## Abstract

Adaptive optics (AO) systems can suppress the signal fade induced by atmospheric turbulence in satellite-to-ground coherent optical communication. The lower bound of the signal fade under AO compensation was investigated by analyzing the pattern of aberration modes for a one-stage imaging AO system. The distribution of the root mean square of the residual aberration is discussed on the basis of the spatial and temporal characteristics of the residual aberration of the AO system. The effectiveness of the AO system for improving the performance of coherent optical communication is presented in terms of the bit error rate and system availability.

© 2014 Optical Society of America

## Corrections

Huang Jian, Deng Ke, Liu Chao, Zhang Peng, Jiang Dagang, and Yao Zhoushi, "Effectiveness of adaptive optics system in satellite-to-ground coherent optical communication: erratum," Opt. Express**22**, 17479-17479 (2014)

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-22-14-17479

## 1. Introduction

In addition to scintillation, wavefront aberration caused by atmospheric turbulence can degrade the information-bearing signal power of coherent optical communication in satellite-to-ground links [1]. An adaptive optics (AO) system, installed in the optical ground station equipped with a large-aperture receiving telescope, can be used to correct the wavefront distortion and further decrease the bit error rate (BER).

This topic has been the subject of much theoretical and experimental work. Fried [1] and Winick [2] have studied the theory of optical heterodyne detection in the presence of atmospheric-turbulence-induced wavefront distortions, and the results were used for numerical simulations to analyze the feasibility of optical coherent transmission [2]. Wavefront aberration compensation can improve the heterodyne efficiency of the downlink and suppress scintillation of the uplink [3], thus reducing the BER of the bidirectional coherent communication. At present, an AO system is being developed for the satellite-to-ground coherent optical communication link of the European Data Relay System [4–6]. The designed residual aberration is near the diffraction limit, and the resulting Strehl ratio for a 20-km horizontal path is better than 0.8 at 1064 nm [7].

In practice, the required performance of the AO system will be determined by other system parameters, especially the signal-to-noise ratio (SNR) and link budget. The effectiveness of an AO system in satellite-to-ground optical coherent communication depends on the instantaneous magnitude and distribution of the residual aberration of the AO system. The diffraction limit and Strehl ratio are insufficient for describing the dynamic characteristics of AO compensation and do not have a clear relationship with the communication performance.

In this paper, we derive the lower bound of the signal fade for a coherent optical communication link after aberration compensation using the homodyne scheme. Spatial and temporal characteristics of the residual aberration of the AO system are analyzed to evaluate the system performance. The availability and mean BER of the communication link under AO compensation are calculated.

## 2. Signal fade after AO compensation

Information-carrying signal fade is defined by the ratio of the signal power in the presence of wavefront aberration to the signal power in the absence of wavefront aberration and is also equal to the loss in the SNR [8]. The relationship can be expressed as

To connect Eq. (1) to the AO system parameters, we must make some approximations. We assume that the demodulation is a homodyne scheme, that ${E}_{lo}(\overrightarrow{r})$ is a plane wave of uniform intensity, and that the scintillation of the signal laser can be disregarded. This yields ${E}_{lo}(\overrightarrow{r})={A}_{lo}{\mathrm{exp}}^{i\omega t}$, and ${E}_{s}(\overrightarrow{r})={A}_{s}{\mathrm{exp}}^{i\left[\omega t+\phi (\overrightarrow{r})\right]}$. Then, from Eq. (1) we obtain

*S*is the area of the aperture, then $\eta $ is the signal fade after AO compensation.

The spatial characteristics of $\phi (\overrightarrow{r})$ (i.e., the magnitude and shape) will determine $\eta $, and so it is reasonable to define the bound of $\eta $ as a function of the root mean square (RMS) of $\phi (\overrightarrow{r})$.

To analyze the spatial characteristics of $\phi (\overrightarrow{r})$, we collected residual wavefront error data from an AO system for bright stars under different atmospheric coherence length conditions. The AO system was that integrated into the 1.8-m telescope operated by the Adaptive Optics Laboratory of the Chinese Academy Of Sciences. The single-stage closed-loop system consists of a tilt mirror, 127-element deformable mirror, and 2000-Hz Hartman wavefront sensor [10]. The Hartmann wavefront sensor operates at 550 nm, and we transformed the residual aberration data at 1550 nm to be consistent with the communication wavelength.

We analyzed the RMS of the coefficients of each Zernike term of the residual aberration. Each set of data was calculated using 10 000 frames of aberration. The results are shown in Fig. 1.The residual aberration contains a large amount of tilt, and the RMS of the tilt coefficient is much larger than other high-order aberration but near the same on X and Y coordinates. The RMS of the coefficients of the higher-order Zernike terms are stable under different turbulence conditions.

We calculated $\eta $ in Eq. (2) numerically by assuming $\phi (\overrightarrow{r})$ represents the first 35 Zernike aberration. We did not calculate the piston error because it is generally analyzed in a phase-lock loop. The results are shown in Fig. 2.

We found that for a residual aberration with a RMS of less than 2 rad the defocus and tilt had a similar effect on degrading the signal power, and they degrade the signal power most .

The tilt degrades the signal power, and the residual tilt is a main part of the overall residual aberration of the AO system. Defining the signal fade induced by a residual aberration as ${\eta}_{c}$ and the signal fade due to tilt as ${\eta}_{t}$, then at the same RMS value ${\eta}_{t}<{\eta}_{c}$.

We calculated and compared the ${\eta}_{t}$ and ${\eta}_{c}$ of the residual aberration, and the results are shown in Fig. 3.

While an analytic relationship between the magnitude of the signal fade and the RMS of the total aberration consisting of multiple orders of Zernike polynomials is difficult to determine, Fig. 3 shows that ${\eta}_{t}$ defines a precise lower bound of the signal fade.

If the residual aberration is only tilt, then

where $r$ and $\theta $ are the polar coordinates, $k=2\pi /\lambda $ is the wave number, and $\beta $ is the tilt angle. Inserting this into Eq. (2) gives ${\eta}_{t}$ asThe residual aberration of the AO system is a “short-term” random variable that determines the maximum signal fade on a time scale of milliseconds. Although Eq. (6) has been derived under the condition that residual tilt is the main aberration, it can be applied even when there is no tilt because ${\eta}_{t}$ is independent of the aberration mode for residual aberration with an RMS of less than 0.5 rad as shown in Fig. 2.

## 3. Dynamic characteristics of signal fade

AO compensation is a random process with respect to time. Theoretically, like the random error in the control loop of the tilt mirror, the Zernike coefficients of the residual tilt in the X and Y directions obey a Gaussian distribution. As the X and Y axes can be chosen arbitrarily, their distribution functions are the same. As a result, the RMS of the total residual tilt ${\sigma}_{t}$ can be expressed as

where ${a}_{2}$ and ${a}_{3}$ are the coefficients of the residual tilt in the X and Y directions. As ${\sigma}_{t}$ has a Rayleigh distribution, its probability density function (PDF) can be written asWithout loss of generality, if the wavefront error is compensated sufficiently, then the Zernike coefficients of the higher-order residuals ${a}_{k}$ can be described as a Gaussian distribution with a mean of zero and a variance of ${\sigma}_{z}^{2}$:

It is known that the square of the Zernike coefficient is equal to the variance of each order of aberration, and therefore, the variance of higher-order aberration ${\sigma}_{h}^{2}$ can be computed by the sum of the squares of ${a}_{k}$. Mathematical statistics reveals that the quantity ${\sigma}_{h}^{2}/{\sigma}_{z}^{2}$ is distributed as a χ^{2} random variable

If the tilt is corrected to the level of the higher-order aberration, then the distribution of $\sigma $ can be described by Eq. (10) and only the number of degrees of freedom needs to be changed.

Using the data in Fig. 1, we calculated the parameters of the AO performance and obtained *b* = 0.65 rad and ${\sigma}_{z}^{}$ = 0.1 rad. We removed the mean values of the coefficients of each Zernike term when computing the histograms. The mean values indicate the system calibration error, which mainly produced a static tilt bias, of about 0.45 rad.

The parameters $b$ and ${\sigma}_{z}^{}$ characterize the compensation performance of the tilt and deformable mirrors, respectively, and they are connected to the AO system through the Taylor and Greenwood frequencies, respectively. These parameters can be used to determine the required control loop bandwidth of the AO system.

Based on the distribution function of the residual aberration, we can produce random variables of the RMS of the residual aberration to numerically calculate the BER of coherent optical communication under AO correction.

In the case of binary phase-shift keying (BPSK) modulation with a Gaussian distribution shot noise, the bit error probability (BEP) can be deduced from the SNR:

where $erf$is the error function. The SNR here is that without wavefront error, and so the mean BER under AO compensation is equal to the mathematical expectation of the BEP.We calculated the PDF of the BEP using the aberration data shown in Fig. 4 for SNR = 49.5. This SNR value corresponds to the best BER of 10^{−12} of the Tesat
laser-communication-terminal-to-ground coherent optical communication link.

In our numerical simulation, the measured aberration are substituted into Eqs. (2) and (6) to calculate the practical value and lower bound of the SNR loss. The corresponding BEP values are computed from Eq. (12), and the PDF of the BEP is analyzed using a frequency histogram method. The differences in the BEPs of the lower bound and practical value for different $\sigma $ values were calculated, and the results are shown in Fig. 5.

The AO system analyzed in this work does not satisfy the requirements of coherent optical communication, i.e., the BEP is greater than 10^{−3} in all cases. The error in the BEP increases with an increase in the RMS of the residual aberration. If, in practice, the RMS of the residual aberration of the AO system reaches the diffraction limit, then the SNR loss due to uncompensated aberration is less than 1 dB, and the BEP error is less than 10^{−10}.

## 4. Availability of the AO system

One method of evaluating the effectiveness of an AO system is to compare the improvement in signal fade before and after aberration compensation, which allows an evaluation of the change in the mean BER of the communication link. However, the timescale of atmospheric fluctuations is about a million times larger than the symbol duration. Millions of bits may arrive continuously during the same period that a large residual aberration caused by imperfect correction of the AO system is present. Hence, there is some debate as to whether the mean BER is a satisfactory indicator.

A large residual aberration in the AO system is similar to a burst error, which can generate momentary outages in the data transfer and cannot be described by the mean BER [11, 12]. The concept of system availability was introduced to evaluate AO systems by computing the probability of the burst error. The availability value depends on the threshold, and generally this threshold is determined according to the link budget; for example, a signal fade larger than 0.5 or 3 dB after aberration compensation.

The availability and mean BER for different *b* and ${\sigma}_{z}$ values were calculated for SNR = 49.5 (Table 1), and communication link outages were defined when the instantaneous
BEP exceeded 10^{−6}.

The availability and mean BER were found to decrease dramatically as the RMS of the residual aberration increased. But a lower communication performance may be associated with smaller mean $\sigma $ values when the aberration contains a large amount of tilt. Thus, the mean value of $\sigma $ is not a suitable indicator of the AO system. The communication performance should be analyzed through the distribution function of $\sigma $, which depends on *b* and ${\sigma}_{z}$.

Burst errors due to insufficient AO system compensation have a different effect depending on the transmitted signal and data rate. The availability and mean BER describe the communication performance for different situations. If the communication link is part of an extended communication system, then the mean BER is a proper specification, but if the link requires strict frame synchronization, then a high availability is essential. Generally, the mean BER is used to specify the communication link performance along with the trade-offs of availability.

## 5. Discussion

The spatial and temporal characteristics of the residual aberration of a single-stage imaging AO system were studied, and the relationship between the AO system performance and the coherent optical communication performance metrics was presented. These results will contribute to the analysis and design of AO systems for coherent optical communication links.

The concepts used for evaluating the performance of conventional imaging AO systems, i.e., the diffraction limit and Strehl ratio, are not as suitable for communication AO systems. The distribution function of the residual aberration is necessary for calculating the BER and communication availability.

Although tilt is fairly easy to correct, residual tilt is a main part of the overall aberration for imaging AO systems. The tilt accounts for about 87% of the total aberration induced by atmospheric turbulence, and the larger stroke of the tilt mirror results in a lower response bandwidth that is insufficient for compensating the tilt to the level of other higher-order aberration. This is not unique to the AO system considered here but is a common characteristic of single-stage AO systems for astronomical observations. Hence, tip-tilt correction is of paramount importance in extreme AO technology.

The AO system is similar to a high-pass filter, and thus, it is difficult to compensate higher-order aberration to a low level. Therefore, the most effective way of improving AO-system performance is to correct the tilt, which can be compensated further by an additional tilt mirror loop with a lower stroke and higher bandwidth.

## Acknowledgments

We gratefully acknowledge the support of the Fundamental Research Funds for the Central Universities (No. ZYGX2011J123), Funds of the Adaptive Optics Laboratory, CAS, (No. LA0F201304), and Natural Science Foundation of China (No. 61308082).

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