Expected performance of the Pyramid wavefront sensor with a laser guide star for 40 m class telescopes

Context. The use of artificial laser guide stars (LGS) is planned for the new generation of giant segmented mirror telescopes in order to extend the sky coverage of their adaptive optics systems. The LGS, being a 3D object at a finite distance, will have a large elongation that will affect its use with the Shack-Hartmann (SH) wavefront sensor. Aims. In this paper, we compute the expected performance for a Pyramid WaveFront Sensor (PWFS) using an LGS for a 40 m telescope affected by photon noise, and also extend the analysis to a flat 2D object as reference. Methods. We developed a new way to discretize the LGS, and a new, faster method of propagating the light for any Fourier filtering wavefront sensors (FFWFS) when using extended objects. We present the use of a sensitivity model to predict the performance of a closed-loop adaptive optic system. We optimized a point-source-calibrated interaction matrix to accommodate the signal of an extended object by computing optical gains using a convolutional model. Results. We find that the sensitivity drop, given the size of the extended laser source, is large enough to make the system operate in a low-performance regime given the expected return flux of the LGS. The width of the laser beam is identified as the limiting factor, rather than the thickness of the sodium layer. Even an ideal, flat LGS will have a drop in performance due to the flux of the LGS, and small variations in the return flux will result in large variations in performance. Conclusions. We conclude that knife-edge-like wavefront sensors, such as the PWFS, are not recommended for use with LGS for a 40 m telescope, as they will operate in a low-performance regime, given the size of the extended object.