  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 are widely used for classical adaptive optics systems. A more
general framework, including both spatial and temporal statistics, can be
useful for the analysis of the existing systems and to support the design of
the future ones. In this paper, we propose an expression of the temporal cross
power spectral densities of the turbulent phases in two distinct beams, that is
from two different sources to two different apertures. We either consider the
phase as it is, without piston, or as its decomposition on Zernike modes. The
general formulas allow to cover 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.
