Parameters characterizing the Atmospheric Turbulence: r 0 ,  0 ,  0

1. Adaptive Optics in the VLT and ELT era Parameters characterizing the Atmospheric Turbulence: r0, 0,0 François Wildi Observatoire de Genève Credit for most slides : Claire Max (UC Santa Cruz)

2. r0 sets the number of degrees of freedom of an AO system • Divide primary mirror into “subapertures” of diameter r0 • Number of subapertures ~ (D / r0)2where r0 is evaluated at the desired observing wavelength • Example: Keck telescope, D=10m, r0 ~ 60 cm at l = 2mm. (D / r0)2 ~ 280. Actual # for Keck : ~250.

3. About r0 • Define r0 as telescope diameter where optical transfer functions of the telescope and atmosphere are equal • r0 is separation on the telescope primary mirror where phase correlation has fallen by 1/e • (D/r0)2is approximate number of speckles in short-exposure image of a point source • D/r0 sets the required number of degrees of freedom of an AO system • Timescales of turbulence • Isoplanatic angle: AO performance degrades as astronomical targets get farther from guide star

4. A simplifying hypothesis about time behavior • Almost all work in this field uses “Taylor’s Frozen Flow Hypothesis” • Entire spatial pattern of a random turbulent field is transported along with the wind velocity • Turbulent eddies do not change significantly as they are carried across the telescope by the wind • True if typical velocities within the turbulence are small compared with the overall fluid (wind) velocity • Allows you to infer time behavior from measured spatial behavior and wind speed:

5. Cartoon of Taylor Frozen Flow • From Tokovinin tutorial at CTIO: • http://www.ctio.noao.edu/~atokovin/tutorial/

6. Order of magnitude estimate • Time for wind to carry frozen turbulence over a subaperture of size r0 (Taylor’s frozen flow hypothesis): t0 ~ r0 / V • Typical values: • l = 0.5 mm, r0 = 10 cm, V = 20 m/sec  t0 = 5 msec • l = 2.0 mm, r0 = 53 cm, V = 20 m/sec  t0 = 265 msec • l = 10 mm, r0 = 36 m, V = 20 m/sec  t0 = 1.8 sec • Determines how fast an AO system has to run

7. But what wind speed should we use? • If there are layers of turbulence, each layer can move with a different wind speed in a different direction! • And each layer has different CN2 V1 Concept Question: What would be a plausible way to weight the velocities in the different layers? V2 V3 V4 ground

8. Rigorous expressions for t0 take into account different layers • fG Greenwood frequency  1 / t0 • What counts most are high velocities V where CN2 is big Hardy§9.4.3

9. Short exposures: speckle imaging • A speckle structure appears when the exposure is shorter than the atmospheric coherence time 0 • Time for wind to carry frozen turbulence over a subaperture of size r0

10. Structure of an AO image • Take atmospheric wavefront • Subtract the least square wavefront that the mirror can take • Add tracking error • Add static errors • Add viewing angle • Add noise

11. atmospheric turbulence + AO • AO will remove low frequencies in the wavefront error up to f=D 2/n, where n is the number of actuators accross the pupil • By Fraunhoffer diffraction this will produce a center diffraction limited core and halo starting beyond 2D/n PSD(f) 2D/n f

12. The state-of-the art in performance: Diffraction limit resolution LBT FLAO PSF in H band. Composition of two 10s integration images. It is possible to count 10diffraction rings. The measured H band SR was at least 80%. The guide star has a mag of R =6.5, H=2.5 with a seeing of 0.9 arcsec V band correcting 400 KL modes

13. Anisoplanatism: how does AO image degrade as you move farther from guide star? • Composite J, H, K band image, 30 second exposure in each band • Field of view is 40”x40” (at 0.04 arc sec/pixel) • On-axis K-band Strehl ~ 40%, falling to 25% at field corner credit: R. Dekany, Caltech

14. More about anisoplanatism: AO image of sun in visible light 11 second exposure Fair Seeing Poor high altitude conditions From T. Rimmele

15. AO image of sun in visible light: 11 second exposure Good seeing Good high altitude conditions From T. Rimmele

16. What determines how close the reference star has to be? Reference Star Science Object Turbulence z Common Atmospheric Path Telescope Turbulence has to be similar on path to reference star and to science object Common path has to be large Anisoplanatism sets a limit to distance of reference star from the science object

17. Expression for isoplanatic angle 0 • Strehl = 0.38 at  = 0 0is isoplanatic angle 0is weighted by high-altitude turbulence (z5/3) • If turbulence is only at low altitude, overlap is very high. • If there is strong turbulence at high altitude, not much is in common path Common Path Telescope

18. Isoplanatic angle, continued • Simpler way to remember 0 Hardy§3.7.2

19. Review • r0 (“Fried parameter”) • Sets number of degrees of freedom of AO system • 0 (or Greenwood Frequency ~ 1 / 0 )  t0 ~ r0 / V where • Sets timescale needed for AO correction • 0 (or isoplanatic angle) • Angle for which AO correction applies

20. How to characterize a wavefront that has been distorted by turbulence • Path length difference Dz where kDz is the phase change due to turbulence • Variance s2 = <(k Dz)2 > • If several different effects cause changes in the phase, stot2 = k2 <(Dz1 + Dz2 + ....)2 > = k2 <(Dz1)2 + ( Dz2 )2 ...) > stot2 = s12 + s22 + s32 + ... radians2 or (Dz)2 = (Dz1)2 + (Dz2)2 + (Dz3)2 + ..... nm2

21. Total wavefront error for an AO system: • List as many physical effects as you can that might contribute to overall wavefront error stot2 stot2 = s12 + s22 + s32 + ...

22. Elements of an adaptive optics system DM fitting error Not shown: tip-tilt error, anisoplanatism error Non-common path errors Phase lag, noise propagation Measurement error

23. Hardy Figure 2.32

24. Wavefront errors due to 0 , 0 • Wavefront phase variance due to t0 = fG-1 • If an AO system corrects turbulence “perfectly” but with a phase lag characterized by a time t, then • Wavefront phase variance due to 0 • If an AO system corrects turbulence “perfectly” but using a guide star an angle  away from the science target, then Hardy Eqn 9.57 Hardy Eqn 3.104

25. Deformable mirror fitting error • Accuracy with which a deformable mirror with subaperture diameter d can remove aberrations sfitting2= m ( d / r0 )5/3 • Constant m depends on specific design of deformable mirror • For segmented mirror that corrects tip, tilt, and piston (3 degrees of freedom per segment) m = 0.14 • For deformable mirror with continuous face-sheet, m = 0.28

26. Image motion or “tip-tilt” also contributes to total wavefront error image motion in radians is indep of l • Turbulence both blurs an image and makes it move around on the sky (image motion). • Due to overall “wavefront tilt” component of the turbulence across the telescope aperture • Can “correct” this image motion either by taking a very short time-exposure, or by using a tip-tilt mirror (driven by signals from an image motion sensor) to compensate for image motion (Hardy Eqn 3.59 - one axis)

27. Scaling of tip-tilt with l and D: the good news and the bad news • In absolute terms, rms image motion in radians is independent of l, anddecreases slowly as D increases: • But you might want to compare image motion to diffraction limit at your wavelength: Now image motion relative to diffraction limit is almost ~ D, and becomes larger fraction of diffraction limit for small l

28. Effects of turbulence depend on size of telescope • Coherence length of turbulence: r0 (Fried’s parameter) • For telescope diameter D  (2 - 3) x r0 : Dominant effect is "image wander" • As D becomes >> r0 : Many small "speckles" develop • Computer simulations by Nick Kaiser: image of a star, r0 = 40 cm D = 2 m D = 8 m D = 1 m

29. Error budget so far stot2 = sfitting2 + sanisop2 + stemporal2+ smeas2 + scalib2 √ √ √ Still need to work on these two

30. Error Budgets: Summary • Individual contributors to “error budget” (total mean square phase error): • Anisoplanatism sanisop2 = ( / 0 )5/3 • Temporal errorstemporal2 = 28.4 (t / t0 )5/3 • Fitting errorsfitting2 = m ( d / r0 )5/3 • Measurement error • Calibration error, ..... • In a different category: • Image motion <a2>1/2 = 2.56 (D/r0)5/6 (l/D) radians2 • Try to “balance” error terms: if one is big, no point struggling to make the others tiny

31. We want to relate phase variance to the “Strehl ratio” • Two definitions of Strehl ratio (equivalent): • Ratio of the maximum intensity of a point spread function to what the maximum would be without aberrations • The “normalized volume” under the optical transfer function of an aberrated optical system

32. Examples of PSF’s and their Optical Transfer Functions Seeing limited OTF Seeing limited PSF 1 Intensity -1  l / D l / r0 r0 / l D / l Diffraction limited PSF Diffraction limited OTF 1 Intensity -1  l / r0 l / D D / l r0 / l

33. Relation between variance and Strehl • “Maréchal Approximation” • Strehl ~ exp(- s2) where s2is the total wavefront variance • Valid when Strehl > 10% or so • Under-estimate of Strehl for larger values of s2