Optimal wave-front reconstruction strategies for multiconjugate adaptive optics

Off-axis point spread function reconstruction for single conjugate adaptive optics

Modern giant segmented mirror telescopes (GSMTs) such as the Extremely Large Telescope, which is currently under construction, depend heavily on adaptive optics (AO) systems to correct for atmospheric distortions. However, a residual blur always remains in the astronomical images corrected by single conjugate AO (SCAO) systems due to fitting and bandwidth errors, which can mathematically be described by a convolution of the true image with a point spread function (PSF). Due to the nature of the turbulent atmosphere and its correction, the PSF is spatially varying, which is known as an anisoplanatic effect. The PSF serves, e.g., as a quality measure for science images and therefore needs to be known as accurately as possible. In this paper, we present an algorithm for PSF reconstruction from pupil-plane data in directions apart from the guide star direction in an SCAO system. Our algorithm is adapted to the needs of GSMTs focused on estimating the contribution of the anisoplanatic and generalized fitting error to the PSF. Results obtained in an end-to-end simulation tool show a qualitatively good reconstruction of the PSF compared to the PSF calculated directly from the simulated incoming wavefront as well as stable performance with respect to imprecise knowledge of atmospheric parameters.

Fast iterative tomographic wavefront estimation with recursive Toeplitz reconstructor structure for large-scale systems

Tomographic wavefront reconstruction is the main computational bottleneck to realize real-time correction for turbulence-induced wavefront aberrations in future laser-assisted tomographic adaptive-optics (AO) systems for ground-based giant segmented mirror telescopes because of its unprecedented number of degrees of freedom, N, i.e., the number of measurements from wavefront sensors. In this paper, we provide an efficient implementation of the minimum-mean-square error (MMSE) tomographic wavefront reconstruction, which is mainly useful for some classes of AO systems not requiring multi-conjugation, such as laser-tomographic AO, multi-object AO, and ground-layer AO systems, but is also applicable to multi-conjugate AO systems. This work expands that by Conan [Proc. SPIE9148, 91480R (2014)PSISDG0277-786X10.1117/12.2054472] to the multi-wavefront tomographic case using natural and laser guide stars. The new implementation exploits the Toeplitz structure of covariance matrices used in an MMSE reconstructor, which leads to an overall O(N log N) real-time complexity compared with O(N²) of the original implementation using straight vector-matrix multiplication. We show that the Toeplitz-based algorithm leads to 60 nm rms wavefront error improvement for the European Extremely Large Telescope laser-tomography AO system over a well-known sparse-based tomographic reconstruction; however, the number of iterations required for suitable performance is still beyond what a real-time system can accommodate to keep up with the time-varying turbulence.

Increasing the field of view of adaptive optics scanning laser ophthalmoscopy

An adaptive optics scanning laser ophthalmoscope (AO-SLO) set-up with two deformable mirrors (DM) is presented. It allows high resolution imaging of the retina on a 4°×4° field of view (FoV), considering a 7 mm pupil diameter at the entrance of the eye. Imaging on such a FoV, which is larger compared to classical AO-SLO instruments, is allowed by the use of the two DMs. The first DM is located in a plane that is conjugated to the pupil of the eye and corrects for aberrations that are constant in the FoV. The second DM is conjugated to a plane that is located ∼0.7 mm anterior to the retina. This DM corrects for anisoplanatism effects within the FoV. The control of the DMs is performed by combining the classical AO technique, using a Shack-Hartmann wave-front sensor, and sensorless AO, which uses a criterion characterizing the image quality. The retinas of four healthy volunteers were imaged in-vivo with the developed instrument. In order to assess the performance of the set-up and to demonstrate the benefits of the 2 DM configuration, the acquired images were compared with images taken in conventional conditions, on a smaller FoV and with only one DM. Moreover, an image of a larger patch of the retina was obtained by stitching of 9 images acquired with a 4°×4° FoV, resulting in a total FoV of 10°×10°. Finally, different retinal layers were imaged by shifting the focal plane.

Comparative Study of Neural Network Frameworks for the Next Generation of Adaptive Optics Systems

Many of the next generation of adaptive optics systems on large and extremely large telescopes require tomographic techniques in order to correct for atmospheric turbulence over a large field of view. Multi-object adaptive optics is one such technique. In this paper, different implementations of a tomographic reconstructor based on a machine learning architecture named "CARMEN" are presented. Basic concepts of adaptive optics are introduced first, with a short explanation of three different control systems used on real telescopes and the sensors utilised. The operation of the reconstructor, along with the three neural network frameworks used, and the developed CUDA code are detailed. Changes to the size of the reconstructor influence the training and execution time of the neural network. The native CUDA code turns out to be the best choice for all the systems, although some of the other frameworks offer good performance under certain circumstances.

Comparison of methods for the reduction of reconstructed layers in atmospheric tomography

For the new generation of extremely large telescopes (ELTs), the computational effort for adaptive optics (AO) systems is demanding even for fast reconstruction algorithms. In wide-field AO, atmospheric tomography, i.e., the reconstruction of turbulent atmospheric layers from wavefront sensor data in several directions of view, is the crucial step for an overall reconstruction. Along with the number of deformable mirrors, wavefront sensors and their resolution, as well as the guide star separation, the number of reconstruction layers contributes significantly to the numerical effort. To reduce the computational cost, a sparse reconstruction profile which still yields good reconstruction quality is needed. In this paper, we analyze existing methods and present new approaches to determine optimal layer heights and turbulence weights for the tomographic reconstruction. Two classes of methods are discussed. On the one hand, we have compression methods that downsample a given input profile to fewer layers. Among other methods, a new compression method based on discrete optimization of collecting atmospheric layers to subgroups and the compression by means of conserving turbulence moments is presented. On the other hand, we take a look at a joint optimization of tomographic reconstruction and reconstruction profile during atmospheric tomography, which is independent of any a priori information on the underlying input profile. We analyze and study the qualitative performance of these methods for different input profiles and varying fields of view in an ELT-sized multi-object AO setting on the European Southern Observatory end-to-end simulation tool OCTOPUS.

Efficient reconstruction method for ground layer adaptive optics with mixed natural and laser guide stars

The imaging quality of modern ground-based telescopes such as the planned European Extremely Large Telescope is affected by atmospheric turbulence. In consequence, they heavily depend on stable and high-performance adaptive optics (AO) systems. Using measurements of incoming light from guide stars, an AO system compensates for the effects of turbulence by adjusting so-called deformable mirror(s) (DMs) in real time. In this paper, we introduce a novel reconstruction method for ground layer adaptive optics. In the literature, a common approach to this problem is to use Bayesian inference in order to model the specific noise structure appearing due to spot elongation. This approach leads to large coupled systems with high computational effort. Recently, fast solvers of linear order, i.e., with computational complexity O(n), where n is the number of DM actuators, have emerged. However, the quality of such methods typically degrades in low flux conditions. Our key contribution is to achieve the high quality of the standard Bayesian approach while at the same time maintaining the linear order speed of the recent solvers. Our method is based on performing a separate preprocessing step before applying the cumulative reconstructor (CuReD). The efficiency and performance of the new reconstructor are demonstrated using the OCTOPUS, the official end-to-end simulation environment of the ESO for extremely large telescopes. For more specific simulations we also use the MOST toolbox.

First on-sky results of the CO-SLIDAR C(2)(n) profiler

COupled SLope and scIntillation Detection And Ranging (CO-SLIDAR) is a recent profiling method of the vertical distribution of atmospheric turbulence strength (C(2)(n) profile). It takes advantage of correlations of slopes and of scintillation, both measured with a Shack-Hartmann wavefront sensor on a binary star. In this paper, we present the improved CO-SLIDAR reconstruction method of the C(2)(n) profile and the first on-sky results of the CO-SLIDAR profiler. We examine CO-SLIDAR latest performance in simulation, taking into account the detection noise bias and estimating error bars along with the turbulence profile. The estimated C(2)(n) profiles demonstrate the accuracy of the CO-SLIDAR method, showing sensitivity to both low and high altitude turbulent layers. CO-SLIDAR is tested on-sky for the first time, on the 1.5 m MeO (Métrologie Optique) telescope at Observatoire de la Côte d'Azur (France). The reconstructed profiles are compared to turbulence profiles estimated from meteorological data and a good agreement is found. We discuss CO-SLIDAR's contribution in the C(2)(n) profilers' landscape and we propose some improvements of the instrument.

Solar tomography adaptive optics

Conventional solar adaptive optics uses one deformable mirror (DM) and one guide star for wave-front sensing, which seriously limits high-resolution imaging over a large field of view (FOV). Recent progress toward multiconjugate adaptive optics indicates that atmosphere turbulence induced wave-front distortion at different altitudes can be reconstructed by using multiple guide stars. To maximize the performance over a large FOV, we propose a solar tomography adaptive optics (TAO) system that uses tomographic wave-front information and uses one DM. We show that by fully taking advantage of the knowledge of three-dimensional wave-front distribution, a classical solar adaptive optics with one DM can provide an extra performance gain for high-resolution imaging over a large FOV in the near infrared. The TAO will allow existing one-deformable-mirror solar adaptive optics to deliver better performance over a large FOV for high-resolution magnetic field investigation, where solar activities occur in a two-dimensional field up to 60'', and where the near infrared is superior to the visible in terms of magnetic field sensitivity.

Feasibility study of a layer-oriented wavefront sensor for solar telescopes

Solar multiconjugate adaptive optics systems rely on several wavefront sensors, which measure the incoming turbulent phase along several field directions to produce a tomographic reconstruction of the turbulent phase. In this paper, we explore an alternative wavefront sensing approach that attempts to directly measure the turbulent phase present at a particular height in the atmosphere: a layer-oriented cross-correlating Shack-Hartmann wavefront sensor (SHWFS). In an experiment at the Dunn Solar Telescope, we built a prototype layer-oriented cross-correlating SHWFS system conjugated to two separate atmospheric heights. We present the data obtained in the observations and complement these with ray-tracing computations to achieve a better understanding of the instrument's performance and limitations. The results obtained in this study strongly indicate that a layer-oriented cross-correlating SHWFS is not a practical design to measure the wavefront at a high layer in the atmosphere.

Static and predictive tomographic reconstruction for wide-field multi-object adaptive optics systems

Multi-object adaptive optics (MOAO) systems are still in their infancy: their complex optical designs for tomographic, wide-field wavefront sensing, coupled with open-loop (OL) correction, make their calibration a challenge. The correction of a discrete number of specific directions in the field allows for streamlined application of a general class of spatio-angular algorithms, initially proposed in Whiteley et al. [J. Opt. Soc. Am. A15, 2097 (1998)], which is compatible with partial on-line calibration. The recent Learn & Apply algorithm from Vidal et al. [J. Opt. Soc. Am. A27, A253 (2010)] can then be reinterpreted in a broader framework of tomographic algorithms and is shown to be a special case that exploits the particulars of OL and aperture-plane phase conjugation. An extension to embed a temporal prediction step to tackle sky-coverage limitations is discussed. The trade-off between lengthening the camera integration period, therefore increasing system lag error, and the resulting improvement in SNR can be shifted to higher guide-star magnitudes by introducing temporal prediction. The derivation of the optimal predictor and a comparison to suboptimal autoregressive models is provided using temporal structure functions. It is shown using end-to-end simulations of Raven, the MOAO science, and technology demonstrator for the 8 m Subaru telescope that prediction allows by itself the use of 1-magnitude-fainter guide stars.

Kaczmarz algorithm for multiconjugated adaptive optics with laser guide stars

We recently introduced the Kaczmarz algorithm for solving the atmospheric tomography problem in multiconjugate adaptive optics (MCAO). This iterative method solves the problem significantly faster than the standard matrix vector multiplication. We present the algorithm as well as an extension, which includes the effects of laser guide stars, such as the cone effect, tip/tilt indetermination, and spot elongation. We show that we can successfully cope with these effects and that the algorithm is suited for an MCAO system for the future generation of extremely large telescopes.

Distributed Kalman filtering compared to Fourier domain preconditioned conjugate gradient for laser guide star tomography on extremely large telescopes

This paper discusses the performance and cost of two computationally efficient Fourier-based tomographic wavefront reconstruction algorithms for wide-field laser guide star (LGS) adaptive optics (AO). The first algorithm is the iterative Fourier domain preconditioned conjugate gradient (FDPCG) algorithm developed by Yang et al. [Appl. Opt.45, 5281 (2006)], combined with pseudo-open-loop control (POLC). FDPCG's computational cost is proportional to N log(N), where N denotes the dimensionality of the tomography problem. The second algorithm is the distributed Kalman filter (DKF) developed by Massioni et al. [J. Opt. Soc. Am. A28, 2298 (2011)], which is a noniterative spatially invariant controller. When implemented in the Fourier domain, DKF's cost is also proportional to N log(N). Both algorithms are capable of estimating spatial frequency components of the residual phase beyond the wavefront sensor (WFS) cutoff frequency thanks to regularization, thereby reducing WFS spatial aliasing at the expense of more computations. We present performance and cost analyses for the LGS multiconjugate AO system under design for the Thirty Meter Telescope, as well as DKF's sensitivity to uncertainties in wind profile prior information. We found that, provided the wind profile is known to better than 10% wind speed accuracy and 20 deg wind direction accuracy, DKF, despite its spatial invariance assumptions, delivers a significantly reduced wavefront error compared to the static FDPCG minimum variance estimator combined with POLC. Due to its nonsequential nature and high degree of parallelism, DKF is particularly well suited for real-time implementation on inexpensive off-the-shelf graphics processing units.

Increased sky coverage with optimal correction of tilt and tilt-anisoplanatism modes in laser-guide-star multiconjugate adaptive optics

Laser-guide-star multiconjugate adaptive optics (MCAO) systems require natural guide stars (NGS) to measure tilt and tilt-anisoplanatism modes. Making optimal use of the limited number of photons coming from such, generally dim, sources is mandatory to obtain reasonable sky coverage, i.e., the probability of finding asterisms amenable to NGS wavefront (WF) sensing for a predefined WF error budget. This paper presents a Strehl-optimal (minimum residual variance) spatiotemporal reconstructor merging principles of modal atmospheric tomography and optimal stochastic control theory. Simulations of NFIRAOS, the first light MCAO system for the thirty-meter telescope, using ~500 typical NGS asterisms, show that the minimum-variance (MV) controller delivers outstanding results, in particular for cases with relatively dim stars (down to magnitude 22 in the H-band), for which low-temporal frame rates (as low as 16 Hz) are required to integrate enough flux. Over all the cases tested ~21 nm rms median improvement in WF error can be achieved with the MV compared to the current baseline, a type-II controller based on a double integrator. This means that for a given level of tolerable residual WF error, the sky coverage is increased by roughly 10%, a quite significant figure. The improvement goes up to more than 20% when compared with a traditional single-integrator controller.

An ANN-based smart tomographic reconstructor in a dynamic environment

In astronomy, the light emitted by an object travels through the vacuum of space and then the turbulent atmosphere before arriving at a ground based telescope. By passing through the atmosphere a series of turbulent layers modify the light's wave-front in such a way that Adaptive Optics reconstruction techniques are needed to improve the image quality. A novel reconstruction technique based in Artificial Neural Networks (ANN) is proposed. The network is designed to use the local tilts of the wave-front measured by a Shack Hartmann Wave-front Sensor (SHWFS) as inputs and estimate the turbulence in terms of Zernike coefficients. The ANN used is a Multi-Layer Perceptron (MLP) trained with simulated data with one turbulent layer changing in altitude. The reconstructor was tested using three different atmospheric profiles and compared with two existing reconstruction techniques: Least Squares type Matrix Vector Multiplication (LS) and Learn and Apply (L + A).

Using artificial neural networks for open-loop tomography

Modern adaptive optics (AO) systems for large telescopes require tomographic techniques to reconstruct the phase aberrations induced by the turbulent atmosphere along a line of sight to a target which is angularly separated from the guide sources that are used to sample the atmosphere. Multi-object adaptive optics (MOAO) is one such technique. Here, we present a method which uses an artificial neural network (ANN) to reconstruct the target phase given off-axis references sources. We compare our ANN method with a standard least squares type matrix multiplication method and to the learn and apply method developed for the CANARY MOAO instrument. The ANN is trained with a large range of possible turbulent layer positions and therefore does not require any input of the optical turbulence profile. It is therefore less susceptible to changing conditions than some existing methods. We also exploit the non-linear response of the ANN to make it more robust to noisy centroid measurements than other linear techniques.

Generalized aliasing and its implications in modal gain optimization for multi-conjugate adaptive optics

The error of generalized aliasing associated with the limited sampling of the atmospheric turbulence volume due to the finite number of wavefront sensing directions in wide-field-of-view adaptive optics is formally defined. Following a modal approach, we extend the direct problem formulation of star-oriented multi-conjugate adaptive optics (MCAO) to model and quantify this error analytically. We show that the turbulence estimation with the least-squares reconstructor is subject to strong generalized aliasing, in particular affecting the badly seen modes, whereas with the minimum-mean-square-error reconstructor the estimation is little affected. Finally, we show that the application of modal gain optimization techniques in closed-loop MCAO systems is jeopardized by the generalized aliasing error.

Real-time turbulence profiling with a pair of laser guide star Shack-Hartmann wavefront sensors for wide-field adaptive optics systems on large to extremely large telescopes

Real-time turbulence profiling is necessary to tune tomographic wavefront reconstruction algorithms for wide-field adaptive optics (AO) systems on large to extremely large telescopes, and to perform a variety of image post-processing tasks involving point-spread function reconstruction. This paper describes a computationally efficient and accurate numerical technique inspired by the slope detection and ranging (SLODAR) method to perform this task in real time from properly selected Shack-Hartmann wavefront sensor measurements accumulated over a few hundred frames from a pair of laser guide stars, thus eliminating the need for an additional instrument. The algorithm is introduced, followed by a theoretical influence function analysis illustrating its impulse response to high-resolution turbulence profiles. Finally, its performance is assessed in the context of the Thirty Meter Telescope multi-conjugate adaptive optics system via end-to-end wave optics Monte Carlo simulations.

Wide field adaptive optics laboratory demonstration with closed-loop tomographic control

HOMER, the new bench developed at ONERA devoted to wide field adaptive optics (WFAO) laboratory research, has allowed the first experimental validations of multi-conjugate adaptive optics (MCAO) and laser tomography adaptive optics (LTAO) concepts with a linear quadratic Gaussian (LQG) control approach. Results obtained in LTAO in closed loop show the significant gain in performance brought by LQG control, which allows tomographic reconstruction. We present a calibration and model identification strategy. Experimental results are shown to be consistent with end-to-end simulations. These results are very encouraging and demonstrate robustness of performance with respect to inevitable experimental uncertainties. They represent a first step for the study of very large telescope (VLT) and extremely large telescopes (ELT) instruments.

Linear quadratic Gaussian control for adaptive optics and multiconjugate adaptive optics: experimental and numerical analysis

We present a comprehensive analysis of the linear quadratic Gaussian control approach applied to adaptive optics (AO) and multiconjugated AO (MCAO) based on numerical and experimental validations. The structure of the control law is presented and its main properties discussed. We then propose an extended experimental validation of this control law in AO and a simplified MCAO configuration. Performance is compared with end-to-end numerical simulations. Sensitivity of the performance regarding tuning parameters is tested. Finally, extension to full MCAO and laser tomographic AO (LTAO) through numerical simulation is presented and analyzed.

Tomographic reconstruction for wide-field adaptive optics systems: Fourier domain analysis and fundamental limitations

Several wide-field-of-view adaptive optics (WFAO) concepts such as multi-conjugate AO (MCAO), multi-object AO (MOAO), and ground-layer AO (GLAO) are currently being studied for the next generation of Extremely Large Telescopes (ELTs). All these concepts will use atmospheric tomography to reconstruct the turbulent-phase volume. In this paper, we explore different reconstruction algorithms and their fundamental limitations, conducting this analysis in the Fourier domain. This approach allows us to derive simple analytical formulations for the different configurations and brings a comprehensive view of WFAO limitations. We then investigate model and statistical errors and their effect on the phase reconstruction. Finally, we show some examples of different WFAO systems and their expected performance on a 42 m telescope case.

Distributed modal command for a two-deformable-mirror adaptive optics system

The design of future single-altitude conjugated adaptive optics (AO) systems may include at least two deformable mirrors (DMs) instead of one as in the current AO system. Each DM will have to correct for a specific spatial frequency range. A method is presented to derive a DM modal basis based on the influence functions of the DM. The modal bases are derived such that they are orthogonal to a given set of modes that restrict the DM correction to a spatial frequency domain. The modal bases have been tested on the woofer-tweeter test bench at the University of Victoria. It has been shown that the rms amplitude of the woofer DM and tweeter DM stroke can be reduced by factors of 3 and 9, respectively, when making the transition from a zonal-driven closed loop to a modal-driven closed loop with the same performance in both cases.

Performance study of Kalman filter controller for multiconjugate adaptive optics

We compare the performance of the Kalman filter (KF)-based and the minimum variance (MV) control algorithms for a zonal adaptive optics with a phase temporal prediction step included for effective compensation of the errors attributable to latencies in the system. The main goal is to evaluate the performance achievable by the computationally more expensive KF approach, which explicitly accounts for the atmospheric turbulence temporal behavior through a first-order autoregressive evolution model, and the simpler MV algorithm, with and without temporal prediction. For a representative example, the Gemini-South 8 m telescope multiconjugate adaptive optics system performance of the KF and the MV controllers has been compared with respect to their turbulence compensation capability. We show that the KF algorithm, as expected, shows superior performance to that of the MV algorithm, especially for extremely low sampling rates and large control latencies. We also show that for moderate control latencies the MV algorithm with a temporal prediction step added to it approaches the performance of the KF technique.

Optimization of star-oriented and layer-oriented wavefront sensing concepts for ground layer adaptive optics

Multiconjugate adaptive optics is one of the major challenges in adaptive optics. It requires the measurement of the volumic distribution of the turbulence. Two wavefront sensing (WFS) concepts have been proposed to perform the wavefront analysis for such systems: the star-oriented and layer-oriented approaches. We give a performance analysis and a comparison of these two concepts in the framework of the simplest of the multi-guide-star adaptive optics systems, that is, ground layer adaptive optics. A phase-related criterion is proposed to assess analytically the performance of both concepts. This study highlights the main advantages and drawbacks of each WFS concept and shows how it is possible to optimize the concepts with respect to the signal to noise ratio on the phase measurement. A comparison of their optimized versions is provided and shows that one can expect very similar performance with the two optimized concepts.

Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography

By 'atmospheric tomography' we mean the estimation of a layered atmospheric turbulence profile from measurements of the pupil-plane phase (or phase gradients) corresponding to several different guide star directions. We introduce what we believe to be a new Fourier domain preconditioned conjugate gradient (FD-PCG) algorithm for atmospheric tomography, and we compare its performance against an existing multigrid preconditioned conjugate gradient (MG-PCG) approach. Numerical results indicate that on conventional serial computers, FD-PCG is as accurate and robust as MG-PCG, but it is from one to two orders of magnitude faster for atmospheric tomography on 30 m class telescopes. Simulations are carried out for both natural guide stars and for a combination of finite-altitude laser guide stars and natural guide stars to resolve tip-tilt uncertainty.

Scintillation and phase anisoplanatism in Shack-Hartmann wavefront sensing

Adaptive optics provides a real-time compensation for atmospheric turbulence that severely limits the resolution of ground-based observation systems. The correction quality relies on a key component, that is, the wavefront sensor (WFS). When observing extended sources, WFS precision is limited by anisoplanatism effects. Anisoplanatism induces a variation of the turbulent phase and of the collected flux in the field of view. We study the effect of this phase and scintillation anisoplanatism on wavefront analysis. An analytical expression of the error induced is given in the Rytov regime. The formalism is applied to a solar and an endoatmospheric observation. Scintillation effects are generally disregarded, especially in astronomical conditions. We shall prove that this approximation is not valid with extended objects.

Sky coverage estimates for adaptive optics systems from computations in Zernike space

A sky coverage model for laser guide star adaptive optics systems is proposed. The atmosphere is considered to consist of a finite number of phase screens, which are defined by Zernike basis polynomials, located at different altitudes. These phase screens are transformed to the aperture plane, where they are converted to laser and natural guide star wavefront sensing measurements. These transformations incorporate the cone effect due to guide stars at finite heights, anisoplanatism due to guide stars off axis with respect to the science object, and adaptive optics systems with multiple guide stars. The wavefront error is calculated tomographically with minimum variance estimators derived from the transformation matrices and the known statistical properties of the atmosphere. This sky coverage model provides fast Monte Carlo simulations over random natural guide star configurations, irrespective of telescope diameter. The Monte Carlo simulations outlined show that inclusion of a finite outer scale for the atmosphere significantly reduces the median wavefront error, that increasing the number of laser guide stars in the asterism reduces the median wavefront error, and that a larger natural guide star patrol field provides a smaller median wavefront error when there is a low star density in the field.

Tip-tilt reconstruction with a single dim natural guide star in multiconjugate adaptive optics with laser guide stars

A solution to the problem of detecting the tip-tilt modes in multiconjugate adaptive optics (MCAO) with laser guide stars (LGS) is presented. This solution requires the presence of only a single relatively dim natural guide star (NGS) within the reconstructed field of view (FoV). The dim NGS is used for the reconstruction of the tip-tilt modes on the entire FoV, while the tomographic reconstruction of second-order and higher-order modes is made possible by having an LGS constellation with LGSs at different heights. Due to the relatively low brightness required for the tip-tilt NGS and the large corrected FoV (as compared with the case of conventional adaptive optics) the presented solution provides a means to achieve near-diffraction-limited performance of a 10-m-class telescope in the near infrared over a large portion of the sky. Sky coverage calculations assuming median seeing conditions indicate that this technique could be applied to 75% (95%) of the sky, achieving corrections with an average Strehl ratio approximately 0.42(approximately 0.33) in the 2.2 microm K band across the 1.5' reconstructed FoV.

Geometric view of adaptive optics control

The objective of an astronomical adaptive optics control system is to minimize the residual wave-front error remaining on the science-object wave fronts after being compensated for atmospheric turbulence and telescope aberrations. Minimizing the mean square wave-front residual maximizes the Strehl ratio and the encircled energy in pointlike images and maximizes the contrast and resolution of extended images. We prove the separation principle of optimal control for application to adaptive optics so as to minimize the mean square wave-front residual. This shows that the residual wave-front error attributable to the control system can be decomposed into three independent terms that can be treated separately in design. The first term depends on the geometry of the wave-front sensor(s), the second term depends on the geometry of the deformable mirror(s), and the third term is a stochastic term that depends on the signal-to-noise ratio. The geometric view comes from understanding that the underlying quantity of interest, the wave-front phase surface, is really an infinite-dimensional vector within a Hilbert space and that this vector space is projected into subspaces we can control and measure by the deformable mirrors and wave-front sensors, respectively. When the control and estimation algorithms are optimal, the residual wave front is in a subspace that is the union of subspaces orthogonal to both of these projections. The method is general in that it applies both to conventional (on-axis, ground-layer conjugate) adaptive optics architectures and to more complicated multi-guide-star- and multiconjugate-layer architectures envisaged for future giant telescopes. We illustrate the approach by using a simple example that has been worked out previously [J. Opt. Soc. Am. A 73, 1171 (1983)] for a single-conjugate, static atmosphere case and follow up with a discussion of how it is extendable to general adaptive optics architectures.

Closed-loop stability and performance analysis of least-squares and minimum-variance control algorithms for multiconjugate adaptive optics

Recent progress has been made to compute efficiently the open-loop minimum-variance reconstructor (MVR) for multiconjugate adaptive optics systems by a combination of sparse matrix and iterative techniques. Using spectral analysis, I show that a closed-loop laser guide star multiconjugate adaptive optics control algorithm consisting of MVR cascaded with an integrator control law is unstable. Tosolve this problem, a computationally efficient pseudo-open-loop control (POLC) method was recently proposed. I give a theoretical proof of the stability of this method and demonstrate its superior performance and robustness against misregistration errors compared with conventional least-squares control. This can be accounted for by the fact that POLC incorporates turbulence statistics through its regularization term that can be interpreted as spatial filtering, yielding increased robustness to misregistration. For the Gemini-South 8-m telescope multiconjugate system and for median Cerro Pachon seeing, the performance of POLC in terms of rms wave-front error averaged over a 1-arc min field of view is approximately three times superior to that of a least-squares reconstructor. Performance degradation due to 30% translational misregistration on all three mirrors is approximately a 30% increased rms wave-front error, whereas a least-squares reconstructor is unstable at such a misregistration level.

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.

Optimal control law for classical and multiconjugate adaptive optics

Classical adaptive optics (AO) is now a widespread technique for high-resolution imaging with astronomical ground-based telescopes. It generally uses simple and efficient control algorithms. Multiconjugate adaptive optics (MCAO) is a more recent and very promising technique that should extend the corrected field of view. This technique has not yet been experimentally validated, but simulations already show its high potential. The importance for MCAO of an optimal reconstruction using turbulence spatial statistics has already been demonstrated through open-loop simulations. We propose an optimal closed-loop control law that accounts for both spatial and temporal statistics. The prior information on the turbulence, as well as on the wave-front sensing noise, is expressed in a state-space model. The optimal phase estimation is then given by a Kalman filter. The equations describing the system are given and the underlying assumptions explained. The control law is then derived. The gain brought by this approach is demonstrated through MCAO numerical simulations representative of astronomical observation on a 8-m-class telescope in the near infrared. We also discuss the application of this control approach to classical AO. Even in classical AO, the technique could be relevant especially for future extreme AO systems.

Preconditioned conjugate gradient wave-front reconstructors for multiconjugate adaptive optics

Multiconjugate adaptive optics (MCAO) systems with 10(4)-10(5) degrees of freedom have been proposed for future giant telescopes. Using standard matrix methods to compute, optimize, and implement wavefront control algorithms for these systems is impractical, since the number of calculations required to compute and apply the reconstruction matrix scales respectively with the cube and the square of the number of adaptive optics degrees of freedom. We develop scalable open-loop iterative sparse matrix implementations of minimum variance wave-front reconstruction for telescope diameters up to 32 m with more than 10(4) actuators. The basic approach is the preconditioned conjugate gradient method with an efficient preconditioner, whose block structure is defined by the atmospheric turbulent layers very much like the layer-oriented MCAO algorithms of current interest. Two cost-effective preconditioners are investigated: a multigrid solver and a simpler block symmetric Gauss-Seidel (BSGS) sweep. Both options require off-line sparse Cholesky factorizations of the diagonal blocks of the matrix system. The cost to precompute these factors scales approximately as the three-halves power of the number of estimated phase grid points per atmospheric layer, and their average update rate is typically of the order of 10(-2) Hz, i.e., 4-5 orders of magnitude lower than the typical 10(3) Hz temporal sampling rate. All other computations scale almost linearly with the total number of estimated phase grid points. We present numerical simulation results to illustrate algorithm convergence. Convergence rates of both preconditioners are similar, regardless of measurement noise level, indicating that the layer-oriented BSGS sweep is as effective as the more elaborated multiresolution preconditioner.

Numerical simulations of multiconjugate adaptive optics wave-front reconstruction on giant telescopes

We present sample Monte Carlo simulation results to illustrate the trends in multiconjugate adaptive optics (MCAO) performance as the telescope aperture diameter increases from 8 to 32 m with all other first-order system parameters held constant. The MCAO system considered includes three deformable mirrors, a 1-arc min square field of view, and five wave-front-sensing references consisting of either natural guide stars or laser guide stars at a range of either 30 or 90 km. The rms residual wave-front error decreases slowly with increasing aperture diameter with natural guide stars, whereas performance degrades significantly with increasing aperture diameter for laser guide stars at 30 km if the number of guide stars is held fixed. Performance with laser guide stars at 90 km is a weak function of telescope aperture diameter in the range from 8 to 32 m, with rms wave-front errors no more than 20% greater than the corresponding natural guide-star case for the same level of wave-front sensor's measurement noise.

Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques

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.

Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction

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.