SAXO+[1] is the name of the proposed second-stage adaptive optics module to increase the performance of the Spectro-Polarimetric High-contrast Exoplanet REsearch instrument (SPHERE[2]) at the Very Large Telescope (Chile). This upgrade aims at improving the raw contrast (up to 10-5, goal 10-6) close to the optical axis, enabling the observation of fainter and redder targets. In order to define the main requirements for the second-stage module design, a few trade-offs need to be carried out. We propose in this paper the use of an analytical approach, based on spatial PSD analysis, to study some error terms affecting the SAXO+ system and to directly compute the AO residual phase screens. For this purpose, a second-stage correction filter in the spatial frequency domain has been implemented in PAOLA[3], a software tool for the analytical modeling of AO systems. The results are then validated by comparing them to the output of a full end-to-end simulation tool, COMPASS[4], running with the same input parameters.

[1] F. Vidal et Al, Adaptive Optics Systems VIII (2022)
[2] J.-L. Beuzit et al., Astronomy & Astrophysics 631, A155 (2019)
[3] L. Jolissaint, J. Europ. Opt. Soc. Rap. Public. 5 , 10055 (2010)
[4] F. Ferreira et al., 2018 HPCS, Orléans, France, pp. 180-187 (2018)