1.1

Free Space Optical Communications

As improvements to satellite communication continue, the benefits of expanding from radio frequencies to optical signals become more important. Free space optical (FSO) communications with satellites transmit line-of-sight signals using modulated laser beams between different types of links; ground station to satellite, satellite to ground station, and horizontal links between ground station and ground station, or between satellite and satellite.

Using optical lasers for FSO communications has a much higher data rate than when using radio frequencies,¹ making it more feasible to transfer the large quantities of data that are becoming more and more prevalent. Additionally, there are less power requirements for optical laser links, since the beam spread is narrower than that of radio frequencies.² This also results in higher security data transmission, as there is inconsiderable interception of the line-of-sight laser signals.³ Currently, there are less regulations and restrictions on frequencies due to high demand than with radio signals. Therefore, interference with other links is reduced.

However, the use of optical wavelengths for satellite communication has specific challenges of signal distortion due to atmospheric turbulence which does not affect radio signals. These are a significant source of error, and are discussed further in Section 1.1.1. Another practical element of signal transmission is that any jittering of the transmitter or receiver due to environmental interference contributes to the degradation of the signal.⁴

Due to the orbital motion of these satellites, the uplink (ground station to satellite) and downlink (satellite to ground station) signals are spatially separated by the point-ahead angle, shown in Figure 1. This angle is defined by 2v/c,⁵ where c is the speed of light and v is the orbital velocity of the satellite. The point-ahead angle of a geostationary satellite is approximately 4 arcseconds, and that of a satellite in low Earth orbit is dependent on the orbital radius, however with the example of the International Space Station (ISS), has an angle of around 10 arcseconds.⁶ The point-ahead angle results in the uplink and downlink laser beams to propagate through different regions of the atmosphere, and thus experience different atmospheric turbulence effects, as discussed further in Section 1.1.2.

Figure 1:

The point ahead angle of the uplink signal with respect to the downlink signal. The two signals must travel through different regions of atmospheric turbulence.⁶

1.1.2

Tip-Tilt and the Isokinetic Angle

The first order aberration of tip-tilt causes the direction of propagation of the light beam to deviate. These changes to the angle of the incoming signal offset the location of an image on the observation screen or detector. This causes the previously described beam wandering effect, which can result in the signal moving around on the detector, or even missing it entirely. It is important to correct for such aberrations in order to minimise the BER by ensuring the signal is always reaching the detector, without having to use detectors with unrealistically large fields of view. Since tip-tilt aberrations cause the movement of a signal around the area of the detector, the magnitude of tip-tilt can be expressed as the variance in the detected beam location.

Regions of the atmosphere that have the same atmospheric turbulence effects on light are called isoplanatic. Within an isoplanatic angle, the aberrations on different light beams are considered to be correlated. The tip-tilt isoplanatic region, or the isokinetic region, is where the tip-tilt effects on light are correlated.¹²

The isokinetic angle varies across the atmosphere due to factors like temperature and different air pressure systems, all of which are characterised by the refractive index structure constant, . Additionally, the elevation of a beam also impacts the isokinetic angle. Transmission closer to the horizon results in higher turbulence effects due to longer propagation through denser layers of the atmosphere. As a result, the elevation of sources is of interest when detecting signals. The elevation is also important when applying the point ahead angle required for satellite communications, since the laser links are each travelling through different areas of the atmosphere which could each have different characteristics of turbulence. If the point-ahead angle is larger than the isokinetic angle, the laser links will acquire different tip-tilt effects.⁵

A focus on correcting for tip-tilt emerges from the fact that lower order modes have the highest contribution to the total wavefront error, around 85%.⁶ Additionally, the isokinetic angle is typically larger than that of the higher order aberrations.⁵ Hence, there is a higher possibility that the point-ahead angle is within the isokinetic angle, increasing the likelihood that the signals experience correlated tip-tilt effects.

Adaptive optics (AO) can be applied to satellite optical communications to minimise the errors due to atmospheric turbulence, like tip-tilt.⁵ The light propagation of a laser guide star (LGS) through the atmosphere in both directions (up into the atmosphere and then returning to ground) results in the tip-tilt effect to be cancelled out once it reaches the ground based detector.¹³ As a result, conventional AO using LGSs cannot be used to correct wavefront slope effects due to tip-tilt. Instead, natural guide stars must be used, which are natural stars of known constant flux that are used as a reference source for AO. In order to correct for this tip-tilt effect in satellite communications where there are no natural guide stars available, this project proposes that the downlink signal could be used as a reference source to correct for these atmospheric low order modes in the uplink signal. This research follows from the work of Alaluf et al. (2021),⁶ with the purpose of undertaking a similar study in the southern hemisphere. In order to assess the feasibility of this process, correlated tip-tilt effects need to be confirmed to determine if the two laser beams are traveling through the same isokinetic area.⁵ Such correlation would enable the turbulence in downlink signals to be used to characterise the pre-compensation needed to be applied to uplink signals to minimise BER.

For this project, double stars with different separation angles were used to determine if the uplink and downlink signals could be in the same isokinetic region. Double stars consist of two stars close together from the perspective of Earth-based observations, hence making them an appropriate analogue for the bidirectional signals from satellites, as shown in Figure 2. The separation between the components of double stars mimics the different orbital radii of bidirectional communications satellites; the closer the components of the double stars, the larger the orbital radius of the satellite.

Figure 2:

Double stars are can be analysed as being equivalent to the uplink and downlink signals of satellite to ground station free space optical communications.

The methodology in Section 2 outlines the data processing methods written in Python code to calculate the tip-tilt of any double star data sets. These methods enabled the analysis of the atmospheric tip-tilt effects to determine the correlation between the two light sources.

2.1

Removal of Background Noise

Background noise and isolation of each of the stars are performed prior to the testing of several centroiding methods as described in Section 2.2. Due to the difference in magnitudes between the primary and secondary stars, each one was isolated and processed separately. Figure 3 shows an example where the reader can see the clear difference in magnitude between the two selected stars.

Figure 3:

Isolating the primary and secondary stars so that binary masks with different threshold values can be applied to each one separately.

Firstly, a binary image of the star was created based on a chosen threshold, to use as a mask. This binary mask was then multiplied by the original in order to produce a background-noise removed image where only relevant star pixels have values greater than zero, an example of which is shown in Figure 4(b). The centre of each star was found to be more accurate by following this procedure.

Figure 4:

The effect of threshold values on the background-removed images. The primary star of H 3 7 used as an example.

An appropriate threshold value was characterised as being large enough to remove any low valued background noise pixels that don’t contribute to the structure of the star being observed, as in the comparison between Figures 4(a) and 4(b). Additionally, the threshold must not be too large to result in the discarding of dimmer yet still important features of the star. Figure 4(c) demonstrates how an incorrect choice of threshold value will cut off the outer pixels of the star.

In order to select the most appropriate threshold value for each star, verification tests were performed to quantify the extent to which threshold values affect the calculated centre for a given star. A range of threshold values were implemented and the resulting centres were obtained and examined by eye. Additionally, the average change in centre of each star over all the frames of data was determined. This enabled the comparison of change of the centroid values between frames to that of the change of centroid values due to variations in the chosen threshold value. If the size of the changes between frames were found to be much larger than the order of the changes due to thresholding, then less care would need to be taken than if the converse were true.

To determine the effect that the threshold value has on the tip-tilt analysis, the centroid of each star was found through the range of threshold values. Across the range of threshold values, the standard deviation of the star’s distance from the origin was recorded. This was done for both the primary and secondary stars. The range of threshold values where the output of the analysis was stable was used to guide the appropriate choice. Threshold values were chosen to be in this region to ensure consistent results, where small differences in chosen threshold value do not significantly impact the centroiding algorithm.

The stable regions were compared with the observations by eye to determine the most appropriate threshold value to use when creating the binary mask.

2.2

Centroiding

Various methods for calculating the location of a star in an image were tested to determine the most accurate method for these data sets. A centre of mass method for finding the centre of the primary and secondary stars was determines to be the most appropriate for this application. This method involved using the different intensities of each pixel to find a centre of mass of the light detected from the star. In this way, the contribution of each pixel to the centre of the star was weighted by its intensity value, as expressed in Equation 1, where the pixels x_j were weighted by their intensities I_j.

2.3

Tip-Tilt Analysis

To analyse the tip-tilt of the wavefront, two types of movement of the double stars were analysed; relative and absolute. These two analysis methods both made use of the centroids of the primary and secondary stars found for each frame of data using methods in Section 2.2.

3.1

Tip-tilt Analysis

The standard deviations of the differential tip-tilt across all the frames of data was found for each double star, to observe how the apparent distance between the stars shift due to the tip-tilt aberrations. Additionally, for each double star a correlation matrix was used to determine if the tip-tilt of the primary and secondary stars was correlated. Tip-tilt effects that are exactly equal have a correlation value of 1, and that of uncorrelated stars is 0. This demonstrates the effects of elevation and separation of the two sources of light in determining how correlated the tip-tilt aberrations are

Figure 5(a) demonstrates the relationship of the standard deviation of the differential tip-tilt to the separation of the double stars. As the separation between the two stars increases, so does the differential tip-tilt, due to the larger difference in atmosphere that each signal was travelling through. This is similarly demonstrated in Figure 5(b), where the correlation of measured tip-tilt between stars is at its highest when the stars are closest together. These results confirm that smaller correlation coefficients of the tip-tilt effects of each star are observed as their separation approaches or exceeds the isokinetic angle.

Figure 5:

Differential tip-tilt and correlation coefficients for double stars of different separations, observed at approximately 80° from the horizon. Blue line corresponds to the Modified Hufnagel-Valley Model prediction.

Additionally, a single double star, DUN 126, was observed at three different elevations, as shown in Figure 6. This enabled an investigation into the effect of the elevation from the horizon of a signal’s path through the atmosphere. Figure 6(a) demonstrates a decrease in differential tip-tilt as the stars are observed closer to zenith. This is supported by the correlation coefficient data in Figure 6(b). Due to the fact that the isokinetic angle is also smaller for observations made with telescope elevations closer to the horizon, the tip-tilt was more highly decorrelated for angles further from zenith.

Figure 6:

Differential tip-tilt and correlation coefficients for the same double star, DUN 126, observed at different elevations from the horizon. Blue line corresponds to the Modified Hufnagel-Valley Model prediction.

In comparison to the work of Alaluf et al.,⁶ a similar relationship between differential tip-tilt and both elevation and separation was obtained. The Modified Hufnagel-Valley Model was used to predict the expected differential tip-tilt at Mount Stromlo Observatory; the fitted curves in Figures 5(a) and 6(a) present these results, showing a stronger decorrelation of tip-tilt for a change in elevation, than that due to an increase in separation of the signals. Further expansion of the data set used in this paper should continue to verify these results and the comparison. Overall, for observations closer to the horizon, larger deviation in tip-tilt was observed as the light has to travel through thicker atmosphere. Larger separations between the two components of the double stars have the same effect, due to the difference in atmosphere that the two light sources propagate through.

3.2

Measurement of Seeing

Seeing is not known exactly at Mount Stromlo Observatory (MSO) where the observational data was taken, as the atmosphere has not been characterised. The seeing can be estimated using the point spread functions (PSFs) of observed stars. The full width at half maximum (FWHM) of these PSF is inversely proportional to the Fried’s parameter r₀, and thus an expression of the seeing.¹⁴ By fitting a Gaussian curve to the cross section of the star, the PSF can be approximated. The FWHM of this fitted Gaussian is .

The PSF of the double stars outlined in Appendix A were computed for both stars in each frame, and the average FWHM of all PSFs were recorded. It was clear that the seeing declined at telescope elevations closer to the horizon, which was expected.

Since seeing is conventionally defined at zenith, an overall value for seeing at the MSO site was found by using trigonometry to convert the FWHM values to that of seeing at zenith. The following equation was used:

where ε₀ is the seeing, σ_FWHM is the FWHM of the PSF, and α is the observed angle of the star from zenith. Applying Equation 2 gives the results in Table 1.

Table 1:

Approximated seeing values for two nights of observation found from data of double stars.

Combining the results from Table 1, the seeing at MSO was experimentally estimated to be 2″.

Additionally, this estimation of seeing enabled an estimation of the Fried parameter using Equation 3. An experimentally estimated Fried parameter of 5cm at 550nm was obtained.

4.1

Uplink Tip-Tilt Correction

Applying the developed methodology to a range of double stars enables an overview of the way in which atmospheric tip-tilt changes across the atmosphere. By comparing the results of tip-tilt determination across double stars of different separations, the decrease in coherence between the two stars was clear, as in Section 3.1. Where two signals have greater spatial separation, the further they are from being within the isokinetic angle, resulting in low correlation between the tip-tilt effects that each of the signals exhibit. Additionally, the different elevations of observations of the same double star result in signals travelling through different regions and thicknesses of the atmosphere. It follows from the dependence of on height and the intuitive sense of propagation at lower elevations meaning propagation through larger distance of atmosphere, that the isokinetic angle also decreases closer to the horizon, and that the tip-tilt correlation was the lowest towards this end of the range. Despite these atmospheric effects, the values in Figures 5(b) and 6(b) demonstrate high levels of correlation of the observed tip-tilt between the primary and secondary stars of the three double stars. This suggests promising potential for feasibility in using the downlink signal as a reference to correct for tip-tilt in the uplink signal.

The results can be verified with the work of Alaluf et al.,⁶ in which similar data was taken in the northern hemisphere. The fitted curves of the observational data in this paper show a clear comparison of the differential tip-tilt measured and simulated by Alaluf et al. This verification of results will be expanded on as more observational data is obtained.

The elevation of the double stars is clearly analogous to the position of satellites in the sky, and differences in the separation of the stars are comparable to the point ahead angle between downlink and uplink signals due to the orbital motion of satellites. As such, the comparison of the results of the double stars in this project to the FSO satellite communications applications is clear.

Signals with separations as large as 26.3″ show correlation values of tip-tilt larger than 0.75. Further observational data sets are needed to verify these preliminary findings. The results suggest a possibility that the downlink signal has tip-tilt effects similar enough to the uplink signal such that it can be used as a reference source for tip-tilt sensing in adaptive optics.