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Sensitivity of a pyramidic Wave Front sensor in closed loop Adaptive Optics

The Pyramidic wavefront has a very interesting property: its SNR has a significant increase once the loop is closed, similarly to the Curvature Sensor and contrasting with the Shack-Hartmann one. This result in a gain comparing the Pyramidic and the Shack-Hartmann sensors. It is shown by a combination of descriptive and numerical arguments that this gain is of the order of (D/r0) for low order modes and drift linearly down to the Shack–Hartmann case at the highest spatial frequency sampled. A discussion of the overall gain (including a first estimation of the magnitude gain) and procedures to force a true system to work in closed loop fashion are sketched.