Abstract
The use of Fourier methods in wave-front reconstruction can significantly reduce the computation time for large telescopes with a high number of degrees of freedom. However, Fourier algorithms for discrete data require a rectangular data set which conform to specific boundary requirements, whereas wave-front sensor data is typically defined over a circular domain (the telescope pupil). Here we present an iterative Gerchberg routine modified for the purposes of discrete wave-front reconstruction which adapts the measurement data (wave-front sensor slopes) for Fourier analysis, fulfilling the requirements of the fast Fourier transform (FFT) and providing accurate reconstruction. The routine is used in the adaptation step only and can be coupled to any other Wiener-like or least-squares method. We compare simulations using this method with previous Fourier methods and show an increase in performance in terms of Strehl ratio and a reduction in noise propagation for a 40×40 SPHERE-like adaptive optics system. For closed loop operation with minimal iterations the Gerchberg method provides an improvement in Strehl, from 95.4% to 96.9% in K-band. This corresponds to ~ 40 nm improvement in rms, and avoids the high spatial frequency errors present in other methods, providing an increase in contrast towards the edge of the correctable band.
Highlights
The performance of large ground-based telescopes depends on the ability of adaptive optics (AO) systems to correct for phase distortions caused by atmospheric turbulence
To avoid the errors induced by the imposition of the pupil the wave-front sensor data must be extended outside the aperture, conforming to the periodic conditions required by the Fast Fourier Transform (FFT) and the characteristics of the wave-front sensor data [6, 15]
We propose a method for extending wave-front sensor data outside of the aperture imposed by the telescope pupil based on an iterative Gerchberg routine
Summary
The performance of large ground-based telescopes depends on the ability of adaptive optics (AO) systems to correct for phase distortions caused by atmospheric turbulence. The routine employs the use of the Laplacian of the data, ∇2 = ∂2/∂ x2 + ∂2/∂ y2, in the continuous Fourier domain leading to a simple, complex multiplication for reconstruction of the phase φˆ = κxsx + κysy /(2iπ|κ|2) where φˆ is the reconstructed phase in Fourier space, κ is the spatial-frequency vector and sare slopes in Fourier space This fails to take into account a more realistic measurement process, the discrete nature of the wave-front measurement in terms of lenslet sampling and the pixelated WFS spots on the detector, and to justify the use of the Laplacian as the solution of a least-squares minimisation of the reconstruction process [8]. To observe the anticipated gains in contrast we require accurate reconstructors which perform well across the correction band, and avoid errors at high spatial frequencies
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