Optimal noise-weighted reconstruction with elongated Shack–Hartmann wavefront sensor images for laser tomography adaptive optics

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.