Spatial frequency domain model for adaptive optics compensation of segmented mirror misalignments and figure errors

In addition to their essential function of providing atmospheric turbulence compensation, astronomical Adaptive Optical (AO) systems also supplement the role of active optics (aO) by providing some additional correction of the wavefront aberrations introduced by mirror mounting, alignment, thermal distortion and/or fabrication errors. This feature is particularly desirable for segmented mirror telescopes such as the Thirty Meter Telescope (TMT), but wavefront discontinuities across segment boundaries are challenging to properly sense and correct. In this paper we describe a fast, analytical, frequency domain model which may be used to study and quantify the above effects, and discuss a range of sample results obtained to support the development of the top-level requirements for the TMT primary mirror. In general, AO compensation of mirror segment piston errors is not particulary useful unless the deformable mirror (DM) interactuator spacing is equivalent to no more than one-half of a mirror segment diameter (when both of these dimensions are expressed in the same pupil plane). Effective AO compensation of mirror segment tip/tilt errors, or low order segment figure errors such as astigmatism, typically requires 3-4 DM actuators per mirror segment. These results illustrate the importance of quantifying and minimizing uncorrectable telescope wavefront errors when developing performance predictions for adaptive optical systems.