Wavefront Reconstruction

Wavefront Reconstruction Lisa A. Poyneer Lawrence Livermore ational Laboratory Center for Adaptive Optics 2008 Summer School University of California, Santa Cruz August 5, 2008 LLL-PRES-405137 This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore ational Laboratory under Contract DE-AC52-07A27344. 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 s φ Phase in pupil Slope vector: both x- and y-slopes Actuator vector 3 Method 1: zonal matrix reconstruction The slope vector s contains x- and y-slopes for all valid subapertures in the pupil The phase vector φ contains all controllable actuators The basis set for reconstruction is the actuators We model the WFS measurement process as s = Wφ With the matrix pseudo-inverse E = W +, the reconstruction is obtained by a matrix-vector multiplication ˆφ = 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 M and synthesis M 1 matrices 6

Modal reconstruction, continued ow the slope measurement process is s = WM 1 c And the modal reconstruction is ĉ = MW + s 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 ˆφ = Es, where E = M 1 GMW + 7 Suppressing local waffle via matrix The regular error criterion, which produces SVD, is J = (s Wˆφ) T (s Wˆφ) We add a term which penalizes certain actuator patterns J = (s Wˆφ) T (s Wˆφ) + ˆφ T Vˆφ Using the weighting matrix V to penalize local waffle SVD modes ew 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 M and M 1 are simply the 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 True phase Do nothing Extension Reconstruction Edge Correction 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] Φ[k, l] + ˆΦ[k, l] G wy [k, l] + Q y [k, l] WFS V y [k, l] Reconstruction 11 Fourier Transform Reconstruction WFS x-slopes WFS y-slopes Desired phase (actuators) Solve boundary problem FFT FFT FFT -1 ˆΦ = G wxx + G wyy Recon. G wx 2 + filter G wy 2 Complex-valued Fourier coefficients Other filters 124

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

20 Fitting a phase shape 10 Phase (AU) 0-10 -20 Phase =8 =12 =16 =24 =32 =48-30 0 10 20 30 40 Actuator (=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 F is the flux from the guide star, the number of electrons received per subaperture is e F d2 f AO Following Guyon, the intensity at a PSF location is I 2 f 2 F 16

1x10-3 1x10-4 PSF intensity with system size, noise only 24 32 40 48 Intensity 1x10-5 1x10-6 1x10-7 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: O(n 2 ) for n actuators We could just rely on Moore s Law... Original Keck AO computer (~1997): 16 Intel i860 processors, 1.35 ms latency Keck extgen WC (~2007): 3 TigerSharc DSPs, 0.081 ms latency Determine an efficient algorithm to solve the matrix equation: O(n log n) Fourier reconstruction, which uses FFTs: O(n log n) O(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) O(n) Multi-grid for least-squares [Vogel, 2006]: Many other proposals Fractal iterative preconditioning for MV (FRIM) [Béchet, 2006]: O(n) Local reconstruction [MacMartin, 2003]: O(n 3/2 ) or O(n 4/3 ) Sparse reconstruction [Shi, 2002]: O(n 2 log n) 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 MFLOPs per time step 1x10 3 100 10 1 0.1 VMM FTR MGCG Keck GPI TMT PFI 0.01 10 100 x system size Actual FLOPs counts for algorithms 20

Advanced matrix techniques: Keck AO Instead of using the pseudo inverse, the Keck AO control matrix is generated using prior information in a Bayesian formulation The covariance matrix of Kolmogorov turbulence is C φ The relative noise matrix for the subapertures is The control matrix is E = (W T 1 W + αc 1 φ ) 1 W T 1 This is based on the open-loop statistics of the wavefront, but achieves good results in Keck AO closed-loop See M.!A. van Dam, D.!L. Mignant, and B.!A. Macintosh, Performance of the Keck Observatory adaptive-optics system, Appl. Opt. 43, 5458 5467 (2004). 21 Minimum Variance Unbiased model The slopes now have noise: s = Wφ + v The error that we wish to minimize is set by the difference between the phase (given viewing angle) and the actuator commands (given DM response): ɛ = H φ φ H a a The best actuator commands are obtained from the slopes â = Cs Where the control matrix is given by C = FE, where E = (W T P 1 v W + P 1 φ ) 1 W T P 1 v F = (H T a BH a + w T w + ki) 1 H T a BH φ See B.!L. Ellerbroek, Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques, J. Opt. Soc. Am. A 19, 1803 1816 (2002). 22

MVU details MVU incorporates statistical information and other knowledge: the atmospheric covariance P φ the WFS noise covariance P v the DM response H a the target angle (to deal with anisoplanatism) and error weighting matrix B a constraint matrix, based on DM and weighting a regularization parameter k This model is based on open-loop statistics. As such, to be used in closed-loop, the pseudo-open loop measurements are first generated from the slopes by adding in the DM shape The structure of MVU can be solved efficiently H φ w 23 Conjugate gradient (CG) methods CG and variants (FD-PCG, MG-PCG, etc) are fast ways to iteratively solve a large AO matrix equation, provided A is SPD Ax = b CG for AO uses the MVU model (or variants) to generate the AO equation. Instead of a matrix-multiplication, the unknown phase/actuator vector is solved for The fundamental concept of CG, whether used for multivariate optimization or the solution of linear systems, is that at each iteration you search/update your estimate in a conjugate (perpendicular) direction. 24

How CG works - simple example We ll use CG to solve the equation To produce our estimate x 1 = The error of this estimate is [ 8 2 [ 7/3 1 ( 3 1 1 1 For the next iteration, we ll search along ) [ 8 x = 2 ] [ 4, we searched along 0 ] ( 3 1 1 1 ) [ 7/3 1 [ ] ] [ = 0 4/3 ] 0 4/3 ] ] x 0 x 1 x 2 25 Most AO CG algorithms are preconditioned Convergence of CG depends on condition number of matrix Convergence can be sped up by preconditioning Preconditioning is accomplished with a matrix P If P 1 P = I, then we solve P 1 AP 1 Px = P 1 b Performance now depends on condition not of A, but of P 1 AP What might be a good P? How about the DFT matrix? 26

Using statistical information in FTR If we know nothing about signal and noise, the filter is ˆΦ = G wxx + G wyy G wx 2 + G wy 2 If we know the signal power σφ 2 and noise power σv 2 for all Fourier modes, we have the Wiener filter ( ) 1 G wx X + G ˆΦ wyy = 1 + SR G wx 2 + G wy 2 SR = σ2 v σ 2 φ 1 G wx 2 + G wy 2 If you use Fourier modes in MVU, the answer is the same 27 Gain and prediction filters The Wiener gain is implemented simply as a real-valued gain filter after reconstruction A real-valued gain filter, based on temporal optimization, not just SR, is used in Optimized-gain Fourier Control Another easy filter to implement is a shift filter. A linear-phase complex exponential shifts the phase estimate This concept forms the basis of Predictive Fourier Control, where the Kalman filter to predict a multi-layer atmosphere uses shift filters 28

System design: summary More actuators means a better fit to the phase, but... More actuators require more (too much) computation Many computationally efficient methods for solving a matrix equation, which may involve statistical priors MVU formulation and CG solution Fourier reconstruction filtering is also efficient, and deals with statistical priors via filtering 29 What happens in a real AO system? * Al Gore is not affiliated with the CfAO Computer simulation Real-world AO system How do we obtain the control matrix? How do we get the filters and use FTR? How do we adjust for the response of the DM? Images from Time Magazine and Gemini Observatory 30

Step through the Altair process The slopes are arrayed x 0, y 0, x 1, y 1, The actuators are arrayed φ 0, φ 1, φ 2, A theoretical model of the WFS and DM are used to generate the slopes that are measured when each actuator is poked ordering x-slopes go down-up, down-up y-slopes go down-down, up-up 31 Altair interaction matrix W Poke an actuator and measure the slopes Those slopes become a specific column in the interaction matrix ote that due to influence function model, only close-by subapertures measure a poke Each row corresponds to an x- or y- slope 1 column per actuator 32

Altair control matrix E Altair matrix is obtained via the modal method, with some modes suppressed ote that the control matrix is not particularly sparse 1 column per x- or y-slope Each row corresponds to an actuator 33 Examples of non-matching systems Though many AO systems have matched subapertures and actuators, some do not. A control matrix can deal with this. UC Davis AO OCT Figure from Zawadzki, SPIE 6429 (2007) 34

Pre-compensating for DM Continuous-surface DMs are not perfect interpolators Linear coupling between actuators causes amplification or attenuation all spatial frequencies on-linear effects (e.g. thin plate behavior) may also contribute significantly A theoretical matrix can incorporate the data-based model of the mirror s influence function to pre-compsenate A measured matrix is taken by executing the previous steps in the AO system, as opposed to in a simulated model 35 Fourier method primarily model-based Fourier reconstruction requires the filters G wx and G wy They can be obtained from a theoretical model of the WFS behavior. This requires very accurate system alignment Grid of WFS spots to CCD pixels: shifted by < 0.05 pixels, magnified by < 1.00045 and rotated < 7.2 arcsec Lenslet subapertures to MEMS actuators: shifted by < 1.25%, magnified < 1.004 This is the default operation for FTR, as used at the LAO ExAO testbed at UCSC They can be measured by poking each actuator on the DM and recording the slope measurements This measurement itself can be noisy, and we are still refining this method 36

Point-based Shack-Hartmann model Shack-Hartmann φ[m, n] φ[m + 1, n] Shack-Hartmann φ[m, n] φ[m + 0.5, n] φ[m + 1, n] x[m, n] y[m, n] x[m, n] φ[m, n + 0.5] φ[m + 1, n + 0.5] y[m, n] φ[m, n + 1] φ[m + 1, n + 1] φ[m, n + 1] φ[m + 0.5, n + 1] φ[m + 1, n + 1] Fried model Modified-Hudgin model Fried model, which suffers from excessive waffle x[m, n] = 1 (φ[m + 1, n] φ[m, n] + φ[m + 1, n + 1] φ[m, n + 1]) 2 G wx [k, l] = [ exp ( j2πl ) ] [ + 1 exp ( j2πk ) ] 1 37 A better point-based model Shack-Hartmann φ[m, n] φ[m + 1, n] Shack-Hartmann φ[m, n] φ[m + 0.5, n] φ[m + 1, n] x[m, n] y[m, n] x[m, n] φ[m, n + 0.5] φ[m + 1, n + 0.5] y[m, n] φ[m, n + 1] φ[m + 1, n + 1] φ[m, n + 1] φ[m + 0.5, n + 1] φ[m + 1, n + 1] Fried model Modified-Hudgin model Modified-Hudgin model is accurate, but has less noise, no problems with waffle or local waffle x[m, n] = φ[m + 1, n + 0.5] φ[m, n + 0.5] ( ) [ ( ) ] jπl j2πk G wx [k, l] = exp exp 1 38

A complete Fourier Optics model Based on a continuous-time model of average derivative x[m, n] = m+1 n+1 x=m When sampled with correct grid spacing and converted into a filter, it becomes G wx [k, l] = {[ cos 2πl j {[ cos 2πl ( ) ] 2πl 1 sin ỹ=n ( ) ] [ 2πk 1 cos [ ] d φ( x, ỹ) d x dỹ d x ( ) [ 2πk + cos ( ) ] 2πl 1 sin ( ) ] 2πk 1 sin ( ) 2πk sin ( )} 2πl + ( )} 2πl 39 Fourier filter for DM compensation Our work at the LAO has shown that the MEMS mirrors have only a small amount of non-linearity The response of the MEMS can be pre-compensated by using a filter which is the inverse of the mirror s response This filter is simply the appropriately-sample Fourier transform of the measured influence function Influence function Frequency domain 40

Real systems: summary In a real operating AO system, the AO control must be aware of system characteristics and alignment a matrix method allows direct measurement of the system Fourier reconstruction usually requires very precise alignment and calibration Example of Altair s model control matrix Examples of Fourier reconstruction filters The physical response of the DM must be accounted for Measured matrix deals with the directly Theoretical matrix can incorporate a model of DM Fourier reconstruction uses a filter based on a model (or measurement) of the influence function 41 Moving beyond on-axis, single conjugate Given multiple WFS measurements, the phase can be reconstructed either in layers or in a volume This enables the AO system to use one or multiple mirrors to improve correction across a field of view Strehl maps from MAD on-sky test of Omega Centauri. SCAO - 1 GS, MCAO - 3 GS. Figure from ESO: 19d/07 42

ew concepts and algorithms MAD: 3 GSs, 2 DMs, one altitude conjugated. Matrix formulation to optimize a uniform Strehl across FoV. Proposed FIRAOS for TMT: 6 LGS + GS, 2 DMs, correction over a 1-2 arcmin FoV candidate algorithm is FD-PCG in MCAO MVU formulation [Yang, 2006] and [Vogel & Yang, 2006] LAO s MCAO test bench 3 simulated LGSs, 3 DMs, Fourier-domain tomography [Gavel, 2004] Ground layer AO: using one mirror and many WFSs, correct the common ground layer across a very wide field ESO s GRAAL: 4 LGS, 1 DM, 7.5 arcmin FoV in the IR ESO s GALACSI: 4 LGS, 1 DM, 1 arcmin FoV in the visible 43 Reconstruction summary The WFS does not measure the phase. So we need to reconstruct it. This is usually done with a matrix, though other methods such as Fourier reconstruction also work While many actuators improve phase fitting, they require computationally efficient algorithms Both matrix and Fourier methods can incorporate statistical information to improve performance Covered the process of developing a control matrix of reconstruction filter for a specific AO system and DM. 44

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