Fast iterative tomographic wavefront estimation with recursive Toeplitz reconstructor structure for large-scale systems

Abstract

Tomographic wavefront reconstruction is the main computational bottleneck to realize real-time correction for turbulence-induced wavefront aberrations in future laser-assisted tomographic adaptive-optics (AO) systems for ground-based giant segmented mirror telescopes because of its unprecedented number of degrees of freedom, N, i.e., the number of measurements from wavefront sensors. In this paper, we provide an efficient implementation of the minimum-mean-square error (MMSE) tomographic wavefront reconstruction, which is mainly useful for some classes of AO systems not requiring multi-conjugation, such as laser-tomographic AO, multi-object AO, and ground-layer AO systems, but is also applicable to multi-conjugate AO systems. This work expands that by Conan [Proc. SPIE9148, 91480R (2014)PSISDG0277-786X10.1117/12.2054472] to the multi-wavefront tomographic case using natural and laser guide stars. The new implementation exploits the Toeplitz structure of covariance matrices used in an MMSE reconstructor, which leads to an overall O(N log N) real-time complexity compared with O(N) of the original implementation using straight vector-matrix multiplication. We show that the Toeplitz-based algorithm leads to 60 nm rms wavefront error improvement for the European Extremely Large Telescope laser-tomography AO system over a well-known sparse-based tomographic reconstruction; however, the number of iterations required for suitable performance is still beyond what a real-time system can accommodate to keep up with the time-varying turbulence.

Advanced control laws for the new generation of AO systems

“…Rodoplhe Conan points out the exact multiplication of a vector by a Toeplitz matrix in the Fourier-domain 49 a work later extended in. 50 On the same vein we also note the annular shapes of telescope apertures are not adapted to the use of bidimensional Fourier transforms. As argued elsewhere, the Fourier modes are shown to be the eigenmodes of the bi-harmonic equation on a rectangular domain.…”

Misalignment estimation for active telescopes

“…As initially proposed for multi-conjugated adaptive optics [7,16,20], the minimum mean square error (MMSE) estimator ̂𝜓 of is defined as the one that minimizes the quadratic distance:…”

PEPITO: atmospheric Profiling from short-Exposure focal Plane Images in seeing-limiTed mOde

“…Atmospheric characterization for a ground-based telescope has become a key step in the design of instrumentation to correct for wave-front aberrations introduced by atmospheric turbulence. Adaptive Optics (AO) compensates the wave-front aberrations in real-time and the benefits gained from knowledge of the atmospheric profile, or C 2 n (h), include operating the tomographic turbulence compensation in wide-field using multiple Guide Stars (GS) (Ono et al 2018(Ono et al , 2016Correia et al 2015;Vidal et al 2010), enabling phase predictive control Males & Guyon 2018;Juvénal et al 2016;Sivo et al 2015;Rudy et al 2014;Petit et al 2014) for optimum AO performance, or providing a comprehensive analysis of AO residuals (Ferreira et al 2018;Martin et al 2017). High altitude layers play a major role in the spatial phase decorrelation (Roddier 1981) and has an impact on wide-field AO performance (Costille & Fusco 2012) and extreme AO (Cantalloube et al 2018).…”