Multi-conjugate Adaptive Optics for the Swedish ELT Investigation of the Effects of Laser Guide Stars at a finite Distance

In a recent paper we presented a procedure for optimal analytical control of several deformable mirrors based on only the measured guide star wavefronts. The expected performance was evaluated for various guide star configurations using up to three deformable mirrors. For simplicity the guide stars were assumed to be infinitely distant. In this paper we address the effect of using laser guide stars in 90 km altitude for the wavefront measurements. As a consequence of this, the image of a distant science star will not be adequately corrected by counteracting the wavefront error measured by a laser guide star in the same direction as the science star. This problem will be more pronounced the larger is the telescope diameter. We present a modified version of the mirror control algorithm taking the finite distance of the laser guide stars into account and some preliminary performance evaluations with special emphasis on the consequences for ELTs.