Effects of the LGS geometry on the Shack-Hartmann wavefront sensor and the Pyramid wavefront sensor

In this work we study the effects of the laser guide star (LGS) on the measurements of both the Shack-Hartmann wavefront sensor (SHWFS) and the Pyramid wavefront sensor (PWFS). We started by describing the LGS geometry and the general effects on each wavefront sensor. Then, we introduced a statistical analysis to predict the centroiding variance for the SHWFS when using an LGS, which we tested for read-out noise and photon noise. We found good agreement between end-to-end simulations and the predictions of the model. We found that the centroiding variance, as expected, follows closely the elongation of the LGS, with the X and Y centroiding evolving each according to the LGS geometry. For the PWFS, we used a convolutional model to compute sensitivity maps. With these maps we could observe that the size of the LGS greately decreases the sensitivity in the low frequencies. We could also obtain a better definition of the size of the LGS, which takes into account the depth of field of the telescope, which can be used to predict the sensitivity of the instrument by computing an equivalent modulation radius equivalent to the LGS size.