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A lower bound on unbiased estimates of wavefront errors (WFE) is presentedfor the linear regime of small perturbation and active control of ahigh-contrast region (dark hole). Analytical approximations and algorithms forcomputing the closed-loop covariance of the WFE modes are provided fordiscrete- and continuous-time linear WFE dynamics. Our analysis applies to bothimage-plane and non-common-path wavefront sensing (WFS) withPoisson-distributed measurements and noise sources (i.e., photon-countingmode). Under this assumption, we show that recursive estimation benefits frominfinitesimally short exposure times, is more accurate than batch estimationand, for high-order WFE drift dynamical processes, scales better than batchestimation with amplitude and star brightness. These newly-derived contrastscaling laws are a generalization of previously known theoretical and numericalresults for turbulence-driven Adaptive Optics. For space-based coronagraphs, wepropose a scheme for combining models of WFE drift, low-order non-common-pathWFS (LOWFS) and high-order image-plane WFS (HOWFS) into closed-loop contrastestimates. We also analyze the impact of residual low-order WFE, sensor noise,and other sources incoherent with the star, on closed-loop dark-holemaintenance and the resulting contrast. As an application example, our modelsuggests that the Roman Space Telescope might operate in a regime that isdominated by incoherent sources rather than WFE drift, where the WFE drift canbe actively rejected throughout the observations with residuals significantlydimmer than the incoherent sources. The models proposed in this paper makepossible the assessment of the closed-loop contrast of coronagraphs withcombined LOWFS and HOWFS capabilities, and thus help estimate WFE stabilityrequirements of future instruments.

Information-theoretical Limits of Recursive Estimation and Closed-loop Control in High-contrast Imaging