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We explore adaptive optics (AO) pre-compensation for optical communication between Earth and geostationary (GEO) satellites in a laboratory experiment. Thus, we built a rapid control prototyping breadboard with an adjustable point-ahead angle where downlink and uplink can operate both at 1064 nm and 1550 nm wavelength. With our real-time system, beam wander resulting from artificial turbulence was reduced such that the beam hits the satellite at least 66% of the time as compared to merely 3% without correction. A seven-fold increase of the average Strehl ratio to (28 ± 15)% at 18 μrad point-ahead angle leads to a considerable reduction of the calculated fading probability. These results make AO pre-compensation a viable technique to enhance Earth-to-GEO optical communication.
© 2016 Optical Society of America
1. Introduction
Means of adaptive optics (AO) have been successfully applied to mitigate atmospheric turbulence and, in doing so, increase the efficiency of Earth-to-space optical communication downlinks [1–4]. In particular, theoretical studies indicate a decrease of bit-error-rate (BER) with increasing AO correction efficiency commonly quantified by the Strehl ratio [5]. There are two possible configurations for AO compensation: post-compensation corrects the beam after passing through the atmosphere and pre-compensation corrects before distortions from turbulence take place. In a transceiver, the received and transmitted beam can be simultaneously post- and pre-compensated if they pass through the same AO-assisted telescope. Depending on the link geometry, it can be beneficial to apply AO correction at one or both communication partners [6].
While post-compensation has been widely demonstrated in experiments [1–4], pre-compensation is less common. Pre-compensation is based on the reciprocity of optical propagation through atmospheric turbulence [7, 8]. Reciprocity basically states that the Green’s function for propagation between two planes is symmetric, assuming that turbulence does not change during propagation. As a result, a light field traveling from one plane to another will be exactly recovered when the beam received at the second plane is propagated back to the initial plane.
The idea of pre-compensation is to exactly reproduce the received field and propagate it back to the transmitter such that the undistorted profile is retrieved after passing through the atmosphere. However, pre-compensation performance was predicted to decrease with increasing point-ahead angle because of the different beam paths [9, 10]. There is not much experimental data on pre-compensation in optical communication [11,12] and all studies are limited to static laboratory setups.
In this article, we will report on laboratory experiments exploring the efficiency of dynamic AO pre-compensation based on a real-time architecture which additionally takes the point-ahead angle into account. We chose Earth-to-GEO optical communication to be our baseline scenario which is particularly interesting for optical feeder links [13]. Transmitter diversity has been proposed to stabilize such links but increases the overall laser power which is limited by eye-safety requirements [14]. A theoretical study, which focused on encoding techniques for Earth-to-GEO communication with AO pre-distortion, anticipates data rates of up to 1 Tbit/s [15]. It should be noted that pre-compensation needs to be performed on ground because AO correction at the satellite is physically impossible. Thus, we propose AO pre-compensation at the optical ground station which uses the downlink beam emitted by the satellite to sense wavefront distortions.
For this purpose, we designed, implemented, and tested a breadboard that closely mimics the geometry of our baseline scenario. Artificial turbulence is introduced by a single phase screen according to the Kolmogorov theory and fitting the local seeing conditions of our baseline scenario. Our AO system consists of a Shack-Hartmann wavefront sensor to detect atmospheric distortions as well as a tip/tilt mirror and a deformable mirror to correct for the measured distortions. The setup operates at two wavelengths, λ = 1550 nm for the uplink and λ = 1064 nm for the downlink, as these wavelengths represent the most promising options for future applications. The wavelength is also used to separate both beams and to avoid blinding of the downlink wavefront sensor by back-reflections of the uplink beam. To account for the low intensity of the downlink beam in the real scenario, we built a custom wavefront sensor working at low incoming power. The wavefront sensor has a high sensitivity at both wavelengths and a sampling rate in the kHz range. Furthermore, a custom deformable mirror matching the requirements of the baseline scenario was realized. The control system was established following a rapid-control-prototyping approach. In addition, the point-ahead angle is adjustable such that its influence on system performance can be measured.
The outline of the paper is as follows: Section 2 describes the design of the system, from the baseline scenario and local seeing conditions to the optical design. Additionally, the implementation of the point-ahead angle is explained in detail. In Section 3 the individual components of the breadboard and their performance are specified. The measurement procedure and data evaluation are detailed in Section 4. The results are presented in Section 5 where we particularly examine the influence of the point-ahead angle on AO system performance and the fading probability. Finally, the results are discussed and summarized in Section 6.
2. System development
2.1. Baseline scenario
The application of our breadboard is AO pre-compensation in Earth-to-GEO optical communication as schematically shown in Fig. 1 (a). A GEO satellite sends a downlink beam to the optical ground station (OGS) which is equipped with an AO system. The AO system measures the wavefront of the received downlink beam, pre-compensates the uplink beam and sends the latter back to the satellite. Thanks to pre-compensation, the uplink beam’s profile at the satellite is improved as compared to the case without AO. To define the AO system parameters, we adopted technical data of a representative ground station and satellite which are already operational. For the GEO satellite we used Alphasat’s technical data [16] such as the telescope diameter of 13.5 cm, downlink wavelength of 1064 nm and approximate output power of 1 W. The ground station is assumed to be ESA’s 1 m telescope on Tenerife, Canary Islands, which emits an uplink laser beam at 1550 nm wavelength [17]. In addition, the OGS has to incorporate a point-ahead angle of approximately 18 μrad to account for the relative velocity of the satellite and the ground station.
Fig. 1 (a) Baseline scenario: a satellite sends a downlink beam which is used at the optical ground station (OGS) to measure wavefront distortions caused by turbulence. The OGS sends an uplink beam pre-compensated by the AO system back to the satellite. The OGS needs to incorporate a point-ahead angle (PAA). (b) Definition of the Strehl ratio S: ratio of the central irradiance of the distorted beam (red dashed curve) Idistorted and the central irradiance of the diffraction-limited beam (blue curve) Iideal.
2.2. Local seeing conditions
Apart from the technical data of OGS and satellite, the local seeing conditions play an important role in system development. Our figure of merit in system design will be the Strehl ratio S which is often used to quantify the impact of aberrations on optical systems. It can take values between 0% and 100%, where 100% is the ideal, unperturbed value. The Strehl ratio at the receiver is defined as the central irradiance value of the distorted beam divided by the central irradiance of the diffraction-limited profile [18, p. 408]
Please note that our definition of the Strehl ratio includes higher and lower order (tip/tilt) aberrations as illustrated in Fig. 1(b). With this definition of the Strehl ratio, we immediately find the local irradiance IGEO of the distorted uplink beam at the GEO satellite using only the diffraction-limited peak irradiance I0,GEO at the GEO orbit: For a telescope with a clear circular aperture telescope of diameter D, the diffraction-limited irradiance I0 can be calculated according to [19]: where P0 is the output power of the uplink laser beam at wavelength λ, R ≈ 36 · 106 m the altitude of the GEO satellite, and T ≈ 0.7 the transmission of the atmosphere. From the OGS diameter of DTel = 1 m and from the wavelength of the uplink beam equal to 1550 nm, we can estimate I0,GEO at the GEO satellite to be 176 μW/m2. Further, we can use Alphasat’s telescope diameter DGEO = 13.5 cm and the downlink wavelength of 1064 nm with Eq. (3) to find the diffraction-limited downlink irradiance at the OGS to be 6.8 μW/m2, and thus a total received power of 5.4μW. Therefore, the wavefront sensor needs to have a high sensitivity.For relatively low wavefront distortions, the Strehl ratio can be approximated using the residual mean-squared wavefront error σ2 [19]:
Three characteristic quantities describing turbulence determine the achievable Strehl ratio of our system as well as the requirements on its individual components: the Fried parameter r0, the isoplanatic angle θ0, and the coherence time τ0. The Fried parameter is the characteristic length scale of atmospheric phase distortions. The ratio of the telescope diameter and Fried parameter DTel/r0 influences the required number of actuators and the stroke of the tip/tilt and deformable mirror. Additionally, it determines the required wavefront sensor accuracy and dynamic range. The isoplanatic angle represents the maximum angular separation of two beams such that they experience almost the same turbulence. As a result, pre-compensation cannot work if the point-ahead angle is much larger than the isoplanatic angle. The temporal behavior of turbulence is quantified by the coherence time or the Greenwood frequency fG = 0.134/τ0 [20] from which we determine the control loop frequency. The local night-time seeing conditions for the OGS on Tenerife were determined by long-term SCIDAR measurements of the refractive index structure constant height profile from which the aforementioned turbulence parameters were calculated [21]. We scaled these values of r0, θ0, and τ0 with the wavelength and zenith angle ξ = 56° using [18]:
Table 1 summarizes the calculated values. From the value of the isoplanatic angle at the uplink wavelength of 1550 nm, (24.7 ± 10.1) μrad, we conclude that pre-compensation can be applied in our baseline scenario because it is of the same order as the point-ahead angle (18 μrad). As already mentioned, the scaled values of r0 and τ0 were used to estimate the required AO system parameters according to [19]: 15 to 36 deformable mirror actuators with a stroke of at least 1.2–2.9 μm, 14 × 14 micro lenses, and a disturbance rejection with at least 50 Hz in the closed loop system. In the real application scenario, the system should be available both during night and daytime. Unfortunately, there are very few publications on the local seeing conditions by day which are generally stronger than by night. To account for this fact, we have incorporated slightly stronger turbulence in our artificial turbulence generator than given in Table 1 which will be discussed in detail in Section 3.4.
Table 1. Local Seeing Conditions Derived from Long-term Measurements Reported in [21]
2.3. Translation of baseline scenario into a breadboard setup
The main idea of our breadboard setup is to mimic the actual link geometry as closely as possible while fitting its size to a laboratory scale. Our system is designed to be compact and could be installed in the OGS e.g. at the Coudé focus. When scaling such a system, it is important that the ratio of the telescope diameter and the Fried parameter DTel/r0 remains constant. In other words, if the system aperture is reduced, so is the Fried parameter. As will be discussed in Section 2.5, the downlink beam is collimated such that an almost plane wave with a homogeneous intensity profile enters the system (before the atmosphere), as one would expect from the downlink beam coming from a GEO satellite.
2.4. Influence of the point-ahead angle
It should be noted that with our system parameters the point-ahead angle contributes the most to the residual wavefront error after AO pre-compensation. Fried investigated the performance of an AO system under the influence of a point-ahead angle θ in detail [9]. In our setup, we use a single phase element, the aberration emulator, to introduce atmospheric turbulence. The wavefront error caused by the point-ahead angle [9]
is translated into a lateral displacement Δ on the phase element between the uplink and the downlink beam. We need to ensure that the wavefront error imparted by the displacement Δ [9] is equal to . By combining Eqs. (8) and (9), we find the relation between the point-ahead angle and the beam displacement Δ Note that Δ needs to be scaled by the miniaturization factor of our breadboard DAE/DTel as described in Section 2.3 to obtain the displacement in the breadboard setup ΔAE given by: We acknowledge that this single phase screen model represents a simplified approach to describing turbulence. In a more realistic simulation, several phase screens should be used, the positions and Fried parameters of which should be chosen according to the local refractive index structure constant profile.2.5. Optical design
The goal of our breadboard setup is to simulate the baseline scenario in a laboratory environment. For this reason, the corresponding optical design shown in Fig. 2 contains not only the AO system but also optical elements accounting both for turbulence and the laser propagation between OGS and satellite.
Fig. 2 Sketch of the optical design layout: The downlink (red line) is collimated, passes the aberration emulator and enters the system at the aperture stop. After passing the tip/tilt and deformable mirror, it is split into two parts for wavefront sensing and possibly communication purposes. The uplink beam (blue lines) passes the system in the opposite direction starting from the fiber and ending at the uplink camera. When the uplink fiber is shifted, the beam leaves the system at a different point-ahead angle (dark blue dotted line).
The AO system shown in Fig. 2 consists of a wavefront sensor, deformable mirror, and tip/tilt mirror which are all positioned in planes conjugate to the entrance aperture. To this end, three subsequent telescopes re-image the aperture stop plane onto the aforementioned components. Since downlink and uplink operate at different wavelengths, achromatic lenses were used to minimize the shift of the conjugate planes caused by chromatic aberrations. In fact, the optical system provides diffraction-limited performance at both wavelengths. The beam diameters, which are given by the telescopes’ magnifications, were chosen to best fit the apertures of the optical components and are listed in Table 2. A rotating diffractive optical phase element, the aberration emulator (AE), introduces predefined dynamic wavefront distortions that simulate the effect of atmospheric turbulence within the breadboard.
Table 2. Beam Diameters at Respective Optical Elements
A fiber-coupled 1064 nm laser source (Thorlabs BF-A64-0180-P5A) provides the downlink signal (red line in Fig. 2). The fiber output passes a dichroic mirror and is collimated by the lens L1. The large respective F-number of 30 ensures that only a small central part of the source output enters the system in order to provide a homogeneous irradiance distribution with negligible aberrations, similar to the downlink beam coming from the satellite. After collimation, the downlink beam passes the AE which introduces turbulence distortions. Then, the downlink enters the AO system at the aperture stop and is reimaged to the tip/tilt mirror, the deformable mirror, and finally to the wavefront sensor. The measured wavefronts are used to control the deformable and tip/tilt mirror. In fact, not all of the downlink light enters the wavefront sensor. A beam splitter separates the downlink beam into two parts - the first for wavefront measurements and the second for communication purposes, e.g. to couple into a fiber.
The uplink signal passes the system in reversed order as compared to the downlink (light and dark blue line in Fig. 2). The fiber output from a tunable laser source at 1550 nm (New Focus 6529-LN) is collimated by the lens L2 and reflected by the deformable and tip/tilt mirror which pre-compensate the beam. The pre-distorted beam leaves the system through the entrance aperture and is “compensated” by the AE. After leaving the AE it should have a plane wavefront thanks to pre-compensation. The combination of the lenses L1 and L4 focuses the uplink beam onto an IR camera (Xenics Xeva). The large respective focal length of 1.7 m allows for recording the far field irradiance distribution with adequate spatial sampling.
An essential goal of the breadboard is the simulation of the point-ahead angle θ in order to investigate its effect on uplink pre-compensation efficiency. Within our setup, atmospheric wavefront distortions are introduced by the AE, a thin phase plate. As discussed in Section 2.4, the different paths for the uplink and the downlink beam can only be realized by displacing their positions at the AE by ΔAE, see Eq. (11). To this end, we shift the uplink fiber laterally (blue dotted line). As a result, the uplink beam passes the aperture stop at exactly the same position as the downlink beam but with a different angle of incidence - the point-ahead angle. If we were to place the AE immediately at the aperture stop, the beams would not have a displacement but merely a different angle of incidence. To achieve the desired displacement ΔAE, we position the AE at a distance from the aperture plane. Figure 2 illustrates how a shifted uplink laser position results in a displaced uplink beam at the satellite. It should be emphasized that the lateral beam position at the tip/tilt and the deformable mirror is not affected by a shift of the uplink fiber thanks to their conjugate location with respect to the aperture stop.
3. Breadboard setup
3.1. Wavefront sensor
The Shack-Hartmann wavefront sensor (WFS) of our demonstration breadboard was specially designed and assembled to meet the requirements of the baseline scenario. In particular, we needed an infrared detector with windowing capability, high frame rate (up to 1 kHz), high quantum efficiency at 1064 nm and 1550 nm, and relatively low noise level. The appropriate detector was selected based on a numerical simulation of the precision of the centroid determination; photon shot-noise and pixel read-out noise were included within the frame of this model. We chose the Xenics Bobcat-640 CL camera because it achieved a centroid precision of about 30 μrad when matched with a micro-lens array with F-number 35 and 200 μm pitch (Flexible Optical B.V., Netherlands). With this WFS configuration, the maximum detectable tilt angle of our designed WFS was estimated to be 9 mrad which is much larger than the expected peak-to-valley wavefront tilt of 2.1 mrad. Simulations and subsequent laboratory experiments confirmed that the best centroid precision was achieved by operating the camera in low gain mode with an input irradiance close to the pixel saturation level. This corresponds to a total incoming power of 2.5 μW which represents about 50% of the estimated downlink power collected by the OGS, see also Section 2.2. Considering that additional losses occur in the optical train of the telescope, this configuration would leave a small margin for the downlink communication channel. We notice however that the use of a significantly more cost-intensive, state-of-the-art cooled, scientific-grade infrared detector would improve the sensitivity of the WFS. Alternatively, a higher output power from the GEO satellite would be required to widen the communication link margin (up to 2.2 W output are technically possible from Alphasat [22]).
3.2. Deformable mirror
To apply correction for higher-order modes, we designed and manufactured a 40-actuator unimorph deformable mirror that consists of a glass substrate (B33) and a piezoelectric disc (PZT), both with a diameter of 50 mm. The PZT-disc (PIC 151, PI Ceramic GmbH, Germany) has one ground electrode facing the glass substrate and 40 signal electrodes on the rear side. The mirror was mounted onto a rigid mirror base using 20 flexible mounting elements, so-called compliant cylinders, that were equally spread on a circle as shown in Fig. 3(a). In particular, the cylinders serve for both the mechanical mounting of the mirror and the electrical contacting of the ground electrode. See [23] for more details on this mounting procedure. We want to point out that the mirror base and the mechanical holder had to be carefully designed to prevent vibration-induced deformations of the mirror. Figure 3(a) shows the mirror’s actuator layout and the position of the compliant cylinders. The mirror offers 40 actuator patches, 24 of which are located inside the aperture. The layout features three characteristic actuators that were characterized with the WFS in the aforementioned breadboard. The inner eight actuators achieve a peak-to-valley (PtV) wavefront deformation of (3.28 ± 0.16) μm, the 16 actuators of the first ring offer PtV=(3.98 ± 0.16) μm and the outer ring imposes a PtV=(3.46 ± 0.30) μm deformation within the mirror aperture. The actuators were activated by an electrical field of 0.75 kV/mm which was restricted by the high-voltage supply. Figure 3(b) depicts the finished mirror in its mount.
Fig. 3 Sketch of the deformable mirror actuator layout (a) with aperture (red) and compliant cylinders to mount the mirror on a baseplate. A photograph of the manufactured mirror is shown in (b).
3.3. Tip/tilt mirror
To correct for tip/tilt, a commercial actuator (S330.2SL, Physik Instrumente GmbH, Germany) suited to our baseline scenario was chosen and equipped with a commercial high-quality mirror. The internal servo control loop was used to compensate for inherent hysteresis, creep, and piston movement. In addition, we designed a custom mount to prevent dynamic exaggeration and to further damp any resonances for frequencies up to 1.2 kHz. In doing so, we reduced the rise time of the step response function to 100 μs over the entire actuation rage.
3.4. Aberration emulator
In order to introduce artificial turbulence in our setup, we designed and manufactured an aberration emulator. A random surface profile according to Kolmogorov turbulence theory was numerically generated with the Fried parameter derived from the local seeing conditions, see Section 2.2. Note that the Fried parameter on the AE needed to be scaled to account for the reduced aperture as compared to the 1 m telescope of the OGS, see Section 2.3. The generated profile was lithographically wet-etched to a circular fused silica plate. We specifically chose fused silica because of its low dispersion over the desired wavelength range.
To incorporate a variety of turbulence conditions, we divided the surface profile into three rings with different Fried parameters – derived from the average value given in Table 1 as well as the average value minus one or two standard deviations. However, we only report on results obtained with the ring that introduces the strongest turbulence. We characterized the manufactured plate with help of our WFS (see Section 3.1) by measuring the phase structure function and the Zernike decomposition of the generated wavefront. The fitted Fried parameter was (1.20 ± 0.16) mm at 1064 nm wavelength on the AE which leads to a value of DTel/r0 = 7.0±0.9 at 1064 nm and DTel/r0 = 4.5±0.6 at 1550 nm. As compared to our baseline scenario with DTel/r0 = 3.4±0.8 at 1064 nm, the turbulence conditions of the AE are more severe which ensures that the system works even under the worst night-time turbulence conditions.
By rotating the AE around its axis, an appropriate temporal change is introduced. As already discussed, we displace the position of uplink and downlink beam on the AE to introduce a point-ahead angle.
3.5. Control loop
The developed breadboard enables a rapid control prototyping approach for agile testing and optimization of the control parameters of both the deformable mirror and the tip/tilt mirror. Our control system consists of the following components: wavefront sensor, deformable mirror, tip/tilt mirror, FPGA-card, and real-time computer.
Our rapid control prototyping architecture was introduced in [24] and evaluated in [25]. However, the Xenics Bobcat-640 CL camera substituted the previously used camera and had to be included in the FPGA environment. Furthermore, the FPGA programming was adapted to implement the real-time spot detection and spot ordering algorithms that were derived in [26]. In fact, the calculation of the spot positions on the FPGA and the transmission via PCI Express to the real-time computer is completed 1μs after the last data from the CameraLink interface is received. Therefore, the time needed for data transmission and spot detection is kept very low. The FPGA-card was integrated into a real-time Linux control computer. The computer uses the spots evaluated by the FPGA-card to calculate the required voltages at the deformable mirror, and tip/tilt mirror and sends these commands to the high-voltage amplifiers.
To compensate for the actuator coupling of the deformable mirror, we captured an actuator influence function by applying a certain voltage vDM to one actuator and measuring the spot deflection at the wavefront sensor. By repeating this procedure for each actuator of the deformable mirror, we obtained a linear mapping ϕ = MDM vDM from vDM, the voltages at the mirror, to ϕ, the slopes at the wavefront-sensor. In a next step, we applied this procedure for the two actuators of the tip-tilt mirror, obtaining a second mapping MTT. These two non-square matrices can be inverted by using the Moore-Penrose pseudoinverse M+, allowing us to calculate the needed voltages to achieve a certain spot deflection. In order to decouple the two mirrors, we used the scheme displayed in Fig. 4. The slopes obtained from the wavefront sensor are converted into the coordinates of the tip/tilt mirror and passed to the corresponding controller. Additionally, we subtract the tip/tilt modes from the slopes used for calculating the DM voltages in order to achieve a decoupling between the two mirrors.
Consequently, we may design the controllers for each channel individually by using the following model
where s denotes the Laplace variable and GWFS(s) the transfer function for the wavefront-sensor and GDM(s) for the deformable mirror accordingly. The wavefront sensor is represented by a time-delay of one sample period of 1ms which is mainly caused by the image transmission. For the model of the deformable mirror, we applied a voltage step to one actuator and fitted a second-order transfer function. The Bode plot for the overall system G(s) can be seen in Fig. 5 denoted as solid blue line.Fig. 5 Bode plot: the solid line denotes the frequency response of the mirror in open loop configuration Sd(s). The disturbance rejection G(s) is displayed as dash-dotted line, showing the ability of the closed-loop to attenuate disturbances up to approximately 72Hz.
In the next step, we designed a PI-Controller to minimize the effect of disturbances which can be represented by the following transfer function
where C(s) represents the transfer function of the controller. The controller parameters were calculated using the Control System ToolboxTM from MATLAB with the goal of maximizing the frequency of the disturbance rejection. The resulting disturbance sensitivity function Sd(s) can be seen as a red dashed line in Fig. 5. This sensitivity function displays the ability of the controller to attenuate disturbances. An amplitude below one means that the effect of the disturbance is reduced. From the Bode plot in Fig. 5, we conclude that disturbances with a frequency up to ≈ 72Hz will be attenuated. This means that we are able to fulfill the specification given in Section 2.2.4. Measurements
4.1. Measurement procedure
The measurements reported in the following section were taken for our worst case turbulence conditions that correspond to more severe turbulence than derived in Section 2.2 with DTel/r0 = 4.5 ± 0.6 at 1550 nm and a wind speed of 10 m/s, see also Table 1. For the isoplanatic angle, we assumed a value of 13.6 μrad, measured at the uplink wavelength of 1550 nm and zenith angle of 56° which corresponds to the average value in Table 1 minus one standard deviation. We present the results of the uplink beam performance taken with the following measurement routine: first a dark image of the downlink wavefront sensor camera was recorded and all spot detection parameters were set (e.g. power threshold of microlens and pixel). Then, the actuator influence function was recorded both for deformable and tip/tilt mirror. Afterwards, a reference wavefront was recorded while the tip/tilt mirror was set to its central position and no voltage was applied at the deformable mirror. Next, the aberration emulator was installed and the controller applied. We recorded images with the uplink far field camera at a low frame rate as well as the centroids measured by the downlink wavefront sensor. By displacing the uplink beam on the aberration emulator, we adjusted the point-ahead angle.
4.2. Data evaluation
Data processing was performed off-line by analyzing the recorded image sequences of the uplink far field. In the first step, the total power of each image (i.e. the sum of the measured values of each pixel) was used to calculate the expected peak irradiance of a diffraction-limited beam spot, which was assumed to have the shape of the Airy pattern. The measured image was normalized by the diffraction-limited peak irradiance. In the second step, the pointing direction of the measured beam spot was calculated from the center of mass for each image. As a result, we obtained a time sequence of pointing directions which were used to quantify the residual beam wandering. In the last step, we calculated the average beam position from the measured time sequence of centroids. This position determines the reference axis of the optical system at which the Strehl ratio is measured. Indeed, the time sequence of the Strehl ratio is simply obtained from the normalized irradiance of the pixel located at the average beam position of the individual images.
5. Results
To evaluate the capabilities of our AO system, we analyzed the far field of the uplink beam as described in Section 4.2. By displacing the uplink and downlink beam on the aberration emulator, we were able to scan the point-ahead angle and, by that, investigate its influence on the compensation efficiency of our system. The displacement could be applied either parallel or anti-parallel to the artificial wind caused by rotating the AE. The maximum point-ahead angle was approximately twice as large as in our baseline scenario and was only limited by the vignetting of our optical setup. In the following, we will report on the improvement achieved with our system either without a point-ahead angle or at the point-ahead angle for Earth-to-GEO communication θGEO. Since we are able to apply the point-ahead either parallel or anti-parallel to the wind, we will always give results both at +θGEO and −θGEO where the sign should merely indicate the two possible directions of the shift.
5.1. Beam wander
First, we analyzed beam wander which is given by the distance of the irradiance centroid to the optical axis. For better comparison with our baseline scenario, we have converted this distance on the uplink camera into an angular beam wander β at the satellite given in μrad. We calculated the cumulative distribution function (CDF) as a function of β which gives the percentage of data points with a value equal or below β. Whether or not this distribution is acceptable, depends on the extent of the uplink beam; for example a highly divergent beam will allow for a higher value of beam wander while a very narrow beam will not. As our threshold, we have chosen the half-width at half maximum (HWHM) of the diffraction-limited beam given by
which can be calculated from the Airy pattern (diffraction-limited beam) by setting the irradiance to 0.5 and solving for the angle β. As a result, the satellite should receive at least 50% of the maximum irradiance as long as the beam wander is below βHWHM in case of an otherwise undistorted beam profile. In our baseline scenario it has a value of approximately 0.79 μrad.Figure 6 shows the CDF calculated from our measured data. The dashed line indicates the aforementioned angle βHWHM. Without compensation, only 3% of the data points have a beam wander below βHWHM. In contrast, with our system, we achieve a percentage of 73% (+θGEO) and 66% (−θGEO). As one might expect, an even higher value of 78% can be achieved at vanishing point-ahead angle.
Fig. 6 Cumulative distribution function (CDF) of the measured beam wander. The blue line shows the compensated beam at vanishing point-ahead angle while the yellow and red curve denote the case at +θGEO and −θGEO, respectively. The case without compensation is shown by the green curve.
5.2. Strehl ratio improvement
From the far field measurements, we also calculated the Strehl ratio of the beam. Before the aberration emulator was installed in the setup, we measured an uplink Strehl ratio of S = (60 ± 1)% with the tip/tilt mirror centered and the deformable mirror without voltages applied. This corresponds to the wavefront reference of the control loop. During our measurements, we observed a strong dependence of the achievable Strehl ratio on this reference which defines the ideal wavefront the control algorithm tries to achieve. Interestingly, reference spots leading to a high Strehl ratio of the downlink beam were not necessarily optimal for the uplink beam. We believe that a further optimization of the reference spots could still increase the system performance beyond this upper boundary of 60%.
After this measurement, we installed the aberration emulator in the setup. Figure 7 shows the temporal evolution of the Strehl ratio. Without correction, the Strehl ratio is practically zero – S = (4 ± 4)%. However, with the AO system a Strehl ratio of (48 ± 17)% was achieved at zero point-ahead angle. At the point-ahead angle for Earth-to-GEO communication, the Strehl ratio was either (32 ± 15)% or (28 ± 15)% for +θGEO and −θGEO, respectively. We want to point out that the large spread in Strehl ratios is most likely caused by the natural spreading of Fried parameters, see also Section 3.4.
Fig. 7 Temporal evolution of Strehl ratio (a) without compensation and with compensation at vanishing point-ahead angle and (b) with compensation at point-ahead angle +θGEO.
The improvement of the Strehl ratio can be seen immediately in the far field as measured by our camera. Figure 8 shows three examples of camera images: (a) without and (b,c) with AO correction. Without AO correction, a broad speckle pattern caused by higher-order turbulence distortions can be seen. Because of the temporal evolution of turbulence, the speckle pattern changes with time such that the satellite will frequently lie in one of the irradiance minima, again leading to a loss of signal. In contrast, the far field exhibits a single pronounced peak when AO correction is applied as shown in Fig. 8(b) and (c). Note that the brightness scaling in Fig. 8 is not equal in (a–c) to improve contrast.
Fig. 8 Far field (a) without, (b) with pre-compensation at 0 point-ahead angle and (c) with pre-compensation at θGEO; (a) shows a broad speckle pattern whereas in (b) and (c) a narrow peak can be seen. Please note that the differences of the image center are only caused by the necessity to move the uplink camera between measurements.
5.3. Fading probability
The impact of these improvements on laser communication can be appreciated at best from estimates of the fading probability of the communication link. It is defined as the probability to receive an irradiance below a certain threshold IT which results from the receiver’s characteristics, e.g. the sensitivity [18]
Since the Strehl ratio is directly proportional to the received irradiance, see Eq. (2), we were able to calculate the fading probability from the Strehl ratio time series shown in Fig. 7. We set the detection threshold equivalent to S = 4% (average Strehl without AO correction) which corresponds to a collected photon flux of ΦRX = 4.16 · 1011 photons/s (output power of OGS laser P0 = 1 W, telescope diameter of GEO satellite DGEO = 13.5 cm). With these assumptions, we found that AO pre-compensation reduces the fading probability by more than 10 times, from 63% (without AO) to a mere ∼ 4%. At a maximum transmission rate of 1.8 Gbit/s [22], this flux is equivalent to a detection threshold of ∼ 230 collected photons per bit.5.4. Point-ahead angle
Finally, we scanned the point-ahead angle and evaluated the Strehl ratio for each measurement sequence. The results are shown in Fig. 9 where the blue points represent the average over the measured Strehl ratios. Theoretical values were adapted from [9] where both the influence of the point-ahead angle and the finite aperture of the telecope were taken into account. In fact, this theory is a more elaborate version of Eq. (8). According to our values of DTel/r0 = 4.5 ± 0.6 at 1550 nm and for each value of the point-ahead angle, we took the appropriate value from [9, Table 1]. We assumed an isoplanatic angle of 13.6 μrad which is the average value from Table 1 minus one standard deviation to ensure a working system also at more severe night-time turbulence. The only modification of the theory from [9] was an overall scaling of the curve to match our maximum Strehl ratio of 48% which is smaller than in [9] because of imperfections in our system. As in the previous section, the measured Strehl ratio has a high standard deviation which is also reflected in the error bars of Fig. 9. Again, we explain this behavior by the strong spreading of the Fried parameters in the AE. As expected from theory [9], see also Section 2.4, the Strehl ratio is highest when the point-ahead angle vanishes and it decays slowly as the point-ahead angle increases. Our results are in close agreement with the theoretical evolution represented by the green curve.
Fig. 9 Comparison of achievable Strehl ratio for different point-ahead angles: blue points represent the average value and the error bars correspond to the standard deviation. The theoretical decay as tabulated in [9] is given as the green solid line.
6. Conclusion
In summary, we have presented a laboratory AO system which emulates the pre-compensation of an uplink beam for Earth-to-GEO optical communication. Wavefront measurements to control the AO system are obtained from the downlink beam. As our baseline scenario, we have analyzed communication between a geostationary satellite with a 13.5 cm telescope diameter and a 1 m telescope on Tenerife which has excellent seeing. Our results were generated assuming DTel/r0 = 4.5 ± 0.6 at 1550 nm and an isoplanatic angle of 13.6 μrad which corresponds to more severe turbulence than expected from night-time measurements at the OGS. Furthermore, we have assumed an output power of 1 W at both telescopes and the point-ahead angle for Earth-to-GEO communication of θGEO = 18 μrad.
The system corrects for tip/tilt such that the number of data points staying within the beam profile (HWHM) was increased from 3% to at least 66%. We have shown that the beam quality at the receiver can be increased from an almost vanishing Strehl ratio of S = (4 ± 4)% to S = (28 ± 15)% at the required point-ahead angle. This Strehl ratio is linearly connected to the power received at the GEO satellite. Our system smooths the speckle pattern and reduces beam wander at the satellite’s position. Thus, the fading probability of communication between the satellite and the ground station has been substantially reduced from 63% to 4% where a receiver sensitivity of ΦRX = 4.16 · 1011 photons/s has been assumed.
The setup features an achromatic design which allows the use of two different wavelengths, in our case 1064 nm and 1550 nm. Please note that it is possible to use the measurements at one wavelength to compensate for the uplink at the other wavelength. In principle, it would also be possible to choose either 1064 nm or 1550 nm for both communication channels. However, slight adjustments of the setup would be necessary to avoid blinding of the wavefront sensor by back-reflections of the uplink beam.
In contrast to previously demonstrated pre-compensation systems [11, 12], we have developed and tested a dynamic system. The real-time control was achieved by using a custom-made fast wavefront sensor with a centroid determination algorithm implemented on an FPGA. By choosing an appropriate camera, we were able to measure at light levels as low as expected from the baseline scenario.
Additionally, we have shown that despite the required point-ahead angle, pre-compensation can be successful. Within our laboratory setup, we were able to scan the point-ahead angle to investigate its influence on the pre-compensation performance. As expected, the Strehl ratio was highest at zero point-ahead angle and decayed according to theory as the point-ahead angle increased. Our measurements suggest that communication between ESA’s ground station on Tenerife and Alphasat can be improved with our system.
Acknowledgments
We thank Matthias Heinzig and Thomas Peschel for valuable help and fruitful discussions on laser sources and laser telescopes. We gratefully acknowledge the support by Zoran Sodnik and Peter de Maagt during the project. Work reported in this article was funded by ESA under contract number 4000112766/14/NL/MH.
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