An approach is presented for evaluating the performance achieved by a closed-loop adaptive-optics system that is employed with an astronomical telescope. This method applies to systems incorporating one or several guide stars, a wave-front reconstruction algorithm that is equivalent to a matrix multiply, and one or several deformable mirrors that are optically conjugate to different ranges. System performance is evaluated in terms of residual mean-square phase distortion and the associated optical transfer function. This evaluation accounts for the effects of the atmospheric turbulence C"2(h) and wind profiles, the wave-front sensor and deformablemirror fitting error, the sensor noise, the control-system bandwidth, and the net anisoplanatism for a given constellation of natural and/or laser guide stars. Optimal wave-front reconstruction algorithms are derived that minimize the telescope's field-of-view-averaged residual mean-square phase distortion. Numerical results are presented for adaptive-optics configurations incorporating a single guide star and a single deformable mirror, multiple guide stars and a single deformable mirror, or multiple guide stars and two deformable mirrors.

The complexity of computing conventional matrix multiply wave-front reconstructors scales as O(n3) for most adaptive optical (AO) systems, where n is the number of deformable mirror (DM) actuators. This is impractical for proposed systems with extremely large n. It is known that sparse matrix methods improve this scaling for least-squares reconstructors, but sparse techniques are not immediately applicable to the minimum-variance reconstructors now favored for multiconjugate adaptive optical (MCAO) systems with multiple wave-front sensors (WFSs) and DMs. Complications arise from the nonsparse statistics of atmospheric turbulence, and the global tip/tilt WFS measurement errors associated with laser guide star (LGS) position uncertainty. A description is given of how sparse matrix methods can still be applied by use of a sparse approximation for turbulence statistics and by recognizing that the nonsparse matrix terms arising from LGS position uncertainty are low-rank adjustments that can be evaluated by using the matrix inversion lemma. Sample numerical results for AO and MCAO systems illustrate that the approximation made to turbulence statistics has negligible effect on estimation accuracy, the time to compute the sparse minimum-variance reconstructor for a conventional natural guide star AO system scales as O(n3/2) and is only a few seconds for n = 3500, and sparse techniques reduce the reconstructor computations by a factor of 8 for sample MCAO systems with 2417 DM actuators and 4280 WFS subapertures. With extrapolation to 9700 actuators and 17,120 subapertures, a reduction by a factor of approximately 30 or 40 to 1 is predicted.

Multi-Conjugate Adaptive Optics (MCAO) holds the promise of moderate to large adaptively compensated field of view with uniform image quality. This paper is a first effort to analyse the fundamental limitations of such systems, and that are mainly related to the finite number of deformable mirrors and guide stars. We demonstrate that the ultimate limitation is due to the vertical discretization of the correction. This effect becomes more severe quite rapidly with increasing compensated field of view or decreasing wavelength, but does not depend at first order on the telescope aperture. We also discuss limitations associated with the use of laser guide stars and ELT related issues.

The Gemini Multi-conjugate adaptive optics System (GeMS) at the Gemini South telescope in Cerro Pachón is the first sodium-based multi-Laser Guide Star (LGS) adaptive optics system. It uses five LGSs and two deformable mirrors to measure and compensate for atmospheric distortions. The GeMS project started in 1999, and saw first light in 2011. It is now in regular operation, producing images close to the diffraction limit in the near infrared, with uniform quality over a field of view of two square arcminutes. The present paper (I) is the first one in a two-paper review of GeMS. It describes the system, explains why and how it was built, discusses the design choices and trade-offs, and presents the main issues encountered during the course of the project. Finally, we briefly present the results of the system first light.

Laser guide star (LGS) atmospheric tomography is described in the literature as integrated minimum-variance tomographic wavefront reconstruction from a concatenated wavefront-sensor measurement vector consisting of many high-order, tip/tilt (TT)-removed LGS measurements, supplemented by a few low-order natural guide star (NGS) components essential to estimating the TT and tilt anisoplanatism (TA) modes undetectable by the TT-removed LGS wavefront sensors (WFSs). The practical integration of these NGS WFS measurements into the tomography problem is the main subject of this paper. A split control architecture implementing two separate control loops driven independently by closed-loop LGS and NGS measurements is proposed in this context. Its performance is evaluated in extensive wave optics Monte Carlo simulations for the Thirty Meter Telescope (TMT) LGS multiconjugate adaptive optics (MCAO) system, against the delivered performance of the integrated control architecture. Three iterative algorithms are analyzed for atmospheric tomography in both cases: a previously proposed Fourier domain preconditioned conjugate gradient (FDPCG) algorithm, a simple conjugate gradient (CG) algorithm without preconditioning, and a novel layer-oriented block Gauss-Seidel conjugate gradient algorithm (BGS-CG). Provided that enough iterations are performed, all three algorithms yield essentially identical closed-loop residual RMS wavefront errors for both control architectures, with the caveat that a somewhat smaller number of iterations are required by the CG and BGS-CG algorithms for the split approach. These results demonstrate that the split control approach benefits from (i) a simpler formulation of minimum-variance atmospheric tomography allowing for algorithms with reduced computational complexity and cost (processing requirements), (ii) a simpler, more flexible control of the NGS-controlled modes, and (iii) a reduced coupling between the LGS-and NGS-controlled modes. Computation and memory requirements for all three algorithms are also given for the split control approach for the TMT LGS AO system and appear feasible in relation to the performance specifications of current hardware technology.

We introduce a multigrid preconditioned conjugate-gradient (MGCG) iterative scheme for computing open-loop wave-front reconstructors for extreme adaptive optics systems. We present numerical simulations for a 17-m class telescope with n = 48756 sensor measurement grid points within the aperture, which indicate that our MGCG method has a rapid convergence rate for a wide range of subaperture average slope measurement signal-to-noise ratios. The total computational cost is of order n log n. Hence our scheme provides for fast wave-front simulation and control in large-scale adaptive optics systems.

Adaptive optics (AO) is a technology used in ground-based astronomy to correct for the wavefront aberrations and loss of image quality caused by atmospheric turbulence. Provided some difficult technical problems can be overcome, AO will enable future astronomers to achieve nearly diffractionlimited performance with the extremely large telescopes that are currently under development, thereby greatly improving spatial resolution, spectral resolution and observing efficiency which will be achieved. The goal of this topical review is to present to the inverse problems community a representative sample of these problems. In this review, we first present a tutorial overview of the mathematical models and techniques used in current AO systems. We then examine in detail the following topics: laser guidestar adaptive optics, multi-conjugate and multi-object adaptive optics, high-contrast imaging and deformable mirror modeling and parameter identification.

Multiconjugate adaptive optics (MCAO) is a technique for correcting turbulence-induced phase distortions in three dimensions instead of two, thereby greatly expanding the corrected field of view of an adaptive optics system. This is accomplished with use of multiple deformable mirrors conjugate to distinct ranges in the atmosphere, with actuator commands computed from wave-front sensor (WFS) measurements from multiple guide stars. Laser guide stars (LGSs) must be used (at least for the forseeable future) to achieve a useful degree of sky coverage in an astronomical MCAO system. Much as a single LGS cannot be used to measure overall wave-front tilt, a constellation of multiple LGSs at a common range cannot detect tilt anisoplanatism. This error alone will significantly degrade the performance of a MCAO system based on a single tilt-only natural guide star (NGS) and multiple tilt-removed LGSs at a common altitude. We present a heuristic, low-order model for the principal source of tilt anisoplanatism that suggests four possible approaches to eliminating this defect in LGS MCAO: (i) tip/tilt measurements from multiple NGS, (ii) a solution to the LGS tilt uncertainty problem, (iii) additional higher-order WFS measurements from a single NGS, or (iv) higher-order WFS measurements from both sodium and Rayleigh LGSs at different ranges. Sample numerical results for one particular MCAO system configuration indicate that approach (ii), if feasible, would provide the highest degree of tilt anisoplanatism compensation. Approaches (i) and (iv) also provide very useful levels of performance and do not require unrealistically low levels of WFS measurement noise. For a representative set of parameters for an 8-m telescope, the additional laser power required for approach (iv) is on the order of 2 W per Rayleigh LGS.