Nouvelle génération d’optique adaptative pour l’astronomie. Problème Inverse à grands nombres de degrés de liberté

New generation adaptive optics for astronomy. An inverse problem with large number of degrees of freedom

Page:

319-327

OPEN ACCESS

Abstract:

Observations of fainter and fainter astrophysical objects in the universe require an increasing flux-collecting area, which means larger and larger telescopes. However, the optical turbulence in the atmosphere damages the resolution of the images provided by the ground-based telescopes. In order to combine high sensitivity and angular resolution large ground-based telescopes are nowadays associated with adaptive optics systems (OA)[1]. OA must provide real-time correction of the atmospherical perturbation thanks to a servo-loop system including at least one wavefront sensor (AFO), a controller and one deformable mirror (MD) to apply the compensation, as illustrated by figure 1.

The OA design must evolve for the future telescopes with mirror diameters of several tenths of meters planned for 2017. On the one hand, the sensed part of the wavefront (WF) will not always match the part to be corrected for the astrophysical observations, so that the criterion of performance should be expressed in the WF space. On the other hand, the next generation of OA will have to cope with several thousands of actuators, to be controlled in real-time, which requires new and fast algorithms.

The outline of this paper can be decomposed in three parts. A first section introduces the statistics and the particularities of the studied signals in OA. Next, the optimization of a criterion related to astrophysical images quality leads to a command expression including a maximum a posteriori reconstructor [3] and an internal model control [4].

A fast iterative algorithm, FrIM [2], involving a fractal preconditioning is then used to compute in real-time such a control.

We consider here a classical OA system as illustrated by figure 1. From Kolmogorov’s model of the turbulence in the atmosphere, the kinetic energy is transferred from large scale vortices towards the smallest structures. This leads to stochastic spatial and temporal variations of the refractive index of the air, distorting the incident WF and producing local phase delays on the pupil. The statistics of the WF in the pupil plane can be described by the structure function in equation (1), where <.>*ρ’*represents the expectation over coordinate vector *ρ’*, and r0 is Fried’s parameter which characterizes the turbulence strength [6]. The statistics of the WF perturbations w have fractal characteristics since they are invariant by a change of scale.

The WFs are sampled on a regular two-dimensional grid larger than the pupil, and are denoted by vectors **w** and **w ^{c}** in R

The aim of the OA is to control a multi-inputs multi-outputs system with several thousands of degrees of freedom

Image quality is assessed for the astrophysical observations thanks to the Strehl ratio, which is optimal for the minimum variance of the residual WF distortions in equation (5).

The details of the computation are studied for the most frequent design, which makes matrices

In order to apply

preconditioner. The estimation of

It must also be noticed that the coherence time of the turbulent WF is assumed to be greater than the loop period, so that the approximation

FrIM has already demonstrated to be the fastest WF reconstruction method for large N [2]. The turbulence dynamics leads to even less computations when the method is involved in a closed-loop OA. The number of iterations required to converge depends on both the signal to noise ratio [2] and the evolution speed of the turbulence. The iterative method in closed-loop OA starts with the best estimate

As a conclusion, the optimization of the closed-loop AO correction in the WF space led to a internal model control law involving a turbulent WF reconstruction. The computation of the command for large N is then possible in real-time thanks to the use of FrIM method for the reconstruction step. The closed-loop configuration accelerates the convergence of this iterative algorithm thanks to the temporal coherence of the turbulence. This method decreases the computational cost of the command of a factor of about 100 in comparison with the classical matrix-vector multiplication approach, on a system with N = 10

Les images astrophysiques issues des télescopes au sol sont dégradées par la turbulence de l’atmosphère terrestre. Un système d’Optique Adaptative (OA) doit corriger en temps réel ces perturbations. L’étude d’une nouvelle génération de télescopes, de plus de 30 mètres de diamètre, avec de nouveaux concepts d’OA ayant de 10

Keywords:

Adaptive optics, inverse problem, internal model control, conjugate gradients, preconditioner, fast algorithm

Optique adaptative, problème inverse, commande par modèle interne, gradients conjugués, préconditionnement, algorithme rapide

1. Introduction

2. Caractériser La Phase Perturbée Par La Turbulence De L’atmosphère

3. L’optimisation Du Critère

4. Algorithme Rapide FrIM, Et Simulations

5. Conclusion

References

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[5] V.I. TATARSKI, Wave propagation in a turbulent medium. Dover Publications, Inc. New York, 1961.

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