In the context of multiconjugate adaptive optics, the optimum linear estimation of a wave-front phase for a target object using the phases of several surrounding natural guide stars (NGS's) is studied. A Wiener-filtertype estimator is constructed. The minimum residual wave-front-phase (tomographic) error depends on the turbulence vertical profile, and for typical profiles it is almost insensitive to the presence of strong layers, contrary to current belief. Tomographic error is characterized by a new parameter ␦ K , equivalent profile thickness, which depends on the NGS number K (typically ␦ 5 ϭ 0.5 km). The angular radius of the NGS configuration must not exceed r 0 /␦ K. Exact profile knowledge is not required. When the optimized filters are constructed from some model profile, the loss of the field size is within 10% with respect to exact profile knowledge. Moreover, a method to measure turbulence profile using wave-front-sensor data is outlined. Noise propagation in the restoration algorithm is significant, but not dramatic. Noise increases with increasing size of NGS constellation. Practically, guide stars for tomography should be at least as bright as those for classical adaptive optics.

Abstract. The performance of modal Multi-Conjugate Adaptive Optics systems correcting a finite number of Zernike modes is studied using a second-order statistical analysis. Both natural and laser guide stars (GS) are considered. An optimized command matrix is computed from the covariances of atmospheric signals and noise, to minimize the residual phase variance averaged over the field of view. An efficient way to calculate atmospheric covariances of Zernike modes and their projections is found. The modal covariance code is shown to reproduce the known results on anisoplanatism and the cone effect with single GS. It is then used to study the error of wave-front estimation from several off-axis GSs (tomography). With increasing radius of the GS constellation Θ, the tomographic error increases quadratically at small Θ, then linearly at larger Θ when incomplete overlap of GS beams in the upper atmospheric layers provides the major contribution to this error, especially on low-order modes. It is demonstrated that the quality of turbulence correction with two deformable mirrors is practically independent of the conjugation altitude of the second mirror, as long as the command matrix is optimized for each configuration.

We study the performance of an adaptive optics (AO) system with four laser guide stars (LGSs) and a natural guide star (NGS). The residual cone effect with four LGSs is obtained by a numerical simulation. This method allows the adaptive optics system to be extended toward the visible part of the spectrum without tomographic reconstruction of three-dimensional atmospheric perturbations, resolving the cone effect in the visible. Diffraction-limited images are obtained with 17-arc ms precision in median atmospheric conditions at wavelengths longer than 600 nm. The gain achievable with such a system operated on an existing AO system is studied. For comparison, performance in terms of achievable Strehl ratio is also computed for a reasonable system composed of a 40 x 40 Shack-Hartmann wave-front sensor optimized for the I band. Typical errors of a NGS wave front are computed by use of analytical formulas. With the NGS errors and the cone effect, the Strehl ratio can reach 0.45 at 1.25 microm under good-seeing conditions with the Nasmyth Adaptive Optics System (NAOS; a 14 x 14 subpupil wave-front sensor) at the Very Large Telescope and 0.8 with a 40 x 40 Shack-Hartmann wave-front sensor.

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In this paper, we study the performance of an adaptive optics system with 4 laser guide stars (LGS) and a natural guide star (NGS), and compare it to the system with 1 LGS, both installed on an 8-rn telescope. Focus anisoplanatism is obtained with a numerical simulation. The typical errors of NGS wavefront are computed with analytical formulae. The entire system is studied to obtain its performance in terms of achievable Strehl ratio.This 4-LGS method allows to push adaptive optics system towards the visible part of the spectrum without tomographic reconstruction of 3D atmospheric perturbation. The cone effect is two times smaller with 4-LGS than with 1-LGS, allowing to reach almost a Strehl ratio of 0.5 at 500 nm. Considering the NGS errors and the focus anisoplanatism, the Strehl ratio (SR) can reach 0.45 at 1.25 im under good seeing (GS) conditions with the Nasmyth Adaptive Optics System (14 x 14 sub-pupils wavefront sensor) at the Very Large Telescope.