## 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${\theta}_{A}$and${\theta}_{B}$, far-field correlation factor$\delta $depend on RMS of deformation error$rms$, which decline with a increasing$rms$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.

© 2015 Optical Society of America

## 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.

The intensity peak of tracking beam is origin${O}_{A}\left(0,0\right)$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${P}_{B}\left({x}_{P},{y}_{P}\right)$, 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*, which can be expressed as

_{B}The light intensity distribution in receiver plane can be represented as

*O*and

_{B}*P*can be obtained according to the Eq. (4), then deformation pointing-tracking errors ${\theta}_{A}$and ${\theta}_{B}$can be calculated.

_{B}## 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$p\left({X}_{p},{Y}_{p}\right)$and point$q\left({X}_{q},{Y}_{q}\right)$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.

$u\left(p,t\right)$and$u\left(q,t\right)$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

*O*is average of time in receiver plane that is obtained as the followingWhere $\u3008\u3009$denote time averaging. Then we take Eq. (5) into the Eq. (6) as

_{R}*O*and selective time. Thus, correlation function can be written in the form

_{R}*O*.

_{R}#### 3.2 Correlation factor

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

*O*superimposed two tracking beams is relevant with the correlation function and correlation factor.

_{R}## 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$\mathrm{exp}\left(j\varphi \right)$. 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

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

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.

When$rms$is$0.10\lambda $, the deformation pointing-tracking error due to Tilt is about$1.4\mu \text{rad}$, and$0.8\mu \text{rad}$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.

The relationship curves of correlation factors for${\theta}_{B}=0.5\mu rad,1.0\mu rad,1.5\mu rad$ are given in Fig. 3 when$rms$value of terminal A is$0.10\lambda $. 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.

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${\theta}_{A}$. And${\theta}_{B}$can be obtained using the same method.

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

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.

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$rms$, 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.

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