## Abstract

In intersatellite optical communications it is important to obtain the most efficient performance of acquisition system with respect to acquisition time for a given probability. In this paper a novel approach to the analytical optimization of spatial acquisition is presented as an alternative of the conventional Monte Carlo simulation. First, an analytical expression of estimating mean acquisition time (MAT) is derived as a function of the desired acquisition probability, taking into account the distribution function of satellite position, field of uncertainty (FOU), beam divergence angle and dwell time. Accordingly, the analytical expression of multi-scan, which is always adopted by practical optical terminals to ensure the acquisition success, is also presented. Then, by minimizing the MAT of multi-scan, the optimum ratio of the FOU ${\theta}_{U}$to the pointing error deviation *σ* is obtained, which is ${\theta}_{U}/\sigma =1.3$. An example for a practical intersatellite acquisition between a Low Earth Orbit Satellite and a Geostationary Earth Orbit Satellite is given. And the theoretical result calculated by the proposed analytical expression is approximately equal to the result by Monte Carlo simulation. The results can be used in designing acquisition system for the intersatellite optical communications.

©2011 Optical Society of America

## 1. Introduction

The need for real-time high-speed data communication from any place on Earth is ever increasing due to increasing commercial and military applications. One solution to meet this requirement is to network satellites together through optical links to provide global coverage access and enhanced communication capacity [1]. Intersatellite optical links (IOLs) offer the potential advantages over microwave links of smaller size and weight of the terminal, less transmitter power, high immunity to interference and especially larger data rate [2–5]. However, the narrow laser beamwidth involved results in accurate pointing, acquisition and tracking (PAT) requirements especially when there is a long distance between satellites. PAT subsystems are critical to optical terminals for IOLs [6,7].

The first step to establish an IOL is acquisition, which is to compensate for large pointing errors at the beginning of the process and to achieve the communication line of sight (LOS) [7,8]. In intersatellite optical communications it is important to obtain the most efficient performance of acquisition system with respect to acquisition time for a given probability. The typical acquisition process involves the transmitter scans with a narrow laser beacon over an uncertainty area until it is detected and locked on, while the receiver stares with the telescope field of view (FOV), which is illustrated in Fig. 1 [9]. The size of uncertainty area scanned for the partner satellite is the transmitter field of uncertainty (FOU), which usually depends on the deviation of partner satellite’s position and decides the probability of acquisition. Because payloads limit the size and power of terminals in space, the beam divergence angle is relative smaller than ground station’s, or even communication laser is taken as beacon which has a rather smaller divergence angle. In practice, multiple scans must be performed to bring the probability of acquisition to an acceptable level for optical IOLs. So it is of concern to know what the optimum size of FOU should be for the production of the minimum acquisition time performance.

The concept of optical intersatellite communication is described in Ref [10]. Numerous studies have focused on the performance and algorithms for acquisition [11–14]. Various spatial acquisition patterns were proposed, such as raster scan, spiral scan, raster spiral scan, lissajo scan and rose scan, and the result shows that spiral scan is more efficient than the others [15]. Several cooperative methods were studied, which is, scan/scan stare/stare and scan/stare [15,16], in which stare/scan requires less power and has been verified reliable by OICETs [17]. Based on the result of Monte Carlo acquisition simulation, C. Hindman and L. Robertson verified the feasibility of using the narrow communication laser beam for acquisition as well as proposed the necessary of multiple scans without further research [18]. Currently, the acquisition time is always conducted through simulation [15,16,18,19]. However, the main problem connected with simulation is complex, time consuming and inconvenient to analyze the relation between system parameter and acquisition time. To the best of author’s knowledge, an analytical study of the acquisition time, correlated with the distribution function of satellite position, the size of the uncertainty area, beam divergence angle, dwell time and so on, has not been carried out in the literature.

In this paper first we derived the analytical expression for estimating the mean acquisition time (MAT) of single-scan based on spiral scan, taking into consideration of the factors mentioned above. And correspondingly, the prediction of the multi-scan MAT was also presented as a function of FOU and satellite distribution deviation. Furthermore, the optimum relation between the FOU and the satellite distribution deviation was derived by using an analytic approximation for the minimum average acquisition time of multi-scan.

## 2. Mathematical models

Initiation of communications between the two optical terminals requires open-loop pointing of the two terminals towards each other based on the predicted orbital positions of the satellites. There is an initial pointing error derives from the difference of the satellite position with the known Ephemeris primarily because of the uncertainty of satellite attitude. There is also error caused by satellite vibration, which can be eliminated by a two dimensional filter [15]. The purpose of acquisition is to compensate for the initial pointing error.

The procedure of acquisition is a statistical process. In order to define the acquisition time, we must first present an appropriate mathematical model to describe the various aspects involved. This includes the distribution function of satellite position and the scan time of spiral scan.

#### 2.1 Probability distribution function of satellite position

For a practical IOL, pointing error caused by satellite distribution can be modeled as Gaussian distributed random variables in vertical and horizontal [15]. And the probability density function (PDF) is defined as

We assume the vertical and the horizontal error follows the identical distribution [15,16], which is zero-mean Gaussian variable with variance of ${\sigma}^{2}$, and independent with each other. Then the radial deviation error is Rayleigh distributed with PDF

Then the acquisition probability which depends on the size of FOU, can be expressed as:

In Fig. 2 we can see ${P}_{acq}$ as a function of ${\theta}_{U}/\sigma $ according to Eq. (3). In Fig. 2 each scattered spot represents a possible location of partner satellite in transmitter satellite’s FOU. And the solid line represents the corresponding acquisition probability for various ${\theta}_{U}/\sigma $. From this figure it is seen that for almost all of the position of satellite depicted, ${\theta}_{U}=3\sigma $is large enough to acquisition them with a high probability approximately to 98.9%. $3\sigma $level is usually adopted by acquisition system design for single-scan [17,18,20].

#### 2.2 Scanning pattern

Among the various spatial acquisition methods, spiral scan is more efficient than others, since it starts from the highest probability center towards the edge of lower probability with a constant linear velocity, and it is easier to implement by the system [14,16,17]. The trace of spiral scan can be described in polar coordinates as

where ${I}_{\theta}$ is the step length related with beacon beam divergence angle ${\theta}_{b}$, which is ${I}_{\theta}={\theta}_{b}\left(1-\alpha \right)$ and*α*is the overlap factor, considering the overlap between the illumination areas in order to ensure the effective coverage; and the radius of spiral scan is limited by FOU, which is ${r}_{s}\le {\theta}_{U}$.

The scanning pattern of spiral scan is illustrated in Fig. 3 . It is shown that the spiral scan by Eq. (4) can cover the FOU efficiently with constant step length determined by beacon beam divergence angle. The dashed line is the edge of FOU for searching, and the solid line is the trace of beacon beam, in which the point means the location of each step. It is drawn from Fig. 3 that the larger the FOU is, the more steps would be taken to search, and accordingly, the scan time of acquisition is increased. Also as the scanning beam size increases, the step length increases and the acquisition is made faster. But in order to produce large and wide divergence beam need more power and a large telescope, which increase the complexity of the optical terminal [18].

The scan time from the initial point $\left(0,0\right)$to point of $\left({r}_{s},{\theta}_{s}\right)$ is described approximately as

where $\Delta t$ is the dwell time on each spot, which includes the twice time of light propagation across the link and the response time of receiver acquisition system.Where- ${T}_{R}$ is the response time of receiver acquisition system
*l*is the link distance between optical transmitter and receiver*c*is the velocity of light travels in space

## 3. Mean acquisition time of single-scan

Based on models in section 2, we derive an analytic expression of MAT with relation to the probability of acquisition by the factor of FOU.

We evaluate the statistical acquisition time according to the expected value. Therefore the single scan acquisition time should be averaged with respect to the PDF of the satellite position and is given by

Substituting Eqs. (5) and (2) into Eq. (8) yields

Equation (9) shows how the acquisition time of single-scan varies with the dwell time, step length of scan, deviation of initial pointing error, and FOU. It is an analytical expression. So it is easy to analyze the effects of these parameters on the acquisition time. When the IOL environment is designated, the other system parameters are settled except the FOU which is of concern in this paper.

Given the typical parameters $\Delta t=0.1s$, ${I}_{\theta}=0.6mrad$and $\sigma =1,2,4mrad$, this gives rise to the plot shown in Fig. 4
which illustrates the impact on $E{T}_{S}$ with respect to deviation *σ* for various FOU values. It is seen in Fig. 4 that $E{T}_{S}$ is increasing with rising in FOU.

We should also notice that the size of FOU is dependent on the required acquisition probability of single scan ${P}_{S}$ which is equal to ${P}_{acq}$. And the relation is described as

Obviously, acquisition time of $E{T}_{S}$ is related to the acquisition probability of single scan ${P}_{S}$via the size of FOU ${\theta}_{U}$. And it is clear that reducing the ${\theta}_{U}$ results in shorter acquisition time of single-scan, however, results in a lower acquisition probability. It seems like that to gain a higher acquisition probability is in price of time. Actually, the balance can be achieved by multi-scan considering the combined effect of them. And there should be an optimum ${\theta}_{U}$ that leads to minimum MAT and high acquisition probability.

## 4. MAT of multi-scan acquisition and analytical optimization

In practice, the acquisition process adopts multi-scan to ensure the acquisition probability approximately to 1. In this section we develop the acquisition probability and the corresponding MAT of multi-scan.

#### 4.1 MAT of multi-scan acquisition

In this paper we research the multi-scan with the assumption of that the position of receiver satellite is static relative to the FOU of transmitter satellite and the dynamic error can be eliminated by a two dimensional filter. In multi-scan mode, a successful acquisition occurs only when the receiver satellite appears in the FOU of transmitter satellite. In multi-acquisition let the $i\text{th}$acquisition involving scan the FOU relative to the $i\text{th}$initial pointing point. We let ${A}_{i}$ and $\overline{{A}_{i}}$ denote that the receiver satellite locates in the $i\text{th}$FOU and it doesn’t locate in the $i\text{th}$FOU respectively. Hence, a successful acquisition of multi-scan can be described as

Then the probability of multi-scan acquisition is given by

In each new scan, since it begins with a new initial pointing according to the Ephemeris, the $i\text{th}$scan is independent with the others. Thus, the optimization FOU of each scan is identical, and Eq. (12) is expressed as

where ${P}_{M}$ is the acquisition probability of multi-scan, and*n*denotes the total number of scan areas until the acquisition is accomplished. It is obvious to obtain from Eq. (13) that when $n\to \infty $, the acquisition probability ${P}_{M}\to 1$.And accordingly, the MAT of multi-scan can be defined as

For the case of multi-scan, if $0<{P}_{S}<1$ is satisfied, the expected value of acquisition time can be simplified as

Substituting Eqs. (3), (7) and (9) into Eq. (16) gives the analytical expression of multi-scan MAT as

Now one can find the relation of system parameters with acquisition time

In Fig. 5
, the multi-scan MAT is plotted as a function of FOU and deviation of satellite position, taking $\Delta t=0.1s$and ${I}_{\theta}=0.6mrad$. It is easily seen in Fig. 5 that for each *σ* there is an optimum ${\theta}_{U}$to minimize the multi-scan MAT.

#### 4.2 Optimum FOU for multi-scan

Theoretically, the optimum ${\theta}_{U}$can be derived by

If we define a new variable $\epsilon ={\theta}_{U}{}^{2}/2{\sigma}_{}^{2}$, Eq. (18) becomes

In practice, the optimum ${\theta}_{U}$should not be 0, and then the problem is simplified to

The approximately analytical result of Eq. (20) is $\epsilon =0.8426$, then the optimum FOU ${\theta}_{U}$ should be expressed as

Figure 6
shows the variation of $E{T}_{M}$ as a function of *σ* at different level of FOU. Both of them increase with increasing *σ*. And obviously the curve of ${\theta}_{U}=1.3\sigma $is always below that of ${\theta}_{U}=3\sigma $, and the deviation between them increases with rises in *σ*. That means the optimum FOU level is better than the traditional $3\sigma $level, and the effect is more significant for a larger *σ*.

The optimum FOU we got in Eq. (21) with respect to initial pointing error deviation for the minimum MAT of multi-scan, will aid optical terminal acquisition system design.

## 5. Practical situation

In this section we simulate the process of multi-scan acquisition in which considering an optical communication link between a LEO satellite and a GEO satellite. Monte Carlo simulation is adopted to investigate the MAT.

The simulation is done in Matlab. In simulation the distance between LEO and GEO is 36,000 km, and the control frequency bandwidth of the fine tracking system in LEO is 200 Hz. Hence, the dwell time should be $\Delta t=0.245s$. The position of satellite is estimated to have attitude knowledge with variance of 2$mrad$ [18]. The beacon beam divergence angle is 300$\mu rad$, and the corresponding step length is 200$\mu rad$. Simulation has been carried out on each FOU for 5000 different satellite positions, and the result is shown in Fig. 7 .

The scattered points in Fig. 7 represent the calculated MAT of multi-scan by simulation for different size of FOU, compared with the theoretical value according to analytical expression Eq. (17). Figure 7 illustrates the effect of the proposed analytical expression Eq. (17) in estimating an MAT approximately equal to the MAT of the Monte Carlo simulation results. And it is also proved that there is an optimum size of FOU associated with satellite position deviation with production of a minimal MAT.

## 6. Conclusions

In this paper, we have developed and validated a novel approach to the analytical expression of estimating average acquisition time for intersatellite optical communications instead of the complex simulation. This analytical model accounts for the major parameters relation with acquisition time, including beam divergence angle, dwell time, as well as initial pointing error and size of the uncertainty area. Considering the acquisition probability, multi-scan acquisition is discussed, and the corresponding MAT is also presented. Then the analytic expression for the optimum ratio of FOU to deviation of initial pointing error is derived for the production of the least MAT, which is obtained to be ${\theta}_{U}/\sigma =1.3$. The comparison of theoretical and simulation results shows consistent, and also indicates that the optimum FOU is more efficient than the conventional $3\sigma $level. This methodology is applied in intersatellite optical system with relative narrow beacon beam especially the system using communications beam for acquisition. The result obtained here will be useful in parametric performance estimation and optimization of acquisition system design.

## Acknowledgements

The authors are grateful to the National Natural Science Foundation of China (NSFC) for financial support under Projects Nos. 10374023 and 60432040

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