Reconstruction Methods

Swapnil Prabhudesai, Instrumentation Lab. IUCAA Why do we need wavefront ?

a b c d/2 d d/2

Plane WF Aberrated WF DM Front Sensing

• Zonal wavefront sensing Wavefront is expressed in terms of OPD over a small spatial area (zone/sub-aperture) Wave Front Sensing

• Modal Wavefront Sensing Wavefront is expressed in terms of some polynomial expansion over the entire pupil (i.e. PM) Zernike Polynomials • WF can be expressed as Zernike polynomials • Orthogonality of Zernike polynomials over unit circle • Annular Zernike Polynomials • Examples: z1 1 piston

z2  2r cos x

z3  2r sin y tilt 2 z4  3(2r 1) defocus 2 z5  6r sin 2 astigmatism 2 z6  6r cos2 astigmatism Zonal Wavefront Sensing

• Sub-aperture size & Fried parameter: does not change over this region for the time interval of the order of , for example, 10 ms. • Local tilts are measured. • Centroid of intensity distribution is equal to shifted origin of the image plane; shift being proportional to the tilt (fig. (c) in next slide) , i.e. local tilt shifts centroid. Zonal sensing

(a) (b)

k R Rk/a a k/a (c) Zonal sensing

2 1 3 4 Quadrant detector lenslet  Ids  Ids  Ids   Ids 1 4 2 3 xcentroid   Ids i i  Ids  Ids  Ids   Ids 1 2 4 3 ycentroid   Ids i i Shack-Hartmann Sensor

s(x, y) (x, y) z WF surface from slope measurements

Original WF

Zonal slopes (one axis)

Reconstructed WF

O/P to WF corrector Wavefront Reconstruction (Phase/OPD from WF slopes) • OPD at various points on WF is determined from WF slopes at other points on the WF • Number of WF slope measurements (M) is generally > Number of unknown phase points (N) • Problem of phase determination from WF slopes depends on geometry of arrangement of actuators w.r.t. phase points on WF Wavefront Reconstruction

Main 3 Geometries :

–Hudgin –Fried –Southwell Hudgin Geometry

Node/phase /actuator w16 location

w9

w5 w6 w7 w8

w1 w2 w3 w4 1 Sub-aperture Fried Geometry

Slope Model  w  w   w  w   2 6    1 5   s x 2 2 w5 w6     w1 w2 w3 w4  w  w   w  w   5 6    1 2   s y  2   2  Waffle Mode (Fried geometry) Two actuators on one diagonal H L H L are equally high and two actuators on the other diagonal are equally low. L H L H Centroid is undisturbed relative to centroid of flat wavefront. H L H L Removal of waffle mode: w5 w6 L H L H w1 w2 w  w w  w 2 5  1 6  0 2 2 Southwell Geometry

Slope Model s x  s x 1 2  w  w 2 2 1 w4 s y  s y w1 w2 w3 1 4  w  w 2 4 1 Slope Model (Hudgin Geometry)

x y w2  w1  s1 w5  w1  s1 x y w3  w2  s2 w6  w2  s2

w16

w9

w5 w6 w7

w1 w2 w3 w4 x s1    1 1  x   s2  1 1    w      1   w       2      x     sM        y     s1  1 1       y   s2  1 1                             wN  s y     M   0   1 1    1 1   SM 1  AM N WN1

Slope measurements From geometry To be estimated from SH sensor Least Squares Method

[A][W]  [S]

 Residual vector[ r] [A][W][S]

 r  AW  S  AW  S for all W

W  (AT A)1 AT S

Difficulty: Every time A T A cannot be inverted. Least squares method

M N 2 2     Si   Aik Wi  i1  k1 

  2  0 wi

1 W  AT A AT S Use of SVD • Every MxN matrix can be factorized into 3 matrices

[A]  [U ][D][V T ] where

d1   d   2  [D]             dn  [A ]  [V ][D1][U T ] Psuedo-inverse:  1     d1   1   d  [A ]  [V ] 2 [U T ]          1     d N 

1 1 [D ]  if di  t di  0 otherwise where t is a small threshold. Finally we get the solution as:

[W ]  [A ][S]  [V ][D1][U T ][S] References • Adaptive Optics for Astronomical Telescopes; John W. Hardy; OUP 1998

• Linear Algebra; Gilbert Strang; Thomson/Brooks,Cole

• Wave-front Reconstruction from wave-front slope measurements; W.H. Southwell; J.Opt.Soc.Am.,Vol.70, August 1980