Radial thresholding to mitigate laser guide star aberrations on centre-of-gravity-based Shack–Hartmann wavefront sensors

O. Lardie`re et al. 0 0 AO Laboratory, Mechanical Engineering Department, University of Victoria , PO Box 3055 STN CSC, Victoria, BC , V8W 3P6, Canada A B S T R A C T Sodium laser guide stars (LGSs) are elongated sources due to the thickness and the finite distance of the sodium layer. The fluctuations of the sodium layer altitude and atom density profile induce errors on centroid measurements of elongated spots, and generate spurious optical aberrations in closed-loop adaptive optics (AO) systems. According to an analytical model and experimental results obtained with the University of Victoria LGS bench demonstrator, one of the main origins of these aberrations, referred to as LGS aberrations, is not the centreof-gravity (CoG) algorithm itself, but the thresholding applied on the pixels of the image prior to computing the spot centroids. A new thresholding method, termed 'radial thresholding', is presented here, cancelling out most of the LGS aberrations without altering the centroid measurement accuracy. 1 I N T R O D U C T I O N Sodium laser guide star (LGS) adaptive optics (AO) systems allow in theory a full sky coverage; however, there are several limitations. The artificial star is elongated due to the sodium layer thickness (about 10 km) and finite distance (90 km). Consequently, if the laser is launched from the secondary mirror holder, the spots of a Shack Hartmann wavefront sensor (SHWFS) are radially elongated from the centre to the edge of the pupil (Fig. 1). The spot elongation is proportional to the telescope diameter and can reach several arcseconds for extremely large telescopes (ELTs). Moreover, the sodium layer is not static, but fluctuates with a time-scale of about 1 min or less (Davis et al. 2006). Fluctuations of the sodium layer altitude and atom density profile induce errors on centroid measurements of elongated spots and generate spurious aberrations on the wavefront in closed-loop AO systems. These aberrations, referred to as LGS aberrations, can reach several hundred nanometres peak-to-valley (PtV) for ELTs with the classical centre-of-gravity (CoG) centroiding algorithm (Clare, van Dam & Bouchez 2007; Lardie`re et al. 2008). Some authors proposed new sophisticated centroiding algorithms to mitigate LGS aberrations, such as the matched filtering (Gilles & Ellerbroek 2008; Conan et al. 2009) or the correlation (Poyneer 2003; Thomas et al. 2006, 2008). However, thanks to the LGS-bench demonstrator built at the University of Victoria (UVic) for ELT projects (Lardie`re et al. 2008; Conan et al. 2009), we found out that one of the main sources of the LGS aberrations was not the CoG algorithm itself, but simply the threshold applied on the pixels of the SHWFS images before the centroid computation. Section 2 reviews the known possible origins of LGS aberrations. A model of the aberrations generated by thresholding is presented in Section 3, as well as a simple new thresholding method, termed radial thresholding, which mitigates most of LGS aberrations. The experimental results obtained with the UVic bench with the radial thresholding show that the CoG algorithm is still well suited for LGS wavefront sensing on ELTs (Section 4). 2 O R I G I N S O F L G S A B E R R AT I O N S If the LGS spots are radially elongated from the pupil centre, as shown on Fig. 1, the sodium layer fluctuations induce centrosymmetric aberrations, such as focus (Z4) and spherical aberrations (Z11, Z22, etc.), and also square symmetric aberrations, such as tetrafoils (Z14, Z26, etc.). The focus is due to a variation of the sodium layer altitude. This error is not an artefact and must be compensated by refocussing the LGS on the WFS with zoom optics and by updating the offsets of the LGS WFS with a natural guide star (NGS) focus sensor (Herriot et al. 2006). Non-common path errors of the LGS optical train, including the zoom optics, can vary with the sodium layer distance, i.e. the zenithal angle, and induce variable aberrations on the science path too. We assume that these systematic aberrations can be calibrated and virtually negated. Consequently, aberrations beyond focus are mainly artefacts of the wavefront sensing. According to a model from Clare et al. (2007) and to the first experimental results obtained with the UVic LGS bench (Lardie`re et al. 2008), the spherical aberrations arise due to a truncation of asymmetric LGS spots by a circular field-stop, while square symmetric aberrations are likely due to (i) a spot truncation by a square field-stop or by pixel boundaries, horizontally and vertically elongated spots being more truncated than diagonally elongated spots, (ii) a spot overlap for square-grid lenslet arrays, horizontally and vertically elongated spots being more prone to overlapping, and (iii) quad-cell or sampling effects on centroid measurements. Both kinds of LGS aberrations have been reproduced and characterized in laboratory on the UVic bench with a time series of 88 real sodium profiles (Fig. 6). Beyond the focus, the most significant LGS aberrations detected are the spherical aberration Z11 up to 100 nm PtV (30 nm rms), and the tetrafoil Z14 with 40 nm PtV (10 nm rms). Moreover, a correlation between the spherical aberration and the profile asymmetry was empirically established (Lardie`re et al. 2008). Square-symmetric aberrations, such as Z14, should be mitigated by using a polar-coordinate CCD array (Beletic et al. 2005; Thomas et al. 2008). However, we discovered later that Z11 mode disappears if no threshold was applied on the pixels of the LGS WFS images before the computation of centroids. The thresholding discards the two extremities of each elongated spot, and consequently truncates radially each spot, as an optical circular field-stop would do. The spot truncation caused by the field-stop is negligible compared to the truncation induced by the pixel thresholding if the field of view (FOV) of the outermost lenslets is wide enough to make an image of a 20-km-thick sodium profile. With such a large FOV and a polar-coordinate CCD array, as expected for the 30 m telescope (TMT) LGS AO facility (Ellerbroek et al. 2008), the thresholding is likely the main source of the LGS aberrations and deserves a specific study. 3 M O D E L L I N G T H E A B E R R AT I O N S I N D U C E D B Y T H R E S H O L D I N G Basically, a thresholding must be applied on the image pixels prior to computing the spot centroids in order to minimize the contribution of the detector read-out noise RON, or of the sky background. The thresholding is generally uniform over the pupil and is implemented as follows: It(x, y) = if I (x, y) Thres if I (x, y) < Thres , with I and It the raw and the thresholded images, respectively. Thres is the intensity level of the threshold expressed in detector counts, i.e. in analog-to-digital units (ADU). Generally, the threshold is defined from the read-out noise (at 3 RON for instance). The remainder of this section demonstra (...truncated)