1. Tomography for Multi-guidestar Adaptive Optics An Architecture for Real-Time Hardware Implementation Donald Gavel, Marc Reinig, and Carlos Cabrera UCO/Lick Observatory Laboratory for Adaptive Optics University of California, Santa Cruz Presentation at the SPIE Optics and Photonics Conference 5903-15 San Diego, CA June 3, 2005

2. Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug. 20052 Outline of talk Introduction: The problem of real-time AO tomography for extremely large telescopes (ELTs): Real-time calculations grow with D 4 An alternative approach using a massively parallized processor (MPP) architecture Performance study results –Experiment –Simulation Conclusions

3. Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug. 20053 AO systems are growing in complexity, size, ambition –MOAO Up to 20 IFUs each with a DM 8-9 LGS 3-5 TTS –MCAO 2-3 conjugate DMs 5-7 LGS 3 TTS

4. Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug. 20054 Extrapolating the conventional vector-matrix-multiply AO reconstructor method to ELTs is not feasible Online calculation requires P x M matrix multiply –M = 10,000 subaps x 9 LGS –P = 20,000 acts (MCAO) or 100,000 acts (MOAO) –f s = 1 kHz frame rate  ~10 11 calcs x 1 kHz = ~10 5 Gflops = ~10 5 Keck AO processors! Offline calculation requires O( M 3 ) flops to (pre)compute the inverse ~10 15 calcs --10 6 sec (12 days) with 1Gflop machine “Moore’s Law” of computation technology growth: processor capability doubles every 18 months. To get a 10 5 improvement takes 25 years growth. Let’s say we use 100 x more processors; a 10 3 improvement takes 15 years. Least-squares solution Minimum variance solution General form H = actuator to sensor influence function matrix

5. Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug. 20055 Preliminary definitions L = number of layers N gs = number of guide stars M = total number of measurements = Ngs x n 0 n L = number of voxels in the metapupil at any layer L n i = number of voxels in the layer p i = number of actuators on a DM L...21L...21 N = total number of voxels = N DM = number of DMs P = total number of DM actuators =

6. Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug. 20056 Alternative: massively parallel processing Advantages –Many small processors each do a small part of the task – not taxing to any one processor –Modularity: each processor has a stand-alone task – possibly specialized to one piece of hardware (WFS or DM) –Modularity makes the system easier to diagnose – each part has a “recognizable” task –Modularity makes system design easier – each subsection depends only on parameters associated with it, as opposed to global optimization of a monolithic design Requires –Lots of small processors, with high speed data paths –Iteration to solution – but what if 1 iteration took only 1  s? – then we would have time for 1000 iterations per 1 ms data frame cycle!

7. Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug. 20057 1. Wavefront sensor processing Hartmann sensor: s = Gy –s = vector of slopes –y = vector of phases –G = gradient operator Problem is overdetermined (more measurements than unknowns), assuming no branch points High speed algorithms are well known e.g. FFT based algorithm by Poyneer et. al. JOSA-A 2002 is O ( n 0 log ( n 0 ))

8. Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug. 20058 Weiner solution of the wavefront sensor slope-to- phase problem in the Fourier domain  = spatial frequency ~ indicate Fourier transform r 0 = Fried’s parameter  n = meas. Noise d a = subap diameter C  = Kolmogorov spectrum C nn = noise spectrum

9. Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug. 20059 2. Tomographic reconstruction where y = vector of all WFS phase measurements x = value of  OPD at each voxel in turbulent volume A is a forward propagation operator (entries = 0 or 1) x is an N -vector y is an M -vector A is M x N The problem in underdetermined – there are more unknowns than measurements Guidestars probe the atmosphere:

10. Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug. 200510 Inverse tomography algorithms A T is the back propagation operator C is the “preconditioner” affects convergence rate only P,N is the “postconditioner” determines the type of solution: P=I, N=0  least squares P =, N =  min variance  = constant feedback gain f (.) = 1 st order regression (and other hidden details of the CG algorithm) Linear feedback Preconditioned conjugate gradient -or-

11. Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug. 200511 Compute count for inverse tomography A and A T are massively parallelizable over transverse dimension, guidestars A T is massively parallelizable over layers Optional Fourier domain preconditioning and postconditioning: per iteration OperationCPUMPPU Fourier Transform M log(M)Log(M) per iteration

12. Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug. 200512 Prototype implementation on an FPGA A Single Voxel Processor An Array of Voxel Processors

13. Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug. 200513 Preliminary Results for MPP Timing and Resource Allocation on an FPGA Timing Basic clock speed supported: 50 MHz (Xilinx Vertex 4) Total number of states per iteration: 36 ElementCurrent ValueDerived FormulaComment Load Measured Value123n 0 Done once per msec Forward Propagate27N GS (2L + 1) Compare11 Back Propagate11 Calculate New Estimate73N GS + 4 Parameters (current Value) L = Layers (4) N GS = Guide Stars (3) n 0 = Sub Apertures (4) A single iteration takes T = 4N GS + 2LN GS + 6 clock cycles Currently this is 36 50MHz clocks = 720 nsec. Per iteration Note: algorithm parallelizes over guidestars For reasons of simplicity and debuging of this first implementation we have not done this yet Chip count This implementation: Vertex 4 chip is 20% utilized (2996 of 15360 available logic cells employed) Scaling to a system with 10,000 subapertures (such as for the 30 meter telescope ) would require 500 of these chips Standard packing density is ~50 chips/board, this equates to 10 circuit boards

14. Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug. 200514 Simulation: extrapolation to the full ELT spatial scale to estimate convergence rates 7800 subapertures per guidestar 5 guidestars 7 layer atmosphere Fixed feedback gain iteration A and A T implemented in the spatial domain Initial atmospheric realizations were random with a Kolmogorov spatial power spectrum. Convergence to 3 digits accuracy in 1ms

15. Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug. 200515 3. Projection and fitting to DMs MCAO –Requires filtering and weighted integral over layers for each DM –Filters and weights chosen to minimize “Generalized Anisoplanatism” (Tokovinin et. al. JOSA-A 2002) –Massively parallelizable over the Fourier domain and over DMs - L steps to integrate MOAO –Requires integral over layers for each science direction (DM) –Massively parallelizable over Spatial or Fourier domain and over DMs – L steps to integrate DM fitting –Deconvolution – massively parallelizable given either spatially invariant or spatially localized actuator influence function –PCG suppresses aperture affects in 2-3 iterations

16. Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug. 200516 Conclusions The architecture: massive parallel computation Conceptually simple Tested with a commercial FPGA; evaluated with simulations – it’s feasible with today’s technology Under study: FD-PCG – extra computation per iteration traded off against faster convergence rate