AN ALIGNMENT STRATEGY FOR THE ATST M2 Implementing a standalone correction strategy for ATST M2

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AN ALIGNMENT STRATEGY FOR THE ATST M2
Implementing a standalone correction strategy
for ATST M2
Robert S. Upton NIO/AURA February 11,2005
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BACKGROUND

• NSO hired ORA to perform a sensitivity analysis
  of the ATST
• Optical performance is most sensitive to M2
  misalignments
• Image and pupil boresight error correctible with
  M3 and M6 tilts
• ORA have defined an alignment strategy using
  their AUT optimization routine in CODE V
• NSO would like a standalone reconstruction/optim
  ization control strategy that can restore optical
  performance subject to ?M2

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OUTLINE

• Statement of work
• Analysis
• a. Pupil and image boresight
• b. Zernike coefficients
• c. Linear mathematical analysis.
• 3. Correction strategy
• 4. Summary
• 5. Other thoughts

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STATEMENT OF WORK

• Develop an understanding of the problem
• Develop a suitable optical model
• Perform analysis to develop suitable alignment
  strategy
• Test the strategy
• Comment of potential future areas of analysis and
  development

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ATST OPTICAL MODEL

• Use CODE V macro capability to perturb ATST,
  develop boresight relations, Zernike sensitivity
  analysis, and test correction strategy

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ANALYSIS

• Characterize the pupil and image bore sight
  sensitivities
• Characterize the higher-order optical
  sensitivities ?M2
• Determine ATST system linearity
• ATST linear analysis
• Alignment strategy. Linear reconstruction and
  optimization

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M3 AND M6 MOTION SENSITIVITY ANALYSIS
Maintaining pupil and image boresight

• Determine the angular motions of M3 and M6 that
  maintain pupil and image alignment subject to
  changing M2
• Used CODE V optimizer with gut ray position
  constraint
• Determined that ? and ? rotation are most
  sensitive for M3 and M6
• Second-order angular contributions and
  cross-term contributions have significance

c subscript denotes compensator motions
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PREALIGNMENT TEST

• Apply boresight equations to actual
  perturbation test
• Data arranged to provide all combinations of
  decenters and tilts, except rotation about Z

?X
?X
??
??

?X
?X
?Z
?Z
?Z
?Z
?Y
?Y
??
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DEFINE THE PRE-ALIGNMENT CORRECTIONPupil and
Image motion

• Perturb the telescope by a total of 400 ?m in
  decenters and 0.4 degrees in tilts

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HIGHER-ORDER OPTICAL SENSITIVITY FOR ?M2

• Perturb the M2 through its 6 DOF and calculate
  the resulting Zernike (rms) coefficients at three
  field locations
• The rms Zernike coefficients Z4, Z5, Z6, Z7,
  Z8, Z9, and Z10 are calculated
• These Zernike coefficients quantify
  astigmatism, focus, trefoil and coma
• M2 is decentered through 20 values from 0 to 2
  mm
• M2 is decentered through values from 0 to 0.2
  degrees

• The Zernike coefficient sensitivities are
  determined whilst correcting the boresight error

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HIGHER-ORDER OPTICAL SENSITIVITY FOR ?M2

• The Zernike coefficients are fit to a
  second-order vector polynomial resulting in
  matrix coefficients C0, C1, C2
• Linear algebraic analysis is performed on the
  linear matrix coefficient C1. Determines linear
  independence
• M2 reconstruction is demonstrated in the linear
  limit

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DETERMINE ATST SYSTEM LINEARITY

• ATST system linearity is encapsulated in C1
• If system is largely linear then a large range
  of elegant linear algebraic tools can be used to
  restore optical performance for the perturbed
  ATST
• In other words,

WFS Modes
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ATST SYSTEM LINEARITY
Z4(y)
Z4(?)
Z5(y)
Z5(z)
Z5(?)
Z6(x)
Z6(?)

• Most dominant aberrations and DOF are linear

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ATST LINEAR ANALYSIS

• System linearity for dominant contributions
  provides an elegant solution space for analysis
  and reconstruction (correction)
• Classical solution to linear problem is
  least-square fit
• Should work. RIGHT?
• Not quite. The LSQ solution requires C1 to be
  full rank (i.e. columns in C1 are linearly
  independent).
• The ATST does not have linear independence in
  WFS modes or DOF
• Use Moore-Penrose pseudo-inverse (Barrett and
  Myers Foundation of Image Science)

ATST ?M2

• Pseudo-inverse algorithms make use of singular
  value decomposition (SVD)

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ATST LINEAR ANALYSIS What SVD does for you

• SVD is a matrix factorization scheme
• The matrix V contains orthonormal columns that
  define a vector subspace in WFS space
• The matrix U contains orthonormal columns that
  define a vector subspace in DOF space
• The matrix ? contains singular values along its
  diagonal in decreasing magnitude. The number of
  values equals the rank of C1
• ATST M2 has a rank of 5 (6 M2 DOF)
• SVD reconstructs d in a non-unique way (minimum
  norm solution)

Uncoupled representation of ATST
One M2 DOF is a combination of 5 others
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ATST LINEAR ANALYSIS What SVD does for you

• Linear reconstruction
• d is a minimum norm solution
• The mirror DOF are reconstructed from d
• The ATST optical performance is reconstructed
  in a non-unique way

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CORRECTION STRATEGY Linear reconstruction and
optimization
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CORRECTION STRATEGY Monte Carlo results

• Plot merit function for 101 trials with random
  perturbations
• Linear reconstruction restores the optical
  performance of the ATST to diffraction limited
  performance

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CORRECTION STRATEGYLimitations of correction

• The linear reconstruction technique requires C0
  to be known every time the control loop is used
• For ?T ?g the telescope prescription changes
  resulting in ?C0
• ?C0 results in ?C1 and ?d
• Reconstructor requires updating
• Use of simplex optimizer can help provide a
  least-squares solution even if C0 is not well
  known
• Other wavefront sensors/fiducials are required
  to distinguish the motions of mirrors from
  changes due to ?T ?g

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SUMMARY

• Statement of work has been completed
• Understand the problem
• b. Develop a suitable model
• c. Define standalone correction strategy
• d. Correction strategy works over large range of
  motions. Average rms error is corrected by a
  factor of 400
• The ATST is linear for the dominant aberrations
  (Focus, Astigmatism Coma)
• The 6 M2 DOF are not linearly independent.
  Rank(C1)5. Suspect the non-full rank C1 has to
  do with Z-rotation of M2 about center of parent
  vertex
• Simplex development
• Report delivered by COB 02/11/05

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FURTHER STUDY

• Finish the simplex optimizer in MATLAB
• Study the field dependence of aberrations that
  arise from ATST perturbation. Zernike
  polynomials are not the most appropriate basis
  set. Use SVD
• For bending modes of mirror the reconstructor
  has to be extended
• Investigate the effects of ?T and ?g on C0.
  Efficacy of linear solution
• Effects of noise on WFS. Atmospheric effects.
  Local mirror seeing and etc
• Develop a true alignment test using focus of
  primary, Gregorian focus, and pupil masks to
  establish the optical axis w.r.t. mechanical axis
• Extend optical model to incorporate control
  architecture. Mapping between mirror modes and
  actuator modes