Performance comparison of the Shack-Hartmann and pyramid wavefront sensors with a laser guide star for 40 m telescopes

Context. The new giant segmented mirror telescopes will use laser guide stars (LGS) for their adaptive optics (AO) systems. Two options to use as wavefront sensors (WFS) are the Shack-Hartmann wavefront sensor (SHWFS) and the pyramid wavefront sensor (PWFS). Aims. In this paper, we compare the noise performance of the PWFS and the SHWFS. We aim to find which of the two WFS is the best to use in a single or tomographic configuration. Methods. To compute the noise performance we extended a noise model developed for the PWFS to be used with the SHWFS. To do this, we expressed the centroiding algorithm of the SHWFS as a matrix-vector multiplication, which allowed us to use the statistics of noise to compute its propagation through the AO loop. We validated the noise model with end-to-end simulations for telescopes of 8 and 16 m in diameter. Results. For an AO system with only one WFS, we found that, given the same number of subapertures, the PWFS outperforms the SHWFS. For a 40 m telescope, the limiting magnitude of the PWFS is around 1 magnitude higher than the SHWFS. When using multiple WFS and a Generalized least squares estimator to combine the signal, our model predicts that in a tomographic system, the SHWFS performs better than the PWFS having a limiting magnitude 0.3 magnitudes higher. If using sub-electron RON detectors for the PWFS, then the performances are almost identical between the two WFSs Conclusions. We conclude that when using a single WFS with LGS, the PWFS is a better alternative than the SH. However, for a tomographic system, either would have almost the same performance.