Optimal wavefront control for adaptive segmented mirrors

A ground-based astronomical telescope with a segmented primary mirror will suffer image-degrading wavefront aberrations from at least two sources: (1) atmospheric turbulence and (2) segment misalignment or figure errors of the mirror itself. This paper describes the derivation of a mirror control feedback matrix that assumes the presence of both types of aberration and is optimum in the sense that it minimizes the mean-squared residual wavefront error. Assumptions of the statistical nature of the wavefront measurement errors, atmospheric phase aberrations, and segment misalignment errors are made in the process of derivation. Examples of the degree of correlation are presented for three different types of wavefront measurement data and compared to results of simple corrections.