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 San Diego, CA June 3, 2005

Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug 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

Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug 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

Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug 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 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

Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug 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 =

Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug 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!

Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug 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 ))

Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug 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

Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug 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:

Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug 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-

Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug 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

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

Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug 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 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

Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug 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

Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug 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

Gavel, Tomography for Multi-guidestar AOSPIE Optics and Photonics, San Diego, Aug 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