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Over the past decade, adaptive optics (AO) has become an established method for overcoming the effects of atmospheric turbulence on both astronomical imaging and spectroscopic observations. These systems are now beginning to make extensive use of laser guide star (LGS) techniques to improve performance and provide increased sky coverage. Sodium LGS AO employs one or more lasers at 589-nm wavelength to produce an artificial guide star through excitation of sodium atoms in the mesosphere (90 km altitude). Because of its dependence on the abundance and distribution of sodium atoms in the mesosphere, this approach has its own unique set of difficulties not seen with natural stars. The sodium layer exhibits time-dependent variations in density and altitude, and since it is at a finite range, the LGS images become elongated due to the thickness of the layer and the offset between the laser projection point and the subapertures of a Shack‐Hartmann wavefront sensor (SHWFS). Elongation causes the LGS image to be spread out resulting in a decrease in the signal-to-noise ratio which, in turn, leads to an increase in SHWFS measurement error and therefore an increased error in wavefront phase reconstruction. To address the problem of elongation, and also to provide a higher level of readout performance and reduced readout noise, a new type of charge-coupled device (CCD) is now under development for Shack‐Hartmann wavefront sensing called the polar coordinate CCD. In this device, discrete imaging arrays are provided in each SHWFS subaperture and the size, shape and orientation of each discrete imaging array are adjusted to optimally sample the LGS image. The device is referred to as the polar coordinate CCD because the location of each imager is defined by a polar coordinate system centred on the laser guide star projection point. This concept is especially suited to Extremely Large Telescopes (ELTs) where the effect of perspective elongation is a significant factor. In this paper, we evaluate the performance of centroiders based on this CCD geometry by evaluating the centroid error variance and also the linearity issues associated with LGS image sampling and truncation. We also describe how we will extend this work to address the problems presented by the time variability of the sodium layer and how this will impact SHWFS performance in LGS AO systems.

Analytical theory is combined with extensive numerical simulations to compare different flavours of centroiding algorithms: thresholding, weighted centroid, correlation, quad cell (QC). For each method, optimal parameters are defined in function of photon flux, readout noise and turbulence level. We find that at very low flux the noise of QC and weighted centroid leads the best result, but the latter method can provide linear and optimal response if the weight follows spot displacements. Both methods can work with average flux as low as 10 photons per subaperture under a readout noise of three electrons. At high-flux levels, the dominant errors come from non-linearity of response, from spot truncations and distortions and from detector pixel sampling. It is shown that at high flux, centre of gravity approaches and correlation methods are equivalent (and provide better results than QC estimator) as soon as their parameters are optimized. Finally, examples of applications are given to illustrate the results obtained in the paper.