  A lower bound on unbiased estimates of wavefront errors (WFE) is presented
for the linear regime of small perturbation and active control of a
high-contrast region (dark hole). Analytical approximations and algorithms for
computing the closed-loop covariance of the WFE modes are provided for
discrete- and continuous-time linear WFE dynamics. Our analysis applies to both
image-plane and non-common-path wavefront sensing (WFS) with
Poisson-distributed measurements and noise sources (i.e., photon-counting
mode). Under this assumption, we show that recursive estimation benefits from
infinitesimally short exposure times, is more accurate than batch estimation
and, for high-order WFE drift dynamical processes, scales better than batch
estimation with amplitude and star brightness. These newly-derived contrast
scaling laws are a generalization of previously known theoretical and numerical
results for turbulence-driven Adaptive Optics. For space-based coronagraphs, we
propose a scheme for combining models of WFE drift, low-order non-common-path
WFS (LOWFS) and high-order image-plane WFS (HOWFS) into closed-loop contrast
estimates. We also analyze the impact of residual low-order WFE, sensor noise,
and other sources incoherent with the star, on closed-loop dark-hole
maintenance and the resulting contrast. As an application example, our model
suggests that the Roman Space Telescope might operate in a regime that is
dominated by incoherent sources rather than WFE drift, where the WFE drift can
be actively rejected throughout the observations with residuals significantly
dimmer than the incoherent sources. The models proposed in this paper make
possible the assessment of the closed-loop contrast of coronagraphs with
combined LOWFS and HOWFS capabilities, and thus help estimate WFE stability
requirements of future instruments.
