2.1.

ATMOSPHERIC TURBULENCE PROFILE

For a given site and at a particular time, the atmospheric turbulence is described by the so-called refractive-index structure, , which characterize the strength of the turbulence with respect to the altitude, h, in the atmosphere. Since the atmosphere is denser at low altitudes, the contribution of the low-altitude layers are generally rather important.

Several models have been proposed to describe the behaviour of the atmosphere (depending on the location of the observatory, the number of turbulence layers considered, the importance given to each layer etc.). Among them, the Hufnagel-Valley profile (HV) [4] is the most commonly used but is only valid at sea level. Given that the Optical Ground Station (OGS) is located at H_OGS = 2393 meters over the sea, modification to the atmospheric model are necessary. Just truncating the HV model would lead to an underestimation of the turbulence strength. Therefore, a modification of the model is used to consider the ground station altitude. This model is called the modified Hufnagel-Valley (MHV) and is given by the following equation [5]:

where v stands for the RMS cross wind velocity, and A₀ is a reference sea-level turbulence value for scaling according to day- and night-time. Note that this model is defined for h > H_OGS. In this paper, these parameters have been set to A₀ = 6 10^–16m^–2/3 and v = 21 m/s as suggested in [5] for the OGS. It is worth mentioning that such a simple empirical model gives an order of magnitude of the expected turbulence and may be rather inaccurate because it does not account for instantaneous meteorological conditions, local topographic characteristics nor seasonal influences.

2.2.

BEAM WANDER

The beam wander is due to large-scales turbulence structures, appearing close to the ground, resulting in random fluctuations of the beam path with respect to its initial direction. This phenomenon is even more critical for optical uplinks since atmospheric turbulence happens close to the Earth’s surface where small angular deviations can lead to large pointing errors (beam displacement up to several hundred meters). Since the beam size is often smaller than typical turbulence eddy size, the beam wander is a major concern for optical communications. The standard deviation of the beam wander is given by [6]:

where λ is the operating wavelength, W₀ is the initial beam radius, r₀ is the Fried parameter (i.e the atmospheric-coherence length) and is given by [7]:

where γ is the zenith angle. Observe that since r₀ is proportional to λ^6/5, σ_BW is constant regardless of the wavelength. The previous equation shows that the Fried parameter decreases as the zenith angles increases. This means that the wavefront can be considered as coherent over a smaller area at low elevations than for elevations close to the zenith. This in turn implies that the beam wander is more important at low elevations.

It is also worthwhile to point out that diffraction causes the diameter, D, of a collimated beam to increase in size over distance according to the beam divergence angle:

Therefore, if the beam wander is larger than the divergence of the beam, Φ, strong fading will occur.

Eq. (2) shows that large diameter transmitters are slightly less impacted by beam wander than small apertures. However, one observe from Eq. (4) that the divergence experienced by a beam decreases with the increase of the transmitter aperture, which makes the optical link much more sensitive for large diameter transmitters. On the other hand, small aperture transmitters involve large divergence, hence large power loss, but are more robust against beam wandering.

2.3.

POINT-AHEAD ANGLE AND ISOPLANATIC ANGLE

Due to the relative motion between the satellite and the ground station, the returned signal (i.e from Earth to the satellite) needs to be angularly offset in order to reach the satellite. This pointing offset is called the point-ahead angle and depends exclusively on the differential velocity between the two communicating parties perpendicular to their line of sight. The point-ahead angle can range between few arcseconds (typically 4 arcsec for GEO satellites) up to ~10 arcseconds for LEO satellites.

The isoplanatic angle corresponds to the angle over which the atmospheric turbulence can be considered as unchanged. Therefore, the larger the isoplanatic angle, the better is the pre-compensation based on the downlink signal. The isoplanatic is defined as follows [8]:

it takes into account the whole distortion of the wavefront (low and high order modes).

In order to decouple the contribution stemming from the tip-tilt modes (i.e those related to the beam wander effect), the tilt-isoplanatic angle or the isokinetic angle, is defined as the angle between two sources, at which one of them jitters with a magnitude of half of its diffraction-limited divergence, with respect to the other one [10]:

Observe that the isokinetic angle depends on the zenith angle, the operating wavelength as well as the diameter of the receiver D. Therefore, working at large wavelengths and with large receiver apertures, allows to increase the isokinetic angle. Note that the contribution of the low-order modes (tip-tilt) accounts for ≈85% of the total mean-square wavefront error, and that the low-order modes have effectively larger isoplanatic angle and decorrelation time than higher modes.

The relationship between the isokinetic angle and the point-ahead angle is of prime importance for optical communications. Indeed, if the angular offset between the signal emitted by the ground station and the signal received from the satellite (i.e., the point-ahead angle) remains smaller than the isokinetic angle, the downlink signal can be used to pre-compensate the uplink signal.

From the previous definitions, it turns out that the challenge lies in increasing the diameter of the beam while actively precompensating the beam wander induced by the atmosphere.

3.1.

THE OPTICAL GROUND STATION

The test campaign was carried out in May 2018 at the ESA’s Optical Ground Station (OGS) located at the “Observatorio del teide” at Izaña, in Tenerife. The site is located at an altitude of 2393 meters above sea level, well above the first inversion layer, and provides exceptional atmospheric conditions over long time periods throughout the year.

The telescope is a Zeiss 1m-diameter Ritchey-Chrétien telescope, supported by an English mount. It can be used in a Cassegrain or Coudé focus configuration. A 20 cm-diameter guider telescope, with a larger field of view, is fixed and coaligned to the main telescope (Figure 1). The results presented in this paper have been obtained using the 1m-aperture telescope, with an Andor Ixon EMCCD camera fixed at the Cassegrain focus. The main parameters of the telescope and the camera are summarized in Table 1.

Figure 1:

1 meter Zeiss telescope of the OGS and 20 cm guider telescope.

Table 1:

Main parameters of the telescope and the Andor camera.

3.2.

ACQUISITION PARAMETERS

In the framework of this test campaign, only double stars with the following characteristics were considered for measurements: (i) a magnitude smaller than 8 (ideally 6), (ii) a magnitude difference smaller than 3 (ideally 2) and (iii) an angular separation between 4 and 60 arcseconds. Two optical filters have been used in front of the camera, one low-pass filter and one high pass-filter, to limit the optical spectrum to a range between 550 nm and 650 nm.

Given the focal length of the telescope, 13.3 meters, and the pixel size of the Andor camera, 8x8 μm², it turns out that the plate scale corresponds to 0.124 arcsec/pixel. Obviously, the smaller the plate scale, the higher the resolution, and the better the centroiding algorithm will perform. On the other hand, a high resolution involves a large number of pixels to process, which in turn limits the acquisition frequency.

Each double star measurement consisted of a sequence of 10 000 short exposures frames. The integration time was set between 3 ms and 6 ms per frame depending on the star magnitude, which is well below typical tip-tilt coherence time (> 10 ms) and long enough to reach an acceptable SNR. Therefore, the acquisition time of each 10 000-frames sequence lies between 30 and 60 seconds. The recording time of the entire sequence was chosen short enough that the statistical properties of the motion remain constant. Indeed, under typical conditions, the atmosphere is considered to maintain the same statistical properties for about 1 minute.

4.1.

DATA SELECTION

As a first step, all measurements were presented on a map according to the elevation and angular separation of the double stars. Then, four sets of double stars were identified for post-processing. The selection was based on the following criteria: (i) the possibility to deduce a trend from a given set, and (ii) the amount of data available during the same night. This map is presented on Figure 2. For the sake of clarity, Table 2 and Table 3 give another presentation of the same information. Before processing the data, a visual inspection of each selected sequence was performed individually in order to discard the sequences that were not relevant or too noisy. Note that most of the measurements were performed several times within a short time interval (i.e several minutes). This allows to check the repeatability of the measurement as well as the accuracy of the centroiding algorithm. This also gives an indication of the atmosphere’s stability within a short time interval.

Figure 2:

Map of the measurements that were performed on 26^th, 27^th, 28^th and 29^th of May 2018. The gray symbols have not yet been processed. Three rectangles show the group identified for analysis. Note that most of the measurements were performed several times (therefore they are superimposed on the map). The measurements are listed in Table 2, Table 3 and Table 4.

Table 2:

36 measurements of the double stars HIP 71762 were performed at different elevations. The superscript index indicate the number of measurements for a given elevation. The evolution of the tilt anisoplanatism with respect to the elevation is presented on Figure 6 a).

Table 3:

20 measurements of double featuring an angular separations between 4.69 arcseconds and 31 arcseconds were performed (at an almost constant elevation). The superscript index indicate the number of measurements for the same double stars. The evolution of the tilt anisoplanatism with respect to the angular separation is presented on Figure 6 b).

At the time of writing this paper, 63 sequences have been post-processed: 36 sequences to determine the impact of the elevation on the tilt anisoplanatism; 20 sequences to determine the impact of the angular separation of the double stars on the tilt anisoplanatism, and 7 sequences to evaluate the jitter of the telescope.

4.2.

CENTROIDING METHOD

The binarization is the first step of the centroiding process. It consists of applying a threshold on each frame of a given sequence in order turn the gray-scale image (encoded on 16 bits) into a binary image (i.e a mask). The mask is then applied on the initial frame as shown on Figure 3. This operation makes it possible to generate a new image where the background around the star is rejected.

Figure 3:

a) The initial frame, b) after thresholding, the gray-scale image is turned into a binary image, c) result of the multiplication of a) with b).

Particular care must be taken when choosing the threshold as an inadequate value may lead to a wrong result. Under-estimating the threshold will generally not reject most of the background of the frame, whereas over-estimating the threshold may lead to an unphysical segmentation of a star in a number of independent areas. These scenarios are illustrated hereafter. Figure 4 a) presents the initial frame; Figure 4 b) shows the outcome of the algorithm when a (low) threshold of 970 and 975 are applied to the initial frame. From Figure 4 b), the following can be observed: (i) the algorithm cannot find the stars, and (ii) the result is strongly impacted by a very small change of the threshold value. Figure 4 c) shows a proper choice of the threshold value, the stars are well identified. Finally, Figure 4 d) presents a case where, one of the star is truncated, because the threshold is too large, which obviously leads to the wrong estimation of the center of gravity (CoG).

Figure 4:

a) Initial frame; b) two different thresholds are applied: 970 and 975. The red rectangles show the four areas identified by the algorithm (one per star and per threshold). The largest rectangle is similar for both thresholds (the crosses are superimposed). If the threshold passes from 970 to 975, the algorithm gives a very different result; c) the algorithm converges, it identifies correctly the stars; d) the threshold is too high, one of the stars is over-segmented.

For a given threshold, one can easily find, for each frame of the sequence, the intensity-weighted centroid, as follow:

where m and n stand for the size of the region (in pixels), respectively in the x and y directions. Note that the intensity-weighted method is the most commonly used, but other centroiding algorithm could be defined. As an example one can quote the so-called “squared weighting centroid method” where the intensity of each pixel is squared in the double sum [9].

Once the centroid of each star is identified, it is straightforward to deduce the distance between them, as well as the standard deviation of the distance, calculated over the entire sequence. It is worthwhile to point out that, because it is a differential method, the double star relative motion is inherently independent on the jitter of the telescope.

From the foregoing discussion, follows that the estimation of the CoG depends on the applied threshold. In order to have reliable results, the standard deviation of the distance between the two stars (i.e between their CoG) should be independent of the value of threshold. In fact, it has been observed that between those extreme “threshold-forbidden” regions, one can find a stable region where the standard deviation is constant, regardless the applied threshold. As an example, Figure 5 shows the evolution of the standard deviation with respect to the threshold, for a given sequence. One can clearly identify the region where the position of the centroid is impacted by the noise of the frame (the left part of the plot). Similarly, on the right hand side of the plot, the region associated with the over-segmentation of the stars is identified, which results in a divergence of the standard deviation. A stable region which does not depend on the threshold, is found between the two extremes previously identified. Note that (i) in some cases (≈ 2%), no convergence has been found, in which case the sequence has been discarded, (ii) a frame containing at least one saturated pixel was systematically rejected and (iii) the method has been found very robust with regard to the number of frames considered in the sample. Indeed, thanks to the large number of frames, keeping only 10% of them gives a very good approximation (i.e >90% accuracy) of the standard deviation determined over the entire sequence.

Figure 5:

Evolution of the standard deviation of the relative motion, with respect to the threshold. The algorithm converges once the threshold is higher than the background. A stable region is found between ≈1200 and ≈1900. If the threshold is too high, the algorithm may diverge.

A curve similar to that presented on Figure 5 was generated for each sequence, making it possible to determine the threshold that is required as an input for the centroiding algorithm. Note that for a given sequence, the threshold was kept identical over all the frames.

4.3.

ONE-AXIS RMS TILT CORRELATION

To a good approximation, the atmospheric turbulence may be seen as a static shape that is transported by the wind through the aperture of the telescope. This approximation is called the Taylor’s frozen-atmosphere approximation. According to this model, the uncompensated atmosphere introduces a one-axis rms tilt error, given by the following equation [10]:

If the wavefront tilt measured in a given line of sight is used to estimate the tilt in another direction, then the one-axis rms tilt error that is due to anisoplanatism is given by [11] [12] :

where C_x(θ) and C_y(θ) are the correlations of the tilts at two points that are separated by an angular distance θ.

The x axis is parallel to the line joining the two stars, whereas the y axis is perpendicular to this line and characterize the transverse relative motion. Squaring σ_TA;x and a_TA;y and averaging gives a useful single measure of the one-axis rms tilt error that is due to tilt anisoplanatism:

The correlation functions are given by:

where the x and y indices in C_xy(θ) respectively refer to - and + sign in the integral.

Numerical approximations of A_0,2(s), with less than 0.2% error, can be calculated by the following [11] [12]:

By combining Eqs. (1), (3) and (8) to (13), it is possible to deduce, if not an exact value, at least an order of magnitude of the rms tilt error that is due to tilt anisoplanatism. Furthermore, its evolution with respect to the elevation, σ_ξ, and with regard to the angular separation between the double stars, σ_θ, can also be determined. The following section compares the numerical simulations of the rms tilt error due to tilt anisoplanatism to the experimental results obtained with the centroiding algorithm.

4.4.

PRELIMINARY RESULTS

Figure 6 a) shows the numerical prediction and an experimental measure of the rms tilt error due to anisoplanatism, for various elevations of the double stars HIP 71762 (featuring an angular separation of 5.42 arcseconds). The measurements were performed between 00h00 and 6h am, on the 28^th and 29^th of May (all the measurements are listed in Table 2). One observes that the experimental results are in very good agreement with the numerical simulations for both days. However, even though the experimental values are very close to the numerical prediction, the figure suggests, an overestimation of the model (or an underestimation of the experimental results) for elevations smaller than 30°. Of course, this discrepancy can also stem from a change in the atmospheric turbulence over the night. Unfortunately, it has not been possible to recover the instantaneous atmospheric parameters for these days. Therefore, r₀(γ) was determined by means of Eq. (3).

Figure 6:

a) Evolution of the rms error due to tilt anisoplanatism, for various elevations (the angular separation is 5.42 arcseconds). The measurements were performed on 29^th and 28^th of May 2018 between 00h00 and 6 am (see Table 2). b) Evolution of the rms error du to tilt anisoplanatism, for various angular separations (the elevation is about 32°) and for three values of r₀ (note that the value predicted by Eq.(3) is r₀ = 16.2 cm at this elevation). The measurements were performed on 29^th of May 2018 between 00h00 and 6 am (see Table 3).

Figure 6 b) presents the experimental measure of the rms tilt error due to anisoplanatism, for a fixed elevation of ≈32° and several angular separations (listed in Table 3). The measurements are compared to the numerical prediction for r₀ = 10 cm, r₀ = 15 cm and r₀ = 20cm (note that the value predicted by Eq.(3) is r₀ = 16.2 cm at this elevation). The experimental results are still clearly in the same order of magnitude as the theoretical model, and in the same way as for Figure 6 a), overestimated by the model at such a low elevation. The discrepancy may indicate the modified HV model described by Eq. (1) overestimates the altitude of the turbulent layer. It is, however, interesting to note that the numerical model follow a trend similar to that measured experimentally.

It is important to mention that even though the tilt contribution statistically account for ≈85% of the full rms error, the experimental measurements, acquired with the 1m telescope, also suffer from higher-order aberrations. Conversely, the numerical predictions have been performed considering only the tilt contribution of the full error due to anisoplanatism. Therefore, a more representative measurement should be performed with the smaller 20 cm aperture telescope, over which the wavefront could be considered as almost coherent. However, the amount of photons gathered by the telescope would be 25 times smaller than with the 1m telescope. An alternative lies in increasing the operating wavelength. Indeed, according to Eq. (8), the tilt should remain unchanged, while the Fried parameter should increase proportionally to λ^6/5 as shown by Eq. (3).