Atmospheric Turbulence: rAtmospheric Turbulence: r00, , 00, , 00

François WildiObservatoire de Genève

Credit for most slides : Claire Max (UC Santa Cruz)

Adaptive Optics in the VLT and ELT eraAdaptive Optics in the VLT and ELT era

rr00 sets the number of sets the number of

degrees of freedom of an AO degrees of freedom of an AO systemsystem

• 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 = m. (D / r0)2 ~ 280. Actual # for Keck : ~250.

About rAbout r00

• 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

What about temporal behavior of What about temporal behavior of turbulence?turbulence?

• Questions: – What determines typical timescale without

AO?– With AO?

A simplifying hypothesis about A simplifying hypothesis about time behaviortime 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 FlowCartoon of Taylor Frozen Flow

• From Tokovinin tutorial at CTIO:

• http://www.ctio.noao.edu/~atokovin/tutorial/

Order of magnitude estimateOrder of magnitude estimate

• Time for wind to carry frozen turbulence over a

subaperture of size r0 (Taylor’s frozen flow

hypothesis):

00 ~ r ~ r00 / V / V

• Typical values:

– = 0.5 m, r0 = 10 cm, V = 20 m/sec 0 = 5 msec

– = 2.0 m, r0 = 53 cm, V = 20 m/sec 0 = 265

msec

– = 10 m, r0 = 36 m, V = 20 m/sec 0 = 1.8 sec

• Determines how fast an AO system has to run

But But whatwhat wind speed should we wind speed should we use?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

ground

V1

V4

V2

V3

Concept Question:Concept Question:What would be a plausible way to

weight the velocities in the different layers?

Concept Question:Concept Question:What would be a plausible way to

weight the velocities in the different layers?

Rigorous expressions for Rigorous expressions for 0 0 take take into account different layersinto account different layers

• fG Greenwood frequency 1 / 0

• What counts most are high velocities V where CN2 is big

0 fG 1 0.102 k 2 sec dz C N

2 (z) V (z)5 / 3

0

3 / 5

6 / 5

0 ~ 0.3 r0

V

where V

dz CN2 (z) V (z)

5 / 3

dz CN2 (z)

3 / 5

Hardy § 9.4.3

Short exposures: speckle imagingShort exposures: speckle imaging

• A speckle structure appears when the exposure is shorter than the atmospheric coherence time 0

• Time for wind to carryfrozen turbulence overa subaperture of size r0

0 ~ 0.3(r0 /V wind )

Structure of an AO imageStructure 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 + AOatmospheric 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

2D/n f

PSD()

atmospheric turbulence + AO IIatmospheric turbulence + AO II

• Spatially Filtered SH – Optimization of the spatial

filter size – Study of BB impact

» NB filters for optimal results » BB [0.5 – 0.9]m : perf acceptable => OK for “faint” GS (mag

9)

– WCOG : confirmation of the gain in perf (simul AND experimentation)

(see Pres. T. Fusco)

WFS band : 0.5-0.9 0.7-0.9 monochromatic

Detectivity curves 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 bystatic speckle

Limited bynoise

Single differenceSingle imageDouble differenceOur approach

• 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

Anisoplanatism: how does AO image Anisoplanatism: how does AO image degrade as you move farther from degrade as you move farther from guide star?guide star?

credit: R. Dekany, Caltech

More about More about anisoplanatisanisoplanatism:m:

AO image of AO image of sun in visible sun in visible lightlight

11 second 11 second exposureexposure

Fair SeeingFair Seeing

Poor high Poor high altitude altitude conditionsconditions

From T. From T. RimmeleRimmele

AO image of sun AO image of sun in visible light:in visible light:

11 second 11 second exposureexposure

Good seeingGood seeing

Good high Good high altitude altitude conditionsconditions

From T. RimmeleFrom T. Rimmele

What determines how close What determines how close the reference star has to be?the reference star has to be?

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

Reference Star ScienceObject

Telescope

Turbulence

z

Common Atmospheric

Path

Expression for isoplanatic angle Expression for isoplanatic angle 00

• 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

0 2.914 k 2(sec )8 / 3 dz CN2 (z) z5 / 3

0

3 / 5

Telescope

Common Path

Isoplanatic angle, continuedIsoplanatic angle, continued

• Isoplanatic angle 0 is weighted by z5/3 CN2(z)

• Simpler way to remember 0

0 0.314 cos r0

h

where h

dz z5 / 3CN2 (z)

dz CN2 (z)

3 / 5

Hardy § 3.7.2

ReviewReview

• r0 (“Fried parameter”)

– Sets number of degrees of freedom of AO system

0 (or Greenwood Frequency ~ 1 / 0 )

00 ~ r ~ r00 / V / V where

– Sets timescale needed for AO correction

0 (or isoplanatic angle)

– Angle for which AO correction applies

V dz CN

2 (z) V (z)5 /3

dz CN2 (z)

3/5

0 0.3 r0

h

where h dz z5 /3CN

2 (z)dz CN

2 (z)

3/5

• Part 2:Part 2:

• What determines the total wavefront error for an AO system

How to characterize a wavefront How to characterize a wavefront that has been distorted by that has been distorted by turbulenceturbulence

• Path length difference z where kz is the phase change due to turbulence

• Variance 2 = <(k z)2 >

• If several different effects cause changes in the phase,

tot2 = k2 <(zz)2 >

= k2 <(z)2(z)2) >

tottot22 = = 11

2 2 + + 222 2 + + 33

22radiansradians22

or (or (z)z)2 2 = (= (zz11))22((zz22))22((zz33))2 2 nmnm22

tottot22 = = 11

2 2 + + 222 2 + + 33

22radiansradians22

or (or (z)z)2 2 = (= (zz11))22((zz22))22((zz33))2 2 nmnm22

QuestionQuestion

• List as many physical effects as you can that might contribute to overall wavefront error tottot

22

Total wavefront errorTotal wavefront errortottot

22 = = 112 2 + + 22

2 2 + + 3322

Total wavefront errorTotal wavefront errortottot

22 = = 112 2 + + 22

2 2 + + 3322

Elements of an adaptive optics Elements of an adaptive optics systemsystem

Phase lag, noise

propagation

DM fitting error

Measurement error

Not shown: tip-tilt error,

anisoplanatism error

Non-common path errors

Hardy Figure 2.32

• 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

• 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

Wavefront errors due to Wavefront errors due to 00 , , 00

timedelay2 28.4

0

5 / 3

angle2

0

5 / 3

Hardy Eqn 9.57

Hardy Eqn 3.104

Deformable mirror fitting errorDeformable mirror fitting error

• Accuracy with which a deformable mirror with subaperture diameter d can remove aberrations

fittingfitting22 = = ( d / r ( d / r00 ) )5/35/3

• Constant depends on specific design of deformable mirror

• For segmented mirror that corrects tip, tilt, and piston (3 degrees of freedom per segment) = 0.14

• For deformable mirror with continuous face-sheet, = 0.28

Image motion or “tip-tilt” also Image motion or “tip-tilt” also contributes to total wavefront contributes to total wavefront errorerror

• 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

Angle of arrival fluctuations 2 0.364 D

r0

5 / 3D

2

0 D-1/3 (units : radians2)

(Hardy Eqn 3.59 - one axis) image motion in radians is indep of

Scaling of tip-tilt with Scaling of tip-tilt with and D: and D: the good news and the bad newsthe good news and the bad news

• In absolute terms, rms image motion in radians is independent of anddecreases slowly as D increases:

• But you might want to compare image motion to diffraction limit at your wavelength:

Now image motion relative todiffraction limit is almost ~ D, and becomes larger fraction of diffraction limit for small

2 1/ 20.6

D

r0

5 / 6D

0 D-1/6 radians

2 1/ 2

/D ~

D5/6

• “Seeing limited”: Units are radians

• Seeing disk gets slightly smaller at longer wavelengths:

FWHM ~ / -6/5 ~ -1/5

• For completely uncompensated images, wavefront error

2uncomp = 1.02 ( D / r0 )5/3

0

98.0)(r

FWHM

Long exposures, no AO correctionLong exposures, no AO correction

Scaling of tip-tilt for Scaling of tip-tilt for uncompensated or “seeing uncompensated or “seeing limited” imageslimited” images

• Image motion is larger fraction of “seeing disk” at longer wavelengths

2 1/ 2

( /r0)

0

1/ 5 1/ 5

Correcting tip-tilt has relatively Correcting tip-tilt has relatively large effect, for seeing-limited large effect, for seeing-limited imagesimages

• 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 “Tip-tilt is the single largest contributor to wavefront error”largest contributor to wavefront error”

But you have to be careful if you But you have to be careful if you want to apply this statement to want to apply this statement to AO correctionAO 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 Effects of turbulence depend on size of depend on size of telescopetelescope

• 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 = 1 m D = 2 m D = 8 m

Effect of atmosphere on long and Effect of atmosphere on long and short exposure images of a starshort exposure images of a star

Hardy p. 94

Vertical axis is image size in units of /r0

FWHM = /D

Image motion only

Correcting tip-tilt only is optimum for D/r0 ~ 1 - 3

Error budget concept (sum of Error budget concept (sum of 2 2 ’s)’s)

tottot22 = = 11

2 2 + + 222 2 + + 33

22

• 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 anisopanisop2 2 = (= ( / / 00 ) )5/35/3

– Temporal error temporaltemporal2 2 = 28.4 (= 28.4 ( / / 00 ) )5/35/3

– Fitting error fittingfitting2 2 = = ( d / r ( d / r00 ) )5/35/3

– Need to find out:» Measurement error (wavefront sensor)» Non-common-path errors (calibration)» .......

Error budget so farError budget so far

tottot22 = = fittingfitting

2 2 + + anisopanisop2 2 + + temporaltemporal

22 measmeas2 2 calibcalib

22

Still need to work on these two

√ √ √

We want to relate phase variance We want to relate phase variance to the “Strehl ratio”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 OTFaberrated ( fx , fy )dfxdfy

OTFun aberrated ( fx , fy )dfxdfy

where OTF( fx , fy ) Fourier Transform(PSF)

Examples of PSF’s and their Examples of PSF’s and their Optical Transfer FunctionsOptical Transfer Functions

Seeing limited PSF

Diffraction limited PSF

Inte

ns

ity

Inte

ns

ity

Seeing limited OTF

Diffraction limited OTF

/ r0

/ r0 / D

/ D r0 / D /

r0 / D /

-1

-1

1

1

Relation between variance and Relation between variance and StrehlStrehl

• “Maréchal Approximation”

– Strehl ~ exp(- 2)

where 2 is the total wavefront variance

– Valid when Strehl > 10% or so

– Under-estimate of Strehl for larger values

of 2

Relation between Strehl and Relation between Strehl and residual wavefront varianceresidual wavefront variance

Dashed lines: Strehl ~ (r0/D)2

for high wavefront variance

Strehl ~ exp(-2)

Error Budgets: SummaryError Budgets: Summary

• Individual contributors to “error budget” (total mean square phase error):– Anisoplanatism anisopanisop

2 2 = (= ( / / 00 ) )5/35/3

– Temporal error temporaltemporal2 2 = 28.4 (= 28.4 ( / / 00 ) )5/35/3

– Fitting error fittingfitting2 2 = = ( d / r ( d / r00 ) )5/3 5/3

– Measurement error– Calibration error, .....

• In a different category: – Image motion <<22>>1/21/2 = 2.56 (D/r = 2.56 (D/r00))5/6 5/6 ((/D) /D)

radiansradians22

• Try to “balance” error terms: if one is big, no point struggling to make the others tiny