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We propose ground-layer adaptive optics (GLAO) to improve the seeing on the 42?m European Extremely Large Telescope. Shack–Hartmann wavefront sensors (WFSs) with laser guide stars (LGSs) will experience significant spot elongation due to off-axis observation. This spot elongation influences the design of the laser launch location, laser power, WFS detector, and centroiding algorithm for LGS GLAO on an extremely large telescope. We show, using end-to-end numerical simulations, that with a noise-weighted matrix-vector-multiply reconstructor, the performance in terms of 50% ensquared energy (EE) of the side and central launch of the lasers is equivalent, the matched filter and weighted center of gravity centroiding algorithms are the most promising, and approximately 10×10 undersampled pixels are optimal. Significant improvement in the 50% EE can be observed with a few tens of photons/subaperture/frame, and no significant gain is seen by adding more than 200 photons/subaperture/frame. The LGS GLAO is not particularly sensitive to the sodium profile present in the mesosphere nor to a short-timescale (less than 100?s) evolution of the sodium profile. The performance of LGS GLAO is, however, sensitive to the atmospheric turbulence profile.

The subaperture images in Shack–Hartmann wavefront sensors are elongated when using sodium laser guide stars, due to the parallax effect and the nonzero thickness of the sodium layer. This elongation increases the measurement error in each subaperture and introduces a correlation between the measurements in the orthogonal x and y pixel axes. We show, via end-to-end numerical simulations, that including these errors in a matrix-vector-multiply maximum a posteriori reconstructor significantly improves the expected performance of a laser tomography adaptive optics system for the 42 m diameter European Extremely Large Telescope. In particular, including this noise-weighted reconstruction shows the detected photon requirement is similar for both central and side launch of the sodium lasers.