gary chanan department of physics and astronomy university of california, irvine 4 february 2000
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Gary Chanan
Department of Physics and AstronomyUniversity of California, Irvine
4 February 2000
1. Atmospheric optics
• A brief introduction to turbulence
• A guide to the relevant mathematics
• Derived optical properties of the atmosphere
2. Wavefront sensing
• Shack-Hartmann wavefront sensing
• Curvature sensing
Outline
Wind Tunnel Experiments
Time (sampling units)
v (
arbi
trar
y un
its)
543210-4
-3
-2
-1
0
1
2
3
4
5
Frequency (s-1)
Pow
er (
arbi
trar
y un
its)
Slope = -5/3
L = 72 mRe = ~ 10
V = 20 msec
VL
7
Gagne (1987)
Kolmogorov’s Law (1941)
Energy cascades down to smaller spatial scales r, higher spatial frequencies = 2/r .
In inertial range: l0 r L0
Power spectrum: () -5/3v
() -11/3v
(cm-1)
Slope = -5/3
Pow
er (
arbi
trar
y un
its)
or
10 cm air jet
(Champagne 1978)
For atmosphere: ~ 1 mm 10 m - 10 km
<~ <~
inner scale outer scale
Oboukov’s Law (1949)
Fluctuations in scalar quantities associated with this flow (passive conservative additives) inherit this same power spectrum.
s () -5/3
Kolmogorov’s Law describes behavior in Fourier space; we are often interested in real space. Consider structure function:
There is a Fourier-like relation between s and s :
s ( r ) = 2 s ( ) ( 1 - e i • r ) d 3
Structure Function
s ( r ) = < | s ( + r ) - s ( ) | >2
mean squared fluctuation
We write s ( r ) = C r s2 2/3
structureconstant
• Power laws predominate.
• Integral of a power law is a power law.
• Power law indices are easy to calculate; numerical coefficients are hard.
Thus:
s ( ) <=> s ( r ) r-5/3 2/3
Mathematical Notes
n - 1 = 79 x 10 -3 PT in Kelvins
in atmospheres
n = -79 x 10-3 P TT2 [Neglect pressure fluctuations.]
For typical night-time atmosphere (0.1 to 10 km):
C ~ 10 m2T
N
-2/3-4
-16C ~ 10 m2 -2/3
Thus on meter scales T ~ 10 mK , n ~ 10 ! -8
Index FluctuationsApplication of Oboukov’s Law to the (pre-existing) large scale temperature gradients in the atmosphere =>
- law for T fluctuations =>
- law for n fluctuations
2323
10-20
10-19
10-18
10-17
10-16
10-15
0.01 0.1 1 10 100
Height (km)
Cn
(m
- 2/3)
2Cn Profile2
The index fluctuations are well-characterized. What are the corresponding fluctuations in the accumulated phase?
h
D = 2R
n ( r ) = Cn r2 2/3
Central Problem
We will do a first-order treatment, which gives a surprisingly good accounting of the typical astronomical situation (esp. for large telescopes):
All points on the wavefront travel straight down, but are advanced or retarded according to:
(x,y) = n(x,y,z) dz2
First Order Treatment
This neglects diffraction effects. Valid when:
For large telescopes,
Characteristic vertical scale of atmosphere is h ~ 104 meters; so the near field approximation is usually well-satisfied in practice.
R2
Typical diffraction angle
( ) h << R => h << R
R2
lateral displacement of ray
Near - Field Approximation
106 meters.
Note that it is precisely these diffraction effects which give rise to scintillation or twinkling.
Different parts of the diffracted wavefront eventually interfere with one another.
Thus there is no scintillation in the near-field approximation. But for the dark-adapted eye R ~ 4 mm and:
~ 30 meters
The inequality turns around and the stars appear to twinkle.
R2
Scintillation
( r ) ~ ( ) Cn h r
Our central propagation problem can be elegantly stated in Fourier space:
Given the 3-dim spectral density of n, what is the corresponding 2-dim spectral density of the phase , which is proportional to the integral of n?
…and elegantly solved by the following theorem:
( x , y ) = 2 h n ( x , y , 0)ndz
The phase structure function follows directly:
2 5/3
2 2
( r ) = 6.88 ( )
We write:
5/3
rro
where ro is the diameter of a circle over which rms phase variation is ~ 1 radian.
Fried’s Parameter r0
( ) Cn h2
2 2
r ~ { } -5/3
Fried’s parametero
For Cn ~ 10-16 m-2/3
h ~ 104 m
~ 0.5 m
2
we have ro ~ 10 cm.
r0 - Related Parameters
r0 6/5 Fried parameter 20 cm 120 cm
(coherence diameter)
0 ~ 6/5 coherence time 20 ms 120 ms
0 ~ 6/5 isoplanatic angle 4" 24"
fwhm~ -1/5 image diameter 0.50" 0.38"
Nact~ 12/5 required no. of actuators 2500 70
S ~ -12/5 uncorrected Strehl ratio 4x10-4 0.014
r0
v
r0
D2
r02
2r0
D2
r0
h
Quantity Scaling Name Value(at 0.5 m)
Value(at 2.2 m)
Expansion of the Phase in Zernike Polynomials
An alternative characterization of the phase comes from expanding in terms of a complete set of functions and then characterizing the coefficients of the expansion:
(r,) = am,n Zm,n(r,)
piston
tip/tilt
focus
astigmatism
astigmatism
Z0,0 = 1
Z1,-1 = 2 r sin
Z1,1 = 2 r cos
Z2,-2 = 6 r2 sin2
Z2,0 = 3 (2r2 - 1)
Z2,2 = 6 r2 cos 2
Z1,-1 Z1,1
Z0,0
Z2,-2
Z2,0
Z2,2
Z3,-3
Z3,-1 Z3,1
Z3,3
Z4,-4
Z4,-2
Z4,0
Z4,4
Z4,2
Atmospheric Zernike Coefficients
Zernike Index
RM
S Z
erni
ke C
oeff
icie
nt
(D
/ro)
5/6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 5 10 15 20 25 30 35 40 45
Shack Hartmann IShack-Hartmann Test
Shack Hartmann IIShack-Hartmann Test, continued
UFS Ref Beam
Ultra FinescreenReference BeamExposure
UFS Image (before)
Ultra Fine Screen Image(Segment 8)
(0.44 arcsec RMS)
UFS C. offsets (before)Centroid Offset Summary Info:
Translation from Ref : -4.87 -0.02
Rotation from Ref (rad) : 0.271E-03 Scale change from Ref : 1.017
KEY: 0.140 arcseconds per pixel
Scale Error : -2.75 80% Enclosed Energy : 10.93 50% Enclosed Energy : 7.66
RMS Error : 3.14
Max Error (pixels) : 11.11 Subimage With Max Error : 209
Centroid Offset Display
15 pixels
Curvature Sensing Concept(F. Roddier, Applied Optics, 27, 1223-1225, 1998)
Laplacian normal derivative at boundary
2r
RI+ I
I+ + I
I ( r )I+ ( r )
(r)
Difference ImageZ1,-1
Z2,-2 Difference Image
Z2,0 Difference Image
Z3,-3 Difference Image
Z4,0 Difference Image
Difference ImageRandom Zernikes
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