1. Parameters characterizing the Atmospheric Turbulence: r0, 0, 0
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)2 where r0 is evaluated at the desired observing wavelength
Example: Keck telescope, D=10m, r0 ~ 60 cm at l = 2 mm. (D / r0)2 ~ 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)2 is 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:
/~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. credit: R. Dekany, Caltech
Anisoplanatism: how does AO image degrade as you move farther from guide star?
credit: R. Dekany, Caltech
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

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?
Science
Object
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
Common
Atmospheric
Path
Telescope
Turbulence z

17. Expression for isoplanatic angle 0
Strehl = 0.38 at  = 0
0 is isoplanatic angle
0 is 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  t0 ~ r0 / V where 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 + Dz )2 >
= k2 <(Dz1)2 + ( Dz2 )2 ...) >
stot2 = s12 + s22 + s radians2
or (Dz)2 = (Dz1)2 + (Dz2)2 + (Dz3) nm2

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

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
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
image motion in radians is indep of l
(Hardy Eqn 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, and decreases 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 = 1 m
D = 8 m

29. Still need to work on these two
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 error stemporal2 = 28.4 (t / t0 )5/3
Fitting error sfitting2 = 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 PSF
Seeing limited OTF 1
Intensity 
-1
l / D
l / r0
r0 / l
D / l
Diffraction limited PSF
Diffraction limited OTF 1
Intensity
-1 
l / D
l / r0
r0 / l
D / l

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