Hartmann modelling in the discrete spatial-frequency domain: application to real-time reconstruction in Adaptive Optics

Adaptive Optical systems (AO) with a very large number of degrees-of-freedom (DoF) need the proper development of reconstruction and control algorithms mingling both increased performance and reduced computational burden. The Hartmann wave-front sensor (HS-WFS) is broadly used in AO, featuring a set of lenslet arrays aligned onto a Cartesian grid. It works by averaging the slope of the wave-front in each sub-aperture. Throughout this paper the suitability of the so-called Hudgin, Fried and Southwell geometries to model the HS are analysed. Methods of exploiting data obtained from the telescope's annular aperture through the DFT are revisited. An alternative approach based upon the discrete Gerchberg iterative algorithm is employed. It inherently solves the extrapolation and circularization. The inverse problem is regularised to form the minimum mean-square error (MMSE) reconstructor in the spatial-frequency domain. Results obtained through Monte-Carlo simulations allow for a comprehensive comparison to the standard vector-matrix multiplies (VMM/VMMr) algorithm. Computational burden is kept O(DoF 1092(DoF)).