Propagation of Light Through Atmospheric
Turbulence
Lecture 6, ASTR 289
Claire Max
UC Santa Cruz
January 24, 2013
Page 1
What is typical timescale for
flow across distance r0 ?
Note: previous
version had
math errors
• Time for wind to carry frozen turbulence over a
subaperture of size r0 (Taylor’s frozen flow hypothesis):
0 ~ r0 / V
• Typical values at a good site, for V = 20 m/sec:
Wavelength (μm)
r0
τ0 = r0 / V
f0 = 1/τ0 = V / r0
0.5
10 cm
5 msec
200 Hz
2
53 cm
27 msec
37 Hz
10
3.6 meters
180 msec
5.6 Hz
Page 2
Review of atmospheric parameters that
are key to AO performance
• r0 (“Fried parameter”)
– Sets number of degrees of freedom of AO system
• τ0 (or “Greenwood Frequency” fG ~ 1 / τ0 )
0 ~ r0 / V
where
é dz CN2 (z) V (z) 5 / 3 ù
ò
ú
Vºê
2
ê
ò dz CN (z) úû
ë
3/ 5
– Sets timescale needed for AO correction
• θ0 (isoplanatic angle)
ær ö
q 0 @ 0.3 ç 0 ÷
èhø
æ dz CN2 (z) z 5 / 3 ö
ò
where h º ç
÷
2
çè ò dz CN (z) ÷ø
3/5
– Angle for which AO correction applies
Page 3
What is “residual wavefront error” ?
AO
System
Telescope
Very distorted
wavefront
Science
Instrument
Less distorted
wavefront
(but still not
perfect)
Page 4
Shape of
Deformable Mirror
Incident wavefront
Image of point source
Corrected wavefront
Image of point source
Credit: James Lloyd, Cornell Univ.
Page 5
Page 6
How to calculate residual wavefront
error
• Optical path difference = Δz where
k Δz is the phase change due to
turbulence
• Phase variance is σ2 = <(k Δz)2 >
• If several independent effects cause
changes in the phase,
2
s tot
= k 2 ( Dz1 + Dz2 + Dz3 + ...)
2
= k 2 ( Dz1 ) + ( Dz2 ) + ( Dz3 ) + ...
2
2
2
• Sum up the contributions from
individual physical effects
independently
Page 7
An error budget can describe wavefront
phase or optical path difference
s
2
tot
2
( Dz1 + Dz2 + Dz3 + ...)
2
( Dz1 ) + ( Dz2 ) + ( Dz3 )
=k
=k
2
2
2
2
+ ...
• Be careful of units (Hardy and I will both use a variety of
units in error budgets):
– For tip-tilt residual errors: variance of tilt angle < 2>
– Optical path difference in meters: OPDm
– Optical path difference in waves: OPDλ = OPDm / λ
– Optical path difference in radians of phase:
 = 2π OPDλ = (2π/λ) OPDm = k OPDm
Page 8
Question
If the total wavefront error is
tot2 = 12 + 22 + 32 + ...
• List as many physical effects as you can think of that
might contribute to the overall wavefront error tot2
Page 9
Elements of an adaptive optics system
DM fitting
error
Not shown:
tip-tilt error,
anisoplanatism
error
temporal
delay, noise
propagation
Non-common
path errors
Measurement error
Page 10
What is an “error budget” ?
1. The allocation of statistical variations and/or error rates
to individual components of a system, in order to satisfy
the full system's end-to-end performance specifications.
2. The term “error budget” is a bit misleading: it doesn’t
mean “error” as in “mistake” - it means performance
uncertainties, and/or the imperfect, “real life”
performance of each component in the system.
3. In a new project: Start with “top down” performance
requirements from the science that will be done.
Allocate “errors” to each component to satisfy overall
requirements. As design proceeds, replace “allocations”
with real performance of each part of system. Iterate.
Page 11
Hardy
Figure 2.32
Page 12
Wavefront errors due to time lags, τ0
• 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
s t = 28.4 (t t 0 )
2
5/3
Hardy Eqn 9.57
• The factor of 28.4 out front is significant penalty:
have to run your AO system faster than τ = τ0
• For στ2 < 1, τ < 0.13 τ0
• In addition, closed-loop bandwidth is usually ~ 10x
sampling frequency  have to run even faster
Page 13
Wavefront variance due to
isoplanatic angle θ0
• If an AO system corrects turbulence “perfectly” but uses a
guide star an angle θ away from the science target, then
s
2
angle
æqö
=ç ÷
è q0 ø
5/3
Hardy Eqn 3.104
• Typical values of θ0 are a few arc sec at λ = 0.5 μm,
15-20 arc sec at λ = 2 μm
• Vision Science: θ0 ~ 1 degree !
(Bedggood et al. J. Biomed. Optics 132, 024008, 2008
Page 14
Deformable mirror fitting error
• Accuracy with which a deformable mirror with subaperture diameter
d can remove aberrations
s DM _ fitting
æ dö
= mç ÷
èr ø
5/3
0
• 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
Page 15
Error budget concept (sum of 2 ’s)
2
s tot
= s 12 + s 22 + s 32 + ... radians2
• There’s not much to be gained by making any particular term much
smaller than all the others: try to roughly equalize all the terms
• Individual terms we know so far:
– Anisoplanatism
2
s angle
= (q q 0 )
– Temporal error
s t = 28.4 (t t 0 )
– Fitting error
s DM _ fitting = m ( d r0 )
2
5/3
5/3
5/3
Page 16
We will discuss other wavefront error
terms in coming lectures
• Tip-tilt errors (this lecture if there is time)
– Imperfect sensing and/or correcting of image wander,
due to atmosphere or telescope motions (wind shake)
• Measurement error
– Wavefront sensor doesn’t make perfect measurements
– Finite signal to noise ratio, optical limitations, …
• Non-common-path errors
– Calibration of different optical paths between science
instrument and wavefront sensor isn’t perfect
Page 17
Error budget so far
s =s
2
tot
2
fitting
√
+s
2
anisopl
√
+s
2
temporal
+s
2
meas
+s
2
calib
+s
2
tip-tilt
+ .....
√
Still need to work
these out
Try to “balance” error terms: if one is big,
no point struggling to make the others tiny
Page 18
Keck AO error budget example
Assumptions:
NGS is very bright (no meas’t error)
10 degree zenith angle
Wavefront sensor bandwidth: 670 Hz
Note that uncorrectable errors in telescope itself are significant
Page 19
We want to relate phase variance <σ2>
to the “Strehl ratio”
• Two definitions of Strehl ratio (equivalent):
1. Ratio of the maximum intensity of a point spread
function to what the maximum would be without any
aberrations:
S º ( I max I max_ no_ aberrations )
2. The “normalized volume” under the optical transfer
function of an aberrated optical system
Sº
ò OTF
aberrated
ò OTF
( fx , fy )df x df y
un-aberrated
( fx , fy )df x df y
where OTF( fx , fy ) = Fourier Transform(PSF)
Page 20
Relation between phase variance and
Strehl ratio
• “Maréchal Approximation”
(
Strehl @ exp -s f
2
)
where σ2 is the total wavefront variance
– Valid when Strehl > 10% or so (σ2 < 2.3 )
– Under-estimates Strehl for larger values of σ2
Page 21
High Strehl Þ PSF with higher peak
intensity and narrower “core”
High
Strehl
Low
Strehl
Medium
Strehl
Page 22
Characterizing image motion or tip-tilt
Page 23
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
a2
æ Dö
= 0.364 ç ÷
èr ø
0
5/3
æ lö
çè ÷ø
D
2
µ l 0 D-1/3 radians 2
(Hardy Eqn 3.59 - one axis)
• Can “correct” this image motion using a tip-tilt mirror
(driven by signals from an image motion sensor) to
compensate for image motion
Page 24
Scaling of tip-tilt with  and D:
the good news and the bad news
• In absolute terms, rms image motion in radians is
independent of   and decreases slowly as D increases:
a
2 1/2
æ Dö
= 0.6 ç ÷
èr ø
0
5 /6
æ lö
çè ÷ø
D
µ l 0 D-1/6 radians
• But you might want to compare image motion to diffraction
limit at your wavelength:
2 1/2
a
D
~
(l / D) l
5/6
Now image motion relative to
diffraction limit is almost ~ D,
and becomes larger fraction of
diffraction limit for smaller 
Page 25
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 ~ λ / λ-6/5 ~ λ-1/5
• For completely uncompensated images, wavefront error
σ2uncomp = 1.02 ( D / r0 )5/3
Page 26
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”
Page 27
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 the near-IR
– Keck, D=10 m, r0 = 60 cm at λ=2 μm, ( 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
Page 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=1m
D=2m
D=8m
Page 29
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  /r0
Page 30
Summary of topics discussed today
• Timescale of turbulence
• Isoplanatic angle
• Deformable mirror fitting error
• Concept of an “error budget”
• Image motion or tip-tilt
• Goal: to calculate <σ2> and thus the Strehl ratio
(
Strehl @ exp -s f
2
)
Page 31