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In the context of adaptive optics for astronomy, one can rely on thestatistics of the turbulent phase to assess a part of the system's performance.Temporal statistics with one source and spatial statistics with two sources arewell-known and are widely used for classical adaptive optics systems. A moregeneral framework, including both spatial and temporal statistics, can beuseful for the analysis of the existing systems and to support the design ofthe future ones. In this paper, we propose an expression of the temporal crosspower spectral densities of the turbulent phases in two distinct beams, that isfrom two different sources to two different apertures. We either consider thephase as it is, without piston, or as its decomposition on Zernike modes. Thegeneral formulas allow to cover a wide variety of configurations, fromsingle-aperture to interferometric telescopes equipped with adaptive optics,with the possibility to consider apertures of different sizes and/or sources ata finite distance. The presented approach should lead to similar results withrespect to existing methods in the Fourier domain, but it is focused ontemporal frequencies rather than spatial ones, which might be convenient forsome aspects such as control optimization. To illustrate this framework with asimple application, we demonstrate that the wavefront residual due to theanisoplanatism error in a single-conjugated adaptive optics system isoverestimated when it is computed from covariances without taking into accountthe temporal filtering of the adaptive optics loop. We also show thisoverestimation in the case of a small-baseline interferometer, for which thetwo beams are significantly correlated.

Spatiotemporal statistics of the turbulent piston-removed phase and Zernike coefficients for two distinct beams