A Statistical Framework for Utilization of Simultaneous Pupil Plane and Focal Plane Telemetry for Exoplanet Imaging, Part II: The Science Camera Image as a Function of the Wavefront Sensor Field
In an effort to transcend the limitations of differential imaging of exoplanets in the era of extremely large telescopes (ELTs), the first paper in this series established a rigorous, fully polarimetric framework for determining the science camera (SC) image given a turbulent wavefront and unknown aberrations in multiple planes the optical system. This article builds on the structure developed in Paper I in order to rigorously express the polarimetric SC image in terms of the field impinging on the wavefront sensor (WFS), thereby providing a direct connection between the measurements made in both subsystems. This formulation allows the SC image to be written as a function of the WFS measurements, including the following unknown quantities which can, in principle, be estimated via statistical inference: the non-common path aberration (NCPA), WFS gain errors, aberrations downstream of the beamsplitter, and the planetary image. It is demonstrated that WFS bias error is mathematically equivalent to NCPA. Thus, with the ability to treat WFS bias and gains, the method should not be overly sensitive to WFS calibration problems. Importantly, this formulation includes stochastic processes that represent noisy measurement of the SC image, noisy WFS measurements, and high-frequency components of the wavefront to which the WFS is insensitive. It is shown that wavefront error due to noise in the WFS measurements has a convenient semi-analytical representation in terms of the WFS measurement operator's singular functions. Further, the first and second order statistics of these processes are specified, thereby setting the stage for the application of statistical inference methods to be describe in later papers in this series.