PSFs for Laser Guide Star Tomographic Adaptive Optics Systems

Transcription

1 PSFs or Laser Guide Star Tomograpic Adaptive Optics Systems Donald Gavel Center or Adaptive Optics UC Santa Cru PSF Reconstruction Worsop, HIA Victoria May 11, 4 UCO Lic Observatory NSF Center or Adaptive Optics Laboratory or Adaptive Optics

2 TMT AO system concepts or te near IR bands MCAO Multi-conjugate adaptive optics Diraction-limited imaging Correction over a wide ield (~3 arcsec) Multiple DMs, Multiple LGS MOAO Multi-object adaptive optics Diraction-limited spectroscopy (IFUs) Correction over small ields (~5 arcsec) at many ocal plane positions Eac unit as a single DM, corrects or te science direction Multiple LGS measure te volume o turbulence PSF Reconstruction Worsop HIA Victoria, May, 4

3 AO Tomograpy using Toovinin s Fourier domain approac 1 ~ ~ s φ + ~ Measurements rom guide stars: ( ) M ( ) (, θ ) ν ( ) = 1, Kn = gs Problem: Find a linear combination o guide star data tat best predicts te waveront in a given science direction, θ ψ~ N ~ g ~ ( ) ~ ( ) ~ T, θ g, θ s ( ) = s = = 1 1 Toovinin, A., Viard, E., Limiting precision tomograpic pase estimation, JOSA-A, 18, 4, Apr. 1, pp PSF Reconstruction Worsop HIA Victoria, May, 4 3

4 Some preliminaries: Te Hartmann waveront sensor model can be ignored Perorm an invertible transormation on te WFS data i x = i ~ s ~ s ~ s i y x y ~ s 1 y i x y x + = i xν x + i yν y φ + i yν x + i xν y Since te nd equation doesn t depend on te data, M = is an appropriate scalar measurement model (te 1 st equation above). Witout loss o generality, we ll assume te WFS as a post-ilter -. Tis allows us to set M=1 (up to some requency cut-o associated wit te WFS spatial sampling). Te noise spectrum is modiied accordingly (1/ spectrum). w.l.g., Assume te waveront sensors measure pase PSF Reconstruction Worsop HIA Victoria, May, 4 4

5 Minimum variance solution or plane waves [ ] * ~ g 1 ~ ( ) = ( A( ) + υ ( ) I) c( ) a~ c~ υ ( ) = M C ( ) exp[ πi ( è è) ] ( ) = M C ( ) exp[ πi ( è è ) ] c = Cn ( ) d ( ) = W ( ) ( W ( ) c ) ν n n d d c c Residual error covariance: W ε W [ *] 1 T ( ) = W ( ) c 1 c ( A + υ I) c 3 11/ 3 ( ) = ( π λ) PSF Reconstruction Worsop HIA Victoria, May, 4 5

6 Te minimum variance solution = bac-projection tomograpy Science waveront solution: ψ~ ~ were 1 [ ] ~ s( ) T (, θ ) = c ( ) A ( ) + Iν ( ) = exp N ( πiθ ) Cn ( ) exp( πiθ ) s ( ) d c = 1 [ ] s( ) ~ s υ 1 ~ ( ) = A ( ) + I ( ) ~ 1 Taing te inverse Fourier transorm: N 1 ψ ( x, θ ) = C ( ) s n ( x ( θ θ ) d c Volumetric estimate: = 1 δn λ πc N ( x, ) = s ( x + θ ) C ( ) = 1 = a sum o bac projections o te iltered waveront measurements n Science waveront estimate: ψ π = d λ ( x, θ ) δn( x θ, ) = te orward propagation along te science direction troug te estimated turbulence volume. PSF Reconstruction Worsop HIA Victoria, May, 4 6

7 Minimum Variance Solution or Sperical Waves Solution is independent o science direction (oter tan te inal orward projection, wic is accomplised by ligt waves in te MCAO optical system) Te ollowing is a least-squares solution or sperical waves (guidestars at inite altitude) N c s a δn est λ πc ( x, ) = s x + è C ( ) = 1 [ ] s( ) ~ s υ 1 ~ s ( ) = A ( ) + I ( ) 11 3 (, ) = ( ( ) C ( ) exp( πi è ) s v 5 3 ( ) = ( ( ) C ( ) exp( πi ( è è ) ) ( ) = v ( ) δ ( c ) n n An approximate solution or inite apertures is obtained by mimicing te bac propagation implied by te ininite aperture solutions ~ FA s ( ) = ~ s ( ) ~ p( ; D) An approximate solution or inite aperture sperical waves (cone beams rom laser guide stars) is obtained by mimicing te sperical wave bac propagations n c d c PSF Reconstruction Worsop HIA Victoria, May, 4 7

8 Sperical Wave Solution Spatial domain Forward propagation Bacward propagation Turbulence at position x at altitude appears at position x θ at te pupil So bac-propagate position x in pupil to position x + at altitude θ Frequency domain Frequencies at altitude scale down to requencies at te pupil Frequencies at te pupil scale up to requencies at altitude PSF Reconstruction Worsop HIA Victoria, May, 4 8

9 PSF Reconstruction Worsop HIA Victoria, May, 4 9 MOAO Residual Covariance or Sperical Waves ( ) ( ) s s s,, ~ 1 * c v A g + = ( ) ( ) ( ) ( ) ( ) ( ) ( ) + = n v g c i i i C g W c 3 11 d, ~ d d exp exp exp, ~ 1, ~ è è è è π π π ε ( ) ( ) 3 11/ = W λ π were

10 For MCAO: project te volume estimates onto a inite number o DMs to minimie generalied anisoplanatism ~ d ( ) g~ DM m (, ) δn~ ( ) = d m, m = 1, K,n DM ~ g ~ b a~ DM DM m DM mm DM 1 DM (, ) = A ( ) b (, ) = J [ πθ( H m ) ] = J [ πθ( H H )] m ~ m Toovinin, A., Le Louarn, M., Sarain, M., Isoplanatism in a multiconjugate adaptive optics system, JOSA-A, 17, 1, Oct., pp PSF Reconstruction Worsop HIA Victoria, May, 4 1

11 Tomograpic Waveront Reconstruction a quic summary Combine onto DMs Propagate ligt rom Science target (wide ield) MCAO Measure ligt rom guidestars Bac- Project* to volume Project along science direction onto one DM Propagate ligt rom Science target (narrow ield) 1 MOAO *ater te all-important iltering step, wic maes te bac projections consistent wit all te data 3 4 PSF Reconstruction Worsop HIA Victoria, May, 4 11

12 TMT Science Requirements or MCAO Potometry Systematic errors in dierential potometry due to PSF residual spatial variability < % or 1 minute integrations, at 1µm over te 3 arc-sec FOV. Astrometry Residual time dependent distortions (ater a it to pysically allowed distortion wit ield stars) sould be no larger tan 1µarcsec or limited by te intrinsic variations caused by te atmospere and te inite integration time. Tis sould be acieved over a 3 FOV PSF Reconstruction Worsop HIA Victoria, May, 4 1

13 PSF Simulations Telescope Diameter D = 3 meters r = cm, CP C n proile (7 layer) LGS = 9 m Tomograpy only (no anisoplanatism or oter error budget terms) PSF Reconstruction Worsop HIA Victoria, May, 4 13

14 7 Natural Guide Stars y y x x Frequency Domain Rejection Frequency Domain Residual Power Spectrum PSF (Log stretc) PSF (Log stretc) PSF Reconstruction Worsop HIA Victoria, May, 4 14

15 7 Laser Guide Stars y y x x Frequency Domain Rejection Frequency Domain Residual Power Spectrum PSF (Log stretc) PSF (Log stretc) PSF Reconstruction Worsop HIA Victoria, May, 4 15

16 5 Natural Guide Stars y y x x Frequency Domain Rejection Frequency Domain Residual Power Spectrum PSF (Log stretc) PSF (Log stretc) PSF Reconstruction Worsop HIA Victoria, May, 4 16

17 5 Laser Guide Stars y y x x Frequency Domain Rejection Frequency Domain Residual Power Spectrum S =.74 PSF (Log stretc) PSF (Log stretc) PSF Reconstruction Worsop HIA Victoria, May, 4 17

18 5 Laser Guide Stars randomied pattern y y x x Frequency Domain Rejection Frequency Domain Residual Power Spectrum S =.75 PSF (Log stretc) PSF (Log stretc) PSF Reconstruction Worsop HIA Victoria, May, 4 18

19 Encircled Energy: A signiicant raction o te ligt is in te alo even at ig Strel 5 LGS S =.74 9 LGS S =.79 PSF Reconstruction Worsop HIA Victoria, May, 4 19

20 Plane wave, 9 guidestars Ininite aperture Finite aperture 65 nm rms 134 nm rms PSF Reconstruction Worsop HIA Victoria, May, 4

21 Sperical Wave, 9 guidestars Ininite Aperture Finite Aperture 75 nm rms 164 nm rms PSF Reconstruction Worsop HIA Victoria, May, 4 1

22 Volume Estimates rom an ininite aperture , m True Estimated.5 PSF Reconstruction Worsop HIA Victoria, May, 4

23 Volume estimates rom a inite aperture 15 m layer estimate, 5 NGS 15 m layer estimate, 5 LGS Metapupil edges cause artiacts at ig altitudes PSF Reconstruction Worsop HIA Victoria, May, 4 3

24 Extension o waveront estimates to outside te aperture Waveront measurement 15 m layer estimate, 5LGS Extension PSF Reconstruction Worsop HIA Victoria, May, 4 4

25 Conclusions Te minimum variance waveront solution = bac-projection tomograpy 1. Estimate te volume o turbulence. Forward propagate in te science direction Finite aperture algoritms mimic teir ininite aperture counterparts Covariance analysis provides MTF Plane wave solution is easy to compute Sperical wave solution is more diicult (it mixes dierent requencies at dierent altitudes) TMT goal is % potometry, 1µ arcsec astrometry will require very accurate nowledge o te PSF, particularly in te wings PSF Reconstruction Worsop HIA Victoria, May, 4 5