Gary Chanan Department of Physics and Astronomy University of California, Irvine 4 February 2000

1. Gary Chanan Department of Physics and Astronomy University of California, Irvine 4 February 2000

2. Outline 1. Atmospheric optics • A brief introduction to turbulence • A guide to the relevant mathematics • Derived optical properties of the atmosphere 2. Wavefront sensing • Shack-Hartmann wavefront sensing • Curvature sensing

3. Wind Tunnel Experiments 5 4 3 Slope = -5/3 2 v (arbitrary units) Power (arbitrary units) 1 0 -1 -2 -3 -4 3 4 5 1 0 2 Frequency (s-1) Time (sampling units) m V = 20 sec VL 7 Re = ~ 10  L = 72 m Gagne (1987)

4. Kolmogorov’s Law (1941) Power (arbitrary units) Slope = -5/3 k (cm-1) < < ~ ~ inner scale outer scale F (k)  k-5/3 F (k)  k-11/3 v v  10 cm air jet (Champagne 1978) Energy cascades down to smaller spatial scales r, higher spatial frequencies k = 2p/r . In inertial range: l0 r L0 For atmosphere: ~ 1 mm 10 m - 10 km Power spectrum: or

5. Oboukov’s Law (1949) Fluctuations in scalar quantities associated with this flow (passive conservative additives) inherit this same power spectrum. s ()  k -5/3

6. Structure Function Kolmogorov’s Law describes behavior in Fourier space; we are often interested in real space. Consider structure function: 2 s ( r ) = < | s ( r + r ) - Ds ( r ) | > mean squared fluctuation There is a Fourier-like relation between s and s : s ( r ) = 2 s ( k ) ( 1 - e ik • r ) d k 3

7. Mathematical Notes • Power laws predominate. • Integral of a power law is a power law. • Power law indices are easy to calculate; numerical coefficients are hard. Thus: -5/3 2/3 s ( k )  k <=> s ( r ) r 2 2/3 We write s ( r ) = C r s structure constant

8. Index Fluctuations 2 2 3 3 in atmospheres P -3 n - 1 = 79 x 10 T in Kelvins -3 P T n = -79 x 10 [Neglect pressure fluctuations.] 2 T For typical night-time atmosphere (0.1 to 10 km): 2 -4 -2/3 C ~ 10 m T 2 -16 -2/3 C ~ 10 m N Application of Oboukov’s Law to the (pre-existing) large scale temperature gradients in the atmosphere => - law for T fluctuations => - law for n fluctuations -8 Thus on meter scales T ~ 10 mK , n ~ 10 !

9. 10-15 10-16 10-17 10-18 10-19 10-20 0.01 0.1 1 10 100 2 Cn Profile Cn (m-2/3) 2 Height (km)

10. Central Problem  h 2 2/3 n ( r ) = Cn r D = 2R The index fluctuations are well-characterized. What are the corresponding fluctuations in the accumulated phase?

11. First Order Treatment We will do a first-order treatment, which gives a surprisingly good accounting of the typical astronomical situation (esp. for large telescopes): All points on the wavefront travel straight down, but are advanced or retarded according to: 2 (x,y) = n(x,y,z) dz 

12. Near - Field Approximation 2 2 R R   This neglects diffraction effects. Valid when: Typical diffraction angle () h << R => h <<  R lateral displacement of ray For large telescopes, Characteristic vertical scale of atmosphere is h ~ 104 meters; so the near field approximation is usually well-satisfied in practice.  106 meters.

13. Scintillation 2 R  Note that it is precisely these diffraction effects which give rise to scintillation or twinkling. Different parts of the diffracted wavefront eventually interfere with one another. Thus there is no scintillation in the near-field approximation. But for the dark-adapted eye R ~ 4 mm and: ~ 30 meters The inequality turns around and the stars appear to twinkle.

14. Our central propagation problem can be elegantly stated in Fourier space: Given the 3-dim spectral density of n, what is the corresponding 2-dim spectral density of the phase , which is proportional to the integral of n? …and elegantly solved by the following theorem:  ( kx , ky ) = 2  h n ( kx , ky , 0) ndz The phase structure function follows directly: 2  ( r ) ~ ( ) Cn h r 2 2 5/3 

15. Fried’s Parameter r0 r ~ { } -5/3 Fried’s parameter o ( ) Cn h 2 2 2  We write:  ( r ) = 6.88 ( ) 5/3 r r o where ro is the diameter of a circle over which rms phase variation is ~ 1 radian. 2 For Cn ~ 10-16 m-2/3 h ~ 104 m l ~ 0.5 mm we have ro ~ 10 cm.

16. Quantity Scaling Name Value (at 0.5 mm) Value (at 2.2 mm) r0l6/5 Fried parameter20 cm 120 cm (coherence diameter) t0 ~ l6/5 coherence time20 ms 120 ms q0 ~ l6/5isoplanatic angle4" 24" qfwhm~ l-1/5image diameter0.50" 0.38" Nact~ l12/5 required no. of actuators2500 70 S ~ l-12/5uncorrected Strehl ratio4x10-4 0.014 r0 l v r0 r0 h D2 2 r0 r0 2 D2 r0 - Related Parameters

17. Expansion of the Phase in Zernike Polynomials An alternative characterization of the phase comes from expanding  in terms of a complete set of functions and then characterizing the coefficients of the expansion: (r,q) = S am,n Zm,n(r,q) Z0,0= 1 Z1,-1= 2 r sinq Z1,1= 2 r cosq Z2,-2= 6 r2 sin2q Z2,0= 3 (2r2 - 1) Z2,2= 6 r2 cos 2q piston tip/tilt astigmatism focus astigmatism

18. Z0,0 Z1,-1 Z1,1

19. Z2,-2 Z2,2 Z2,0

20. Z3,-3 Z3,3 Z3,-1 Z3,1

21. Z4,-4 Z4,4 Z4,0 Z4,-2 Z4,2

22. Atmospheric Zernike Coefficients RMS Zernike Coefficient (D/ro)5/6 Zernike Index

23. Shack Hartmann I Shack-Hartmann Test

24. Shack Hartmann II Shack-Hartmann Test, continued

25. UFS Ref Beam Ultra Finescreen Reference Beam Exposure

26. UFS Image (before) Ultra Fine Screen Image (Segment 8) (0.44 arcsec RMS)

27. UFS C. offsets (before) Centroid Offset Display Centroid Offset Summary Info: Translation from Ref : -4.87 -0.02 Rotation from Ref (rad) : 0.271E-03 Scale change from Ref : 1.017 KEY: 0.140 arcseconds per pixel Scale Error : -2.75 80% Enclosed Energy : 10.93 50% Enclosed Energy : 7.66 RMS Error : 3.14 Max Error (pixels) : 11.11 Subimage With Max Error : 209 15 pixels

28. Curvature Sensing Concept  I+ ( r )  I ( r ) I+ I I++ I  r (F. Roddier, Applied Optics, 27, 1223-1225, 1998)   (r)     2 dR   Laplacian normal derivative at boundary

29. Z1,-1 Difference Image

30. Z2,-2 Difference Image

31. Z2,0 Difference Image

32. Z3,-3 Difference Image

33. Z4,0 Difference Image

34. Random Zernikes Difference Image