1. LSST Wavefront Sensing Bo Xin Systems Analysis Scientist (LSSTPO) Chuck Claver, Ming liang, Srinivasan Chadrasekharan, George Angeli (LSST wavefront sensing team)

2. LSST Active Optics & Wavefront Sensing 2 LSST Active Optics System (AOS): Maintain system alignment Maintain surface figure on three mirrors DOFs controlled by AOS M2 hexapod rigid body positions Camera hexapod rigid body positions M1M3 bending modes M2 bending modes DOFs controlled by AOS M2 hexapod rigid body positions Camera hexapod rigid body positions M1M3 bending modes M2 bending modes

3. Outline 3 WFS Design & Constraints Paraxial Curvature WFS algorithms LSST WFS challenges & algorithmic modifications Large Central Obscuration (61%) Fast f/number (f/1.23) Off-axis Distortion and Vignetting (~1.7 o ) Field Dependence (covering 1.51° to 1.84°) Algorithm Performance Beyond Unit Tests Future Work

4. WFS Design and Constraints 4 Curvature sensing enables significant flexibility in selecting sources due to the large field of view of the area sensors split sensors: because of the fast f- number (f/1.23) and crowded focal plane, using a beam splitter and delay line or physically moving the detector will not work. Use multiple sources to increase S/N, to help average out atmosphere noise, and to alleviate problems due to vignetting. LSST WFS challenges 61% Central Obscuration f/1.23 Off-axis Distortion & Vignetting (~1.7 o ) Field Dependence (covering 1.51° to 1.84°)

5. Curvature WFS algorithm 5 One way of doing wavefront sensing is to solve the transport of intensity equation (TIE): Intensity difference between the two defocused images is proportional to the curvature of the wavefront: Make use of the property of Laplacian in Fourier space Iterative algorithm, involves repeatedly setting the boundary condition Iterative FFT method: (C. Roddier and F. Roddier, J Opt Soc Am A 10, 2277-87 (1993) ) Orthogonal series expansion: (T. E. Gureyev and K. A. Nugent, J. Opt. Soc. Am. A 13, 1670-1682 (1996)) Non-iterative algorithm, involves integrations over the pupil faster than iterative FFT in general

6. Further Improving the Accuracy 6  Both the Iterative FFT and the Series Expansion algorithms are first order approximations valid for highly defocused images. Create or obtain I 1 & I 2 a Cocenter I 1 & I 2 Iterative FFT or Series Expansion curvature wavefront sensing algorithm to determine initial W estimate or W residual Wavefront compensation to create compensated I 1 & I 2 After W residual converges to zero or after a set # of iterations, save (W compensated + W residual ) b c d e The accuracy can be improved by iteratively compensating the effect of the estimated aberrations on the defocused images. Original intra and extra focal images Images after compensating the estimated aberrations (2 waves of 45 o -astigmatism, in this example)

7. Algorithmic Challenges for LSST 7 LSST WFS challenges & algorithmic modifications Large Central Obscuration (61%) Fast f/number (f/1.23) Off-axis Distortion and Vignetting (~1.7 o ) Field Dependent Corrections (covering 1.51° to 1.84°)

8. Test with Paraxial Lens Model 8 2 waves of Z4 (defocus) λ=770nm Annular Zernike numbering following Mahajan 1981 Z22=(6,0) Even with annular Zernikes, large obscuration still makes the wavefront harder to recover, because of reduced pixel sampling

9. What a fast f/# means 9 f p =10.312m f m =9.427m Large curvature on the principal plane! Red: actual marginal rays Blue: paraxial lens marginal rays image

10. No Aberration? Still non-uniform! 10 A non-linear mapping from the pupil to the image When there is no aberration in a fast- beam on-axis system, the intensity distributions on the defocused images are NOT uniform. IntraExtra o If we run our algorithms on a pair of aberration-free images, because of the non- uniform intensity distribution, there will be anormalous focus+spherical signal. o The obscuration ratio on the pupil plane is different from that on the image plane. (61% vs. 58%)

11. Test with LSST On-axis Images 11 1/2 waves of Z11 (spherical aberration) λ=770nm Annular Zernike numbering following Mahajan 1981 Z22=(6,0)

12. What Happens at 1.7 o Off-axis 12 ZEMAX view of LSST Field angle (1.185 o, 1.185 o ) pupilimage

13. Modeling the Off-axis Distortion 13 Aperture Image plane Using paraxial lens model as reference, rayhit coordinate change on the image plane: @center of wavefront chip off-axis distortion <~7pixels 200nm of 45◦− astigmatism <~1/3 pixels

14. LSST Off-axis Tests 14 Single Zernike term: 2λ Z5 (astigmatism 45 o ) Field angle (1.185 o, 1.185 o ) Vignetting: Use vignetted pupil mask Annular Zernikes not orthogonal over pupil Loss of intensity information on the edge Effects are observed to be small for most parts of the wavefront chips.

15. Field-dependent Distortions & Vignetting 15 Intra focalextra focal

16. Interpolation on one wavefront chip 16

17. Other Field Dependent Corrections 17 The intra and extra focal images are formed at different field positions “Migrate” both to the center of wavefront chip using the field- dependent off-axis correction. Properly normalized the intensities. Given the split sensor design, the intra focal and extra focal images are vignetted differently Apply the logical AND of the two pupil masks Vignetting ratio by area (spanning the field angle of the wavefront chips)

18. Algorithm Performance Beyond Unit Tests 18 More systematic tests with ZEMAX images Tests using PhoSim Images Algorithm Linearity Covariance Analysis As part of the LSST Integrated Simulation Prototype algorithms with real data and real telescopes

19. Tests of WFS with Perturbed LSST Model 19

20. LSST Off-axis Tests 20 Intrinsic aberration Field angle (1.185 o, 1.185 o ) 25nm

21. Split Sensor Tests 21 Original intra Original extra Using the logical AND of the two pupil masks

22. Tests using PhoSim Images 22 telescope perturbations and atmospheric turbulence turned off Phosim V3.2.6 April 2013 Field=(1.185,1.185) Field=(1.237,1.237)

23. Algorithmic Linearity 23 Z7 on M1M3 (in wave) Wavelength=770nm

24. Covariance Analysis 24 (Algorithmic + atmospheric) Covariance matrix Total covariance is almost entirely dominated by atmosphere – Diagonal elements (lower left) similar – Singular values (lower right) also similar; some increase for low singular values Variance Singular values

25. Used in LSST Integrated Simulation 25 With atmosphere, mirror gravity print throughs, polishing errors (M1 only for now), thermal shape errors, and camera internal perturbations

26. LSST WFS Pipeline Tests on WIYN pODI pODI Full Field (Sep. 13, 2012) CCD-4 Blended Pair Template Model FitDe-blended Early results show excellent agreement with proven software LSST wavefront software is open source and is designed for general analysis of intra/extra focal image pairs. Early pODI commissioning tests show wavefront estimation in excellent agreement with obsolete unsupported (but proven) software. LSST & ODI teams working together to analyze pODI alignment. De-blending algorithm shown to be effective, maximizing the available sources for wavefront estimation.

27. Conclusions & Future Work 27 Work has been written up and will be submitted to a journal Performance analysis has helped with decisions in wavefront sensor design Code written in MATLAB, and has been converted to Python (Andy Connolly) Algorithm optimization and code restructuring (Robert Lupton & Mario Juric) Image pre-processing software has been developed and will be tested and integrated into DM (ISR, source selection, background removal, wavefront image de-blending, etc.) Algorithm is being used routinely as part of the LSST integrated simulation, where it is being tested with more and more realistic simulated images every day. Prototype algorithms with real data and real telescopes DECam – comparing our algorithm with forward modeling algorithm (Aaron Roodman) WIYN ODI – work in progress (Daniel Harbach) LBT prime focus – f/1.142 (Mark Wagner) Magellan – compare curvature measurements with facility SH (Chris Stubbs)