Zernike mode analysis of an adaptive optics system for horizontal free-space laser communication

The standard practice for coupling power evaluation is the use CE, which is an important parameter in a FSLC system that determines the light power that can be detected in the fiber. The calculation of CE is based on the optical field distributions of the signal beam at the fiber end face and the optical fiber mode. The fundamental mode of power distribution in a fiber can be approximated within a 1% error in a Gaussian beam if the normalized frequency V of the optical fiber is in the range 1.9 < V < 2.4. Based on this condition, the Gaussian approximation can be used to describe the optical fiber mode field distribution characteristics. However, CE cannot accurately describe the energy in the actual fiber, especially for strong turbulence, and moreover, CE ignores the area of the laser-receiving end face, which leads to further inaccuracy in the approximation. PIB not only takes into account the optical field distribution of the beam, but also the size of the fiber end face when calculating the energy receiving efficiency. Thus, in this paper, we adopt the PIB as a more accurate evaluation methodology by which to calculate the coupling power [13].

As is shown in figure 1, an incident beam is focused on the surface of the coupling fiber with a coupling lens. The diameter and focal length of the coupling lens are denoted D and f, respectively. The radius of the coupling fiber is R. The entrance pupil and its focal planes are expressed in polar coordinates (ρ, θ) and (γ, ϕ), respectively.

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The power coupled into the single mode fiber is:

$$\begin{equation}{\text{PIB}} = \int\limits_S I \left( P \right){\text{d}}s\end{equation} \tag{ 1 }$$

where I(P) is the received light intensity of point P on the focal plane and $S$ is the end face area of the fiber. According to the Huygens–Fresnel principle, the light intensity of point $P$ can be described by [15]:

$$\begin{equation}I\left( P \right) = U\left[ {P\left( {\gamma ,\varphi } \right)} \right] \bullet U*\left[ {P\left( {\gamma ,\varphi } \right)} \right]\end{equation} \tag{ 2 }$$

where U[P(γ, ϕ)] represents the complex amplitude of point P:

$$\begin{align}U\left[ {P\left( {\gamma ,\varphi } \right)} \right] &= - \frac{{iA{D^2}}}{{4\lambda {f^{\,2}}}}\int\limits_0^1 {\int\limits_0^{2\pi } {\exp } } \left\{ {i\left[ {\phi \left( {\rho ,\theta } \right) - \frac{D}{{2f}}\rho \gamma \cos (\theta - \varphi } )\right]} \right\}\nonumber\\ &\quad\times\rho {\text{d}}\rho {\text{d}}\theta .\end{align} \tag{ 3 }$$

After substituting equations (2) and (3) into equation (1), the equation for the PIB becomes:

$$\begin{align} {\text{PIB}} &= \zeta \int\limits_s {\exp } \left\{ {i\left[ {\phi \left( {{\rho _1},{\theta _1}} \right) - \frac{D}{{2f}}{\rho _1}\gamma \cos ({\theta _1} - \varphi } )\right]} \right\}{\text{d}}{s_1}\nonumber \\ &\quad\bullet \int\limits_s {\exp } \left\{ {i\left[ { - \phi \left( {{\rho _2},{\theta _2}} \right) + \frac{D}{{2f}}{\rho _2}\gamma \cos ({\theta _2} - \varphi } )\right]} \right\}{\text{d}}{s_2} \nonumber \\ &= \zeta \int\limits_s {\int\limits_s {\exp } } [i\left( {{\phi _1} - {\phi _2}} \right)] \nonumber \\ &\quad\!\!\bullet \exp \left\{ {i\frac{D}{{2f}}\left[ {{\rho _2}\gamma \cos ({\theta _2} \!-\! \varphi } \right)\!\! -\!\! {\rho _1}\gamma \cos ({\theta _1} \!- \!\varphi )]} \right\}{\text{d}}{s_1}{\text{d}}{s_2} \end{align} \tag{ 4 }$$

where $\zeta = {(\frac{{A{D^2}}}{{4\lambda {f^2}}})^2}$, $\phi \left( {\rho ,\theta } \right)$ represents the phase of point (ρ, θ) on the entrance pupil.

Theoretically, PIB can be calculated by equation (4), however, it is challenging to analyze the influence of PIB caused by turbulence. According to Yin's study, the statistical average PIB can be simplified into [13]:

$$\begin{equation}\langle {\text{PIB}}\rangle = \zeta \int\limits_0^2 \left\langle{\int\limits_s {\exp } } [i\phi \left( {\boldsymbol{\rho }} \right)]{\exp ^*}[i\phi \left( {{\boldsymbol{\rho }} + l} \right)]{\text{d}}{\boldsymbol{\rho }}\right\rangle \times \frac{{2R{J_1}\left( {\pi Rl} \right)}}{l}{\text{dl}}.\end{equation} \tag{ 5 }$$

The formula above represents the autocorrelation of the phase in the entrance pupil, where J₁(x) is the first-order Bessel function and $l$ is the distance between points ρ₁ and ρ_2, Φ is the distorted wave front. Generally, the wave front distortion Φ can be described using Zernike modes. As the number of Zernike modes increases, the spatial frequency of the distortion also increases. The wave front distortion can be described by the linear accumulation of all Zernike modes $\phi = \mathop \sum \limits_{j = 1}^J {a_j}{Z_j}$. According to Robert J Noll's study, the low order Zernike modes occupy most of the wave front distortion, the higher the order of Zernike mode, the smaller the proportion of the wave front distortion [15]. Hence, the choice of corrected number of Zernike modes used in this study is from low order to high order based on Noll's theory. Thus, we can obtain the theoretical <PIB> with a measured wave front phase of $\phi$ after the first J Zernike modes are corrected. The normalized PIB may be written as:

$$\begin{equation}\langle{\text{PI}}{{\text{B}}_{{\text{norm}}}}\rangle = \frac{{\langle {\text{PIB}}\rangle }}{{{\text{Powe}}{{\text{r}}_{{\text{No}}{\text{Turbulence}}}}}} = \frac{{\langle {\text{PIB}}\rangle }}{{{A^2}\left( {\frac{{\pi {D^2}}}{4}} \right)}}.\end{equation} \tag{ 6 }$$

To calculate the mean PIB after correction of different Zernike modes by the AO system, it is essential to find the relationship between the mean PIB and the correct modes $J$. According to Noll's study [15], if the first J modes are corrected, the remainder of the wave-front distortion can be expressed as:

$$\begin{equation}\Delta = \int {\text{d}} \rho W\left( \rho \right)\langle{[\phi \left( {R\rho } \right) - {\phi _{\text{c}}}\left( {R\rho } \right)]^2}\rangle\end{equation} \tag{ 7 }$$

where $\phi \left( {R\rho } \right)$ and ${\phi _{\text{c}}}\left( {R\rho } \right)$ denote the distorted wave-front and the correction, respectively. Thus, we can calculate the mean square residual error:

$$\begin{equation}{\Delta _J} = \langle{\phi ^2}\rangle - \sum\limits_{j = 1}^J {\langle {{\left| {{a_j}} \right|}^2}\rangle .} \end{equation} \tag{ 8 }$$

The first ten values of ${\Delta _J}$ are shown in table 1.

Table 1. Residual error after correction of the first ten Zernike modes.

${\Delta _1} = 1.0299{\left( {D/{r_0}} \right)^{5/3}}$	${\Delta _6} = 0.0648{\left( {D/{r_0}} \right)^{5/3}}$
${\Delta _2} = 0.582{\left( {D/{r_0}} \right)^{5/3}}$	${\Delta _7} = 0.0587{\left( {D/{r_0}} \right)^{5/3}}$
${\Delta _3} = 0.134{\left( {D/{r_0}} \right)^{5/3}}$	${\Delta _8} = 0.0525{\left( {D/{r_0}} \right)^{5/3}}$
${\Delta _4} = 0.111{\left( {D/{r_0}} \right)^{5/3}}$	${\Delta _9} = 0.0463{\left( {D/{r_0}} \right)^{5/3}}$
${\Delta _5} = 0.0880{\left( {D/{r_0}} \right)^{5/3}}$	${\Delta _{10}} = 0.0401{\left( {D/{r_0}} \right)^{5/3}}$

When J is large (J > 10), equation (8) can be approximately by [15]:

$$\begin{equation}{\Delta _J} \approx 0.2944{J^{ - \sqrt 3 /2}}{(D/{r_0})^{5/3}}\left[ {{\text{ra}}{{\text{d}}^2}} \right]\,J > 10.\end{equation} \tag{ 9 }$$

Theoretically, we can calculate the PIB with equations (5) and (9) (or table 1), however due to ignoring the fitting error of the DM, the mean square residual error ${\Delta _J}$ deduced by Noll idealized result. Hence, to obtain an accurate result, we also must take into account the fitting error.

Considering the fitting error, Védrenne et al studied the mean square residual error of the first ${n_r}$ orders after correction by the AO system, which can be expressed by [16]:

$$\begin{equation}\sigma _{{\text{fitting}}}^2 = 0.458{({n_r} + 1)^{ - 5/3}}{\left(\frac{D}{{{r_0}}}\right)^{5/3}}.\end{equation} \tag{ 10 }$$

So, the mean square residual error after correcting the first ${n_r}$ orders is:

$$\begin{equation}{\sigma ^2} = {\Delta _J} + \sigma _{{\text{fitting}}}^2 = {\Delta _J} + 0.458{({n_r} + 1)^{ - 5/3}}{\left(\frac{D}{{{r_0}}}\right)^{5/3}}.\end{equation} \tag{ 11 }$$

This equation considers both the spatial distribution characteristics of atmospheric distortion and the fitting error of the DM, which can more accurately describe the effect of AO correction. Theoretically, based on equations (5) and (11), we can get the PIB after correction of the first J Zernike modes. The choice of AO parameters is based on the communication performance, which is described by BER. Different application scenarios have different requirements for BER. For example, video communication requires BER to be lower than 10⁻⁷, audio requires a minimum BER of 10⁻⁶ and a value of 10⁻⁵ is required for basic communication performance. It is, therefore, crucial to establish the relationship between BER and PIB, so that, we can obtain the relationship between the corrected number J of Zernike modes and BER.

To establish the connection between AO parameters J and communication performance, the BER is essential. The average BER can be obtained from the communication energy, thus the BER can be calculated from the average PIB. If we assume the 'zero' bit and 'one' bit are uncorrelated, the optimal reception BER of homenergic binary code can be expressed as:

$$\begin{equation}{\text{BER}} = \frac{1}{2}{\text{erfc}}\left( {\frac{{\sqrt {{\text{SNR}}} }}{2}} \right) = \frac{1}{2}{\text{erfc}}\left( {\sqrt {\frac{{{\text{PIB}}}}{{2{n_0}}}} } \right)\end{equation} \tag{ 12 }$$

where SNR is the signal-to-noise ratio, ${n_0}$ is the energy of noise, and erfc() is the complementary error function. Which can be written as:

$$\begin{equation}{\text{erfc}}\left( x \right) = 1 - \frac{2}{{\sqrt \pi }}\int\limits_0^x {\exp } \left( { - {z^2}} \right){\text{d}}z.\end{equation} \tag{ 13 }$$

For an FSLC system, we can determine the effect of turbulence on the BER. By combining equation (5) with equation (12), we can obtain the relationship between BER and turbulence, then shows that turbulence increases BER by decreasing communication energy PIB, and the noise n₀ remains unchanged. The BER of fiber optical communication is approximately 10⁻⁹, and PIB/2n₀ is approximately 36 according to equations (12) and (13). Thus, when using the fiber to receive the laser, the BER as a function of PIB may be written as [13]:

$$\begin{equation}\langle {\text{BER}}\rangle \!=\! \frac{1}{2}{\text{erfc}}\left( {\sqrt {\frac{{36\langle{\text{PI}}{{\text{B}}_{{\text{norm}}}}\rangle}}{2}} } \right) \!=\! \frac{1}{2}{\text{erfc}}\left( {\sqrt {18\langle{\text{PI}}{{\text{B}}_{{\text{norm}}}}\rangle} } \right)\!.\end{equation} \tag{ 14 }$$

The BER represented by logarithm is near linear to normalized PIB, so the following approximate formula can be used:

$$\begin{equation}N = - 8.234{\text{PI}}{{\text{B}}_{{\text{norm}}}} - 0.8076,\end{equation} \tag{ 15 }$$

where N is the index of BER, i.e.:

$$\begin{equation}{\text{BER}} = {10^N}.\end{equation} \tag{ 16 }$$

Via equation (16), the BER value can be calculated after the first J Zernike modes are corrected by the AO system. Generally, when the BER is lower than 10⁻⁵, the basic communication performance can be maintained, hence, the PIB should be larger than 50% according to equation (14).

As shown in figure 2, the FSLC experiment was conducted with an AO system from one mountaintop to another mountaintop via a horizontal optical link. An 808 nm laser used as a beacon beam and a 1550 nm laser used as a signal beam are emitted by a transmitting terminal from a mountaintop and are received at a receiving terminal located on another mountaintop. The optical link spans a range of 8.9 km at an average altitude of about 110–202 m above sea level.

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Figure 3(a) shows the schematic diagram and the optical setup of the AO system for FSLC. The lasers of 1550 and 808 nm are used as signal and beacon lights, respectively, and are first received by a telescope with an aperture of 150 mm before entering the AO system. The beam is collimated and reflected by a tip–tilt mirror (TTM) and a DM prior to entering beam splitter SW1, which is designed to reflect 80% of incident 808 nm beams and transmit light of 1550 nm and 20% of light of 808 nm. The reflected light of wavelength 808 nm enters the S–H WFS for wave front detection, and the transmitted light is split again by beam splitter SW2 for TTM tracking. The transmitted 1550 nm signal light is coupled into a fiber, and the reflected light with a wavelength of 808 nm is received by a tracking camera. In this AO system, a 145-element continuous surface DM is used to correct high-order aberration. The diameter of the DM is 25 mm, the stroke of the actuators is 3 μm, and the number of effective actuators is 97. The distorted wave-front is detected by a 15 × 15 S–H WFS with a high speed camera at 1.6 kHz frame frequency with a pixel size of 25 μm. The diameter of the S–H WFS is 2.25 mm. The aperture and focal length of the micro-lens are 3.19 mm and 150 μm, respectively. The TTM (FSM-300, Newport) with a tip–tilt angle of ±1.5° and a 25.4 mm aperture is used to correct tip–tilt aberrations. A tracking camera (EoSens GE, Mikrotron) is used to obtain the tracking error with a region of interest area of 90 × 90 pixels and a frame frequency of 500 Hz. Figure 3(b) shows the actual optical configuration of the designed AO system.

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In this study, we made a series of experiments under different atmospheric coherence lengths (r0) and conducted the experiment at noon, nightfall and night, at which the representative atmospheric coherence lengths are 1.70, 4.43, and 7.51 cm respectively. In this paper, we corrected the first 65 Zernike modes to test the performance of our AO system.

As is shown in figure 4, we calculated and obtained the theoretical and experimental RMS of the distorted wave front after the first J Zernike modes were corrected. By comparing the theoretical and experimental RMS curves, we can find that the theoretical curve dropped to an ideal level (RMS = 0.05 rad) after correcting the tip and tilt. Experimentally, for the conditions of r0 = 4.43 cm and r0 = 7.51 cm, after correcting the first 11 Zernike modes, the measured RMS value dropped to about 0.05 rad and maintained a stable state. As turbulence becomes stronger (r0 = 1.7 cm), the experimental RMS remains large (about 0.35 rad) even after correcting the first 65 Zernike modes.

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According to equation (6), we can calculate the PIB after correcting the first J Zernike modes with a measured wave front. Figure 5 indicated the relationship between PIB (theoretical and actual) and corrected Zernike modes. Theoretically, the PIB can reach about 70% for r0 = 4.43 cm and r0 = 7.51 cm after correcting the first 22 Zernike modes. When r0 = 1.7 cm, the PIB reached 60% after correcting the first 22 Zernike modes. In the experimental measurements, due to the influence of turbulence, the PIB reached 70% after correcting the first 30 Zernike modes for r0 = 7.51 cm. When r0 = 4.43 cm, the PIB reached and maintained about 55% after correcting the first 22 Zernike modes. For stronger turbulence (r0 = 1.70 cm), the PIB value remained at about 20%, even after correcting more Zernike modes.

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According to equation (13), we obtained the BER after correcting the first J Zernike modes for different r0 values. As shown in figure 6, theoretically, the BER can drop about to 10⁻⁶ after correcting the first 22 Zernike modes for r0 = 1.70 cm, 4.43 and 7.51 cm. In our experimental data, for r0 = 7.51 cm, the BER dropped to around 10⁻⁷ after correcting the first 30 Zernike modes. For r0 = 4.43 cm, the BER dropped to below 10⁻⁵ after correcting the first 22 Zernike modes, and for r0 = 1.70 cm, the BER is above 10⁻³ even after correcting more Zernike modes.

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In figure 7, the experimental images recorded by S–H WFS are shown. The number of corrected Zernike modes is fixed at 65 for different conditions of r0. For r0 = 4.43 and r0 = 7.51, all the light spots in the WFS can be distinguished after AO correction and the brightness is significantly increased, suggesting that the wave front was accurately measured and that AO correction had a positive impact on image quality. For r0 = 1.70 cm, the light spots in the WFS image are severely distorted. Although there are obvious improvements in the imaging of light spots after AO correction, only parts of the light spots can be distinguished. The performance of our AO system is poor under this condition.

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In figures 4–6, there are differences observed between the theoretical curves and experimental curves. There are several possible reasons for these differences. First, there are wave front detection errors, unlike astronomical observations, FSLC has strong turbulence scintillation, and the accuracy of wave front reconstruction needs to be improved, as also shown in figure 7. More advanced wave front detection technology needs to be further studied [17]. Second, the finite bandwidth of the DM will lead to time delay errors [18]. Generally, the turbulence changes faster under strong turbulence, and the Greenwood frequency will reach more than 50 Hz, a higher bandwidth corrector must be used to ensure real-time compensation. Moreover, there are fitting errors during the DM correction, and, thus, thus, a high-order large-stroke corrector is needed, the control algorithm also needs to be further optimized. Therefore, the high-speed high-order large-stroke corrector should be added for strong turbulence, such as spatial light modulator or digital mirror device [19]. Although the theoretical and experimental results show deviations, these deviations are acceptable and the trend of change is consistent. The results show that with the increase of corrected modes, the PIB increases, and the RMS of the wave front and the BER decrease. Similarly, with increasing r0, the PIB increases, and the RMS of the wave front and the BER decrease. These results indicate that communication performance can be improved after AO correction. For example, when r0 = 7.51 cm, after correcting the first 30 Zernike modes, the PIB and BER reached about 70% and 10⁻⁷, respectively. When r0 = 4.43 cm, after the first 22 Zernike modes were corrected, the PIB and BER reached 55% and 10⁻⁵, respectively. For r0 = 1.70 cm, the PIB and BER reached and maintained values of about 20% and 10⁻³, respectively, even after correcting more Zernike modes, which can be explained through a combination of factors. First, the measurement accuracy of the wave front is poor when the light spots are severely distorted, as shown in figure 7. Second, the finite bandwidth and stroke of the actuators of the DM cannot meet the requirement of strong turbulence [18]. In addition, there are fitting errors during DM correction, which require further optimization of the control algorithm to overcome. Hence, the PIB and BER show no improvement even after correcting more Zernike modes under the r0 = 1.7 cm condition.

Our experimental results indicate that for the AO system in this study, under turbulence conditions of r0 = 4.43 and 7.51 cm, correction of the first 22 and 30 Zernike modes, respectively, is enough to achieve basic and ideal communication performance. However, when turbulence becomes stronger, (e.g. r0 = 1.70 cm), it is difficult to achieve basic communication performance. Nevertheless, this experiment can prove the correctness of the theoretical formula proposed in this paper, and can provide theoretical guidance for the design of AO system in FSLC. In order to correct stronger turbulence, the AO system requires further study of weak signal wave front detection and calculation algorithms, in addition to high-performance correctors and control methods in the future.