Transcript Document

Adaptive Optics in the VLT and ELT era
Atmospheric Turbulence: r0, 0, 0
François Wildi
Observatoire de Genève
Credit for most slides : Claire Max (UC Santa Cruz)
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 ~ 280. Actual # for Keck : ~250.
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 shortexposure 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
What about temporal behavior of
turbulence?
• Questions:
– What determines typical timescale without AO?
– With AO?
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:
Cartoon of Taylor Frozen Flow
• From Tokovinin
tutorial at CTIO:
• http://www.ctio.noao.edu
/~atokovin/tutorial/
Order of magnitude estimate
• Time for wind to carry frozen turbulence over a
subaperture of size r0 (Taylor’s frozen flow hypothesis):
 0 ~ r0 / V
• Typical values:
– l = 0.5 mm, r0 = 10 cm, V = 20 m/sec  0 = 5 msec
– l = 2.0 mm, r0 = 53 cm, V = 20 m/sec  0 = 265 msec
– l = 10 mm, r0 = 36 m, V = 20 m/sec  0 = 1.8 sec
• Determines how fast an AO system has to run
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:
V2
V3
V4
ground
What would be a
plausible way to
weight the velocities
in the different layers?
Rigorous expressions for 0 take into
account different layers
• fG  Greenwood frequency  1 / 0
3/5
5 / 3 
2

dz CN (z) V (z)
r0 


 0 ~ 0.3   where V  
2
V 

 dz CN (z) 

Hardy § 9.4.3
3 / 5


5 / 3 
1
2
2
 0  fG  0.102 k sec  dz C N (z) V (z) 


0
 l6 / 5
• What counts most are high velocities V where CN2 is big
Short exposures: speckle imaging
• A speckle structure appears when the exposure is
shorter than the atmospheric coherence time  0
 0 ~ 0.3  (r0 /Vwind )
• Time for wind to carry
frozen turbulence over
a subaperture of size r0
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
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
PSD(f)
2D/n
f
• By Fraunhoffer diffraction this will produce a center
diffraction limited core and halo starting beyond 2D/n
atmospheric turbulence + AO II
•
WFS band :
0.5-0.9
0.7-0.9
monochromatic
Spatially Filtered SH
– Optimization of the spatial
filter size
– Study of BB impact
» NB filters for optimal results
» BB [0.5 – 0.9]mm : perf acceptable => OK for “faint” GS (mag 9)
– WCOG : confirmation of the gain in perf (simul AND experimentation)
(see Pres. T. Fusco)
Detectivity curves
• Detectivity at 5: evaluated by variance computation in the
object estimation maps for 100 samples of both aberrations and
noise
The proposed approach
performs better than the
alternatives:
Detectivity increased by a
factor ~10 over the whole
field
Estimated gain in
magnitude difference ~ 2.5
Limited by
static speckle
Single difference
Single image
Double difference
Our approach
Limited by
noise
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
More about
anisoplanatism:
AO image of sun
in visible light
11 second
exposure
Fair Seeing
Poor high
altitude
conditions
From T.
Rimmele
AO image of sun
in visible light:
11 second
exposure
Good seeing
Good high
altitude
conditions
From T. Rimmele
What determines how close the
reference star has to be?
Reference Star
Turbulence has to be similar
Science
Object
on path to reference star
and to science object
Common path has to be
large
Anisoplanatism sets a limit
Turbulence
to distance of reference
star from the science
object
z
Common
Atmospheric
Path
Telescope
Expression for isoplanatic angle
0
• Strehl = 0.38 at  = 0
0 is isoplanatic angle
3 / 5


2
8/3
2
5/3
 0  2.914 k (sec )  dz CN (z) z 


0

0 is weighted by high-altitude turbulence
(z5/3)
• If turbulence is only at low altitude,
overlap is very high.
Common
Path
• If there is strong turbulence at high
altitude, not much is in common path
Telescope
Isoplanatic angle, continued
• Isoplanatic angle 0 is weighted by z5/3 CN2(z)
• Simpler way to remember 0
r0 
0  0.314 cos  
h 
  dz z 5 / 3C 2 (z) 
N


whereh  
2

dz
C
(z)
N
 

Hardy § 3.7.2
3/5
Review
• r0 (“Fried parameter”)
– Sets number of degrees of freedom of AO system
• 0 (or Greenwood Frequency ~ 1 / 0 )
 0 ~ r0 / V
where
 dz CN2 (z) V (z) 5 / 3 


V
2

 dz CN (z) 

3/5
– Sets timescale needed for AO correction
• 0 (or isoplanatic angle)
 r0 
 0  0.3  
h
– Angle for which AO correction applies
 dz z 5 / 3C N2 (z) 

where h  

2
  dz CN (z) 
3/5
• Part 2:
• What determines the total wavefront error for an AO
system
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 2 = <(k Dz)2 >
• If several different effects
cause changes in the phase,
tot2 = k2 <(Dz1 +Dz2 +....)2 >
= k2 <(Dz1)2 +( Dz2)2 ...) >
tot2 = 12 + 22 + 32 +...radians2
or (Dz)2 = (Dz1)2 +(Dz2)2 +(Dz3)2 +.....nm2
Question
Total wavefront error
tot2 = 12 + 22 + 32 +...
• List as many physical effects as you can that might
contribute to overall wavefront error tot2
Elements of an adaptive optics system
DM fitting
error
Not shown: tiptilt error,
anisoplanatism
error
Non-common
path errors
Phase lag,
noise
propagation
Measurement error
Hardy
Figure 2.32
Wavefront errors due to
 , 0
0
• Wavefront phase variance due to 0 = fG-1
– If an AO system corrects turbulence “perfectly” but
with a phase lag characterized by a time ,then
2
 timedelay
  5 / 3
 28.4  
 0 
Hardy Eqn 9.57
• 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
5/3



2
Hardy Eqn 3.104
 angle   
0 
Deformable mirror fitting error
• Accuracy with which a deformable mirror with subaperture
diameter d can remove aberrations
fitting2 = 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
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
Angle of arrival fluctuations  2
D 5 / 3 l 2
 0.364      l0 D-1/3 (units: radians2 )
r0  D 
(Hardy Eqn 3.59 - one axis)
image motion in
radians is indep of l
• 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
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:

2 1/ 2
D 5 / 6 l 
 0.6      l0 D-1/6 radians
r0  D 
• But you might want to compare image motion to
diffraction limit at your wavelength:


2 1/ 2
l /D
~
D
5/6
l
Now image motion relative to
diffraction limit is almost ~ D,
and becomes larger fraction of
diffraction limit for small l
Long exposures, no AO correction
FWHM (l )  0.98
l
r0
• “Seeing limited”: Units are radians
•
Seeing disk gets slightly smaller at longer wavelengths:
FWHM ~ l / l-6/5 ~ l-1/5
• For completely uncompensated images, wavefront error
2uncomp = 1.02 ( D / r0 )5/3
Scaling of tip-tilt for uncompensated or
“seeing limited” images
• Image motion is larger fraction of “seeing disk” at
longer wavelengths

2 1/ 2
l0
1/ 5
 1/ 5  l
( l / r0 )
l

Correcting tip-tilt has relatively large
effect, for seeing-limited images
• For completely uncompensated images
2uncomp = 1.02 ( D / r0 )5/3
• If image motion (tip-tilt) has been completely removed
2tiltcomp = 0.134 ( D / r0 )5/3
(Tyson, Principles of AO, eqns 2.61 and 2.62)
• Removing image motion can (in principle) improve the
wavefront variance of an uncompensated image by a
factor of 10
• Origin of statement that “Tip-tilt is the single largest
contributor to wavefront error”
But you have to be careful if you want to
apply this statement to AO correction
• If tip-tilt has been completely removed
2tiltcomp = 0.134 ( D / r0 )5/3
• But typical values of ( D / r0 ) are 10-50 in near-IR
– VLT, D=8 m, r0 = 50 cm, ( D/r0 ) = 17
2tiltcomp = 0.134 ( 17 )5/3 ~ 15
so wavefront phase variance is >> 1
• Conclusion: if ( D/r0 ) >> 1, removing tilt alone won’t give you
anywhere near a diffraction limited image
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=1m
D=2m
D=8m
Effect of atmosphere on long and short
exposure images of a star
Hardy p. 94
Correcting tip-tilt only is
optimum for D/r0 ~ 1 - 3
Image
motion only
FWHM = l/D
Vertical axis is image size in units of l/r0
Error budget concept (sum of 2 ’s)
tot2 = 12 + 22 + 32 +...
• There’s not much gained by making any particular term much
smaller than all the others: try to equalize
• Individual terms we know so far:
– Anisoplanatism
– Temporal error
– Fitting error
anisop2 = ( / 0 )5/3
temporal2 = 28.4 ( / 0 )5/3
fitting2 = m ( d / r0 )5/3
– Need to find out:
» Measurement error (wavefront sensor)
» Non-common-path errors (calibration)
» .......
Error budget so far
tot2 = fitting2 + anisop2 + temporal2 +meas2 +calib2
√
√
√
Still need to work
on these two
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
S
 OTF
aberrated
 OTF
( f x , f y )df x df y
un  aberrated
( f x , f y )df x df y
where OTF( f x , f y )  Fourier Transform(PSF)
Examples of PSF’s and their
Optical Transfer Functions
Seeing limited OTF
Seeing limited PSF
Intensity
1
Intensity
l/D
l / r0

Diffraction limited PSF
l/D
l / r0

r0 / l
1
-1
D/l
Diffraction limited OTF
r0 / l
D/l
-1
Relation between variance and Strehl
• “Maréchal Approximation”
– Strehl ~ exp(- f2)
where f2 is the total wavefront variance
– Valid when Strehl > 10% or so
– Under-estimate of Strehl for larger values of f2
Relation between Strehl and residual
wavefront variance
Strehl ~ exp(-f2)
Dashed lines:
Strehl ~ (r0/D)2
for high
wavefront
variance
Error Budgets: Summary
• Individual contributors to “error budget” (total mean
square phase error):
– Anisoplanatism anisop2 = ( / 0 )5/3
– Temporal error temporal2 = 28.4 ( / 0 )5/3
– Fitting error
fitting2 = m ( d / r0 )5/3
– Measurement error
– Calibration error, .....
• In a different category:
– Image motion
<2>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