Laser Guide Star Fourier Tomography on Extremely Large Telescopes

1. Laser Guide Star Fourier Tomography on Extremely Large Telescopes Luc Gillesa, Paolo Massionib, Caroline Kulcsárc, Henri-François Raynaudc, Brent Ellerbroekaand Corinne Boyera a TMT Observatory Corporation, Pasadena, California, USA b INSA, Lyon, France c Institute of Optics Graduate School, Paris Orsay, France AO4ELT3 Conference, Florence, Italy, May 26-31 2013

2. Outline of presentation • Minimum Mean Square Error (MMSE) Estimator excluding temporal dynamics • Sparse iterative solution • MMSE Estimator including temporal dynamics (Kalman filter) • Spatially invariant (non-iterative) solution • Computational Cost and Performance Budgets for the Thirty Meter Telescope (TMT) Narrow Field Infra Red Adaptive Optics System (NFIRAOS) • Online tuning of estimators (Cn2 and wind velocity profiles) from Laser Guide Star (LGS) slope covariance

3. Modeled AO Time Lineand Notations : all slopes computed Slope+Tomo+Fitting+Write=1 frame Exposure n+1 n-1 n-2 n Sensor Frame Nb. Corrector Frame Nb. • Static MMSE Estimator + Prediction: • Kalman Filter: end-to-end latency = 2 frames

4. AO Controller Design • Linear Plant Model: , where • For NFIRAOS: dim(s) ~ 35K, dim(x) ~52K-200K, dim(a) ~7K • Wave Front Sensor (WFS) influence matrices and highly sparse • Control goal: minimize residual variance over FoV: • Ray tracing matrices and are highly sparse Measurement noise

5. AO Controller Design (2) • Control split into 2 sequential steps: • Bayesian tomography estimation • Static MMSE estimator, or • Dynamics included via e.g. 1st-order Auto Regressive model: • Least-squares Deformable Mirror (DM) fitting Frozen Flow Process noise

6. Static MMSE Estimator • Steps: • Tomo solution obtained via warm-started iterations applied to linear, sparse matrix system of the form . • Prediction: ( if wind profile is unknown) • Fitting: • Unfiltered DM commands: • Temporal filter (Low Pass Filtering): Optional, traditionally not implemented data Tomo model

7. Dynamic MMSE Estimator (Kalman filter) • Steps: • Tomo solution updated as: • Kalman gain matrix : • Very largeand full: single precision storage is 10-30GB for NFIRAOS (i.e. a factor 10-30 from reconstructor mapping slopes to actuators) • Off-line solution of recursive Riccati equation • Prediction: with • Fitting identical to static case • Optional temporal filtering as for static case Kalman gain matrix Innovation

8. Off-line Kalman gain computation • Requires solution of Riccati equation for steady state covariance matrix of estimation error (covariance propagation) • Kalman gain expressed as • Complexity is for matrix inverses computed via SVD • One attractive solution to overcome computational bottleneck proposed by Massioni et al. [JOSA , A28 (2011)] : solve Riccati in Fourier Domain (FD) !

9. FD Kalman gain computation • Trivial off-line cost: for each spatial frequency, solve system • Spatially invariant controller (Distributed Kalman Filter, DKF) : • LGS WFS elongation noise cannot be incorporated, • Infinite pupil approximation • Cost-effective on-line FD filters: complexity is • Cost/performance for NFIRAOS by Gilles et al. [JOSA , A30 (2013)]

10. Performance Analysis for NFIRAOS • Turbulence predictor modeling frozen flow • Static MMSE tomography via 3 warm-started FD Preconditioned Conjugate Gradient (FDPCG) iterations ( complexity) • 3 iterations required for optimal performance • End-to-end, wave optics NFIRAOS simulations at zenith for nominal observing conditions at Mauna Kea. • Wavefront Error (WFE) averaged over 17’’x17’’ Field of View (FoV)

11. Performance Analysis for NFIRAOS (2) • WFE due to spatial invariance approximation on the order of ~43nm • of which ~80% from infinite pupil approximation • With prediction, WFE decreases by ~57nm, leading to • ~35nm lower WFE than static MMSE estimator+prediction! • ~38nm lower WFE than static MMSE without prediction NFIRAOS performance estimate of 173nm 169nm RMS @ 50% sky coverage @ Galactic Pole with DKF+prediction • Wind profile required to be known to ~10% accuracy

12. Sample Monte Carlo simulation results • Tomo estimation @ 2X Nyquist to reduce spatial aliasing (“oversampling”) 57nm 35nm

13. Computational Cost Analysis for NFIRAOS • pre-computed FD filters • FDPCG : FFTs/iteration/frame, i.e. 36 FFTs/frame for 3 iterations • DKF : i.e. 18 FFTs/frame • Cost ~4X cheaper than 3 iterations of FDPCG : • ~34 Gflopsversus ~138Gflops • Storage ~ FDPCG storage (~37 MB) • Simple division of labor for hardware implementation • no permutation required as for FDPCG • Parallel implementation

14. Note on Latency Considerations for Real Time Implementation • Iterative and Fourier-based controllers can only start once pixel processing has completed (i.e. all slopes are available) • For NFIRAOS: Pixel processing requirementis 510μs, DM write ~60μs 2-frame end-to-end latency dictates Tomo+Fitting ≤ 680μs for iterative or Fourier-based controllers

15. Wind Profiling from Slope Covariances • Generalization of Cn2 profiling from SLODAR [Gilles, JOSA , A27 (2010)] to time delayed covariances • FD (again !) nonlinear measurement model: • has analytic FD expression • Forward model can be solved for wind profile via least-squares fit: • Nonlinear solver (e.g. Newton iteration with Hessian information), or • Linear approximation of complex exponential: where with least-squares solution baseline measurement direction Nonlinear in wind velocity Cn2 weighted wind profile

16. Assessing Estimation Error due to Linear Approximation • Relative estimation error is • Estimate with obtained from the nonlinear model • Estimation error < 1%on sample 6-layer profile using 1-frame delayed slope covariances from 3 LGSs • Next steps: Monte Carlo frozen flow simulations to assess required accumulation time in the presence of noise 1%

17. Takeaway Summary • Including temporal dynamics in AO plant modeling pays off • Off-line FD solution of Riccati equation overcomes computational bottleneck for large scale systems • Cost effective on-line non-iterative FD implementation • ~4X cheaper than iterative MMSE estimator • Simple hardware implementation • No FD permutation required • Parallel implementation • Accurate wind profiler required for optimal performance • Promising preliminary results on SLODAR-based estimator • Despite spatial invariance assumptions, ~35nm RMS WFE improvement compared to static MMSE estimator + prediction !

18. Acknowledgements The authors gratefully acknowledge the support of the TMT partner institutions. They are the Association of Canadian Universities for Research in Astronomy (ACURA), the California Institute of Technology, the University of California, the National Astronomical Observatory of Japan, the National Astronomical Observatories of China and their consortium partners, and the Department of Science and Technology of India and their supported institutes. This work was supported as well by the Gordon and Betty Moore Foundation, the Canada Foundation for Innovation, the Ontario Ministry of Research and Innovation, the National Research Council of Canada, the Natural Sciences and Engineering Research Council of Canada, the British Columbia Knowledge Development Fund, the Association of Universities for Research in Astronomy (AURA), and the U.S. National Science Foundation.