1

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

Robert S. Upton

NIO/AURA

February 11,2005

2

BACKGROUND

• NSO hired ORA to perform a sensitivity analysis of the ATST

1. Optical performance is most sensitive to M2 misalignments

2. Image and pupil boresight error correctible with M3 and M6 tilts

3. ORA have defined an alignment strategy using their AUT optimization routine in CODE V

4. NSO would like a “standalone” reconstruction/optimization control strategy that can restore optical performance subject to M2

3

OUTLINE

1. Statement of work

2. Analysis

a. Pupil and image boresight

b. Zernike coefficients

c. Linear mathematical analysis.

3. Correction strategy

4. Summary

5. Other thoughts

4

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

5

ATST OPTICAL MODEL09:55:23

Y

X

1

Y

X

2

Y

X

3

Y

X

4

Y

X5

Y

X

6

Y

X

7

Y

X8

YX

9

YX

10 YX

CoudeAdjust5 Scale: 0.01 RSU 03-Jan-05

2500.00 MM

+Z +X

+Y

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

6

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

7

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

2

2

222110

222110

1

)()()()()()(

)()()()()()(

M

YXXYX

YXXYX

C

y

x

x

y

x

CCCCCC

CCCCCC

c subscript denotes compensator motions

M3 A M3 B M6 A M6 BC0 0 0 0 0C1X 0.0005 0.0061 0.0014 -0.0784C1Y -0.0083 0.0004 0.0813 0.0011C2X 0 0 0 0C2Y 0 0 0 0.0001C2XY 0 0 0.0001 0C1A -0.3211 0.014 3.162 0.0415C1B -0.0196 -0.2253 -0.0509 2.9C2A -0.0259 -0.0019 0.0074 0.0972C2B -0.0094 0.0003 0.0186 0.0006C2AB -0.0008 -0.0151 -0.1218 -0.0085C2AX 0 0.0004 0.0032 0.0003C2AY -0.0014 0 0.0008 0.0051C2BX 0.0006 0 -0.0016 0C2BY 0 -0.0003 -0.0031 -0.0002

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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

Z Z

Y

X

X

…

Z Z

Y

X

X

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DEFINE THE PRE-ALIGNMENT CORRECTIONPupil and Image motion

Pupil Dx (mm) Dy (mm)Uncorrected 0.25 0.89Corrected 7.64E-04 1.40E-03Improvement Factor 3.27E+02 6.36E+02Image Dx (mm) Dy (mm)Uncorrected 50 2.2Corrected 0.05 6.00E-03Improvement Factor 1000 3.67E+02

• 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

(0, 1.5 arc-min)

(-1.5 arc-min, 0) (1.5 arc-min, 0)

Perturb telescope

Perform pre-alignment

Determine Zernike coefficient

Renew telescope prescription

• 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

2

2

2

2

2

2

210

3,10

2,10

1,10

3,4

2,4

1,4

z

y

x

Cz

y

x

CC

Z

Z

Z

Z

Z

Z

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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,

z

y

x

CC

Z

Z

Z

Z

Z

Z

10

3,10

2,10

1,10

3,4

2,4

1,4

?WFS

Modes

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ATST SYSTEM LINEARITY

0 0.5 1 1.5 2

-20

-10

0

10

20

Z4

y decenter (mm)

Zer

nike

coe

ffici

ent (

)

0 0.05 0.1 0.15 0.2

-20

-10

0

10

20

Z4

tilt (deg)

Zer

nike

coe

ffici

ent (

)

0 0.5 1 1.5 2

-20

-10

0

10

20

Z5

y decenter (mm)

Zer

nike

coe

ffici

ent (

)

0 0.5 1 1.5 2

-20

-10

0

10

20

Z5

z decenter (mm)

Zer

nike

coe

ffici

ent (

)

0 0.05 0.1 0.15 0.2

-20

-10

0

10

20

Z5

tilt (deg)

Zer

nike

coe

ffici

ent (

)

0 0.5 1 1.5 2

-20

-10

0

10

20

Z6

x decenter (mm)

Zer

nike

coe

ffici

ent (

)

0 0.05 0.1 0.15 0.2

-20

-10

0

10

20

Z6

tilt (deg)

Zer

nike

coe

ffici

ent (

)

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

DOFWFS dCCZ

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ATST M2• 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)

01

1

11 CZCCCd WFSTT

DOF

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

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ATST LINEAR ANALYSIS What SVD does for you …

TUVC 1• 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)

DOFT

WFS dUVCZ

0

DOFT

WFST dUCZV

0

dZWFS

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 …

WFSZd 1• 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

du

ud

Tnn

R

nnn

1

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CORRECTION STRATEGY Linear reconstruction and optimization

Perturb telescope

Boresight correction(pre-alignment)

WFS measurement

Zernike sensitivity

data

Perturb telescope

WFS measurement

Align telescope

Boresight correction(pre-alignment)

Define sensitivity data Reconstruction

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CORRECTION STRATEGY Monte Carlo results

20 40 60 80 1000

5

10

15

20

Trial number

Rm

s er

ror f

unct

ion

perturbationcorrection

20 40 60 80 1000

0.02

0.04

0.06

0.08

0.1

0.12

Trial numberR

ms

erro

r fun

ctio

n

• Plot merit function for 101 trials with random perturbations

• Linear reconstruction restores the optical performance of the ATST to diffraction limited performance

10987654 ZZZZZZZf

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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

1. Statement of work has been completed

a. 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

2. The ATST is linear for the dominant aberrations (Focus, Astigmatism & Coma)

3. 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

4. Simplex development

5. 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