Compared performance of different centroiding algorithms for high–pass filtered laser guide star Shack–Hartmann wavefront sensors Olivier Lardi`erea , Rodolphe Conana , Richard Clareb , Colin Bradleya and Norbert Hubinb a Adaptive
Optics Laboratory, Mechanical Engineering Department, University of Victoria, 3800 Finnerty Road, Victoria, BC, V8W 3P6, Canada; b European Southern Observatory, Karl–Schwarzschild-Strasse 2, D-85748 Garching bei M¨ unchen, Germany ABSTRACT
Variations of the sodium layer altitude and atom density profile induce errors on laser–guide–star (LGS) adaptive optics systems. These errors must be mitigated by (i), optimizing the LGS wavefront sensor (WFS) and the centroiding algorithm, and (ii), by adding a high–pass filter on the LGS path and a low–bandwidth natural–guide– star WFS. In the context of the ESO E–ELT project, five centroiding algorithms, namely the centre–of–gravity (CoG), the weighted CoG, the matched filter, the quad-cell and the correlation, have been evaluated in closedloop on the University of Victoria LGS wavefront sensing test bed. Each centroiding algorithm performance is compared for a central versus side–launch laser, different fields of view, pixel sampling, and LGS flux. Keywords: laser guide star, wavefront sensing, extremely large telescopes
1. INTRODUCTION Sodium laser guide stars (LGS) increase the sky coverage of adaptive optics (AO) systems. However, temporal variations of the sodium layer altitude and density profile induce errors on centroids of the elongated LGS spots of Shack–Hartmann wavefront sensors (SH–WFSs), especially for extremely large telescopes (ELTs) as the spot elongation increases with the telescope diameter. In closed loop AO systems, these centroiding errors generate spurious aberrations on the science path, usually termed LGS aberrations. The three main sources of LGS aberrations are the sodium layer itself, the optical system and the LGS WFS (Tab. 1). The aberrations induced by the sodium layer variations are a focus error, and a tip/tilt error if the laser is launched from the side of the primary mirror. These tip/tilt and focus errors are due to the variations of the mean altitude of the sodium layer, resulting either from a vertical shift of the whole layer or from a more subtle variation of the sodium density profile. These errors must be negated optically with a trombone (zoom optics) in order to keep each LGS spot focused and centred on its subaperture.1 In theory, no other LGS aberrations should exist beyond the focus, but higher order LGS aberrations may be introduced by the imperfections of the optical system and the LGS WFS. The optical aberrations of the LGS path (from the primary mirror to the zoom optics) may indeed vary with the sodium layer distance. The finite field of view (FoV) of the LGS WFS and the finite number of pixel of the WFS camera induce a lot of LGS aberrations too. According to models and experimental results,2–5 there are two kinds of LGS aberrations beyond the focus generated by the LGS WFS with central–launch lasers: centro–symmetric (or spherical) aberrations (Z11 , Z22 , etc.) and square symmetric (or quad) aberrations (Z14 , Z26 , etc.). Centro–symmetric aberrations are due to a truncation of asymmetric elongated spots by a circular field–stop or a uniform pixel thresholding,5 while square symmetric aberrations are due to a truncation of the spots by the pixels boundaries of each subaperture, and the sampling of the spots by a finite number of square pixels. If the laser is launched from the side, non centro–symmetric aberrations arise in addition to spherical aberrations, such as astigmatism, coma and trefoil.3, 5 Corresponding author:
[email protected]
Table 1. LGS aberration origins, Zernike modes and mitigations.
Origins
Modes
Mitigations
Sodium layer altitude & vertical profile variations Optical system aberrations varying with LGS distance
tip/tilt∗ , focus
• Zoom optics (trombone)
varied
• Calibration
tip/tilt∗ , focus, astigmatism∗ , coma∗ , trefoil∗ , spherical, quad
• Optimization of the LGS WFS & centroiding algorithm • Filtering with a NGS low–bandwidth WFS
LGS WFS spot truncation & sampling errors ∗
for side–launch laser only
Consequently, all the LGS aberrations beyond the focus are instrumental artifacts and must be mitigated. A solution is to filter out the LGS aberrations in the temporal frequency domain with a low–bandwidth (LB) natural guide star (NGS) WFS and a high–pass filter (HPF) on the LGS path.6 Ideally, the frame rate of the LBWFS must be as low as possible to be able to work with fainter stars and keep the sky coverage unchanged. As the LBWFS frame rate is driven by the amount and the change rate the LGS aberrations, it is important to choose a LGS wavefront sensing system (i.e. sensor and centroiding algorithm) generating the lowest LGS aberrations. The aim of this study is to determine which centroiding algorithm provides the best performance on a SH– WFS with changing elongated spots. An optical test–bed has been built in the Adaptive Optics Laboratory of the University of Victoria (UVic) to evaluate the performance of a complete AO LGS system, including a LBWFS and a HPF LGS WFS, subjected to variable LGS and atmospheric disturbances.4 Five centroiding algorithms have been evaluated, namely the centre–of–gravity (CoG), the weighted CoG (WCoG),7 the quad–cell (QC), the matched filter (MF)8, 9 and the correlation (Cor),10–12 for a central–launch versus side–launch laser, different fields of view, pixel sampling, and LGS flux. Section 2 reviews the principle of the LGS aberration filter. Section 3 presents the five centroiding algorithms considered in this study, while Sec. 4 presents the UVic LGS wavefront sensing test bed used for this study. Section 5 shows some closed–loop results without turbulence in order to characterize the LGS aberrations generated by the different centroiding algorithms, and to check the efficiency of the LBWFS to filter the LGS aberration out. Results with turbulence and a LBWFS are presented in Sec. 6. Lastly, the performances of the algorithms are compared in Sec. 7.
2. PRINCIPLE OF THE LGS ABERRATION FILTER In this section, the theory of discrete–time control systems is applied to a LGS AO–system. Figure 1a shows a simplified scheme and the rejection transfer functions (RTFs) of a LGS AO system subjected to LGS disturbances D(z) induced by the fluctuations of the sodium layer density profile, with z the z–transform variable. As this disturbance is an artifact of the WFS, the LGS aberrations can be termed internal feedback disturbance. These aberrations are fully propagated on the science path C(z) and must be filtered out. As the fluctuations of the sodium layer are quite slow (typical timescale between 1 and 60s13, 14 ), the LGS disturbance corrupts only the low temporal frequencies of the science path. Consequently, a suitable filter can be designed to discard the low frequencies coming from the LGS WFS and to replace them by uncorrupted low frequencies coming from a NGS WFS. Concretely, this filter requires a high–pass filter (HPF) on the LGS path, and a low–bandwidth NGS WFS (LBWFS) in addition to the LGS WFS. In other words, we prefer to use the NGS WFS for sensing the quasi–static aberrations and the slow turbulence, but we still trust (and still need) the LGS WFS to sense the fast turbulence. The LBWFS acts as a low–pass filter (LPF) for the turbulence. The HPF has to be defined complementary to the LPF, such as HPF(z) + LPF(z) = 1. By doing so, both WFSs complete each other and work in tandem
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