is ellipse Gaussian function, and Φ

_{2}is a constant.

*H*(

*x*,

*y*) is the optical field before the optical device,

*exp*(

*j*Φ) is called aberration term caused by the localized distortion.

_{1}, which can be expressed as

*S*denotes the deformation area which is an ellipse with major axis radius

*a*and minor axis radius

*b*. From Eq. (5) we can find that rms proportionably depends on the center deepness

*h*, but has no relation to the radiuses

*a*and

*b*.

## 3. Pointing and tracking errors

10. 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**, 592–602 (2001). [CrossRef] [PubMed]

### 3.1. Pointing error

10. 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**, 592–602 (2001). [CrossRef] [PubMed]

*C*is a constant,

*F*

_{0}is the radius of curvature at the transmitter,

*M*

_{1}(

*x*

_{0},

*y*

_{0}) is transmitter aperture function which is determined by the transmitter antenna with primary mirror radius

*R*

_{1}and secondary mirror radius

*R*

_{2},

*ω*

_{0}is waist radius of the Gaussian beam. The intensity distribution

*I*(

_{re}*x*,

*y*) in the receiver plane is obtained as the following [15]

*λ*is the wavelength,

*z*is the distance of the two communication terminals. For transmitter beam free of aberrations, the peak intensity is at the origin. And for the beam with aberrations, it is at the position of

_{f}*I*(

_{re}*x*,

*y*) |

_{max}=

*I*(

_{re}*x*,

_{max}*y*). In this case, pointing error

_{max}*θ*can be written in the form

_{P}### 3.2. Tracking error

*f*, it is focused on the focal plane, and the intensity is given by [16]

*B*is a constant,

*M*

_{2}(

*x*,

*y*) is receiver aperture function which is determined by the receiver antenna,

*r*

_{1}and

*r*

_{2}are the primary mirror radius and secondary mirror radius, Φ(

*x*,

*y*) is wave-front deformation in receiver plane. Similarly, when there is no aberrations in the optical systems, the gravity center of the received optical power in the focus plane is at the origin. However, normally the center of gravity is at (

*X*,

*Y*) when aberrations exist in the optical systems. By definition tracking error

*θ*can be written as

_{T}*θ*, tracking error

_{P}*θ*also depends on the following parameters: the center deepness

_{T}*h*, the radiuses

*a*and

*b*, and the distance

*d*.

*H*(

*x*) denotes Gaussian beam for pointing error, or plane beam for tracking error.

*F*is the distance between two satellites for pointing error, or the focal length of receiver optical system for tracking error. And

*D*=2

*R*

_{1}for pointing error, or

*D*=2

*r*

_{1}for tracking error. Substituting Eq. (3) into Eq. (14), we can obtain the following equation

*exp*(

*j*Φ

_{1}). It is known that

*exp*(

*j*Φ

_{1})=

*exp*[

*j*(Φ

_{1}+

_{2}

*π*)], namely the aberration term is a periodic function whose period is 2

*π*. Therefore, pointing and tracking errors would vary periodically with the change of Φ

_{1}. When Φ

_{1}=0, pointing and tracking errors are zeros. We know that the wave-front difference for Φ

_{1}=0 and Φ

_{1}=(2

*n*-1)

*π*is the maximum. Therefore, the peaks of pointing and tracking errors would appear around Φ

_{1}=(2

*n*-1)

*π*(

*n*is positive integer). Due to Φ

_{1}being a function of

*x*and

*y*, the peaks should be around rms=(2

*n*-1)

*π*. Furthermore, from the integral region we can conclude that it is the localized deformation, not the whole aperture, which determines the pointing and tracking errors, consequently rms should obtained from localized deformation area (See Eq. (5)). Related to Eq. (6), we can conclude that pointing and tracking errors would change periodically as the center deepnees

*h*increases. Though the radiuses

*a*and

*b*don’t contribute to the rms of Φ

_{1}according to Eq. (6), it determines the aberration area. When

*a*rises, the value of

*h*(

*u*) increases, namely the influence of wave-front deformation increases too. According to the definitions of pointing and tracking errors, they would increase as the distortion becomes wide.

## 4. Numerical results and analysis

*N*=20, 40 and 60, respectively. The results are in the Fig. 4. When

*N*=20 the result of Zernike polynomials is very poor, and when

*N*=40 the result becomes better. When

*N*=60 the result is close to that of Gaussian function. The results show that it does need many terms for Zernike polynomials to express the localized deformation with less error, which will complicate the calculation. Fig. 5 gives the results of Zernike polynomials with

*N*=40 for different

*a*/

*D*. As can be seen that the result is better for large value of

*a*/

*D*than for small value of

*a*/

*D*. In a word, by comparison with Zernike polynomials, ellipse Gaussian model can really simplify the calculation due to its simple expression, especially for small value of

*a*/

*D*.

*D*=2

*R*

_{1}=2

*r*

_{1}=250 mm,

*R*

_{2}=

*r*

_{2}=40 mm,

*λ*=800 nm,

*ω*

_{0}=125 mm, and

*f*=1000 mm. The distance of the two satellites is taken to be

*z*=50,000 km. Fig. 6 shows how pointing and tracking errors vary with the center deepness

_{f}*h*, the radiuses

*a*and

*b*, and the distance

*d*. In calculation we only consider the condition that the Gaussian deformation is totally in the aperture of the antenna, and the center of Gaussian distortion is in

*x*axis. As can be seen from Fig. 6, pointing and tracking errors do not monotonically rise with

*h*increasing as generally expected, but fluctuates like damped oscillation. On the other hand, pointing and tracking errors monotonically increases as

*a*rises. In other words, the wider localized distortion, the stronger influence on pointing and tracking errors. With the distance

*d*increasing, tracking error increases monotonically, while pointing error increases monotonically at first and then decreases secondly. The difference is considered that the beam contributing to tracking error is plane beam, while that contributing to pointing error is Gaussian beam whose intensity decreases with

*d*increasing. Fig. 7 shows clearly the fluctuation of pointing and tracking errors with

*h*and rms rising. The peak appears around

*h*=0.2

*λ*(rms=

*π*),

*h*=0.75

*λ*(rms=3

*π*),

*h*=1.25

*λ*(rms=5

*π*),

*et. al.*. The fluctuation period for rms value is 2

*π*. The results show that to reduce the impact of localized deformation on pointing and tracking errors, the center deepness

*h*should be more less than 0.2

*λ*, namely the machining accuracy of the optical devices should be more greater than 0.2

*λ*. Moreover, the influence of localized deformation is up to 0.7

*µ*rad for pointing error, and 0.5

*µ*rad for tracking error.

*N*increasing. Figs. 10(a) and 10(d) show that Zernike results are better for small value of

*h*than for large value

*h*. The reason is that, for small value of

*h*, the localized deformation plays an important role, and the effect of Zernike error is comparatively weak. With

*h*rising, the influence of the localized deformation reduces, then the impact of Zernike error gradually increases. Furthermore, as shown in Figs. 8(b) and 8(e), Zernike results are obviously worse for small value

*a*than for large value

*a*. The reason is that Zernike error is large for small value

*a*/

*D*than large value

*a*/

*D*, which is shown in Fig. 5.

*a*/

*D*. To weaken the effect of localized deformation on pointing and tracking errors, processing precision of optical devices should be more than 0.2

*λ*. If we have to use the optical devices with localized deformation, we may select them according to the following principles: (1) The deepness

*h*is more less than 0.2

*λ*; (2) The radiuses

*a*and

*b*are small; (3) The center position (

*x*

_{0},

*y*

_{0}) is near by the center of the optical device. In addition, if we know the localized deformation before laser beam transmitting/receiving, we can adjust pointing direction to compensate the pointing and tracking errors caused by localized aberrations.

## 5. Conclusion

*a*/

*D*by comparison with Zernike polynomials. It is found that pointing and tracking errors due to localized deformation are mainly determined by the center deepness

*h*, the radiuses

*a*and

*b*, and the distance

*d*. With the increasing of the deepness

*h*, both of pointing and tracking errors fluctuate like damped oscillation with peak values around

*h*=0.2

*λ*(rms=

*π*),

*h*=0.75

*λ*(rms=3

*π*),

*h*=1.25

*λ*(rms=5

*π*),

*et al.*. The wider the localized deformation is, the more for the influence on pointing and tracking errors being. With the distance

*d*rising, tracking error increases monotonically, while pointing error increases monotonically at first and then decreases monotonically. The effects of localized deformation is up to 0.7urad for pointing error, and 0.5urad for tracking error. To reduce the impact of localized deformation on pointing and tracking errors, the processing accuracy of optical devices should be more greater than 0.2

*λ*. The principle of choosing the optical devices with localized distortion is presented, and the method that adjusts the pointing direction to compensate pointing and tracking errors is given. We hope the conclusion can be used in the design of inter-satellite lasercom systems.

## References and links

1. | F. Cosson, P. Doubrere, and E. Perez, “Simulation model and on-ground performances validation of the PAT system for SILEX program, in Free-Space Laser Communication Technologies III, D. L. Begley and B. D. Seery, eds.,” Proc. SPIE |

2. | B. Laurent and G. Planche, “SILEX overview after flight terminals campaign, in Free-Space Laser Communication Technologies IX, G. S. Mecherle, ed.,” Proc. SPIE |

3. | A. Mauroschat, “Reliability analysis of a multiple-laser-diode beacon for inter-satellite links, in Free-Space Laser Communication Technologies III, D. L. Begley and B. D. Seery, eds.,” Proc. SPIE |

4. | M. Renard, P. Dobie, J. Gollier, T. Heinrichs, P. Woszczyk, and A. Sobeczko, “Optical telecommunication performance of the qualification model SILEX beacon, in Free-Space Laser Communication Technologies VII, G. S. Mecherle, ed.,” Proc. SPIE |

5. | K. Nakagawa and A. Yamamoto, “Engineering model test of LUCE (laser utilizing communications equipment), in Free-Space Laser Communication Technologies VIII, G. S. Mecherle, ed.,” Proc. SPIE |

6. | Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, “Generation of hollow Gaussian beams by spatial filtering,” Opt. Lett. |

7. | Rodrigo J. Noriega-Manez and Julio C. Gutirrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express |

8. | Brian R. Strickland, Michael J. Lavan, Eric Woodbridge, and Victor Chan, “Effects of fog on the bit-error rate of a free-space laser communication system,” Appl. Opt. |

9. | Shlomi Arnon, “Power versus stabilization for laser satellite communication,” Appl. Opt. |

10. | 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 |

11. | J. F. Sun, L. R. Liu, M. J. Yun, and L. Y. Wan, “Mutual alignment errors due to wave-front aberrations in intersatellite laser communications,” Appl. Opt. |

12. | J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. |

13. | W. H. Swantner and W. H. Lowrey, “Zernike-Tatian polynomials for interferogram reduction,” Appl. Opt. |

14. | V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. |

15. | L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, (Bellingham, Washington, SPIE Press, 1998). |

16. | J. W. Goodman, Introduction to Fourier Optics, Second Edition, (New York, McGraw-Hill, 1996). |

17. | M. Katzman, Ed., Laser Satellite Communications, (Englewood Cliffs, N.J., Prentice-Hall, 1987). |

**OCIS Codes**

(010.3310) Atmospheric and oceanic optics : Laser beam transmission

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: April 30, 2008

Revised Manuscript: August 1, 2008

Manuscript Accepted: August 1, 2008

Published: August 15, 2008

**Citation**

Liying Tan, Yuqiang Yang, Jing Ma, and Jianjie Yu, "Pointing and tracking errors due to localized deformation in inter-satellite
laser communication links," Opt. Express **16**, 13372-13380 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13372

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### References

- F. Cosson, P. Doubrere, and E. Perez, "Simulation model and on-ground performances validation of the PAT system for SILEX program, in Free-Space Laser Communication Technologies III, D. L. Begley and B. D. Seery, eds.," Proc. SPIE 1417, 262-276 (1991). [CrossRef]
- B. Laurent and G. Planche, "SILEX overview after flight terminals campaign, in Free-Space Laser Communication Technologies IX, G. S. Mecherle, ed.," Proc. SPIE 2990, 10-22 (1997). [CrossRef]
- A. Mauroschat, "Reliability analysis of a multiple-laser-diode beacon for inter-satellite links, in Free-Space Laser Communication Technologies III, D. L. Begley and B. D. Seery, eds.," Proc. SPIE 1417, 513-524 (1991). [CrossRef]
- M. Renard, P. Dobie, J. Gollier, T. Heinrichs, P. Woszczyk, and A. Sobeczko, "Optical telecommunication performance of the qualification model SILEX beacon, in Free-Space Laser Communication Technologies VII, G. S. Mecherle, ed.," Proc. SPIE 2381, 289-300 (1995). [CrossRef]
- K. Nakagawa and A. Yamamoto, "Engineering model test of LUCE (laser utilizing communications equipment), in Free-Space Laser Communication Technologies VIII, G. S. Mecherle, ed.," Proc. SPIE 2699, 114-120 (1996). [CrossRef]
- Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, "Generation of hollow Gaussian beams by spatial filtering," Opt. Lett. 32, 2076-2078 (2007). [CrossRef] [PubMed]
- Rodrigo J. Noriega-Manez and Julio C. Gutirrez-Vega, "Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere," Opt. Express 15, 16328-16341 (2007). [CrossRef] [PubMed]
- Brian R. Strickland, Michael J. Lavan, Eric Woodbridge, and Victor Chan, "Effects of fog on the bit-error rate of a free-space laser communication system," Appl. Opt. 38, 424-431 (1999). [CrossRef]
- Shlomi Arnon, "Power versus stabilization for laser satellite communication," Appl. Opt. 38, 3229-3233 (1999). [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. Express 9, 592-602 (2001). [CrossRef] [PubMed]
- J. F. Sun, L. R. Liu, M. J. Yun, and L. Y. Wan, "Mutual alignment errors due to wave-front aberrations in intersatellite laser communications," Appl. Opt. 44, 4953-4958 (2005). [CrossRef] [PubMed]
- J. Y. Wang and D. E. Silva, "Wave-front interpretation with Zernike polynomials," Appl. Opt. 19, 1510-1518 (1980). [CrossRef] [PubMed]
- W. H. Swantner and W. H. Lowrey, "Zernike-Tatian polynomials for interferogram reduction," Appl. Opt. 19, 161-163 (1980). [CrossRef] [PubMed]
- V. N. Mahajan, "Zernike annular polynomials for imaging systems with annular pupils," J. Opt. Soc. Am. 71, 75-85 (1981). [CrossRef]
- L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, (Bellingham, Washington, SPIE Press, 1998).
- J. W. Goodman, Introduction to Fourier Optics, Second Edition, (New York, McGraw-Hill, 1996).
- M. Katzman, Ed., Laser Satellite Communications, (Englewood Cliffs, N.J., Prentice-Hall, 1987).

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