Lecture 5: Atmospheric Turbulence: r0, 0, 0

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Lecture 5Atmospheric Turbulence r0, ?0, ?0

• Claire Max
• Astro 289C, UCSC
• January 22, 2008

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Class business

• Homework number 3 due Tues Jan 29 (assignment is
  on the web)
• Reading assignments for Thurs January 24th and
  Tuesday January 29th (on the web)
• Thursday Wavefront Sensing reading (Tyson,
  Chanan, Bauman)
• Tuesday Deformable mirrors (Tyson, Sechaud,
  Morzinski, Arsenault, Miller)
• Lectures, reading assignments, and homework
  assignments are on our original class web page
• PDF files and discussion/chat rooms are on WebCT
• I will email to all of you the instructions for
  how to log in to WebCT

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Atmospheric Turbulence Main Points from the last
lecture

• The dominant locations for index of refraction
  fluctuations that affect astronomers are the
  atmospheric boundary layer and the tropopause
• Atmospheric turbulence (mostly) obeys Kolmogorov
  statistics
• Kolmogorov turbulence is derived from dimensional
  analysis (heat flux in heat flux in turbulence)
• Structure functions derived from Kolmogorov
  turbulence are proportional to r2/3
• All else will follow from these points!

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Outline of lecture

• Part 1 Understand atmospheric turbulence
  parameters
• r0 (Fried parameter)
• Greenwood frequency ? 0 (typical timescale for
  changes in the turbulence)
• Isoplanatic angle ?0 (angle within which
  turbulence is well corrected)
• Part 2 Begin discussion of what determines the
  total wavefront error for an AO system
• Error budget concept
• Tip-tilt correction
• Strehl ratio

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From last lecture evaluated the spatial
coherence function C? (r)
For a slant path you can add factor ( sec ? )5/3
to account for dependence on zenith angle ?
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Given the spatial coherence function, calculate
effect on telescope resolution

• Define optical transfer functions of telescope,
  atmosphere
• Define r0 as the telescope diameter where the two
  optical transfer functions are equal
• Calculate expression for r0

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Define optical transfer function (OTF)

• Imaging in the presence of imperfect optics (or
  aberrations in atmosphere) in intensity units
• Image Object ? Point Spread Function
• I (r) O ? PSF ? ? dx O( x - r ) PSF ( x )
• Take Fourier Transform F ( I ) F (O ) F ( PSF
  )
• Optical Transfer Function is Fourier Transform of
  PSF
• OTF F ( PSF )

convolved with
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Examples of PSFs and their Optical Transfer
Functions
Seeing limited OTF
Seeing limited PSF
Intensity
?-1
?
l / r0
l / D
r0 / l
D / l
Diffraction limited PSF
Diffraction limited OTF
Intensity
?-1
?
l / r0
l / D
D / l
r0 / l
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Now describe optical transfer function of the
telescope in the presence of turbulence

• OTF for the whole imaging system (telescope plus
  atmosphere)
• S ( f ) B ( f ) ? T ( f )
• Here B ( f ) is the optical transfer fn. of the
  atmosphere and T ( f) is the optical transfer fn.
  of the telescope (units of f are cycles per
  meter).
• f is often normalized to cycles per
  diffraction-limit angle (l / D).
• Measure the resolving power of the imaging system
  by
• R ? df S ( f ) ? df B ( f ) ? T ( f )

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Derivation of r0

• R of a perfect telescope with a purely circular
  aperture of (small) diameter d is
• R ? df T ( f ) ( p / 4 ) ( d / l )2
• (uses solution for diffraction from a circular
  aperture)
• Define a circular aperture r0 such that the R of
  the telescope (without any turbulence) is equal
  to the R of the atmosphere alone
• ? df B ( f ) ? df T ( f ) ? ( p / 4 ) ( r0
  / l )2

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Derivation of r0 , continued

• Now we have to evaluate the contribution of the
  atmospheres OTF ? df B ( f )
• B ( f ) C? ( l f ) (to go from cycles per
  meter to cycles per wavelength)

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Derivation of r0 , continued

• Now we need to do the integral in order to solve
  for r0
• ( p / 4 ) ( r0 / l )2 ? df B ( f ) ? df
  exp (- K f 5/3)
• Now solve for K
• K 3.44 (r0 / l )-5/3
• B ( f ) exp - 3.44 ( l f / r0 )5/3 exp -
  3.44 ( ? / r0 )5/3

Replace by r
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Derivation of r0 , concluded
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Scaling of r0

• r0 is size of subaperture, sets scale of all AO
  correction
• r0 gets smaller when turbulence is strong (CN2
  large)
• r0 gets bigger at longer wavelengths AO is
  easier in the IR than with visible light
• r0 gets smaller quickly as telescope looks
  toward the horizon (larger zenith angles ? )

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Typical values of r0

• Usually r0 is given at a 0.5 micron wavelength
  for reference purposes. Its up to you to scale
  it by ?-1.2 to evaluate r0 at your favorite
  wavelength.
• At excellent sites such as Mauna Kea in Hawaii,
  r0 at 0.5 micron is 10 - 30 cm. But there is a
  big range from night to night, and at times also
  within a night.

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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, D10m, r0 60 cm at l
  2 mm. (D / r0)2 280. Actual for Keck 250.

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Several equivalent meanings for 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
• Can you think of others?

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Next All sorts of good things come from knowing
r0

• Timescales of turbulence
• Isoplanatic angle AO performance degrades as
  astronomical targets get farther from guide star

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What about temporal behavior of turbulence?

• Questions
• What determines typical timescale without AO?
• With AO?

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A simplifying hypothesis about time behavior

• Almost all work in this field uses Taylors
  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

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Cartoon of Taylor Frozen Flow

• From Tokovinin tutorial at CTIO
• http//www.ctio.noao.edu/atokovin/tutorial/

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Order of magnitude estimate

• Time for wind to carry frozen turbulence over a
  subaperture of size r0 (Taylors 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

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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
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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
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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

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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 40x40 (at 0.04 arc sec/pixel)
• On-axis K-band Strehl 40, falling to 25 at
  field corner

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More about anisoplanatism AO image of sun in
visible light 11 second exposure Fair
Seeing Poor high altitude conditions From T.
Rimmele
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AO image of sun in visible light 11 second
exposure Good seeing Good high altitude
conditions From T. Rimmele
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What determines how close 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

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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
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Isoplanatic angle, continued

• Isoplanatic angle ?0 is weighted by z5/3 CN2(z)
• Simpler way to remember ?0

Hardy 3.7.2
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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

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• Part 2
• What determines the total wavefront error for an
  AO system

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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 lt(k Dz)2 gt
• If several different effects cause changes in the
  phase,
• stot2 k2 lt(Dz1 Dz2 ....)2 gt
• k2 lt(Dz1)2 ( Dz2 )2 ...) gt

stot2 s12 s22 s32 ... radians2 or (Dz)2
(Dz1)2 (Dz2)2 (Dz3)2 ..... nm2
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Question
Total wavefront error stot2 s12 s22 s32
...

• List as many physical effects as you can that
  might contribute to overall wavefront error stot2

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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
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Hardy Figure 2.32
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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
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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

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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

(Hardy Eqn 3.59 - one axis)
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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

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Long exposures, no AO correction

• Seeing limited Units are radians
• Seeing disk gets slightly smaller at longer
  wavelengths FWHM ? / ?-6/5 ?-1/5
• For completely uncompensated images, wavefront
  error
• s2uncomp 1.02 ( D / r0 )5/3

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Scaling of tip-tilt for uncompensated or seeing
limited images

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

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

• For completely uncompensated images
• s2uncomp 1.02 ( D / r0 )5/3
• If image motion (tip-tilt) has been completely
  removed
• s2tiltcomp 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

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But you have to be careful if you want to apply
this statement to AO correction

• If tip-tilt has been completely removed
• s2tiltcomp 0.134 ( D / r0 )5/3
• But typical values of ( D / r0 ) are 10-50 in
  near-IR
• Keck, D10 m, r0 60 cm, ( D/r0 ) 17
• s2tiltcomp 0.134 ( 17 )5/3 15
• so wavefront phase variance is gtgt 1
• Conclusion if ( D/r0 ) gtgt 1, removing tilt
  alone wont give you anywhere near a diffraction
  limited image

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Effects of turbulence depend on size of telescope

• Coherence length of turbulence r0 (Frieds
  parameter)
• For telescope diameter D ? (2 - 3) x r0
• Dominant effect is "image wander"
• As D becomes gtgt 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
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Effect of atmosphere on long and short exposure
images of a star

• Hardy p. 94
• Vertical axis is image size in units of l/r0

Image motion only
FWHM l/D
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Error budget concept (sum of s2 s)

• ?stot2 s12 s22 s32 ...
• Theres not much gained by making any particular
  term much smaller than all the others try to
  equalize
• Individual terms we know so far
• Anisoplanatism sanisop2 (? / ?0 )5/3
• Temporal error stemporal2 28.4 (t / t0 )5/3
• Fitting error sfitting2 m ( d / r0 )5/3
• Need to find out
• Measurement error (wavefront sensor)
• Non-common-path errors (calibration)
• .......

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Error budget so far

• ?stot2 sfitting2 sanisop2 stemporal2
  smeas2 scalib2

v
v
v
Still need to work on these two
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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

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Examples of PSFs and their Optical Transfer
Functions
Seeing limited OTF
Seeing limited PSF
1
Intensity
?-1
?
l / r0
l / D
r0 / l
D / l
Diffraction limited PSF
Diffraction limited OTF
1
Intensity
?-1
?
l / r0
l / D
D / l
r0 / l
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Relation between variance and Strehl

• Maréchal Approximation
• Strehl exp(- s?2)
• where s?2 is the total wavefront variance
• Valid when Strehl gt 10 or so
• Under-estimate of Strehl for larger values of s?2

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Relation between Strehl and residual wavefront
variance
Strehl exp(-s?2)
Dashed lines Strehl (r0/D)2 for high wavefront
variance
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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 lta2gt1/2 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

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Next Two Lectures

• Thursday Wavefront Sensing
• Tuesday Deformable Mirrors