Tomography for WideField Adaptive Optics

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Tomography for Wide-Field Adaptive Optics Don
Gavel UCO/Lick Observatory Laboratory for
Adaptive Optics University of California, Santa
Cruz ASTRO 289C March 4, 2008
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Outline

• Light propagation through the atmosphere
• Current limitations for AO with single guidestars
• How tomography works
• New limitations for tomographic AO
• Tomography implementation

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Light propagation through the atmosphere

• Huygens integral, Fresnel approximation

• Variation in phase of wavefront is approximately
  delta-OPD

• The rest of the change is due to Huygens
  wavelets (i.e. diffraction)

i.e. spatial frequencies with Fresnel numbers lt 1

• which is significant if

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Weak turbulence

• Original plane waves from space have u(x,y,z) 1
• As waves propagate in air, u picks up high
  spatial frequency variation from the
  index-variation induced phase lags
• But, the entire atmosphere is not enough to make
  the diffractive term significant beyond scales of
  1 cm
• To a very close approximation the wavefront
  change is phase only, due to integrals of dOPD
  over altitude

an approximation we will adopt for the remainder
of this talk
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Wavefront aberration through the atmosphere

• The phase at the ground is the line integral of
  delta-OPD along the ray

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Limitations for AO systems with one guide star

• Isoplanatic Angle

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Limitations for AO systems with one guide star

• Isoplanatic Angle
• Limits the corrected field

r0
q0
h
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Limitations for AO systems with one guide star

• Cone effect

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Limitations for AO systems with one guide star

• Cone effect
• Missing turbulence outside cone
• Spherical wave stretching of wavefront
• Limits the telescope diameter

r0
D0
h
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How tomography works
kZ
kX
Fourier slice theorem in tomography (Kak,
Computer Aided Tomography, 1988)

• Each wavefront sensor measures the integral of
  index variation along the ray lines
• The line integral along z determines the kz0
  Fourier spatial frequency component
• Projections at several angles sample the kx,ky,kz
  volume

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Equivalent layer thickness
r0
Q
Q Dz lt r0
Altitude, z
Dz
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Tomography error for a 30 meter aperture and CP
profile
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How tomography worksdo the math
x

• where
• y vector of all WFS measurements
• x value of dOPD at each voxel in turbulent
  volume

y
A is a forward propagator (entries 0 or 1)

• Assume we measure y (i.e. direct phase
  measurements) well deal with the separate
  issue of estimating phase from slopes (i.e.
  Hartmann sensing) later
• The equations are underdetermined there are
  more unknown voxel values than measured phases Þ
  blind modes

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Types of solutions
x2
y Ax
x1

• The choice of which of the non-unique solutions
  to use is only an a-priori preference of the
  user
• Consistent solutions differ from each other by a
  vector invisible to the measurements i.e. by
  blind modes

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Tomography solutions

• Least squares
• Minimum variance
• Minimum variance with measurement noise

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The back-propagator
x
y
AT is a back propagator along rays back toward
the guidestars
Paths for laser guide stars
and a tip/tilt star
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Pre and Post Conditioners

• Preconditioner
• de cross-correlates the measurements so they
  can be back propagated once to get the final
  answer
• Postconditioner
• Makes the solution minimize an a-priori covariance

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Fourier domain conditioning

• P, C, and N are approximately convolution
  operators
• Transforming to the Fourier domain
• Convolution ltgt Multiplication Þ decouples the
  problem into independent equations for each
  Fourier component, f
• Approximate because boundary conditions due to
  finite aperture are not taken into account

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Iterative algorithms

• AT is the back propagator

• P, C, and N are any positive-definite matrices
• C is the preconditioner affects convergence
  rate only
• C (APATN)-1 Þ convergence in one step
• P is the postconditioner determines the type
  of solution
• PI Þ least squares, PltxxTgt Þ min variance

• g constant feedback gain
• f(.) 1st order regression (and other hidden
  details of the CG algorithm)

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Real-time implementation of AO tomography

• The problem of real-time AO tomography for
  extremely large telescopes (ELTs)
• ? Real-time calculations grow with telescope
  diameter to the 4th power
• An alternative approach using a massively
  parallized processor (MPP) architecture
• Performance study results
• Experiment
• Simulation

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AO systems are growing in complexity, size,
demands on performance

• MCAO
• 2-3 conjugate DMs
• 5-7 LGS
• 3 TTS

• MOAO
• Up to 20 IFUs each with a DM
• 8-9 LGS
• 3-5 TTS

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Extrapolating the conventional vector-matrix-multi
ply AO reconstructor method to ELTs is not
feasible

• Online calculation requires P x M matrix multiply
• M 10,000 subaps x 9 LGS
• N 20,000 acts (MCAO) or 100,000 acts (MOAO)
• fs 1 kHz frame rate
• Þ 1011 calcs x 1 kHz 105 Gflops 105 Keck
  AO processors!
• Offline calculation requires O(M3) flops to
  (pre)compute the inverse 1015 calcs --106 sec
  (12 days) with 1Gflop machine
• Moores Law of computation technology growth
  processor capability doubles every 18 months. To
  get a 105 improvement takes 25 years growth.
  Lets say we use 100 x more processors a 103
  improvement takes 15 years.

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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 ms? then we would have time for
  1000 iterations per 1 ms data frame cycle!

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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(n0 log(n0))

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2. Inverse tomography

• A and AT are massively parallelizable over
  transverse dimension, guidestars
• AT is massively parallelizable over layers

M supaps L layers
per iteration

• Optional Fourier domain preconditioning and
  postconditioning

per iteration
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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

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Prototype implementation on an FPGA
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Preliminary Results for MPP Tomography Timing and
Resource Allocation on an FPGA

• Timing
• Basic clock speed supported 50 MHz (Xilinx
  Vertex 4)
• Total number of states per iteration 36

Prototype demonstrator parameters (current
Value) L Layers (4) NGS Guide Stars (3) n0
Sub Apertures (4) A single iteration takes T
4NGS 2LNGS 6 clock cycles Currently this is
36 50MHz clocks 720 nsec. Per iteration
Note algorithm parallelizes over guidestars For
reasons of simplicity and debugging 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

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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 AT implemented in the spatial domain
• Initial atmospheric realizations were random with
  a Kolmogorov spatial power spectrum.

Convergence to 3 digits accuracy in 1ms
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Tomography Implementation Summary

• The architecture massive parallel computation
• Conceptually simple
• Tested with a commercial FPGA evaluated with
  simulations its feasible with todays
  technology
• Under study
• FD-PCG extra computation per iteration traded
  off against faster convergence rate

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AO Tomography Conclusion

• Multi-guidestar tomography enlarges the field of
  view
• Tomography also solves the cone-effect problem
• Simple scaling laws determine the required guide
  star number and placement
• Mixed guide star types and altitudes are
  accommodated
• Massively parallel implementation is feasible
  (fast converging iterative algorithm)