Gary Chanan Department of Physics and Astronomy University of California, Irvine 4 February 2000

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