Differential focal anisoplanatism in laser guide star wavefront sensing on extremely large telescopes

Laser guide stars (LGSs) aim at increasing the sky coverage of adaptive optics (AO) as this is highly restricted when using only natural guide stars. With such three-dimensional extended objects, spot elongation may limit the measurement accuracy of wavefronts. We evaluate the effect of differential focal anisoplanatism, induced solely by the longitudinal extension of a side-launched LGS, on the slope measurements performed by a Shack-Hartmann for a 40 m class telescope. We also take this effect into account in the wavefront reconstruction and derive estimations of the resulting wavefront error in a multi-LGS AO system. We find an error of 100 nm in the worst case at the subaperture level and a small error of the order of 10 nm for six LGSs after wavefront reconstruction.