Far-field correlation of bidirectional tracking beams due to wave-front deformation in inter-satellites optical communication links

Siyuan Yu; Zhongtian Ma; Jing Ma; Feng Wu; Liying Tan

doi:10.1364/OE.23.007263

Far-field correlation of bidirectional tracking beams due to wave-front deformation in inter-satellites optical communication links

Siyuan Yu, Zhongtian Ma, Jing Ma, Feng Wu, and Liying Tan

Optics Express
Vol. 23,
Issue 6,
pp. 7263-7272
(2015)
•https://doi.org/10.1364/OE.23.007263

Get Citation
Copy Citation Text
Siyuan Yu, Zhongtian Ma, Jing Ma, Feng Wu, and Liying Tan, "Far-field correlation of bidirectional tracking beams due to wave-front deformation in inter-satellites optical communication links," Opt. Express 23, 7263-7272 (2015)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Get PDF (1433 KB)
Set citation alerts
Save article

More Like This

Related Topics
Table of Contents Category
- Optical Communications
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Laser beam transmission (010.3310)
- Optical communications (060.4510)
- Free-space optical communication (060.2605)

About this Article
History
- Original Manuscript: December 18, 2014
- Revised Manuscript: January 28, 2015
- Manuscript Accepted: January 28, 2015
- Published: March 11, 2015
Virtual Issues
Optics Express Advanced Solid-State Lasers 2014 (2015)

Accessible

Open Access

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 errors $θ_{A}$ and $θ_{B}$ , far-field correlation factor $δ$ depend on RMS of deformation error $r m s$ , which decline with a increasing $r m s$ including 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.

1. Introduction

High-speed satellite optical communication technology is developing toward engineering applications gradually and in-orbit tests have been operated in recently years, which needs new technology. Compared with fiber optic communication, inter-satellite optical communication links (IOCLs) possesses lots of merits such as smaller weight and size of terminal, lower transmitter power, higher immunity to interference and larger date rate, however, the laser beam is narrow, and the energy of receiver light is very weak because IOCLs has longer distance. Therefore, a high-performance APT system, and high-precision control accuracy are required [1,2]. The first bidirectional optical tracking communication experiment in the world has been successfully established between OICETS and ARTEMIS, which was completed by JAXA and ESA using a laser beam on December 9, 2005. The holding time of the link is 10mins, however, communication data rate is only 50Mbps (OICETS- ARTEMIS) and 2Mbps (ARTEMIS -OICETS) [3–7]. Inter-satellites optical communication relates to laser beam transmission, which has been extensively studied [8,9]. Due to the small beam divergence and the ultra-long distance of communication link, wave-front deformation strongly affects the spatial pointing and tracking of laser beams. In bidirectional tracking process, the tracking beams will form a weak correlation in far field, but a detailed analysis of the far-field correlation of the bidirectional beams as a result of wave-front aberrations has not yet been reported, which is significant to improve the stability of communication links and extend the holding time of links.

Laser is the main information carrier in IOCLs, and it works under near optical diffraction limit environment with long communication distance and extreme optical detection environments. In order to establish communication links and maintain high rate of data transmission in such harsh environments, optical terminal must possess a high tracking performance [10]. Wherein, the influence of laser beam quality on tracking stability cannot be ignored, and wave-front deformation is an important factor affecting the beam quality [11]. Due to processing errors on surfaces of optical element, optical system adjustment errors, and space environments, there is no ideal laser beam from optical antenna, but with a wave-front deformation. Optical signal transmission will be impacted by wave-front deformation that can change the characteristics of far-field laser beam, resulting in a great impact on far-field correlation of bidirectional tracking beams.

Wave-front deformation is composed of whole wave-front deformation and local wave-front deformation according to spatial scale. The whole wave-front deformation refers to deformation of the aperture over the entire beam such as aberration. The local wave-front deformation refers to local deformation of wave-front laser beam, due to unevenness of surface temperature and processing errors of optical components [12–15]. We primary research whole wave-front deformation effect on far-field correlation of bidirectional tracking beams in this paper.

Deformation pointing-tracking error is defined based on APT principles and the far-field correlation characteristics as a result of wave-front deformation. The far-field correlation model of bidirectional tracking due to wave-front deformation was analyzed and built through the deformation pointing-tracking errors which were detected by the array detector in receiver terminal. Then the theoretical equations of far-field correlation and correlation factor were deduced. And the theoretical results have been verified and modified by the ground experimental simulations, in which the far-field correlation of tracking beams under different compensation effects and deformation pointing-tracking errors was obtained. The experiment results were fit better with the theoretical results.

This paper has the following outline. In section 2 the deformation pointing-tracking error is defined. In Section 3 the far-field correlation model for wave-front deformation is introduced to describe far-field correlation. Section 4 is devoted to numerical analysis. Section 5 summarizes our results.

2 Deformation pointing-tracking errors

Emission optical axis is the optical axis according to the light intensity peak position in receiver plane. Deformation pointing-tracking error is defined as the angle between deformation emission optical axis and emission optical axis without deformation. The process of bidirectional acquisition and tracking between terminal A and terminal B is shown as Fig. 1. Detector field of tracking coordinate system for terminal A is denoted as $O_{A} - x_{A} y_{A}$ , for terminal B is $O_{B} - x_{B} y_{B}$ , we take terminal A for introducing as follow.

Fig. 1 Far-field correlation model for wave-front deformation.

Download Full Size | PPT Slide | PDF

The intensity peak of tracking beam is origin $O_{A} (0, 0)$ in receiver plane of terminal B, when tracking beam emit from terminal A without wave-front deformation. But the peak will shift to the point $P_{B} (x_{P}, y_{P})$ , when tracking beam with wave-front deformation from terminal A. Deformation pointing-tracking error is determined by positions of beam intensity peaks O_B and P_B, which can be expressed as

θ_{A} = \frac{\sqrt{x_{P}^{2} + y_{P}^{2}}}{z}, θ_{B} = \frac{\sqrt{x_{Q}^{2} + y_{Q}^{2}}}{z}

Distance of IOCLs is very far, close to Fraunhofer diffraction conditions, so the light field on receiver plane can be written as

u (x, y) = C_{1} \iint E (x, y) M (x, y) \exp [- \frac{i k}{2 z} (x_{1} x + y_{1} y)] d x d y

Where

C_{1}

is a constant,

2 z

denotes the distance of IOCLs,

M (x, y)

is the aperture function of transmit antenna,

E (x, y)

is the deformation Gaussian beam on the receiver plane. The Influence on light field of tracking beam due to wave-front aberration can be described using a phase factor, which can be shown in the form

E (x, y) = C_{2} \exp [- \frac{x^{2} + y^{2}}{ω_{0}^{2}} - \frac{x^{2} + y^{2}}{ρ_{}^{2}}] \exp [i ϕ (x, y)]

Where

C_{2}

is a constant,

ω_{0}^{}

is the waist radius of Gaussian beam,

ρ

is the curvature radius of Gaussian beam in emitting plane,

ϕ (x, y)

is the phase of Gaussian beam generated by wave-front aberration in emitting plane.

The light intensity distribution in receiver plane can be represented as

I_{r e} (x, y) = u (x, y) u^{*} (x, y) = {| u (x, y) |}^{2}

The positions of light intensity peaks O_B and P_B can be obtained according to the Eq. (4), then deformation pointing-tracking errors

θ_{A}

and

θ_{B}

can be calculated.

3 Far-field correlation model

In bidirectional tracking laser communication links, a weak correlation between two tracking beacon beams exists in far field, the weak correlation has a great impact on tracking stability. Correlation of beams is largely affected by wave-front deformation, satellite platform vibration, spatial environment and other factors, thereby affecting the quality of optical communications [16]. In order to describe the far-field correlation, concepts of correlation function and correlation factor have been presented in there.

3.1 Correlation function

Process of bidirectional tracking between terminals A and B is shown in Fig. 1, far-field correlation cannot be established in any tracking detector filed coordinate system. Therefore, reference tracking detector filed coordinate system $O_{R} - X_{R} Y_{R}$ has been established to analyze correlation characteristics. Point $p (X_{p}, Y_{p})$ and point $q (X_{q}, Y_{q})$ are tracking beam spots (which emit from terminals A and B) in $O_{R} - X_{R} Y_{R}$ . They can be seen as secondary wave sources to research beams correlation at point O_R in far-field.

With the influence of wave-front deformation, $r_{A}$ and $r_{B}$ denote the distances propagating along the optical axis to reference coordinate systems of tracking beams from terminal A and B respectively.

$u (p, t)$ and $u (q, t)$ denote light vibration analytic signals at points $p$ and $q$ with time $t$ . The light vibration analytic signal at point O_R with time $t$ is the superposition of two light waves, which is obtained as the following

u (O_{R}, t) = C_{p} u (p, t - t_{p}) + C_{q} u (q, t - t_{q})

Where

t_{p} = r_{A} / c, t_{q} = r_{B} / c

, and

c

is the speed of light in vacuum.

C_{p}, C_{q}

(Propagation factors) are constants inversely proportional to distance, regardless of time. Light intensity of point O_R is average of time in receiver plane that is obtained as the following

I (O_{R}) = 〈 u (O_{R}, t) u^{*} (O_{R}, t) 〉

Where

〈 〉

denote time averaging. Then we take Eq. (5) into the Eq. (6) as

\begin{array}{l} I (O_{R}) = C_{p}^{2} 〈 u (p, t - t_{p}) u^{*} (p, t - t_{p}) 〉 \\ \begin{matrix} + \end{matrix} C_{q}^{2} 〈 u (q, t - t_{q}) u^{*} (q, t - t_{q}) 〉 \\ \begin{matrix} + C_{p} \end{matrix} C_{q}^{*} 〈 u (p, t - t_{p}) u^{*} (q, t - t_{q}) 〉 \\ \begin{matrix} + C_{p}^{*} \end{matrix} C_{q}^{} 〈 u^{*} (p, t - t_{p}) u (q, t - t_{q}) 〉 \end{array}

We suppose that the light field is stationary, and its statistical properties do not change with time, so time origin can shift without influencing the average value in above equation. In other words, there is no relation between light intensity of point O_R and selective time. Thus, correlation function can be written in the form

Γ_{p q} (τ) = 〈 u (p, t - t_{p}) u^{*} (q, t - t_{q}) 〉 = 〈 u (p, t + τ) u^{*} (q, t) 〉

Where

τ = t_{p} - t_{q}

is relative delay,

Γ_{p q} (τ)

denotes far-filed correlation function of points

p

and

q

with relative delay

τ

at point O_R.

3.2 Correlation factor

When points $p$ and $q$ are superposition, self-correlation function of the points are given as

\begin{array}{l} 〈 u (p, t - τ) u^{*} (p, t) 〉 = Τ_{p p} (τ) = Τ_{p p} (0) \\ 〈 u (q, t - τ) u^{*} (q, t) 〉 = Τ_{q q} (τ) = Τ_{q q} (0) \end{array}

Correlation factor is defined as the normalization of

Γ_{p q} (τ)

, which can be expressed as

δ_{p q} (τ) = \frac{Γ_{p q} (τ)}{{[Γ_{p p} (0) Γ_{q q} (0)]}^{1 / 2}}

Using Cauchy-Schwarz inequality, we can prove that

Γ_{p q} (τ) \leq {[Γ_{p p} (0) Γ_{q q} (0)]}^{1 / 2}

, according to Eq. (10), the range of correlation factor can be described as

0 \leq δ_{p q} (τ) \leq 1

In summary, as the optical communication link is very far, far-field beam can be seen as a uniform plane wave. When the wave-front deformation exists, deformation pointing-tracking errors

θ_{A}

and

θ_{B}

can be calculated using the Eqs. (4) and (1). The distances

r_{A}

and

r_{B}

can be expressed as

r_{A} = z θ_{A}, r_{B} = z θ_{B}

and

τ = (θ_{A} - θ_{B}) z / c

. The light intensity at point O_R superimposed two tracking beams is relevant with the correlation function and correlation factor.

4 Numerical results and analysis

The light field influenced by wave-front deformation of the optical systems can be described that the original light field distribution function multiply by a phase factor $\exp (j ϕ)$ . The whole deformation is obtained using Zernike polynomial expansion.

Zernike polynomial is orthogonality in unit circle, therefore, wave-front deformation phase can be developed into Zernike orthogonal polynomial on circle pupil [17]. Thus, phase factor can be written as

ϕ (x, y) = \frac{2 π}{λ} W (x, y) = \frac{2 π}{λ} \sum_{n = 0}^{\infty} a_{n} Z_{n} (x, y)

Where

a_{n}

is coefficient of Zernike polynomial expansion, and

Z_{n} (x, y)

is Zernike polynomial expansion. Zernike polynomials are corresponding with all levels of aberrations in optical detect systems, and corresponding relationship is shown in Table 1.

Table 1. Zernike polynomials for primary polynomials

View Table

Effects of primary aberrations for deformation pointing-tracking errors and correlation factor have been researched in reflective optical antenna. To facilitate computation, the degree of deformation is represented by using the RMS of wave-front deformation errors, which can be represented as

r m s = \sqrt{\frac{\iint M (x, y) ϕ^{2} (x, y) d x d y}{\iint M (x, y) d x d y}}

Where

M (x, y)

is be written in reflective optical antenna as

M (x, y) = {\begin{matrix} 1 \begin{matrix} r_{2} \leq \sqrt{x^{2} + y^{2}} \leq r_{1} \end{matrix} \\ 0 \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} & \begin{matrix}  \end{matrix} \end{matrix} \end{matrix}

We assume that optical parameters of terminals A and B are same, optical antenna diameter is

D = 0.25 m

. Truncation ratio is

β = 1.0

. Screening ratio is

ς = 0.2

. Laser wavelength is

λ = 847 n m

. Focal length of optical system is

f = 1 m

. And the distance between terminals A and B is

2 z = 30, 000 k m

Deformation pointing-tracking errors with primary aberrations are shown in Fig. 2. Only Tilt ( $Z_{1}, Z_{2}$ ) and Coma ( $Z_{6}, Z_{7}$ ) can cause deformation pointing-tracking errors, which is no effected by other primary aberrations in reflective optical antenna system.

Fig. 2 Deformation pointing-tracking errors with primary aberrations.

Download Full Size | PPT Slide | PDF

When $r m s$ is $0.10 λ$ , the deformation pointing-tracking error due to Tilt is about $1.4 μ rad$ , and $0.8 μ rad$ caused by Coma. Thus, impact of Tilt on deformation pointing-tracking error is larger than Coma.

The relationship between correlation factors and Tilt is shown in Fig. 3.

Fig. 3 Correlation factors with Tilt.

Download Full Size | PPT Slide | PDF

The relationship curves of correlation factors for $θ_{B} = 0.5 μ r a d, 1.0 μ r a d, 1.5 μ r a d$ are given in Fig. 3 when $r m s$ value of terminal A is $0.10 λ$ . The results show that, to reduce the impact of deformation on correlation factors, deformation pointing-tracking errors should be compensated.

Researches on tracking stability focus on improving the control system accuracy of communication terminal, and aim at raising the disturbance rejection. A wide range and high-precision tracking task is completed by the complex axis tracking system. The system is composed of coarse and fine tracking system. The coarse and fine tracking bandwidth designs are given in [18], and the tracking error is less than 1μrad under the high-frequency disturbances. The typical tracking control algorithm is PID control and H_∞ control. Compared with feedback control, feed-forward compensation control method can significantly improve the performance of pointing and tracking [19]. Uncertainty caused by platform vibrations and perturbations can be restrained effectively by using H_∞ control. In research on the impact of wave-front deformation, since the long communication distance, there exist deformation pointing-tracking errors between the satellites. The satellite must take into account additional errors in relaxation-time. Adopting ahead pointing-tracking assembly can effectively restrain its impact.

In order to simulate the process of bidirectional tracking in inter-satellites optical communications, the simulation experiment needs two optical terminals, one dynamic link simulator and two computers. The simulation system and optical system are shown in Fig. 4 and Fig. 5.

Fig. 4 The functional block diagram of tracking test.

Download Full Size | PPT Slide | PDF

Fig. 5 The scheme of the optical system.

Download Full Size | PPT Slide | PDF

Laser beam is launched by Cassegrain optical antenna after collimating through the collimator lens from terminal A. However, due to optical components (Primary and Secondary mirrors) and vibrations, the wave-front deformation is produced, which can cause deformation pointing-tracking errors. Then tracking beam will be detected in receiver CCD plan after optical filter and focus lens group. According to the position of light intensity, we can obtain the deformation pointing-tracking error $θ_{A}$ . And $θ_{B}$ can be obtained using the same method.

And the experiment tracking system is shown in Fig. 6.

Fig. 6 The experiment tracking system.

Download Full Size | PPT Slide | PDF

A correction test has been done in terminal A. And we get the distance of light spot form the center of receiver CCD plan of terminal B as deformation pointing-tracking errors, as shown in Fig. 7 and Fig. 8.

Fig. 7 Deformation pointing-tracking error angle without correction.

Download Full Size | PPT Slide | PDF

Fig. 8 Deformation pointing-tracking error angle with correction.

Download Full Size | PPT Slide | PDF

As shown in Fig. 7, deformation pointing-tracking errors were too large for the load of terminal, which will make optical communication link losing, so the correction is obligatory. The deformation pointing-tracking errors have been corrected, and became smaller shown as Fig. 8. The mean value of deformation pointing-tracking error has been reduced from 145.73″to 25.63″.

5. Conclusion

To research wave-front deformation on bidirectional tracking stability in inter-satellites optical communications, far-field correlation model for wave-front deformation is proposed. It is found that the deformation pointing-tracking errors due to deformation are mainly determined by RMS of wave-front deformation errors, Tilt and Coma. With the increasing of $r m s$ , both of deformation pointing-tracking errors induced by Tilt and Coma are increasing. The method to compensate pointing and tracking errors was given. The experiment of the correction for terminal A has been performed, and average deformation pointing-tracking error has been reduced to 18.12″ (Azimuth) and 17.65″ (Elevation).We hope the conclusion can be used in the design of inter-satellites laser communication systems.

Acknowledgment

This work was supported by excellent Satellite Optical Communications team in Harbin Institute of Technology.

References and links

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]

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]

3. 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]

4. 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]

5. 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]

6. 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]

7. 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).

8. 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]

9. 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]

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

11. A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase aberrations,” Appl. Opt. 32(30), 5893–5901 (1993). [CrossRef] [PubMed]

12. 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]

13. 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(15), 2076–2078 (2007). [CrossRef] [PubMed]

14. R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007). [CrossRef] [PubMed]

15. J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19(9), 1510–1518 (1980). [CrossRef] [PubMed]

16. 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]

17. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71(1), 75–85 (1981). [CrossRef]

18. 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).

19. 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).

References

View by:
Article Order
|
Year
|
Author
|
Publication

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]
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]
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).
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]
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]
V. W. S. Chan, “Optical space communications,” IEEE J. Sel. Top. Quantum Electron. 6(6), 959–975 (2000).
[Crossref]
A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase aberrations,” Appl. Opt. 32(30), 5893–5901 (1993).
[Crossref] [PubMed]
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]
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(15), 2076–2078 (2007).
[Crossref] [PubMed]
R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
[Crossref] [PubMed]
J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19(9), 1510–1518 (1980).
[Crossref] [PubMed]
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]
V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71(1), 75–85 (1981).
[Crossref]
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).
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).

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)

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]

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(15), 2076–2078 (2007).
[Crossref] [PubMed]

R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
[Crossref] [PubMed]

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)

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]

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)

A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase aberrations,” Appl. Opt. 32(30), 5893–5901 (1993).
[Crossref] [PubMed]

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)

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71(1), 75–85 (1981).
[Crossref]

1980 (1)

J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19(9), 1510–1518 (1980).
[Crossref] [PubMed]

Adhikari, P.

Ahmad, M. A.

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(15), 2076–2078 (2007).
[Crossref] [PubMed]

Arai, K.

Bird, A.

Brown, W.

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.

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.

Fields, R. A.

Fujiwara, Y.

Gregory, M.

Gutiérrez-Vega, J. C.

R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
[Crossref] [PubMed]

Hakakha, H.

Heine, F. F.

Jono, T.

Kim, I. I.

Korevaar, E. J.

Kozlowski, D. A.

Kunimori, H.

Lee, S.

Li, X.

Lin, J.

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(15), 2076–2078 (2007).
[Crossref] [PubMed]

Liu, J.

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(15), 2076–2078 (2007).
[Crossref] [PubMed]

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.

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(15), 2076–2078 (2007).
[Crossref] [PubMed]

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.

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(15), 2076–2078 (2007).
[Crossref] [PubMed]

Luna, J. J.

Lunde, C. T.

Ma, J.

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.

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71(1), 75–85 (1981).
[Crossref]

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.

Mokuno, M.

Nakagawa, K.

Noriega-Manez, R. J.

R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
[Crossref] [PubMed]

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.

Siegman, A. E.

A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase aberrations,” Appl. Opt. 32(30), 5893–5901 (1993).
[Crossref] [PubMed]

Silva, D. E.

J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19(9), 1510–1518 (1980).
[Crossref] [PubMed]

Sodnik, Z.

Sun, J.

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]

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.

Toyoshima, M.

Wan, L.

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]

Wandernoth, B. K.

Wang, J. Y.

J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19(9), 1510–1518 (1980).
[Crossref] [PubMed]

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.

Wong, L. R.

Wong, N. M.

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.

Yamawaki, T.

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.

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]

Yura, H. T.

Zhao, H.

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(15), 2076–2078 (2007).
[Crossref] [PubMed]

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)

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)

J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19(9), 1510–1518 (1980).
[Crossref] [PubMed]

A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase aberrations,” Appl. Opt. 32(30), 5893–5901 (1993).
[Crossref] [PubMed]

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]

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)

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71(1), 75–85 (1981).
[Crossref]

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)

R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
[Crossref] [PubMed]

Opt. Lett. (1)

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(15), 2076–2078 (2007).
[Crossref] [PubMed]

Proc. SPIE (6)

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]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

Click here to see a list of articles that cite this paper

Fig. 1 Far-field correlation model for wave-front deformation.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 2 Deformation pointing-tracking errors with primary aberrations.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 3 Correlation factors with Tilt.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 4 The functional block diagram of tracking test.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 5 The scheme of the optical system.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 6 The experiment tracking system.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 7 Deformation pointing-tracking error angle without correction.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 8 Deformation pointing-tracking error angle with correction.

View in Article | Download Full Size | PPT Slide | PDF

Tables (1)

Table 1 Zernike polynomials for primary polynomials

View Table

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

θ_{A} = \frac{\sqrt{x_{P}^{2} + y_{P}^{2}}}{z}, θ_{B} = \frac{\sqrt{x_{Q}^{2} + y_{Q}^{2}}}{z}

u (x, y) = C_{1} \iint E (x, y) M (x, y) \exp [- \frac{i k}{2 z} (x_{1} x + y_{1} y)] d x d y

E (x, y) = C_{2} \exp [- \frac{x^{2} + y^{2}}{ω_{0}^{2}} - \frac{x^{2} + y^{2}}{ρ_{}^{2}}] \exp [i ϕ (x, y)]

I_{r e} (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) 〉

\begin{array}{l} I (O_{R}) = C_{p}^{2} 〈 u (p, t - t_{p}) u^{*} (p, t - t_{p}) 〉 \\ \begin{matrix} + \end{matrix} C_{q}^{2} 〈 u (q, t - t_{q}) u^{*} (q, t - t_{q}) 〉 \\ \begin{matrix} + C_{p} \end{matrix} C_{q}^{*} 〈 u (p, t - t_{p}) u^{*} (q, t - t_{q}) 〉 \\ \begin{matrix} + C_{p}^{*} \end{matrix} C_{q}^{} 〈 u^{*} (p, t - t_{p}) u (q, t - t_{q}) 〉 \end{array}

Γ_{p q} (τ) = 〈 u (p, t - t_{p}) u^{*} (q, t - t_{q}) 〉 = 〈 u (p, t + τ) u^{*} (q, t) 〉

\begin{array}{l} 〈 u (p, t - τ) u^{*} (p, t) 〉 = Τ_{p p} (τ) = Τ_{p p} (0) \\ 〈 u (q, t - τ) u^{*} (q, t) 〉 = Τ_{q q} (τ) = Τ_{q q} (0) \end{array}

δ_{p q} (τ) = \frac{Γ_{p q} (τ)}{{[Γ_{p p} (0) Γ_{q q} (0)]}^{1 / 2}}

0 \leq δ_{p q} (τ) \leq 1

ϕ (x, y) = \frac{2 π}{λ} W (x, y) = \frac{2 π}{λ} \sum_{n = 0}^{\infty} a_{n} Z_{n} (x, y)

r m s = \sqrt{\frac{\iint M (x, y) ϕ^{2} (x, y) d x d y}{\iint M (x, y) d x d y}}

M (x, y) = {\begin{matrix} 1 \begin{matrix} r_{2} \leq \sqrt{x^{2} + y^{2}} \leq r_{1} \end{matrix} \\ 0 \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} & \begin{matrix}  \end{matrix} \end{matrix} \end{matrix}

Type of Aberrations	Term	Zernike polynomials
Tilt	1	$Z_{1} (r, θ) = r \cos θ$
Tilt	2	$Z_{2} (r, θ) = r \sin θ$
Focus	3	$Z_{3} (r, θ) = 2 r^{2} - 1$
Astigmatism	4	$Z_{4} (r, θ) = r^{2} \cos 2 θ$
Astigmatism	5	$Z_{5} (r, θ) = r^{2} \sin 2 θ$
Coma	6	$Z_{6} (r, θ) = (3 r^{2} - 2) r \cos θ$
Coma	7	$Z_{7} (r, θ) = (3 r^{2} - 2) r \sin θ$
Spherical	8	$Z_{8} (r, θ) = 6 r^{4} - 6 r^{2} + 1$