Algorithms for maintaining quasi-
static alignment of the ATST
Robert Upton
1
Outline
• Problem statement
• Operating conditions and figures-of-merit
• Alignment strategy
• ATST linear optical system
• Quasi-static alignment (QSA) strategy using damped
least squares and maximum likelihood reconstructors
• Modeling and simulation results
• Ongoing development and analysis
2
Problem statement
• Design and implement an active alignment strategy that:
– does not rely on least-square minimization embedded within
optical software
3
Operating conditions and
figures-of-merit
• The active alignment strategy must:
– not be sensitive to atmospheric noise and guiding error
– correct 50% encircled energy diameter from ~2 arc-sec in
perturbed state to less than 0.017 arc-sec
– correct for large amplitude and low spatial frequency optical
errors
– be integrated into the ATST wavefront sensing commissioning
strategy
• ATST figures-of-merit:
– RMS wavefront error
– 50 % encircled energy diameter
4
Alignment strategy
Truncated singular values for
• The ATST has up to 44 DOF from Gregorian front end
perturbed degrees-of-freedom DOF Truncated sing val
that can be corrected in a least- 1
2
M1A
M1B
2.01E+03
1.77E+03
norm sense with the QSA 3 M1 XTrefoil 2.45E+01
4 M1C 3.52E+00
– M1: 6 Rigid body modes + 4 5 M2B 3.18E+00
Bending modes 6 M2C 3.18E+00
7 M1 YTrefoil 1.82E+00
– M2: 6 Rigid body modes + 4 8 M2X 1.12E+00
9 M2A 5.20E-01
Bending modes 10 M2Z 5.06E-01
– M4: 6 Rigid body modes 11 M1 45Astig 5.41E-02
12 M2 XTrefoil 4.02E-02
– M7: 6 Rigid body modes 13 M2 YTrefoil 3.83E-02
14 M2Y 2.20E-03
– M10: 6 Rigid body modes 15 M2 XAstig 4.30E-04
– M11: 6 Rigid body modes 16 M1Z 2.09E-04
17 M2 45Astig 1.47E-04
18 M1Y 7.85E-05
19 M1X 1.66E-05
5
20 M1 XAstig 1.47E-05
ATST Perturbed mirrors
M7
M9
M8 M3 M4
M5, M6 Camera
M2 M13
M12
M11
M10
M9
M1
• Perturbed mirrors are in Gregorian front-end and Coude
spaces 6
Alignment strategy
• The alignment compensators are M2 rigid bodies and
M1 low-order bending modes
• Pupil and image boresight are maintained with M3 and
M6 by lookup table
• For noisy systems and large telescope perturbations the
damped-least squares and maximum likelihood
reconstructors are used
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Alignment strategy
• The alignment signal is a matrix multiply between the
wavefront sensor signals and the reconstructor
• Wavefront sensor geometry chosen as an optimum
subset of a 3x3 grid
• Wavefront sensor modes are 8 Zernike RMS coefficients
over three field positions
Y Y
WFS1(1.5,1.5)
X WFS2(0,0)
X
WFS3(1.5,-1.5)
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ATST linear optical
system
• The ATST is linearized to form g Hf h0 n
g Vector of RMS weighted Zernike polynomial coefficients
f Vector of ATST active degrees-of-freedom
h0 Vector of Zernike polynomial coefficients. Baseline optical performance
n Vector of noise contributions to Zernike polynomial coefficients
H ATST active optics interaction matrix
• The subset of columns representing the active optics
compensators in H are chosen in the least-norm inverse
• The compensator matrix is HS
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Reconstructors
• The damped least-squares reconstructor: Tikhonov
filtering of singular values
S
f 2 m
T T
un vn g h0 n
m 1 m
• The maximum likelihood reconstructor: Maximize joint
probability P(f|n) with respect to f in the presence of n
f H S nnT
T 1
H S ff T 1
H
T
S nn T 1
g h0 n
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Reconstructors
• The damped least-squares reconstructor:
– Useful for obtaining linear least-squares solution for large
perturbation amplitudes
– Good formalism for reconstruction with non-optimized noise
control
• The maximum likelihood reconstructor:
– “Optimized” reconstructor in the presence of noise and known
telescope perturbation statistics
– Not optimized for reconstruction of alignment with perturbation of
Coude conics. Covariance of telescope error requires treatment
f H S nnT
T 1
H S ff T 1
H
T
S nn T 1
g h0 n
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Modeling and simulation results:
Estimating atmospheric noise
• The noise contribution is estimated for a Kolmogorov atmosphere
that obeys the Taylor frozen flow hypothesis
• Sample case*:
r0=5cm: =0.5m: h=20 m
bandwidth=10 Hz: Speed=5 m/s
average over 250 samples
Zernike RMS noise
radial order (nm)
N=2 7.95
N=3 6.52
N=4 6.08
• In simulations 10 nm RMS
amplitude noise is added to each
Zernike polynomial coefficient 12
*F.Hill etal, “Site testing for the ATST.” SPIE 6267
Analysis cases: Monte-
Carlo perturbation of ATST
• Case I: Perturbation of Gregorian front-end: Decenters
amplitude = 500 m; Tilts amplitude = 10e-03 degrees;
Surface bending amplitude = 500 nm
• Case II: Perturbation of Gregorian front-end and Coude
optics: Gregorian front-end: As in Case I. Coude optics:
Decenters amplitude = 1.5 mm; Tilts amplitude = 15e-03
degrees
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Damped least-squares
reconstructor results:CASE I
Perturbed Corrected
RMS
wavefront
error
50%
encircled
energy 97% of cases
<17e-03 arc sec
14
Damped least-squares
reconstructor results:CASE II
Perturbed Corrected
RMS
wavefront
error
50%
encircled
100% of cases
energy <17e-03 arc sec
15
Maximum likelihood
reconstructor results:CASE I
Perturbed Corrected
RMS
wavefront
error
50%
encircled
energy 100% of cases
<17e-03 arc sec
16
Maximum likelihood
reconstructor results:CASE II
Perturbed Corrected
RMS
wavefront
error
50%
encircled 95% of cases
energy <17e-03 arc sec
17
Conclusions
• Demonstrated strategy and algorithm for quasi-static
active optical control of ATST using WFS distributed in
FOV
• The damped least squares reconstructor is vital for
controlling ATST non-linear behavior
• Monte-Carlo simulations in the presence of atmospheric
noise and Coude misalignments demonstrated
• Demonstrated reconstruction using maximum likelihood
and Tikhonov filtering exceeds error budget
requirements
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QSA extension
• aO wavefront sensor optical design is complete
• QSA is demonstrated with the aO wavefront sensor
optical design
• QSA and HOAO optical designs can be integrated with
instrument designs to model non-common path errors
19
Bibliography
• Roddier, “Adaptive optics for astronomy.” Cambridge University
Press (2004)
• Conan, “Temporal filtering of atmosphere.”JOSA (1994)
• Barrett and Myers, “Principles of image science.” Wiley (2004)
• Vogel, “Inverse methods.” SIAM (2002)
• Lubliner and Nelson, “Stressed mirror polishing. 1: Atechnique for
producing non-axisymmetric mirrors” App. Opt (1980)
• Upton, Rimmele and Hubbard “Active optical alignment of ATST.”
SPIE (2006)
• Upton, “Optical control of ATST.” App. Opt. (2006)
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