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 statistic al analysis. Both natural and laser guide stars (GS) are considered. An opt imized command matrix is computed from the covariances of atmospheric signa ls and noise, to minimize the residual phase variance averaged over the fie ld of view. An efficient way to calculate atmospheric covariances of Zernik e modes and their projections is found. The modal covariance code is shown to reproduce the known results on anisoplanatism and the cone effect with s ingle 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 constel lation Theta, the tomographic error increases quadratically at small Theta, then linearly at larger Theta when incomplete overlap of GS beams in the u pper atmospheric layers provides the major contribution to this error, espe cially on low-order modes. It is demonstrated that the quality of turbulenc e 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.