Abstract

In some applications of optical communication systems, such as inter-satellites optical communication, the correlation of the bidirectional tracking beams changes in far-field as a result of wave-front deformation. Far-field correlation model with wave-front deformation on tracking stability is established. Far-field correlation function and factor have been obtained. Combining with parameters of typical laser communication systems, the model is corrected. It shows that deformation pointing-tracking errorsθAandθB, far-field correlation factorδdepend on RMS of deformation errorrms, which decline with a increasingrmsincluding Tilt and Coma. The principle of adjusting far-field correlation factor with wave-front deformation to compensate deformation pointing-tracking errors has been given, through which the deformation pointing-tracking error is reduced to 18.12″ (Azimuth) and 17.65″ (Elevation). Work above possesses significant reference value on optimization design in inter-satellites optical communication.

© 2015 Optical Society of America

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References

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    [Crossref]
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2011 (2)

R. A. Fields, D. A. Kozlowski, H. T. Yura, L. R. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandernoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

X. Li, S. Yu, J. Ma, and L. Tan, “Analytical expression and optimization of spatial acquisition for intersatellite optical communications,” Opt. Express 19(3), 2381–2390 (2011).
[Crossref] [PubMed]

2008 (2)

N. Tanzillo, B. Dunbar, and S. Lee, “Development of a lasercom testbed for the pointing, acquisition, and tracking subsystem of satellite-to-satellite laser communications link,” Proc. SPIE 6877, 687704 (2008).
[Crossref]

Y. H. Zheng, Y. Wang, and X. L. Chen, “H∞Control Applied for APT System of Inter-satellite Laser Communications,” Acta Aeronaut. Astronaut. Sin. 29(6), 1619–1625 (2008).

2007 (3)

2006 (1)

X. Liu, L. Liu, J. F. Sun, and Q. X. Xi, “Bandwidth Design of Composite Axis System in Satellite Laser Communication,” Acta Opt. Sin. 26(1), 101–106 (2006).

2005 (1)

2004 (1)

S. Y. Yu, J. Ma, and L. Y. Tan, “Methods of improving acquisition probability of scanning in intersatellite optical communication,” J. Optoelectron. Laser Netw. 16(12), 57–62 (2004).

2001 (2)

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]

I. I. Kim, B. Riley, N. M. Wong, M. Mitchell, W. Brown, H. Hakakha, P. Adhikari, and E. J. Korevaar, “Lessons learned from the STRV-2 satellite-to-ground lasercom experiment,” Proc. SPIE 4272, 1–15 (2001).
[Crossref]

2000 (1)

V. W. S. Chan, “Optical space communications,” IEEE J. Sel. Top. Quantum Electron. 6(6), 959–975 (2000).
[Crossref]

1996 (1)

K. Nakagawa and A. Yamamoto, “Engineering model test of LUCE (laser utilizing communications equipment), in Free-Space Laser Communication Technologies,” Proc. SPIE 2699, 114–120 (1996).
[Crossref]

1993 (1)

1991 (2)

F. R. Cossec, P. Doubrere, and E. Perez, “Simulation model and on-ground performances validation of the PAT system for SILEX program,” Proc. SPIE 1417, 262–276 (1991).
[Crossref]

A. Mauroschat, “Reliability analysis of a multiple-laser-diode beacon for inter-satellite links, in Free-Space Laser Communication Technologies,” Proc. SPIE 1417, 513–524 (1991).
[Crossref]

1981 (1)

1980 (1)

Adhikari, P.

I. I. Kim, B. Riley, N. M. Wong, M. Mitchell, W. Brown, H. Hakakha, P. Adhikari, and E. J. Korevaar, “Lessons learned from the STRV-2 satellite-to-ground lasercom experiment,” Proc. SPIE 4272, 1–15 (2001).
[Crossref]

Ahmad, M. A.

Arai, K.

Y. Fujiwara, M. Mokuno, T. Jono, T. Yamawaki, K. Arai, M. Toyoshima, H. Kunimori, Z. Sodnik, A. Bird, and B. Demelenne, “Optical inter-orbit communications engineering test satellite (OICETS),” Acta Astronaut. 61(1-6), 163–175 (2007).
[Crossref]

Bird, A.

Y. Fujiwara, M. Mokuno, T. Jono, T. Yamawaki, K. Arai, M. Toyoshima, H. Kunimori, Z. Sodnik, A. Bird, and B. Demelenne, “Optical inter-orbit communications engineering test satellite (OICETS),” Acta Astronaut. 61(1-6), 163–175 (2007).
[Crossref]

Brown, W.

I. I. Kim, B. Riley, N. M. Wong, M. Mitchell, W. Brown, H. Hakakha, P. Adhikari, and E. J. Korevaar, “Lessons learned from the STRV-2 satellite-to-ground lasercom experiment,” Proc. SPIE 4272, 1–15 (2001).
[Crossref]

Chan, V. W. S.

V. W. S. Chan, “Optical space communications,” IEEE J. Sel. Top. Quantum Electron. 6(6), 959–975 (2000).
[Crossref]

Chen, X. L.

Y. H. Zheng, Y. Wang, and X. L. Chen, “H∞Control Applied for APT System of Inter-satellite Laser Communications,” Acta Aeronaut. Astronaut. Sin. 29(6), 1619–1625 (2008).

Cossec, F. R.

F. R. Cossec, P. Doubrere, and E. Perez, “Simulation model and on-ground performances validation of the PAT system for SILEX program,” Proc. SPIE 1417, 262–276 (1991).
[Crossref]

Demelenne, B.

Y. Fujiwara, M. Mokuno, T. Jono, T. Yamawaki, K. Arai, M. Toyoshima, H. Kunimori, Z. Sodnik, A. Bird, and B. Demelenne, “Optical inter-orbit communications engineering test satellite (OICETS),” Acta Astronaut. 61(1-6), 163–175 (2007).
[Crossref]

Doubrere, P.

F. R. Cossec, P. Doubrere, and E. Perez, “Simulation model and on-ground performances validation of the PAT system for SILEX program,” Proc. SPIE 1417, 262–276 (1991).
[Crossref]

Dunbar, B.

N. Tanzillo, B. Dunbar, and S. Lee, “Development of a lasercom testbed for the pointing, acquisition, and tracking subsystem of satellite-to-satellite laser communications link,” Proc. SPIE 6877, 687704 (2008).
[Crossref]

Fields, R. A.

R. A. Fields, D. A. Kozlowski, H. T. Yura, L. R. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandernoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

Fujiwara, Y.

Y. Fujiwara, M. Mokuno, T. Jono, T. Yamawaki, K. Arai, M. Toyoshima, H. Kunimori, Z. Sodnik, A. Bird, and B. Demelenne, “Optical inter-orbit communications engineering test satellite (OICETS),” Acta Astronaut. 61(1-6), 163–175 (2007).
[Crossref]

Gregory, M.

R. A. Fields, D. A. Kozlowski, H. T. Yura, L. R. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandernoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

Gutiérrez-Vega, J. C.

Hakakha, H.

I. I. Kim, B. Riley, N. M. Wong, M. Mitchell, W. Brown, H. Hakakha, P. Adhikari, and E. J. Korevaar, “Lessons learned from the STRV-2 satellite-to-ground lasercom experiment,” Proc. SPIE 4272, 1–15 (2001).
[Crossref]

Heine, F. F.

R. A. Fields, D. A. Kozlowski, H. T. Yura, L. R. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandernoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

Jono, T.

Y. Fujiwara, M. Mokuno, T. Jono, T. Yamawaki, K. Arai, M. Toyoshima, H. Kunimori, Z. Sodnik, A. Bird, and B. Demelenne, “Optical inter-orbit communications engineering test satellite (OICETS),” Acta Astronaut. 61(1-6), 163–175 (2007).
[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(11), 592–602 (2001).
[Crossref] [PubMed]

Kim, I. I.

I. I. Kim, B. Riley, N. M. Wong, M. Mitchell, W. Brown, H. Hakakha, P. Adhikari, and E. J. Korevaar, “Lessons learned from the STRV-2 satellite-to-ground lasercom experiment,” Proc. SPIE 4272, 1–15 (2001).
[Crossref]

Korevaar, E. J.

I. I. Kim, B. Riley, N. M. Wong, M. Mitchell, W. Brown, H. Hakakha, P. Adhikari, and E. J. Korevaar, “Lessons learned from the STRV-2 satellite-to-ground lasercom experiment,” Proc. SPIE 4272, 1–15 (2001).
[Crossref]

Kozlowski, D. A.

R. A. Fields, D. A. Kozlowski, H. T. Yura, L. R. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandernoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

Kunimori, H.

Y. Fujiwara, M. Mokuno, T. Jono, T. Yamawaki, K. Arai, M. Toyoshima, H. Kunimori, Z. Sodnik, A. Bird, and B. Demelenne, “Optical inter-orbit communications engineering test satellite (OICETS),” Acta Astronaut. 61(1-6), 163–175 (2007).
[Crossref]

Lee, S.

N. Tanzillo, B. Dunbar, and S. Lee, “Development of a lasercom testbed for the pointing, acquisition, and tracking subsystem of satellite-to-satellite laser communications link,” Proc. SPIE 6877, 687704 (2008).
[Crossref]

Li, X.

Lin, J.

Liu, J.

Liu, L.

X. Liu, L. Liu, J. F. Sun, and Q. X. Xi, “Bandwidth Design of Composite Axis System in Satellite Laser Communication,” Acta Opt. Sin. 26(1), 101–106 (2006).

J. Sun, L. Liu, M. Yun, and L. Wan, “Mutual alignment errors due to wave-front aberrations in intersatellite laser communications,” Appl. Opt. 44(23), 4953–4958 (2005).
[Crossref] [PubMed]

Liu, S.

Liu, X.

X. Liu, L. Liu, J. F. Sun, and Q. X. Xi, “Bandwidth Design of Composite Axis System in Satellite Laser Communication,” Acta Opt. Sin. 26(1), 101–106 (2006).

Liu, Z.

Luna, J. J.

R. A. Fields, D. A. Kozlowski, H. T. Yura, L. R. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandernoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

Lunde, C. T.

R. A. Fields, D. A. Kozlowski, H. T. Yura, L. R. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandernoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

Ma, J.

X. Li, S. Yu, J. Ma, and L. Tan, “Analytical expression and optimization of spatial acquisition for intersatellite optical communications,” Opt. Express 19(3), 2381–2390 (2011).
[Crossref] [PubMed]

S. Y. Yu, J. Ma, and L. Y. Tan, “Methods of improving acquisition probability of scanning in intersatellite optical communication,” J. Optoelectron. Laser Netw. 16(12), 57–62 (2004).

Mahajan, V. N.

Mauroschat, A.

A. Mauroschat, “Reliability analysis of a multiple-laser-diode beacon for inter-satellite links, in Free-Space Laser Communication Technologies,” Proc. SPIE 1417, 513–524 (1991).
[Crossref]

Mitchell, M.

I. I. Kim, B. Riley, N. M. Wong, M. Mitchell, W. Brown, H. Hakakha, P. Adhikari, and E. J. Korevaar, “Lessons learned from the STRV-2 satellite-to-ground lasercom experiment,” Proc. SPIE 4272, 1–15 (2001).
[Crossref]

Mokuno, M.

Y. Fujiwara, M. Mokuno, T. Jono, T. Yamawaki, K. Arai, M. Toyoshima, H. Kunimori, Z. Sodnik, A. Bird, and B. Demelenne, “Optical inter-orbit communications engineering test satellite (OICETS),” Acta Astronaut. 61(1-6), 163–175 (2007).
[Crossref]

Nakagawa, K.

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]

K. Nakagawa and A. Yamamoto, “Engineering model test of LUCE (laser utilizing communications equipment), in Free-Space Laser Communication Technologies,” Proc. SPIE 2699, 114–120 (1996).
[Crossref]

Noriega-Manez, R. J.

Perez, E.

F. R. Cossec, P. Doubrere, and E. Perez, “Simulation model and on-ground performances validation of the PAT system for SILEX program,” Proc. SPIE 1417, 262–276 (1991).
[Crossref]

Riley, B.

I. I. Kim, B. Riley, N. M. Wong, M. Mitchell, W. Brown, H. Hakakha, P. Adhikari, and E. J. Korevaar, “Lessons learned from the STRV-2 satellite-to-ground lasercom experiment,” Proc. SPIE 4272, 1–15 (2001).
[Crossref]

Siegman, A. E.

Silva, D. E.

Sodnik, Z.

Y. Fujiwara, M. Mokuno, T. Jono, T. Yamawaki, K. Arai, M. Toyoshima, H. Kunimori, Z. Sodnik, A. Bird, and B. Demelenne, “Optical inter-orbit communications engineering test satellite (OICETS),” Acta Astronaut. 61(1-6), 163–175 (2007).
[Crossref]

Sun, J.

Sun, J. F.

X. Liu, L. Liu, J. F. Sun, and Q. X. Xi, “Bandwidth Design of Composite Axis System in Satellite Laser Communication,” Acta Opt. Sin. 26(1), 101–106 (2006).

Takahashi, N.

Tan, L.

Tan, L. Y.

S. Y. Yu, J. Ma, and L. Y. Tan, “Methods of improving acquisition probability of scanning in intersatellite optical communication,” J. Optoelectron. Laser Netw. 16(12), 57–62 (2004).

Tanzillo, N.

N. Tanzillo, B. Dunbar, and S. Lee, “Development of a lasercom testbed for the pointing, acquisition, and tracking subsystem of satellite-to-satellite laser communications link,” Proc. SPIE 6877, 687704 (2008).
[Crossref]

Toyoshima, M.

Y. Fujiwara, M. Mokuno, T. Jono, T. Yamawaki, K. Arai, M. Toyoshima, H. Kunimori, Z. Sodnik, A. Bird, and B. Demelenne, “Optical inter-orbit communications engineering test satellite (OICETS),” Acta Astronaut. 61(1-6), 163–175 (2007).
[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(11), 592–602 (2001).
[Crossref] [PubMed]

Wan, L.

Wandernoth, B. K.

R. A. Fields, D. A. Kozlowski, H. T. Yura, L. R. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandernoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

Wang, J. Y.

Wang, Y.

Y. H. Zheng, Y. Wang, and X. L. Chen, “H∞Control Applied for APT System of Inter-satellite Laser Communications,” Acta Aeronaut. Astronaut. Sin. 29(6), 1619–1625 (2008).

Wicker, J. M.

R. A. Fields, D. A. Kozlowski, H. T. Yura, L. R. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandernoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

Wong, L. R.

R. A. Fields, D. A. Kozlowski, H. T. Yura, L. R. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandernoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

Wong, N. M.

I. I. Kim, B. Riley, N. M. Wong, M. Mitchell, W. Brown, H. Hakakha, P. Adhikari, and E. J. Korevaar, “Lessons learned from the STRV-2 satellite-to-ground lasercom experiment,” Proc. SPIE 4272, 1–15 (2001).
[Crossref]

Xi, Q. X.

X. Liu, L. Liu, J. F. Sun, and Q. X. Xi, “Bandwidth Design of Composite Axis System in Satellite Laser Communication,” Acta Opt. Sin. 26(1), 101–106 (2006).

Yamamoto, A.

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]

K. Nakagawa and A. Yamamoto, “Engineering model test of LUCE (laser utilizing communications equipment), in Free-Space Laser Communication Technologies,” Proc. SPIE 2699, 114–120 (1996).
[Crossref]

Yamawaki, T.

Y. Fujiwara, M. Mokuno, T. Jono, T. Yamawaki, K. Arai, M. Toyoshima, H. Kunimori, Z. Sodnik, A. Bird, and B. Demelenne, “Optical inter-orbit communications engineering test satellite (OICETS),” Acta Astronaut. 61(1-6), 163–175 (2007).
[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(11), 592–602 (2001).
[Crossref] [PubMed]

Yu, S.

Yu, S. Y.

S. Y. Yu, J. Ma, and L. Y. Tan, “Methods of improving acquisition probability of scanning in intersatellite optical communication,” J. Optoelectron. Laser Netw. 16(12), 57–62 (2004).

Yun, M.

Yura, H. T.

R. A. Fields, D. A. Kozlowski, H. T. Yura, L. R. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandernoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

Zhao, H.

Zheng, Y. H.

Y. H. Zheng, Y. Wang, and X. L. Chen, “H∞Control Applied for APT System of Inter-satellite Laser Communications,” Acta Aeronaut. Astronaut. Sin. 29(6), 1619–1625 (2008).

Acta Aeronaut. Astronaut. Sin. (1)

Y. H. Zheng, Y. Wang, and X. L. Chen, “H∞Control Applied for APT System of Inter-satellite Laser Communications,” Acta Aeronaut. Astronaut. Sin. 29(6), 1619–1625 (2008).

Acta Astronaut. (1)

Y. Fujiwara, M. Mokuno, T. Jono, T. Yamawaki, K. Arai, M. Toyoshima, H. Kunimori, Z. Sodnik, A. Bird, and B. Demelenne, “Optical inter-orbit communications engineering test satellite (OICETS),” Acta Astronaut. 61(1-6), 163–175 (2007).
[Crossref]

Acta Opt. Sin. (1)

X. Liu, L. Liu, J. F. Sun, and Q. X. Xi, “Bandwidth Design of Composite Axis System in Satellite Laser Communication,” Acta Opt. Sin. 26(1), 101–106 (2006).

Appl. Opt. (3)

IEEE J. Sel. Top. Quantum Electron. (1)

V. W. S. Chan, “Optical space communications,” IEEE J. Sel. Top. Quantum Electron. 6(6), 959–975 (2000).
[Crossref]

J. Opt. Soc. Am. (1)

J. Optoelectron. Laser Netw. (1)

S. Y. Yu, J. Ma, and L. Y. Tan, “Methods of improving acquisition probability of scanning in intersatellite optical communication,” J. Optoelectron. Laser Netw. 16(12), 57–62 (2004).

Opt. Express (3)

Opt. Lett. (1)

Proc. SPIE (6)

I. I. Kim, B. Riley, N. M. Wong, M. Mitchell, W. Brown, H. Hakakha, P. Adhikari, and E. J. Korevaar, “Lessons learned from the STRV-2 satellite-to-ground lasercom experiment,” Proc. SPIE 4272, 1–15 (2001).
[Crossref]

R. A. Fields, D. A. Kozlowski, H. T. Yura, L. R. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandernoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

N. Tanzillo, B. Dunbar, and S. Lee, “Development of a lasercom testbed for the pointing, acquisition, and tracking subsystem of satellite-to-satellite laser communications link,” Proc. SPIE 6877, 687704 (2008).
[Crossref]

F. R. Cossec, P. Doubrere, and E. Perez, “Simulation model and on-ground performances validation of the PAT system for SILEX program,” Proc. SPIE 1417, 262–276 (1991).
[Crossref]

A. Mauroschat, “Reliability analysis of a multiple-laser-diode beacon for inter-satellite links, in Free-Space Laser Communication Technologies,” Proc. SPIE 1417, 513–524 (1991).
[Crossref]

K. Nakagawa and A. Yamamoto, “Engineering model test of LUCE (laser utilizing communications equipment), in Free-Space Laser Communication Technologies,” Proc. SPIE 2699, 114–120 (1996).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Far-field correlation model for wave-front deformation.
Fig. 2
Fig. 2 Deformation pointing-tracking errors with primary aberrations.
Fig. 3
Fig. 3 Correlation factors with Tilt.
Fig. 4
Fig. 4 The functional block diagram of tracking test.
Fig. 5
Fig. 5 The scheme of the optical system.
Fig. 6
Fig. 6 The experiment tracking system.
Fig. 7
Fig. 7 Deformation pointing-tracking error angle without correction.
Fig. 8
Fig. 8 Deformation pointing-tracking error angle with correction.

Tables (1)

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Table 1 Zernike polynomials for primary polynomials

Equations (14)

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θ A = x P 2 + y P 2 z , θ B = x Q 2 + y Q 2 z
u( x,y )= C 1 E( x,y )M( x,y )exp[ ik 2z ( x 1 x+ y 1 y ) ]dxdy
E( x,y )= C 2 exp[ x 2 + y 2 ω 0 2 x 2 + y 2 ρ 2 ]exp[ iϕ( x,y ) ]
I re ( x,y )=u( x,y ) u * ( x,y )= | u( x,y ) | 2
u( O R ,t )= C p u( p,t t p )+ C q u( q,t t q )
I( O R )= u( O R ,t ) u * ( O R ,t )
I( O R )= C p 2 u( p,t t p ) u * ( p,t t p ) + C q 2 u( q,t t q ) u * ( q,t t q ) + C p C q * u( p,t t p ) u * ( q,t t q ) + C p * C q u * ( p,t t p )u( q,t t q )
Γ pq ( τ )= u( p,t t p ) u * ( q,t t q ) = u( p,t+τ ) u * ( q,t )
u( p,tτ ) u * ( p,t ) = Τ pp ( τ )= Τ pp ( 0 ) u( q,tτ ) u * ( q,t ) = Τ qq ( τ )= Τ qq ( 0 )
δ pq ( τ )= Γ pq ( τ ) [ Γ pp ( 0 ) Γ qq ( 0 ) ] 1/2
0 δ pq ( τ )1
ϕ( x,y )= 2π λ W( x,y )= 2π λ n=0 a n Z n ( x,y )
rms= M( x,y ) ϕ 2 ( x,y )dxdy M( x,y )dxdy
M( x,y )={ 1 r 2 x 2 + y 2 r 1 0

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