Linear systems modeling of adaptive optics in the spatial-frequency domain

Spatial-frequency domain techniques have traditionally been applied to obtain estimates for the independent effects of a variety of individual error sources in adaptive optics (AO). Overall system performance is sometimes estimated by introducing the approximation that these individual error terms are statistically independent, so that their magnitudes may be summed in quadrature. More accurate evaluation methods that account for the correlations between the individual error sources have required Monte Carlo simulations or large matrix calculations that can take much longer to compute, particularly as the order of the AO system increases beyond a few hundred degrees of freedom. We describe an approach to evaluating AO system performance in the spatial-frequency domain that is relatively computationally efficient but still accounts for many of the interactions between the fundamental error sources in AO. We exploit the fact that (in the limits of an infinite aperture and geometrical optics) all the basic wave-front propagation, sensing, and correction processes that describe the behavior of an AO system are spatial-filtering operations in the Fourier domain. Essentially all classical wave-front control algorithms and evaluation formulas are expressed in terms of these filters and may therefore be evaluated one spatial-frequency component at a time. Performance estimates for very-high-order AO systems may be obtained in 1 to 2 orders of magnitude less time than needed when detailed simulations or analytical models in the spatial domain are used, with a relative discrepancy of 5% to 10% for typical sample problems.