Adaptive Optics Reconstruction Methods Swapnil Prabhudesai, Instrumentation Lab. IUCAA Why do we need plane wavefront ? a b c d/2 d d/2 Plane WF Aberrated WF DM Wave 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 tilt 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: Refractive Index 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 point/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 WN1 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 i1 k1 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 ][D1][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 ][D1][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.