Optimal Estimators for Astronomical Adaptive Optics

Abstract

An adaptive optics system is a closed-loop servo control system that seeks to maximize PSF Strehl ratio performance by minimizing wavefront distortions. The wavefront is sampled over discrete subapertures and the local slopes are used to estimate the instantaneous wavefront shape which is then used to drive a deformable mirror with a discrete array of actuators. The temporal and spatial performance of the system is embodied in a single mathematical descriptor of the form Γ is a covariance matrix of the error between the atmosphere phase φ(ti+1) at time ti+1 and the deformable mirror figure, ϕdm(ti), derived from measurements at the previous time ti. Boldface quantities are vectors and matrices. The phase reflecting off the mirror is φ(ti+1) – ϕ(ti), presently assuming that ϕdm(ti)≈ϕ(ti), where ϕ(ti) is the estimated wavefront phase. Covariance matrices are a powerful mathematical tool because they contain information, in an ensemble average sense, about the sources of error and correlations present in the system. From Γ, the Strehl ratio, the MTF, time-delay, etc., can be computed. The Strehl ratio in the Marechal approximation is S ~ exp(–σ2), where σ2=Tr(Γ) /Na, for Na actuators within the pupil.