Transcript d / r 0 - Observatoire de Genève

Adaptive Optics in the VLT and ELT era
Characterizing the Atmospheric Turbulence
&
Systems engineering
François Wildi
Observatoire de Genève
Credit for most slides : Claire Max (UC Santa Cruz)
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):
t 0 ~ 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
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 t0 take into
account different layers
• fG  Greenwood frequency  1 / t0
3/5
5 / 3 
2

dz CN (z) V (z)
r0 


t 0 ~ 0.3   where V  
2
V 

 dz CN (z) 

Hardy § 9.4.3
3 / 5


5 / 3 
1
2
2
t 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 t 0
t 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
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
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
• Simpler way to remember 0
r0 
0  0.314 cos  
h 
  dz z 5 / 3C 2 (z) 3 / 5
N


where h  
2

dz
C
(z)
N
 

Hardy § 3.7.2
Review
• r0 (“Fried parameter”)
– Sets number of degrees of freedom of AO system
• t0 (or Greenwood Frequency ~ 1 / t0 )
t 0 ~ r0 / V
where
 dz C N2 (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 C N (z) 
3/5
Systems engineering
Issues for designer of AO systems
• Performance goals:
– Sky coverage fraction, observing wavelength, degree
of compensation needed for science program
• Parameters of the observatory:
– Turbulence characteristics (mean and variability),
telescope and instrument optical errors, availability of
laser guide stars
• In general, residual wavefront error is the quality
criterion in AO
stot2 = s12 + s22 + s32 +...
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
Adaptive Optics wavefront errors parameters
• The wavefront error depends on design:
– The number of degrees do freedom (i.e. +/- nb of actuators)
of the deformable mirror.
– The lag (delay) in the control system
– The noise in the wavefront sensor which depends on WFS
type (and the guide star magnitude)
– The size of the field of view
– Side effects like WFS non-ideality, NCPA, disturbances like
vibrations
• And operations
– integration time on wavefront sensor, wavelength, guide
star mag. & offset
Wavefront errors due to
t , 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
5/3
2
s timedelay
 t 
 28.4  
t 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
s angle   
0 
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 deformable mirror with continuous face-sheet,
m
= 0.28
• For segmented mirror that corrects tip, tilt, and piston (3 degrees
of freedom per segment) m = 0.14. Those mirrors are being
phased out
Dependence of Strehl on l and number of DM
degrees of freedom
 
5/3
S  exp s f2  exp  0.28 d / r0  


r0 l   r0 0.5 m m l / 0.5 m m 
6 /5
5/3
2



d
 0.5 m m 

S  exp 0.28 

 

l
 r0 0.5 m m 


Deformable mirror fitting error only
Reminder #1: Dependence of Strehl on l and
number of DM degrees of freedom (fitting)
• Assume bright
natural guide
star
Decreasing fitting error
Deformable mirror fitting error only
• No meas’t error
or iso-planatism
or bandwidth
error
Classical PIEZO actuators
DSM @ MICROGATE: INSTALLING THE SHELL
SHELL TRANSPORT BOX USED FOR
SHELL INSTALLATION
DSM @ MICROGATE: SYSTEM LEVEL
ELECTROMECHANICAL AND ENVIRONMENTAL (-15°C)
TEST
Existing MEMS mirror
(sufficient for Hybrid-MOAO)
Boston Micromachines
32x32 actuator, 0.6mm
MEMS device, pitch 300mm
30
Basics of wavefront sensing
• Measure phase by measuring intensity variations
• Difference between various wavefront sensor schemes
is the way in which phase differences are turned into
intensity differences
• General box diagram:
Guide
star
Turbulence
Telescope
Wavefront sensor
Optics
Detector
Reconstructor
Computer
Transforms aberrations into
intensity variations
Types of wavefront sensors
• “Direct” in pupil plane: split pupil up into subapertures
in some way, then use intensity in each subaperture to
deduce phase of wavefront. REAL TIME
– Slope sensing: Shack-Hartmann, pyramid sensing
– Curvature sensing
• “Indirect” in focal plane: wavefront properties are
deduced from whole-aperture intensity measurements
made at or near the focal plane. Iterative methods take a lot of time.
– Image sharpening, multi-dither
– Phase diversity
Shack-Hartmann wavefront sensor
concept - measure subaperture tilts
f
CCD
Pupil plane
Image plane
CCD
WFS implementation
• Compact
• Time-invariant
How to reconstruct wavefront from
measurements of local “tilt”
Effect of guide star magnitude
(measurement error)
Because of the photons statistics, some noise is
associated with the read-out of the Shack-Hartmann
spots intensities
 6.3 

SNR 
2
s S2 H  

2


6.3


2
S  exp  s S  H   exp   
 

SNR


1
SNR 
N photons
Effect of guide star magnitude
(measurement error)
Assumes no fitting error or other error terms
INTERLUDE:
---------Ooh ooh ooh.....
[Chord progression 2x
bright
star
with
electric
Fill 1]
Decreaing
measurement error
dim star
Pyramid wavefront sensors
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
5/3
Angle of arrival fluctuations
2
D  l 2
 0.364      l0 D-1/3 (units : radians 2 )
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
5/6
D  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
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
Error budget so far
stot2 = sfitting2 + sanisop2 + stemporal2 +smeas2 +scalib2
√
√
√
Still need to work
on these two
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
<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
Frontiers in AO technology
• New kinds of deformable mirrors with > 5000 degrees
of freedom
• Wavefront sensors that can deal with this many degrees
of freedom
• (ultra) Fast computers
• Innovative control algorithms
• “Tomographic wavefront reconstuction” using multiple
laser guide stars
• New approaches to doing visible-light AO
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
( fx , fy )dfx dfy
un  aberrated
( fx , fy )dfx dfy
where OTF( fx , fy )  Fourier Transform(PSF)
Relation between phase variance and Strehl
• Maréchal Approximation
– Strehl ~ exp(- sf2)
where sf2 is the total wavefront variance
– Valid when Strehl > 10% or so
– Under-estimate of Strehl for larger values of sf2