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

In the context of adaptive optics for astronomy, one can rely on the statistics of the turbulent phase to assess a part of the system's performance. Temporal statistics with one source and spatial statistics with two sources are well known and widely used for classical adaptive optics systems. A more general framework, including both spatial and temporal statistics, can be useful for analysis of the existing systems and to support the design of future ones. In this paper, we propose an expression of the temporal cross power spectral densities of turbulent phases in two distinct beams, which is from two different sources to two different apertures. We consider the phase either as it is, without a piston, or as its decomposition on Zernike modes. The general formulas allow coverage of a wide variety of configurations, from single-aperture to interferometric telescopes equipped with adaptive optics, with the possibility to consider apertures of different sizes and/or sources at a finite distance. The presented approach should lead to similar results with respect to existing methods in the Fourier domain, but it is focused on temporal frequencies rather than spatial ones, which might be convenient for some aspects such as control optimization. To illustrate this framework with a simple application, we demonstrate that the wavefront residual due to the anisoplanatism error in a single-conjugated adaptive optics system is overestimated when it is computed from covariances without taking into account the temporal filtering of the adaptive optics loop. We also show this overestimation in the case of a small-baseline interferometer, for which the two beams are significantly correlated.