Wavefront Sensing Update

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Wave-front Sensing Update
LSST Camera Project Meeting March 23-24

• K.L. Baker, C. Carrano, D. Phillion

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Outline

• I. Overview of previous results
• II. Wave-front sensing techniques investigated
• Phase Retrieval
• Phase Diversity
• Shack-Hartmann Sensing
• Curvature sensing
• III. Wave-front tomographic reconstruction

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Overview of previous results

• Phase retrieval works well at small D/ro but does
  not work well for moderate to high D/ro.
• Phase diversity can handle undersampling from
  Nyquist by a factor of eight at D/ro values in
  the range of 20-40.
• The LSST telescope is undersampled from Nyquist
  by as much as a factor of 80 for the shortest
  wavelength and the worst turbulence conditions.
  (Precludes the use of this technique without
  additional optics to magnify the FPA.)

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Hartmann Sensing
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Hartmann Simulations

• The wave-front measurements for the Hartmann
  sensor average the atmospheric turbulence for 10
  seconds along with the aberrations attributed to
  the mirrors.(In the simulations, the mirror
  aberration occurs in a single plane)
• Kolmogorov phase screens are used to simulate the
  atmosphere.
• The atmospheric turbulence screen is translated
  at 10 m/s and a new atmosphere is sampled every
  0.1 s.

Kolmogorov turbulence screen, ro0.2 m
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Decreased performance with increased atmospheric
turbulence

• An increase in the atmospheric turbulence
  strength causes an increased residual variance in
  the reconstructed wave-front.
• An increase in the atmospheric turbulence
  strength causes an increased residual variance in
  the reconstructed wave-front.

z_coeff 7.5(l in mm)-1(num_zern)1.24
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Distributed Aberrations increase the variance of
the reconstructed phase

• The LSST mirror aberrations are distributed in
  the telescope.
• Measuring the phase in a plane that is not relay
  imaged onto the lenslet array causes an increase
  in the reconstructed variance.

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Hartmann sensor design for the LSST

• Minimize impact of the wave-front sensor on the
  FPA.
• Avoid moving parts.
• A relay lens composed of two triplets is used to
  form an image of the pupil onto a lenslet
  array.(Lynn)
• A single wave-front sensor provides a
  field-of-view of 37.5 arcseconds.

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Design of compound lens for Shack-Hartmann
Wave-front Sensor (Lynn Seppala)

• Lens reimages the pupil onto a Shack-Hartmann
  lenslet array.
• Consists of 9 elements with a NA of 0.4.
• Sees 37.5 Arcsec on the sky.
• Lens diameter is 6.8 mm.

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Focal Plane Structure and Wave-front Sensor
Placement
4 FOV ? 74 cm ?
WFS
FPA Structure design
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Required Photon Flux and Availability of Stars

• The LSST will integrate the sky for 10 sec.
• Placing 24 lenslets across the 4 mm pupil image
  would require 60k photons to put 50
  photons/pixel.
• Hartmann sensor in classical Hartmann mode would
  require a 19.6th mag. star.
• An array of 9 wave-front detectors would take up
  an area of 2 cm by 2 cm on the FPA.

• The field-of-view would be 110 arcseconds.
• That field-of-view would result in a 60 chance
  of seeing a star at a Galactic latitude of 90 deg.

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Curvature Sensing
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Curvature Sensing

• A curvature sensor measures the spatial intensity
  distribution equal distances on either side of
  focus.
• The difference in the intensity is proportional
  to the Laplacian of the phase.

Transport of Intensity Equation
Ref T. E. Gureyev and K. A. Nugent, Phase
retrieval with the transport-of-intensity
equation. II. Orthogonal series
solution for nonuniform illumination, J. Opt.
Soc. Am. A. 13, 1670(1996). Nirmal
Bissonauth et al., Image Analysis Algorithms for
Critically Sampled Curvature Wavefront Sensor
Images in the Presence of Large
Intrinsic Aberrations, in Advanced Software,
Control, and Communication Systems for
Astronomy, Proc. SPIE 5496, 738(2004).
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Curvature Sensing I Annular Zernike
decomposition of the Transport of Intensity
Equation
The starting point for curvature sensing is the
Transport of Intensity Equation, which relates
the perpendicular intensity and phase as the beam
propagates along z.
The TIE can be converted to a system of
algebraic eqs. by assuming an annular Zernike
decomposition of the phase, multiplying the TIE
by the annular Zernike, Zj, and integrating
over the aperture.
The annular Zernike coefficients of the phase are
determined by using singular value decomposition
to solve the system of algebraic equations.
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Reconstruction Results with ro1 m and the
Distance from Focus Equal to 5 mm
Applied Phase at Aperture (Rad)
Phase Difference (Rad)
Reconstructed Phase (Rad)
Intensity at the Intra-focus (-5 mm)
Intensity at the Extra-focus (5 mm)
Intensity Difference
z_coeff 7.5(l in mm)-1(num_zern)1.24
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Results I
Lc 5 mm

• Particular algorithm works best at large
  distances from focus(gt 1mm).
• An increase in the atmospheric turbulence
  strength causes an increased residual variance in
  the reconstructed wave-front.

z_coeff 7.5(l in mm)-1(num_zern)1.24
Z5 2.0, Z7 1.0, Z12 0.5, Z17 -0.5
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Curvature Sensing II Iterative solution using
simplex algorithm

• Represent wave-front with annular Zernike basis
  set.
• Minimize error metric of mean square difference
  of intensities between measured and simulated
  defocused images.
• Iterative solution using simplex (amoeba)
  algorithm to estimate new annular Zernike
  coefficients .
• Propagate new annular Zernike coefficients to
  form intensity images after each iteration.

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Results II Simplex curvature sensing no
atmosphere
Aberration Z5 2 rms rad Z6 1 rms
rad Z10 0.4 Allowed 10 degrees of freedom No
atmosphere, no noise, wl650nm
Zernike mode coefs Estimate vs actual
/- 0.5 mm from focus
Tolerance parameter
Estimate vs actual Phase RMS error 2e-4 rad
iterations
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Effect of defocus distance on reconstruction
results
No atmosphere, 128x128 pixel sized images Z4
2.0 Z9 1.0 Z10 0.5 Z13 -0.5 Did 3600
iterations took 424 seconds Defocus 0.5 mm
rmserr 0.05 rad Defocus 0.4 mm rmserr
0.018 rad Defocus 0.3 mm rmserr 0.31
rad Defocus 0.2 mm rmserr 2.47 rad
Phase aber
Extra-focal images
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Effect of atmosphere on reconstruction results
Exp time 30 sec Dt 20 ms Vwind 10
m/s Wavelength 650 nm Defocus 0.5 mm Z5
2.0 Z7 1.0 Z12 0.5 Z17 -0.5 (Not
incorporating atmosphere into simplex algorithm
itself) Did 3000 iterations took 247 seconds No
atmosphere rmserr 0.18 rad Ro 50 cm
rmserr 0.25 rad Ro 30 cm rmserr 0.31
rad Ro 20 cm rmserr 0.42 rad
Phase aber
Extra-focal images
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Effect of rebinning on reconstruction results (40
pixels across aperture)
Exp time 30 sec Dt 20 ms Vwind 10
m/s Wavelength 650 nm Defocus 0.5 mm Z5
2.0 Z7 1.0 Z12 0.5 Z17 -0.5 Did 3000
iterations took 247 seconds No atmosphere
rmserr 0.17 rad Used a 64x64 pixel screen,
sampled at 40 pixels across aperture (Not
incorporating atmosphere into simplex algorithm
itself) Ro 50 cm rmserr 0.53 rad (57
seconds) Ro 30 cm rmserr 1.13 rad Ro 20
cm rmserr 2.57 rad
Phase aber
Extra-focal images
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Effect of pupil overlap on reconstruction results
Aberration Z5 2 rms rad, Z7 1 rms rad, Z10
0.5, Z17 -0.5 Allowed 17 degrees of freedom No
atmosphere, no noise, wl650nm
RMS error .15 rad
10 intensity, 30 pixel shift
Residual phase is trefoil
20 intensity, 30 pixel shift
RMS error 0.6 rad
40 intensity, 30 pixel shift
RMS error 0.8 rad
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Tomographic Reconstruction Via Zernike
Decomposition
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LSST least squares alignment strategy
The alignment strategy is to first measure the
wavefronts at a set of field angles using the
wavefront sensors, and then to calculate the
mirror deformations and mirror rigid body motions
that will give the best overall wavefront in a
least squares sense.
B1 sub-vector Zernike coefficients representing
the bending modes of the primary, secondary, and
tertiary deformable mirrors B2 sub-vector rigid
body motions for the optics
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LSST least squares alignment strategy matrix
formalism
The j-th wavefront sensor measures the wavefront
from the telescope for a star in the direction
Wj. For small slowly varying bendings of the
deformable mirror motions and small rigid body
motions, higher order terms may be ignored and
only the linear response be considered
Zernike coefficient transformation matrix
tranpose Nmax X 3 Nmax
Zernike coefficient rigid body motion
matrix Nmax X R
Zernike coefficients for wavefront sensor at Wj
field angle Nmax
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LSST least squares alignment strategy SVD
equation
Define
Find the vector B which minimizes the sum
This gives the equation
where
W is a square matrix with zeroes and ones on its
diagonal and zeroes elsewhere. Each zero on the
diagonal represents an unobservable wavefront
sensor Zernike coefficient. For instance, a
piston is always unobservable.
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Tomographic reconstruction status

• Bending modes and rigid body motion of mirrors
  have been implemented.
• Ability to turn off given modes in the
  reconstruction has been implemented.
• Arbitrary location of wave-front sensors is
  possible.
• Testing of accuracy of results is being
  performed.
• After reconstruction fidelity is verified
• Investigate reconstruction quality from periphery
  geometry
• Include wave-front reconstruction inaccuracies
• Combine the wave-front sensor techniques with the
  tomographic reconstruction

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Summary

• We have been investigating a number of wave-front
  sensing techniques for the LSST.
• Phase Retrieval-Problems with large D/ro
• Phase Diversity-Problems with undersampling gt8
• Hartmann Sensing-more widely used then
  curvature-sees relatively small field(37.5
  Arcsec)-slightly more difficult to implement in
  hard.
• Curvature Sensing-Simplex iterative-type
  reconstructor looks most promising-convergence
  time(50 seconds for 64x64 pixels)-sees larger
  field than Hartmann sensing
• Tomographic reconstruction is being implemented
  with the goal of combining the wave-front sensing
  and Zernike reconstruction of the optical
  surfaces.
• A number of issues still need to be resolved
  including
• the best location for the wave-front sensors
• interplay between the tomography and the
  wave-front sensors
• which wave-front sensing technique will be
  implemented