Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics

The Pyramidic wavefront has a very interesting property: its SNR has a sign ificant increase once the: loop is closed, similarly to the Curvature Senso r and contrasting with the Shack-Hartmann one. This result in a gain compar ing the Pyramidic and the Shack-Hartmann sensors. It is shown by a combinat ion of descriptive and numerical arguments that this gain is of the order o f (D/r(0))(2) for low order modes and drift linearly down to the Shack-Hart mann case at the highest spatial frequency sampled. A discussion of the ove rall gain (including a first estimation of the magnitude gain) and procedur es to force a true system to work in closed loop fashion are sketched.