Laser guide stars for extremely large telescopes: efficient Shack-Hartmann wavefront sensor design using the weighted centre-of-gravity algorithm

Over the last few years increasing consideration has been given to the study of laser guide stars (LGS) for the measurement of the disturbance introduced by the atmosphere in optical and near-infrared (near-IR) astronomical observations from the ground. A possible method for the generation of a LGS is the excitation of the sodium layer in the upper atmosphere at approximately 90 km of altitude. Since the sodium layer is approximately 10 km thick, the artificial reference source looks elongated, especially when observed from the edge of a large aperture. The spot elongation strongly limits the performance of the most common wavefront sensors. The centroiding accuracy in a Shack-Hartmann wavefront sensor, for instance, decreases proportionally to the elongation (in a photon noise dominated regime). To compensate for this effect, a straightforward solution is to increase the laser power, i.e. to increase the number of detected photons per subaperture. The scope of the work presented in this paper is twofold: an analysis of the performance of the weighted centre of gravity algorithm for centroiding with elongated spots and the determination of the required number of photons to achieve a certain average wavefront error over the telescope aperture.