Optimized modal tomography in adaptive optics

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.