Theoretical spatiotemporal analysis of angle of arrival induced by atmospheric turbulence as observed with the grating scale monitor experiment

Theoretical investigations of the statistical properties of the wave front perturbed by atmospheric turbulence are presented. They are deduced from the calculation of the two-dimensional spatial covariance and the temporal cross spectrum of the angle-of-arrival fluctuations with a finite outer scale over a pair of circular pupils as in the case of the grating scale monitor or any other Shack–Hartmann-type sensor. Both calculations lead to integral expressions that are numerically evaluated and hold for any baseline vector in the mean wave-front plane. It is proposed to retrieve the wave-front outer scale L0 from estimations of this two-dimensional spatial covariance, normalized by the angle-of-arrival structure function. To eliminate instrument vibration errors, the covariance and the structure function are estimated from measurements obtained by mechanically independent and mechanically coupled devices, respectively. The angle-of-arrival temporal cross spectrum is calculated for any mean wind velocity vector. It is shown that the baseline component in the mean wind direction affects the phase of the angle-of-arrival temporal cross spectrum, whereas the component in the perpendicular direction affects the modulus. From simultaneous measurements of the phase of the angle-of-arrival temporal cross spectrum obtained with two nonparallel baselines, one can calculate the mean wind speed and direction, which allows estimation of the coherence time for techniques of optical observation at high angular resolution through the atmosphere.