Generalized Wave Front Reconstruction Algorithm for ...

Generalized Wave Front Reconstruction Algorithm for slope data Weiyao Zou [email protected]

Advisor: Jannick Rolland School of Optics/CREOL University of Central Florida, Orlando, FL 32816-2700 http://www.creol.ucf.edu W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

1

Outline 1. 2.

3. 4. 5. 6. 7.

Mathematical Model Domain Extension technique & Generalized WFR matrix equations Solutions Error analysis Deviation error removal Summary References

W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

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Motivations: Have you tied of wavefront reconstruction? Especially when the sampling pupil is large and irregular or even changing?

Doing a WF reconstruction is laborious and time-consuming!

Motivations:








Develop an efficient and universal WF reconstruction algorithm for slope data based on linear equation approach, so that This algorithm can deal with various irregular pupil shapes in adaptive optics and optical shop testing. This algorithm is especially useful for large and irregular pupils !

W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

3

1. Mathematical Model Neumann boundary problem


Wavefront reconstruction from slope data is a Neumann problem of Poisson's equation

∇ 2W = f ( y , z ),   ∂W = g( y, z )  ∂n ∂Ω 




( y, z ) ∈ Ω

The solution to this problem is unique except for an additive constant.

W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

4

1. Mathematical Model Postulations





Suppose we have got slope data (gradients/ derivatives /differences) from WFS, and we adopt the Southwell reconstruction geometry

(Southwell, JOSA A 70, 1980)

W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

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1. Mathematical Model Zonal integration for Southwell geometry (D. Su et al, SPIE 2199, pp609-62,1994)

1.

If the slopes (gradients) can be linearly interpolated between two consecutive grids, then ∂W ∂W  y  ∂W = 1 −  + ∂y ∂y i  a  ∂y

So 2.

W i +1 − W i =

1  ∂W 2  ∂y

+ i

∂W ∂y

i +1

y a

 a  i +1 

For interior point i, we have four phase equations W. Zou/ CfAO/2003

  w i − w i −1     w i +1 − w i   w i− t − w i   w i − w i+ t 

School of Optics/CREOL/ FPCE

= = = =

1 ∂W ∂W + ( 2 ∂y i ∂y 1 ∂W ( 2 ∂y 1 ∂W ( 2 ∂z

i −1

+

∂W )a ∂y i

+

∂W )a ∂z i

i +1

i−t

)a

1 ∂W ∂W + ( 2 ∂z i ∂z

)a i+ t

6

2.1 Domain extension technique


The reconstruction matrix changes with pupil shape and sampling grid indexing mode!

W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

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2.1 Domain extension technique (W. Zou and Z. Zhang, Appl. Opt., Vol. 39, No.2, Jan.2000)

In developing a generalized wavefront reconstruction algorithm, steps are included: 1.

2.

3.

Extending the sampling domain Ω0 (exit pupil, simply connected domain or multiple connected domains) to a square domain Ω1 that covers the whole sampling domain Ω0. Indexing the grids in Ω1 serially from 1 to m row by row (or column by column alternatively). Setting the slopes to zero in the Ω0. additive extended regions Ω1\Ω (i.e. Zero padding!!)

W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

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2.2 Generalized WF reconstruction matrix equation (W. Zou and Z. Zhang, Appl. Opt., Vol. 39, No.2, Jan.2000)

 −1                                  1             

where

1 −1

1 p p −1 1

−1

1 −1

1 p p −1 1 p p −1

0

0

l

0

−1

1

0

0

l

0

p p p

−1 p p p p p 1

ci +1,i =

W. Zou/ CfAO/2003

(

a wy + wy i +1 i 2

)

p

0

0

l

1

0

0

and

     c     2,1 w     1  c     3,2 w    2     m     m      c    w − t t 1 ,     t     c  w t +1   t + 2, t +1    m    w  t+2        m m         c  w    − m m 1 ,  2t      = d m  + 1 t 1 ,        1 d  w    − + m 2 t 1 + 2 t 2 ,     −1 1  w    m   m − 2t + 2    p p      m d     t 2 t ,     −1 1  w   m  m−t      w    d  m − t +1    + i i t ,     m  m         w   d − m 1     m − t −1, m −1 p  w     d m   0 −1 m − t ,m     l 0 −1 

d i ,i + t =

a (wz i + wzi +t ) 2

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2.2 Generalized WF reconstruction matrix equation

(W. Zou and Z. Zhang, Appl. Opt., Vol. 39, No.2, Jan.2000)

• Generalized normal equation for WF reconstruction

A AW = A F T

where

where

 E1 − I  ATA =     2 − 1  E1 =    

T

−I E2

−I







−I

E2 −I

−1 3

−1







−1

3

W. Zou/ CfAO/2003

−1

     − I E 1  m × m

  3 −1   − 1 4 − 1  − 1      − I =   E2 =        − 1  − 1 4 − 1   2  t×t  − 1 3  t × t

School of Optics/CREOL/ FPCE

   − 1 t × t

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2.2 Generalized WF reconstruction matrix equation

(W. Zou and Z. Zhang, Appl. Opt., Vol. 39, No.2, Jan.2000)

Summary for Normal equation 1.

This is a sparse band matrix!

2.

No matter how irregular the pupil shapes are, the reconstruction matrices are the same! (except for dimension sizes)!

3.

rank(ATA)=rank(A)=m-1

4.

It has 1-dimension solution space!

W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

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3. Solutions


(W. Zou and Z. Zhang, Appl. Opt., Vol. 39, No.2, Jan.2000)

Least squares solution with a selected piston value:
manually set a zero point for WF

W = ( AT A + µI ) −1 AT F

µ = diag (0,0, 0,1030 ,0, ,0)


Least squares solution with minimum norm (LSMN):
Select a zero point so that the LS solution satisfies m

∑ wi = minimum 2

i =1

W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

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3. Solutions
1.

2.

(W. Zou and Z. Zhang, Appl. Opt., Vol. 39, No.2, Jan.2000)

Which point is a good “zero point” for a wave front? The best constraint point is located at the center of the testing pupil, at which the condition number of normal equation is the smallest.

Definitions Condition number of normal matrix:

[(

cond( A T A) = lub 2 A T A

)

−1

]lub (A A) T

2

and matrix norm:

X T A T AX lub 2 ( A ) = max = ρ( A T A ) T x≠o X X W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

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3. Solutions

(W. Zou and Z. Zhang, Appl. Opt., Vol. 39, No.2, Jan.2000)

How to obtain the Minimum-Norm Least-Squares solution (MNLS)? For most cases in AO and Optical tests, We are looking for the Mini-Norm LS solutions. MNLS can be obtained in following means: 1. Select a “free “ zero point for the wavefront, and in the LS sense we look for the solution for m

∑ wi = 0 2

i =1

2.

3. 4.

Damping least squares solution converges to MNLS solution, when the damping factor is very small. Iterative method, when the initial values are zero. Singular Value Decomposition method (SVD)

W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

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3. Solutions How to solve the normal equation set?

1.

Iterative methods

2.

Singular Value Decomposition method (SVD)

3.

Gaussian elimination method

4.

Cholesky decomposition method

5.

….

W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

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3. Solutions

(W. Zou and Z. Zhang, Appl. Opt., Vol. 39, No.2, Jan.2000)

Numerical methods for solving the Normal equation


Iterative methods 1.

Jacobi method (D. Su at al 2000) wi = w1 =

2.

3.

1 a  ∂W w i − t + w i −1 + w i +1 + w i + t ) +  ( 4 8  ∂y 1 a  ∂W w 2 + w t +1 ) +  ( 2 4  ∂z

+ 1

∂W ∂z

− t +1

+ i −1

∂W ∂y

∂W ∂z −

2

− i+t

∂W ∂z

− i−t

∂W ∂y

  i +1 

∂W   ∂y 1 

Gauss -Seidel method SOR method
most efficient : Optimal relaxation factor is 1.881, corresponding iteration times for convergence is 111

W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

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3. Solutions

(W. Zou and Z. Zhang, Appl. Opt., Vol. 39, No.2, Jan.2000)

Numerical methods for solving the Normal equation
Cholesky

Decomposition Method

The normal equation must be positive 1.

AT A > 0

Decomposition (m3/6 +m square-root computations)

[ ]

A A = a ij T

m× m

[ ]

B = b ij

m ×m

 0 i< j = b ij i ≥ j

i

a ij = ∑ b ik b jk ,

(1 ≤ i ≤ j ≤ m)

k =1

Then the normal equation BBΤW=Y Let U=BΤW, U=(u1,u2,…um)Τ , so BU=Y 2.

Substitution ( m2 computations) w m = u m bm,m

u1 = y1 b11 i −1

u i = (y i − ∑ b ij y j ) b ii , j =1

W. Zou/ CfAO/2003

i = 2,3,l , m

m   wi =  ui − ∑ b ji w j  bii , i = m - 1, m - 2,l,1 j= i +1  

School of Optics/CREOL/ FPCE

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3. Solutions

(W. Zou and Z. Zhang, Appl. Opt., Vol. 39, No.2, Jan.2000)

The memory storage problem in solving the normal equation


In stead of saving it in memory space, we express the matrix coefficients as a function a(i, j), which has only 5 different values.

if i ≥ j if (i, j ) ∈ Ω 4  4,  3, else if (i, j ) ∈ Ω 3  a (i, j) =  2, else if (i, j ) ∈ Ω 2 − 1, else if (i, j ) ∈ Ω −1   0, else else a(i, j) = a(j, i)


 E1 − I  − I E − I  2    AT A =  r r r   − − I E I 2    − I E1  m×m   2 −1  3 −1   − 1 3 − 1 − 1 4 − 1  − 1        E1 =  r r r − I =  r   = E r r r 2      − 1 3 − 1  − 1 t × t − 1 4 − 1    − 1 2  t × t − 1 3  t × t 

Therefore, we only need mx(t+1) elements of memory space for Cholesky decomposition.

W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

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4. Error analysis 1.

(W. Zou and Z. Zhang, Appl. Opt., Vol. 39, No.2, Jan.2000)

Discretization errors of the normal equation For Interior grids ~a

4

, a is the grid interval.

− w′i+1 − w′i+t + 4w′i − w′i−1 − w′i−t = −a2∇ni −

For Boundary grids ~ a

( )

3

i

w i − w i +t

a4  ∂ 4 W′ ∂ 4 W′  + 4 + Ο a6 4   12  ∂y i ∂z i 

i

∂W a3 ∂ 3 W =a + + Ο(a5 ) 3 ∂z i +t 24 ∂z i +t

So, the less complexity the pupil shape is, the higher precision the algorithm can achieve.
Remove the Boundary effect , see Roddier at al, Appl Opt, 1987. W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

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4. Error analysis 2.

(W. Zou and Z. Zhang, Appl. Opt., Vol. 39, No.2, Jan.2000)

Error propagation a4 A AW ′ = (M 3 − 2M 4 ) + aA T N ′ 24 T

3.

Wavefront Error estimate σW′ ≤

(

)

T

Discretization error

4.

[

(

)]

a cond A A aβ k cond A A M M 2 − + 3 4 2 f0 ⋅ t lub2 ( A) 24 ⋅ t lub2 ( A)2 4

T

1 2

σ CCD

Centroiding error

Error estimate for a specific S-H system

(neglect discretization errors, and, m=txt=15x15, a=2mm, f0=184mm, β=7.5 )

σ W ′ ≤ 17.752 W. Zou/ CfAO/2003

β σ CCD = 0.724σ CCD f0 School of Optics/CREOL/ FPCE

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5. Deviation error removal (W. Zou and J Rolland, Submitted to JOSA A, 2003)

1.

Original wavefront Non-domain extension wavefront Ave=-0.1848 micron,Var=0.3829 micron

1

micron

0.5

0

-0.5

-1 30 20

30 20

10

10

0 mm

W. Zou/ CfAO/2003

0 -10

-10

mm

School of Optics/CREOL/ FPCE

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5. Deviation error removal (W. Zou and J Rolland, Submitted to JOSA A, 2003)

2.

The reconstructed wavefront with domain extension technique Iterations=0, Var=0.3596 micron, Ave=-0.1834

1

micron

0.5

0

-0.5

-1 30 20

30 20

10

10

0 mm

W. Zou/ CfAO/2003

0 -10

-10

mm

School of Optics/CREOL/ FPCE

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5. Deviation error removal (W. Zou and J Rolland, Submitted to JOSA A, 2003)

We haveλ λ/3 -λ λ/2 P-V deviation error (λ λ=632.8nm) induced between the two wave fronts!!!

Why??? Iterations=0, Var=0.0412 micron, max=0.1903 micron, min=-0.0917 micron

1

micron

0.5

0

-0.5

-1 30 20

30 20

10

10

0 mm

W. Zou/ CfAO/2003

0 -10

-10

mm

School of Optics/CREOL/ FPCE

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5. Deviation error removal (W. Zou and J Rolland, Submitted to JOSA A, 2003)


Compatibility of Domain extension Since the extended slope data does not satisfy the derivative continuity condition of Neumann problem, so the domain extension is not strictly compatible.


Conclusion: 1.

2.

Zero padding of slopes outside the pupil will introduce deviation errors! We need to extrapolate the slope data outside the pupil instead of zero padding!

W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

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5. Deviation error removal (W. Zou and J Rolland, Submitted to JOSA A, 2003)




We employ an iterative linear equation approach to remove the deviation error induced by domain extension in the Generalized wavefront reconstruction algorithm!

W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

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5. Deviation error removal (W. Zou and J Rolland, Submitted to JOSA A, 2003)

Example 1. Optical pupil without obstruction i=0 ,Rms=0.0412
m 1

1

0.5

0.5

0

-0.5

-1 30

-1 30 30 20

10

20

30

-10

mm

a) Original wavefront

School of Optics/CREOL/ FPCE

10

0

0 -10

20

10

10

0 mm

W. Zou/ CfAO/2003

0

-0.5

20

A pupil without obstruction

Iterations=0, Var=0.0412 micron, max=0.1903 micron, min=-0.0917 micron

micron

micron

Non-domain extension wavefront Ave=-0.1848 micron,Var=0.3829 micron

mm

0 -10

-10

mm

b) Deviation error of the reconstructed WF from the original 26

5. Deviation error removal (W. Zou and J Rolland, Submitted to JOSA A, 2003)

Example 1. Algorithm Convergence process…… Iterations=0, Var=0.0412 micron, max=0.1903 micron, min=-0.0917 micron

Iterations=1, Var=0.0172 micron, max=0.1169 micron, min=-0.0380 micron The difference between the original and the reconstructed after 2th itration,RMS: 0.01micron

0.5

0.5

-0.5 30

0

micron

micron

0

-0.5 30 20

30

20

-10

30 20

10

0 -10

mm

mm

20

10

0

0 -10

mm

20

10

10

0

0

-0.5 30

30

20

10

-10

mm

i=1, Rms=0.02 μm

i=0, Rms=0.04 μm

10

0

0 -10

mm

-10

mm

i=2, Rms=0.01 μm

Iterations=4, Var=0.0062 micron, max=0.0540 micron, min=-0.0169 micron

The difference between the original and the reconstructed after 3th itration,RMS: 0.007micron

Iterations=13, Var=0.0049 micron, max=0.0297 micron, min=-0.0090 micron

1

0.5

1 0.5 0.5

0

0

micron

micron

micron

0.5

-0.5

0

-0.5 -1 30

-0.5 30 20

30 20

10

20

30

0 -10

-10

i=3, Rms=0.007
m W. Zou/ CfAO/2003

mm

20

10

0

10

0

-1 30

20

10 -10

-10

30 20

10

0

10

0 mm

i=4, Rms=0.006


mm

0 -10

-10

mm

i=13, Rms=0.005 μm

School of Optics/CREOL/ FPCE

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5. Deviation error removal (W. Zou and J Rolland, Submitted to JOSA A, 2003)

Example 2. Optical pupil with central obstruction i=0, Rms=0.043
m Original central blocked wavefront Ave=0.0012 micron,Var=0.1466 micron

Iterations=0, Var=0.0434 micron, max=0.0817 micron, min=-0.0690 micron

1

1

0.5 micron

micron

0.5

0

-0.5

-0.5

-1 30

-1 30 20

30 20

10

W. Zou/ CfAO/2003

20

-10

mm

a) Original wavefront

School of Optics/CREOL/ FPCE

20 10

0

0 -10

30

10

10

0 mm

A pupil with central obstruction

0

mm

0 -10

-10

mm

b) Deviation error of the reconstructed WF from the original 28

5. Deviation error removal (W. Zou and J Rolland, Submitted to JOSA A, 2003)

Example 2. Algorithm Convergence process…… Iterations=0, Var=0.0434 micron, max=0.0817 micron, min=-0.0690 micron

Iterations=1, Var=0.0242 micron, max=0.0461 micron, min=-0.0396 micron Iterations=3, Var=0.0101 micron, max=0.0220 micron, min=-0.0205 micron

1

1

1 0.5 0.5 micron

0

0

micron

micron

0.5

0

-0.5

-0.5

-0.5 -1 30

-1 30 20

-1 30

20

30

10

20

10

-10

-10

mm

mm

i=0, Rms=0.043
m

30

10

10

0

0 -10

20

20

10

0 mm

30 0

20 10

0

-10

mm

i=1, Rms=0.024


-10

mm

0 -10

mm

i=3, Rms=0.01
m

Iterations=5, Var=0.0059 micron, max=0.0164 micron, min=-0.0149 micron Iterations=7, Var=0.0047 micron, max=0.0143 micron, min=-0.0132 micron Iterations=10, Var=0.0041 micron, max=0.0132 micron, min=-0.0126 micron

1 1 1

0.5 0.5

0

micron

0

micron

micron

0.5

-0.5 -0.5 -1 30

-0.5

-1 30

20 30

10

20 10

0 mm

-10

-1 30

20

20

30

10

0 -10

20

i=5, Rms=0.006
m W. Zou/ CfAO/2003

mm

-10

mm

i=7, Rms=0.005
m School of Optics/CREOL/ FPCE

20 10

0

0 -10

30

10

10

0 mm

0

mm

-10

0 -10

mm

i=10, Rms=0.004
m

29

5. Deviation error removal (W. Zou and J Rolland, Submitted to JOSA A, 2003)

The deviation achieved with this algorithm is less than λ/150~λ/200 for λ=632.8nm, which is excellent for most optical testing and AO systems. Computation time for our PC

~1sec

~2 sec

Unit in wavelength of 632.8 nm

0.05

Data 1,no obstruction 0.04

data 2,no obstruction 0.03

Data 4,central obstruction

0.02 0.01 0 0

5

10

15

iterations

Deviation errors vs Iteration times Slope data is from a Shack-Hartmann sensor W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

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6. Summary Fourier Transform- based algorithm


1.

2.

Klaus Freischlad and Chris Koliopoulos did a pioneer work for generalized algorithm techniques ( Freischlad and Koliopoulos, JOSA A Vol. 3, Zo. 11, 1986) Roddier et al use a Gerchberg-type iterative algorithm to extrapolate interferogram outside the pupil and obtained a perfect domain extension.

Linear equation set-based algorithm


1.

2.

W. Zou proposed a generalized algorithm with zero padding the slope data outside the domain (W. Zou at al, Appl. Opt., Vol. 39, No.2, Jan.2000) W. Zou and J. Roland improved their previous technique by extrapolating the slope data outside the pupil with iterative linear equation approach ( Submitted to JOSA A in 2003 and patent is in filing…)

W. Zou/ CfAO/2003

School of Optics/CREOL/ FPCE

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7. References 1.

W. H. Southwell, J. Opt. Soc. Am., Vol. 70, No.8, pp.998-1006,1980C.

2.

Roddier and F. Roddier, Appl. Opt., Vol.26,No.9, 1987.2

3.

F. Roddier and C. Roddier, Appl. Opt.,Vol.30, No.11, pp.1325-1327,1991.

4.

W. Zou and Z. Zhang, Appl. Opt., Vol. 39, No.2, Jan.2000.

5.

R. W. Gerchberg and W. O. Saxton, Optik, Vol. 35, pp237-246, 1972.

6.

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School of Optics/CREOL/ FPCE

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