## Abstract

Satellite platform vibration causes the misalignment between incident direction of the beacon and optical axis of the satellite optical communication system, which also leads to the instability of the laser link and reduces the precision of the system. So how to simulate the satellite platform vibration is a very important work in the ground test of satellite optical communication systems. In general, a vibration device is used for simulating the satellite platform vibration, but the simulation effect is not ideal because of the limited randomness. An approach is reasonable, which uses a natural random process for simulating the satellite platform vibration. In this paper, we discuss feasibility of the concept that the effect of angle of arrival fluctuation is taken as an effective simulation of satellite platform vibration in the ground test of the satellite optical communication system. Spectrum characteristic of satellite platform vibration is introduced, referring to the model used by the European Space Agency (ESA) in the SILEX program and that given by National Aeronautics and Space Development Agency (NASDA) of Japan. Spectrum characteristic of angle of arrival fluctuation is analyzed based on the measured data from an 11.16km bi-directional free space laser transmission experiment. Spectrum characteristic of these two effects is compared. The results show that spectra of these two effects have similar variation trend with the variation of frequency and feasibility of the concept is proved by the comparison results. At last the procedure of this method is proposed, which uses the power spectra of angle of arrival fluctuation to simulate that of the satellite platform vibration. The new approach is good for the ground test of satellite optical communication systems.

©2012 Optical Society of America

## 1. Introduction

Compared with satellite microwave communication, satellite optical communication has many advantages, such as higher data rate, lower probability of intercept, less power consumption, less mass [1,2]. Recently, satellite optical communication has been concerned by many countries and it has turned to be a hot topic in the field of satellite communications [3–6].

Pointing, Acquisition and Tracking (PAT) technology is one of key technologies in satellite optical communications. In the process of establishing an optical communication link, satellite platform vibration leads to the misalignment between input direction of the beacon and optical axis of receiving system, and then influences the precision of PAT system [7].

How to simulate the satellite platform vibration is an important technology in the ground test of the satellite optical communication system. In general, to test the performance of PAT system under the condition of vibration, a vibration device is often applied in simulating the satellite platform vibration. The relevant work reported in the reference indicates that satellite vibration has a continuous spectrum [8]. However, the vibration device used in the test just gives several discrete frequencies, so the simulation of this approach is relatively rough.

When a laser beam propagates though atmosphere, atmosphere turbulence causes phase fluctuation of the beam, which induces the effect of angle of arrival (AOA) fluctuation [9–13]. Because angle of arrival fluctuation is natural, not artificial, and it also has a continuous spectrum. It is necessary to discuss whether spectrum characteristic of these two effects is the same or similar, so in this paper, spectrum characteristic of these two effects is compared. The comparison results give justification to the new approach that angle of arrival fluctuation can be taken as a more effective simulation for the satellite platform vibration in ground test process of PAT system.

This paper presents an experiment that measures angle of arrival fluctuation of a laser beam propagating through atmosphere over an 11.16km near horizontal urban laser link. An 11.16km bi-directional free space optical (FSO) link experiment is established to examine the effect of angle of arrival fluctuation, which is introduced in section 2. The experimental data are analyzed in section 3, compared with the power spectra of the typical satellite platform vibration. The theoretical analysis is also conducted in section 3.We propose the new approach to simulate the satellite platform vibration in section 4 and section 5, and the final part summarizes the conclusions.

We use safety goggles for protecting our eyes from being burned by the laser. In addition, the optical link is higher than most of buildings near the optical path and beam directions are adjusted so that the laser can’t injure other residents’ eyes. So this experiment is very safe.

## 2. Experimental setup

An 11.16km bi-directional FSO near horizontal laser link is established to examine the effect of angle of arrival fluctuation. The experimental system owns two optical communication systems (OS1 and OS2), with different receiving apertures (150mm and 250mm). The specification of this experiment is described as follows:

The laser link is established between two buildings in Harbin. One of the two systems is placed on the 28th floor (100m above ground level) of a building in Songbei area of Harbin, and the other is located on the roof of a building in urban area of Harbin.

The aerial photo of the laser link is shown in Fig. 1 . Landscape along the path mainly includes a river and the wet land near it, several roads and building arrays. The path is complex. The total distance of this laser link is 11.16km, which is measured by a GPS.

Two optical communication systems of this experiment have similar optical structures. The configuration is shown in Fig. 2 , and main parameters of the equipment are provided in Table 1 . As shown in Fig. 2, the laser is emitted by the laser source, and reflected onto a color separator. The wavelengths of the output and input lights are different, so the color separator has different effects of total reflection and total transmission on the output and input light, respectively. The output light is reflected by the color separator into a Cassegrain telescope to compress the divergence angle and then transmitted outside. The laser is received by the other optical communication system at the other end of the laser link at last, after propagating through an 11.16km atmosphere turbulent channel.

As received (input) light, the laser is received by the Cassegrain telescope. After transmitting through the color separator, the laser is divided into two beams by a splitter. One of the two beams is focused on an avalanche photodiode (APD) to perform the communication experiment (not mentioned in this paper). The other beam is focused on a CMOS camera. A computer controls the CMOS camera to record its images at 1900 frames/s.

In Fig. 2, an optical filter is placed before the splitter to mitigate the influence of the background light.

During the experiment, a temperature and humidity recorder is used to record the temperature and humidity of the experiment every 2 minutes.

## 3. Data analysis

#### 3.1 Computation of spectrum

The effect of angle of arrival fluctuation is investigated with the recorded data. For one experimental data trial, centroid coordinate (*X*_{i}, *Y*_{i}) of the i-th frame is calculated by the gray centroid algorithm, that is:

*g*is gray-level value of the pixel with coordinate of (

_{xy}*x*,

*y*).

Centroid coordinates of all frames in one data trial lead to two coordinate sequences of *X* = (*X*_{1}, *X*_{2} …*X*_{n}) and *Y* = (*Y*_{1}, *Y*_{2} …*Y*_{n}). The sequence *A* = (*A*_{1}, *A*_{2} … *A*_{n}) for angle of arrival is obtained from the coordinate sequences by the following formula,

*A*is the

_{i}*i-*th element in the sequence of

*A*,

*d*is pixel size of the CMOS,

*f*is focal length of the receiving optical system and

_{L}*M*is enlargement factor of the optical system.

The experiment was performed from April 2009 to August 2009. Partial data set are given in Table 2 . Table 2 summarizes the date and climate conditions for an initial data set, consisting of two data trials measured by OS1 (T1 and T2) and four data trials measured by OS2 (T3-T6), respectively.

#### 3.2 High-frequency spectrum

Figure 3 shows power spectra of angle of arrival fluctuation, which are derived from the Discrete Fourier Transform of data trials T1-T6. The black line is the power spectrum of angle of arrival fluctuation and the blue line represents the power law exponents of high-frequency spectrum, respectively in every subgraph of Fig. 3.

As shown in Fig. 3, power spectra of angle of arrival fluctuation have the characteristic that there is high power in low frequency area and low power in high frequency area. Especially, high-frequency power spectra of angle of arrival fluctuation have the approximate negative exponential power law dependence.

We want to know the scale of the power law exponents of high-frequency spectra, so some experimental data set are used. In Fig. 4 and Fig. 5 , the power law exponents of high-frequency spectra are not fixed values, and mostly they are greater than −11/3, which is more complicated than the theoretical results reported in Ref. [10]. The theoretical model should be revised according to the specific conditions. Although the power law exponents of high-frequency spectra are not constant, the minimal value of the power law exponents is increasing along with the augmentation of variance of AOA, generally.

#### 3.3 Contrastive analysis

Here, we mainly refer to two important power spectrum models of the satellite platform vibration which are often cited in many references [14,15] (shown in Fig. 6 ). In Fig. 6, the blue line is the spectrum model that is used by the European Space Agency (ESA) in the SILEX program, while the green line is given by National Aeronautics and Space Development Agency (NASDA) of Japan.

As shown in Fig. 6, power spectra of the satellite platform vibration have the characteristic that there is high power in low frequency area and low power in high frequency area. Similarly, high-frequency power spectrum has the approximate negative exponential power law dependence, too.

The similarity between the power spectrum of angle of arrival fluctuation and the power spectrum of the satellite platform vibration is analyzed, and it is calculated as [15]:

*d*is the distance of two curves, two curves are expressed as ${f}_{1}(X)$and ${f}_{2}(X)$, respectively;

*n*is the total number of elements compared.

If the curvature of one curve is the same or approximately equal as that of the other one, for comparing the similarity of two curves, one curve needs to be parallel move to superpose the other one in the rectangular coordinate system.

From Eq. (3), we see that the less *d* is, the better the similarity is. From Fig. 7(a)
, in order to analyze the similarity between the two curves, the blue line needs to be parallel move to superpose the black line. The distance of the two curves can be calculated according to Eq. (3). Here *f* is frequency. Where *f* ≥ 50Hz, *d* is 0.6, and where *f *< 50Hz, *d* is 37.9.

In a similar way, the similarity is analyzed in Fig. 7(b) and Fig. 7(c), and the curves need not to be parallel move. From Fig. 7(b), where *f *≥ 50Hz, *d* is 48.5, and where *f* <50Hz, *d*is 1473.1. From Fig. 7(c), where *f *≥ 50Hz, *d* is 128.1, and where *f* < 50Hz, *d *is 1215.5.

Based on the results, where *f* < 1000Hz, the similarity of Fig. 7(a) is the best, and where *f *≥ 50Hz, the similarities of the three figures are good. But where *f* < 50Hz, the similarities of Fig. 7(b) and Fig. 7(c) are relatively worse, which is observed from Fig. 7(b) and Fig. 7(c).

From Fig. 7(a), the power spectrum of angle of arrival fluctuation measured by OS1 (T1) is higher than that of the satellite platform vibration used by ESA. However, the trend of the power spectra of these nearly is the same, and the similarity of the power spectra of these is very high.

In Fig. 7(b), where *f* < 50Hz, the power spectrum of the satellite platform given by NASDA is slightly higher than the power spectrum of AOA measured by OS2 (T4). But where *f* > 50Hz, the trend of the power spectra of these is very similar. The order of magnitude of these two power spectra shown in Fig. 7(b) is the same.

In Fig. 7(c), where *f* < 50Hz, the power spectrum of the satellite platform given by NASDA is slightly higher than the power spectrum of AOA measured by OS2 (T5), too. But where *f* > 50Hz, the trend of the power spectra of these also is very similar. The order of magnitude of these two power spectra shown in Fig. 7(c) is also the same.

In fact, it can be seen in Fig. 7 that spectra of angle of arrival fluctuation are consistent with those of satellite platform vibration. The comparison results confirm the new approach that the effect of angle of arrival fluctuation is taken as an effective simulation of satellite platform vibration in the ground test process of PAT system for a satellite communication optical system.

We discuss the similarity between the spectra of angle of arrival fluctuation and those of satellite platform vibration theoretically. According to Ref. [16], the temporal power spectrum of AOA fluctuation for a plane wave propagating in non-Kolmogorov atmospheric turbulence (${W}_{1}(\alpha ,\omega ,\beta )$) is expressed as [16]

*L*is the transmission distance of the light wave,

*k*is $2\pi /\lambda $, and

*λ*denotes the optical wavelength, ${\omega}_{0}=v{(L/k)}^{-1/2}$, and $v$ denotes the transverse wind velocity;

*b*is a constant (

*b*= 0.4832),

*D*is the receiving aperture of optical communication system,

*β*is the angle between the baseline and the AOA observation axis, and $W(z)$is Whittaker’s confluent hypergeometric function.

When *α* is −11/3, the corresponding variance for a plane wave propagating in non-Kolmogorov atmospheric turbulence [10,16,17] is shown as follows:

In addition, the temporal power spectrum of AOA fluctuation for a spherical wave propagating in non-Kolmogorov atmospheric turbulence (${W}_{2}(\alpha ,\omega ,\beta )$) is given by [16]

When *α* is −11/3, the corresponding variance for a spherical wave propagating in non-Kolmogorov atmospheric turbulence [10,16,17] is shown as follows:

The simulation conditions are based on the parameters of the 11.16km bi-directional FSO near horizontal laser link. Furthermore, when *α* is −11/3, the power spectra for AOA fluctuation obtained by Eq. (4) and Eq. (6) are consistent with the conventional results of Kolmogorov atmospheric turbulence [18]. So the conditions are that *α* is −11/3, *β* is 0, *L* is 11.16km, *D* is 150mm or 250mm, *λ* is 800nm and *v* is 2m/s. According to Eq. (4)-Eq. (7), the temporal power spectra of AOA fluctuation scaled by the corresponding variance are illustrated in Fig. 8
and Fig. 9
. The comparisons between the theoretical values of AOA fluctuation and the power spectra of the satellite platform vibration are shown, with Fig. (8) for a plane wave, Fig. (9) for a spherical wave, respectively.

We compute the similarity between the temporal power spectrum of AOA fluctuation and the power spectrum of the satellite platform vibration by Eq. (3). In Fig. 8, where *f *< 100Hz, the distance between the blue line and the red line is 781.9, and the distance between the black line and the blue line is 782.03.Where *f *< 100Hz, the distance between the green line and the red line is 250.04, and the distance between the green line and the black line is 229.8(here the red line or the black line needs to be parallel move to superpose the green line for computing the similarity in the rectangular coordinate system, respectively). In Fig. 9, where *f *< 100Hz, the distance between the blue line and the sky blue line is 781.9, and the distance between the violet line and the blue line is 782.03.Where *f *< 100Hz, the distance between the green line and the sky blue line is 250.04, and the distance between the green line and the violet line is 229.8(here the violet line or the sky blue line needs to be parallel move to superpose the green line for computing the similarity in the rectangular coordinate system, respectively). We can get the conclusion that the temporal power spectrum of AOA fluctuation for a plane wave or a spherical wave with a larger receiving aperture of optical communication system is relatively similar to the power spectrum of the satellite platform given by NASDA, which is very consistent with the experimental result (shown in Fig. 7(b) and Fig. 7(c)).

From Fig. 8 and Fig. 9, the temporal power spectrum of AOA fluctuation for a plane wave or a spherical wave has the same characteristic that there is high power in low frequency area and low power in high frequency area. Furthermore, high-frequency power spectrum has the approximate negative exponential power law dependence, too. But the order of magnitude of the temporal power spectrum of AOA fluctuation is lower than that of the power spectrum of the satellite platform vibration, and where *f *> 100Hz, the temporal power spectra of AOA fluctuation are not well consistent with the power spectra of the satellite platform vibration. We analyze that because the atmospheric turbulence is very complex, and some factors can affect the results, such as turbulence intensity, temperature gradients, moisture. Furthermore, some simplified conditions are specified in the theoretical models in Ref.16. The experimental data show the randomness of AOA fluctuation due to the variation of the atmospheric turbulence, too.

And next section demonstrates the procedure of the new approach for simulating the satellite platform vibration.

## 4. Procedure of the new method

We need to use the power spectrum experimental data of angle of arrival fluctuation to simulate the power spectrum of the satellite platform vibration. Because the power spectrum of angle of arrival fluctuation of laser is randomness, here the mathematical processing method is the linear model of smooth time sequence, which is symbolized as ARMA(p,q) [19]. As an example, the processing procedure of the data trial T3 is illustrated.

The power spectral density of the data trial T3 is described according to the linear model of smooth time sequence,

where $P(\omega )$represents the power spectral density of the data trial T3, $\omega $ is frequency, A(z) and B(z) are real coefficient polynomial, and${\sigma}_{a}$is discrete white noise variance.A(z) and B(z) can be expressed as follows:

where ${\theta}_{1}$,...,${\theta}_{q}$and ${\varphi}_{1}$,…,${\varphi}_{q}$ are real coefficients.A smooth time sequence is shown as bellows

_{t}is a smooth time sequence, t = 0, ±1, ±2,…,${a}_{t}$is discrete white noise.

According to Eq. (8)-Eq. (10), we obtain a spectral line to simulate the power spectrum of NASDA satellite platform vibration (In fact, we can get many spectral lines by this method, and only one is taken as an example here), illustrated in Fig. 8. From Fig. 8, we see that this simulation spectral line is very similar to the power spectrum of NASDA satellite vibration, so the simulation approach is effective. By Eq. (11), a smooth time sequence (${\text{S}}_{t}$) is derived from this simulation spectral line by inverse Fourier transform, which is utilized to simulate the satellite vibration equivalently in the time domain, shown in Fig. 9.

## 5 Discussion

Although the power spectrum of satellite platform vibration varies considerably based on satellite platform design, the satellite platform should meet the requirement of the satellite optical system, such as the order of magnitude and frequency of vibration, the orbit determination precision and so on. Because the spatial experiments have been done by ESA and NASDA, respectively, the two representational power spectra are cited here, which are used by ESA and NASDA, separately. From the comparison results, we see that the effect of angle of arrival fluctuation can be taken as an effective simulation of satellite platform vibration because of the similarity. Of course, we should also see that atmospheric effects vary considerably depending on environmental conditions, which can lead to the alteration of power spectrum of angle of arrival fluctuation. The randomness of angle of arrival fluctuation is the basis of the new approach. The new approach needs an experiment similar to that described in this paper, in order to test the performance of satellite optical systems. In the environment of atmospheric fluctuation, the performance of satellite optical systems can be verified adequately. And the effect of angle of arrival fluctuation needs to be evaluated by the method mentioned in section 4. If the power spectrum of angle of arrival fluctuation is higher than that of the satellite platform vibration, the redundancy of PAT system can be measured, which offers a significant reference for the satellite platform design. And then success of PAT process in the environment of angle of arrival fluctuation on ground can guarantee equivalently the establishment and maintenance of the satellite laser link in the environment of satellite platform vibration.

In addition, according to the experimental results, the application principles of this method are that the suitable distance of the laser link is at least 11km, and the optimum test time interval is from April to June.

## 6. Conclusions

An 11.16km bidirectional free space optical link experiment is conducted to examine the effect of angle of arrival fluctuation. An initial data set is complied, which consists of six cases collected including spring and summer months. High-frequency power spectrum of angle of arrival fluctuation has the approximate negative exponential power law dependence, and the power law exponents of high-frequency spectra are not fixed values, and mostly they are greater than −11/3, which is more complicated than the theoretical results. In this paper, the spectrum characteristic of satellite platform vibration is introduced, referring to the model used by ESA in the SILEX program and the model given by NASDA of Japan. And spectrum characteristic of these two effects is compared. Comparison results show that spectra of both effects have the characteristic of high power in low frequency area and low power in high frequency area. High-frequency power spectra of both effects have the approximate negative exponential power law dependence. In addition, power spectra of angle of arrival fluctuation are coincident with those of the satellite vibration so that angle of arrival fluctuation can be taken as an equivalent simulation for the satellite platform vibration. Compared with the limitation of traditional vibration devices, it is considered that angle of arrival fluctuation is taken as a more effective simulation of the satellite platform vibration in ground test process of PAT system. This paper also proposes the procedure of this new approach. This work is benefit for the free space optical communication system design, and it also gives justification to the satellite platform design suitable for satellite optical systems.

## Acknowledgments

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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