Wavefront Reconstruction Lisa A. Poyneer Lawrence Livermore - PDF document

Wavefront Reconstruction Lisa A. Poyneer Lawrence Livermore National Laboratory Center for Adaptive Optics 2008 Summer School University of California, Santa Cruz August 5, 2008 LLNL-PRES-405137 This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. What is wavefront reconstruction? • Most* wavefront sensors do not measure the wavefront phase directly, but instead measure the average derivative • Most* wavefront correctors are used to conjugate that phase on the mirror’s surface • We must reconstruct the phase from the WFS slopes, achieving the most accurate, lowest noise estimate possible in the least amount of computational time • We’ll start with 1 sensor and the 2D phase... * I’m ignoring direct phase sensors as well as curvature sensors. I’m also ignoring AO systems which operate without a WFS or that do not conjugate the phase 2

Mapping subapertures and actuators φ s Slope vector: both x- and y-slopes Actuator vector Phase in pupil 3 Method 1: zonal matrix reconstruction • The slope vector contains x- and y-slopes for all valid s subapertures in the pupil • The phase vector contains all controllable actuators φ • The basis set for reconstruction is the actuators • We model the WFS measurement process as s = W φ • With the matrix pseudo-inverse , the reconstruction is E = W + obtained by a matrix-vector multiplication ˆ φ = Es 4

Examples of modal basis sets • Actuators • Zernike modes • Fourier modes 5 Method 2: modal matrix reconstruction • We first define a orthogonal modal basis set to represent the actuators � m k , m l � = 0 , for k � = l • We can analyze the phase in terms of modal coefficients with the inner product c k = � φ , m k � • The phase is synthesized from the modal coefficients as n − 1 m k � φ = c k � m k , m k � . k =0 • We can put the modes into rows or columns to produce modal analysis and synthesis matrices M − 1 M 6

Modal reconstruction, continued • Now the slope measurement process is s = WM − 1 c • And the modal reconstruction is c = MW + s ˆ • With modes, we can think about weighting or manipulating them. For example, If we choose the Zernike modes for a basis set, we can easily remove piston, tip and tilt (or other Zernikes) by zeroing the correct coefficients before converting back to actuators, using matrix G ˆ E = M − 1 GMW + φ = Es , where 7 Suppressing local waffle via matrix • The regular error criterion, which produces SVD, is φ ) T ( s − W ˆ J = ( s − W ˆ φ ) • We add a term which penalizes certain actuator patterns T V ˆ φ ) T ( s − W ˆ J = ( s − W ˆ φ ) + ˆ φ φ • Using the weighting matrix to penalize local waffle V SVD modes New modes Figures from D. � T. Gavel, “Suppressing anomalous localized waffle behavior in least squares wavefront reconstruction,” Proc. SPIE 4839 , pp. 972–980 (2002). 8

Method 3: Fourier reconstruction • The zonal perspective: the slopes and phase are spatial signals. If we describe the WFS process with a filter, we can simply inverse filter to obtain the phase • The modal perspective: the Fourier modes (sines and cosines) form a basis set. The matrices and are simply the M − 1 M DFT matrices if we embed the aperture in a square grid • How do we deal with this circular aperture/square grid problem? • What to do with slopes that equal zero? 9 Essential to solve boundary problem Do nothing Do nothing Do nothing Extension Extension Edge Correction True phase True phase True phase Reconstruction Reconstruction • Without slope management, region outside pupil will be forced flat, making phase estimate incorrect • Two methods for fixing this: Extension and Edge Correction 10

Fourier reconstruction, continued • The spatial domain / frequency domain pair is x [ m, n ] ⇐ ⇒ X [ k, l ] • Fourier modes are eigenfunctions of LSI systems - for each mode the filter is simply multiplication by a complex number V x [ k, l ] G wx [ k, l ] Q x [ k, l ] + ˆ Φ [ k, l ] + Φ [ k, l ] G wy [ k, l ] Q y [ k, l ] + WFS Reconstruction V y [ k, l ] 11 Fourier Transform Reconstruction Desired phase WFS x-slopes (actuators) WFS y-slopes Solve boundary problem FFT FFT FFT -1 Φ = G ∗ wx X + G ∗ wy Y Recon. filter Other filters ˆ | G wx | 2 + | G wy | 2 Complex-valued Fourier coefficients 12 4

Reconstruction: summary • Matrix reconstruction is widely used and familiar • It allows easy measurement of arbitrary system alignments and geometries (e.g. mis-matched WFS-actuator grids as in many vision AO systems) • Many well-established mathematics techniques can be used to formulate more sophisticated control • Fourier transform reconstruction is a computationally efficient method which uses the Fourier basis set • Many well-established signal processing techniques can be used to formulate more sophisticated control 13 Fundamental design decisions • The number of actuators in the pupil affects performance and system design • Incorporation of statistical information about the signal and noise for better reconstruction performance 14

Fitting a phase shape Fitting a phase shape Fitting a phase shape Fitting a phase shape Fitting a phase shape Fitting a phase shape Fitting a phase shape 20 20 20 20 20 20 20 10 10 10 10 10 10 10 Phase (AU) Phase (AU) Phase (AU) Phase (AU) Phase (AU) Phase (AU) Phase (AU) 0 0 0 0 0 0 0 -10 -10 -10 -10 -10 -10 -10 -20 -20 -20 -20 -20 -20 -20 Phase Phase Phase Phase Phase Phase Phase N=24 N=32 N=16 N=48 N=8 N=12 -30 -30 -30 -30 -30 -30 -30 0 0 0 0 0 0 0 10 10 10 10 10 10 10 20 20 20 20 20 20 20 30 30 30 30 30 30 30 40 40 40 40 40 40 40 Actuator (N=48) Actuator (N=48) Actuator (N=48) Actuator (N=48) Actuator (N=48) Actuator (N=48) Actuator (N=48) More actuators = a better fit 15 More actuators = more noise • More actuators require more WFS measurements in order to properly sample the phase • These smaller subapertures receive less light each, resulting in more noisy measurements • Where is the flux from the guide star, the number of F electrons received per subaperture is e ∝ Fd 2 f AO • Following Guyon, the intensity at a PSF location is I ∝ N 2 f 2 F 16

PSF intensity with system size, noise only 1 x 10 -3 24 32 40 48 1 x 10 -4 Intensity 1 x 10 -5 1 x 10 -6 1 x 10 -7 0.1 1 Arcsec Shack-Hartmann noise halo 17 More actuators = more computation • The computational cost of implementing a full matrix-vector multiplication is prohibitive for systems with thousands of actuators: for actuators O ( n 2 ) n • We could just rely on Moore’s Law... • Original Keck AO computer (~1997): 16 Intel i860 processors, 1.35 ms latency • Keck NextGen WC (~2007): 3 TigerSharc DSPs, 0.081 ms latency • Determine an efficient algorithm to solve the matrix equation: O ( n log n ) O ( n ) • Fourier reconstruction, which uses FFTs: O ( n log n ) 18

Fast reconstruction algorithms • Pseudo open-loop minimum variance unbiased (MVU) formulation can be solved with several different techniques • Sparse matrix techniques [Ellerbroek, 2002]: O ( n 3 / 2 ) • Conjugate gradient algorithm [Gilles, 2002] and variations by Vogel, Yang: O ( n log n ) • Open-loop reconstructions • MV conjugate gradient with multi-grid [Gilles, 2003]: O ( n ) • Multi-grid for least-squares [Vogel , 2006]: O ( n ) • Many other proposals • Fractal iterative preconditioning for MV (FRIM) [Béchet, 2006]: O ( n ) • Local reconstruction [MacMartin, 2003]: or O ( n 3 / 2 ) O ( n 4 / 3 ) O ( n 2 log n ) • Sparse reconstruction [Shi, 2002]: • Fourier demodulation from WFS CCD spots [Ribak, 2006], [Glazer, 2007]: O ( n log n ) 19 MGCG from Gilles, 2003. FTR from Poyneer, 2007. Computational costs - reconstruction Computational costs - reconstruction Computational costs - reconstruction 1 x 10 3 1 x 10 3 1 x 10 3 TMT PFI TMT PFI TMT PFI VMM VMM VMM FTR FTR MGCG 100 100 100 MFLOPs per time step MFLOPs per time step MFLOPs per time step GPI GPI GPI 10 10 10 1 1 1 Keck Keck Keck 0.1 0.1 0.1 0.01 0.01 0.01 10 10 10 100 100 100 N x N system size N x N system size N x N system size Actual FLOPs counts for algorithms 20