Phase estimation at the point-ahead angle for AO pre-compensated ground to GEO satellite telecoms

We present a new method to estimate the off-axis adaptive optics pre-compensation phase of a ground to GEO satellite telecom link suffering from point-ahead anisoplanatism. The proposed phase estimator relies on the downlink phase and log-amplitude measurements that are available at the optical ground station. We introduce the analytical tools, extended from the literature, to build the estimator as well as a general modal formalism to express the reciprocal residual phase covariance matrix resulting from any estimation linear with measurements. We use this residual phase covariance matrix to generate independent coupled flux samples thanks to a pseudo-analytical approach and study the gain offered by the proposed estimator on the coupled flux statistics, in various atmospheric conditions. The estimator is shown to reduce the anisoplanatic residual phase variance by at least 35%, and 46% at best, with a greater impact on the lower modes, especially on the tip and tilt residual phase variances. The phase variance reduction brings a gain up to 15 dB on the cumulative density function at probability 10^-3. This gain should allow to relax the power constraints on the link budget at the OGS and renews the interest in large aperture diameter (60 cm class telescopes) for GEO Feeder links by reducing the atmospheric turbulence impact on the uplink coupled signal.