Jean-Marc Conan

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.

Wave-front temporal spectra in high-resolution imaging through turbulence

A general formulation is given for the derivation of theoretical temporal power spectra of quantities related to turbulent wave-front phase. These temporal power spectra and their asymptotic power laws and cutoff frequencies are presented for various quantities of interest in the field of interferometry (differential piston), wave-front sensing (Shack–Hartmann and curvature sensor), adaptive optics (Zernike polynomials), and seeing monitoring (differential angle of arrival). We show that the differential piston spectrum has two cutoff frequencies and exhibits a very steep decrease at high frequencies. The curvature sensor is shown to be much less sensitive than the Shack–Hartmann sensor to the low temporal frequencies. A study of the Zernike temporal power spectra shows that their cutoff frequencies increase with the polynomial radial degree. Both single-layer and multilayer plane and spherical waves are considered. The effect of wind direction is also taken into account. We point out the influence of the cone effect on the temporal power spectra when Rayleigh or sodium laser guide stars are used for wave-front sensing. The cone effect results in a temporal decorrelation between natural and laser guide star wave fronts. Finally, we demonstrate that in adaptive optics systems low-order modes require higher servoloop bandwidths than do high-order modes in order for the residual variance to be balanced between the corrected modes. The same conclusion applies to fringe tracking in large telescope interferometers equipped with adaptive optics systems.

Mistral: a myopic edge-preserving image restoration method, with application to astronomical adaptive-optics-corrected long-exposure images

Deconvolution is a necessary tool for the exploitation of a number of imaging instruments. We describe a deconvolution method developed in a Bayesian framework in the context of imaging through turbulence with adaptive optics. This method uses a noise model that accounts for both photonic and detector noises. It additionally contains a regularization term that is appropriate for objects that are a mix of sharp edges and smooth areas. Finally, it reckons with an imperfect knowledge of the point-spread function (PSF) by estimating the PSF jointly with the object under soft constraints rather than blindly (i.e., without constraints). These constraints are designed to embody our knowledge of the PSF. The implementation of this method is called MISTRAL. It is validated by simulations, and its effectiveness is illustrated by deconvolution results on experimental data taken on various adaptive optics systems and telescopes. Some of these deconvolutions have already been used to derive published astrophysical interpretations.

Optimal wave-front reconstruction strategies for multiconjugate adaptive optics

We propose an optimal approach for the phase reconstruction in a large field of view (FOV) for multiconjugate adaptive optics. This optimal approach is based on a minimum-mean-square-error estimator that minimizes the mean residual phase variance in the FOV of interest. It accounts for the C2n profile in order to optimally estimate the correction wave front to be applied to each deformable mirror (DM). This optimal approach also accounts for the fact that the number of DMs will always be smaller than the number of turbulent layers, since the C2n profile is a continuous function of the altitude h. Links between this optimal approach and a tomographic reconstruction of the turbulence volume are established. In particular, it is shown that the optimal approach consists of a full tomographic reconstruction of the turbulence volume followed by a projection onto the DMs accounting for the considered FOV of interest. The case where the turbulent layers are assumed to match the mirror positions [model-approximation (MA) approach], which might be a crude approximation, is also considered for comparison. This MA approach will rely on the notion of equivalent turbulent layers. A comparison between the optimal and MA approaches is proposed. It is shown that the optimal approach provides very good performance even with a small number of DMs (typically, one or two). For instance, good Strehl ratios (greater than 20%) are obtained for a 4-m telescope on a 150-arc sec x 150-arc sec FOV by using only three guide stars and two DMs.

Optimal control, observers and integrators in adaptive optics.

The fundamental issue of residual phase variance minimization in adaptive optics (AO) loops is addressed here from a control engineering perspective. This problem, when suitably modeled using a state-space approach, can be broken down into an optimal deterministic control problem and an optimal estimation problem, the solution of which are a linear quadratic (LQ) control and a Kalman filter. This approach provides a convenient framework for analyzing existing AO controllers, which are shown to contain an implicit phase turbulent model. In particular, standard integrator-based AO controllers assume a constant turbulent phase, which renders them prone to the notorious wind-up effect.

First laboratory validation of vibration filtering with LQG control law for Adaptive Optics

We present a first experimental validation of vibration filtering with a Linear Quadratic Gaussian (LQG) control law in Adaptive Optics (AO). A quasi-pure mechanical vibration is generated on a classic AO bench and filtered by the control law, leading to an improvement of the Strehl Ratio and image stability. Vibration filtering may be applied to any AO system, but these results are of particular interest for eXtrem AO, and for instance for the SPHERE AO design, where high performance is required.

Retrieving parameters of the anisotropic refractive index fluctuations spectrum in the stratosphere from balloon-borne observations of stellar scintillation

Scintillation effects are not negligible in the stratosphere. We present a model based on a 3D model of anisotropic and isotropic refractive index fluctuations spectra that predicts scintillation rates within the so-called small perturbation approximation. Atmospheric observations of stellar scintillation made from the AMON-RA (AMON, Absorption par les Minoritaires Ozone et NO(x); RA, rapid) balloon-borne spectrometer allows us to remotely probe wave-turbulence characteristics in the stratosphere. Data reduction from these observations brings out values of the inner scale of the anisotropic spectrum. We find metric values of the inner scale that are compatible with space-based measurements. We find a major contribution of the anisotropic spectrum relative to the isotropic contribution. When the sight line plunges into the atmosphere, strong scintillation occurs as well as coupled chromatic refraction effects.

Myopic deconvolution of adaptive optics images by use of object and point-spread function power spectra

Adaptive optics systems provide a real-time compensation for atmospheric turbulence. However, the correction is often only partial, and a deconvolution is required for reaching the diffraction limit. The need for a regularized deconvolution is discussed, and such a deconvolution technique is presented. This technique incorporates a positivity constraint and some a priori knowledge of the object (an estimate of its local mean and a model for its power spectral density). This method is then extended to the case of an unknown point-spread function, still taking advantage of similar a priori information on the point-spread function. Deconvolution results are presented for both simulated and experimental data.

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.

On the optimal reconstruction and control of adaptive optical systems with mirror dynamics

In adaptive optics (AO) the deformable mirror (DM) dynamics are usually neglected because, in general, the DM can be considered infinitely fast. Such assumption may no longer apply for the upcoming Extremely Large Telescopes (ELTs) with DM that are several meters in diameter with slow and/or resonant responses. For such systems an important challenge is to design an optimal regulator minimizing the variance of the residual phase. In this contribution, the general optimal minimum-variance (MV) solution to the full dynamical reconstruction and control problem of AO systems (AOSs) is established. It can be looked upon as the parent solution from which simpler (used hitherto) suboptimal solutions can be derived as special cases. These include either partial DM-dynamics-free solutions or solutions derived from the static minimum-variance reconstruction (where both atmospheric disturbance and DM dynamics are neglected altogether). Based on a continuous stochastic model of the disturbance, a state-space approach is developed that yields a fully optimal MV solution in the form of a discrete-time linear-quadratic-Gaussian (LQG) regulator design. From this LQG standpoint, the control-oriented state-space model allows one to (1) derive the optimal state-feedback linear regulator and (2) evaluate the performance of both the optimal and the sub-optimal solutions. Performance results are given for weakly damped second-order oscillatory DMs with large-amplitude resonant responses, in conditions representative of an ELT AO system. The highly energetic optical disturbance caused on the tip/tilt (TT) modes by the wind buffeting is considered. Results show that resonant responses are correctly handled with the MV regulator developed here. The use of sub-optimal regulators results in prohibitive performance losses in terms of residual variance; in addition, the closed-loop system may become unstable for resonant frequencies in the range of interest.