  The use of artificial Laser Guide Stars (LGS) is planned for the new
generation of giant segmented mirror telescopes, 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. In this paper, we compute the expected performance for a
Pyramid WaveFront Sensor (PWFS) using a LGS for a 40 m telescope affected by
photon noise, and also extend the analysis to a flat 2D object as reference. 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 means of computing optical gains using a convolutional model. We
found 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, rather than
the thickness of the sodium layer was identified as the limiting factor. 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. We conclude that knife-edge-like wavefront sensors, such as the
PWFS, are not recommended for their use with LGS for a 40 m telescope, as they
will operate in a low-performance regime, given the size of the extended
object.
