adaptive optics in the vlt and elt era atmospheric turbulence

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

Atmospheric Turbulence Atmospheric Turbulence

François WildiObservatoire de Genève

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

Atmospheric Turbulence EssentialsAtmospheric Turbulence Essentials• We, in astronomy, are essentially interested in the effect of

the turbulence on the images that we take from the sky. This effect is to mix air masses of different index of refraction in a random fashion

• The dominant locations for index of refraction fluctuations that affect astronomers are the atmospheric boundary layer , the tropopause and for most sites a layer in between (3-8km) where a shearing between layers occurs.

• 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 r2/3

Fluctuations in index of refraction are Fluctuations in index of refraction are due to temperature fluctuationsdue to temperature fluctuations

• Refractivity of air

where P = pressure in millibars, T = temp. in K, in micronsn = index of refraction. Note VERY weak dependence on

• Temperature fluctuations index fluctuations

(pressure is constant, because velocities are highly sub-sonic. Pressure differences are rapidly smoothed out by sound wave propagation)

N (n 1) 106 77.6 17.52 10 3

2

PT

N 77.6 (P / T 2 )T

Turbulence arises in several placesTurbulence arises in several places

stratosphere

Heat sources w/in dome

boundary layer~ 1 km

tropopause10-12 km

wind flow around dome

Within dome: “mirror seeing”Within dome: “mirror seeing”

• When a mirror is warmer than dome air, convective equilibrium is reached.

• Remedies: Cool mirror itself, or blow air over it, improve mount

To control mirror temperature: dome air conditioning (day), blow air on back (night)

credit: M. Sarazin credit: M. Sarazin

convective cells are bad

Local “Seeing” -Local “Seeing” -Flow pattern around a telescope domeFlow pattern around a telescope dome

Cartoon (M. Sarazin): wind is from left, strongest turbulence on right side of dome

Computational fluid dynamics simulation (D. de Young) reproduces features of cartoon

Boundary layers: day and nightBoundary layers: day and night

• Wind speed must be zero at ground, must equal vwind several hundred meters up (in the “free” atmosphere)

• Boundary layer is where the adjustment takes place, where the atmosphere feels strong influence of surface

• Quite different between day and night– Daytime: boundary layer is thick (up to a km),

dominated by convective plumes– Night-time: boundary layer collapses to a few hundred

meters, is stably stratified. Perturbed if winds are high.

• Night-time: Less total turbulence, but still the single largest contribution to “seeing”

Real shear generated turbulence (aka Kelvin-Real shear generated turbulence (aka Kelvin-Helmholtz instability) measured by radarHelmholtz instability) measured by radar

• Colors show intensity of radar return signal. • Radio waves are backscattered by the turbulence.

Kolmogorov turbulence, cartoonKolmogorov turbulence, cartoon

Outer scale L0

ground

Inner scale l0

hconvection

solar

h

Wind shear

Kolmogorov turbulence, in wordsKolmogorov turbulence, in words

• Assume energy is added to system at largest scales - “outer scale” L0

• Then energy cascades from larger to smaller scales (turbulent eddies “break down” into smaller and smaller structures).

• Size scales where this takes place: “Inertial range”.

• Finally, eddy size becomes so small that it is subject to dissipation from viscosity. “Inner scale” l0

• L0 ranges from 10’s to 100’s of meters; l0 is a few mm

Assumptions of Kolmogorov turbulence Assumptions of Kolmogorov turbulence theorytheory

• Medium is incompressible

• External energy is input on largest scales (only), dissipated on smallest scales (only)

– Smooth cascade

• Valid only in inertial range L0

• Turbulence is– Homogeneous – Isotropic

• In practice, Kolmogorov model works surprisingly well!

Questionable

Concept QuestionConcept Question

• What do you think really determines the outer scale in the boundary layer? At the tropopause?

• Hints:

Outer Scale ~ 15 - 30 m, from Generalized Outer Scale ~ 15 - 30 m, from Generalized Seeing Monitor measurementsSeeing Monitor measurements

• F. Martin et al. , Astron. Astrophys. Supp. v.144, p.39, June 2000• http://www-astro.unice.fr/GSM/Missions.html

Atmospheric structure functionsAtmospheric structure functions

A structure function is measure of intensity of fluctuations of a random variable f (t) over a scale

Df() = < [ f (t + ) - f ( t) ]2 >

With the assumption that temperature fluctuations are carried around passively by the velocity field (for incompressible fluids), T and N have structure functions like

• DT ( r ) = < [ T (x ) - v ( T + r ) ]2 > = CT2 r 2/3

• DN ( r ) = < [ N (x ) - N ( x + r ) ]2 > = CN2 r 2/3

CN2 is a “constant” that characterizes the strength of the variability of N. It varies

with time and location. In particular, for a static location (i.e. a telescope) CN2 will

vary with time and altitude

Typical values of CTypical values of CNN22

• Index of refraction structure function

DN ( r ) = < [ N (x ) - N ( x + r ) ]2 > = CN2 r 2/3

• Night-time boundary layer: CN2 ~ 10-13 - 10-15 m-2/3

10-14

Paranal, Chile, VLT

Turbulence profiles from SCIDARTurbulence profiles from SCIDAR

Eight minute time period (C. Dainty, Imperial College)

Siding Spring, Australia Starfire Optical Range, Albuquerque NM

Atmospheric Turbulence: Main PointsAtmospheric Turbulence: Main Points

• 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 r2/3

• All else will follow from these points!

Phase structure function, Phase structure function, spatial coherence and rspatial coherence and r00

Definitions - Structure Function and Definitions - Structure Function and Correlation FunctionCorrelation Function

• Structure functionStructure function: Mean square difference

• Covariance functionCovariance function: Spatial correlation of a random variable with itself

D (r ) (

x ) (

x r ) 2 dx

(x ) (

x r ) 2

B (r ) (

x

r )(

x ) dx

(x

r )(

x )

Relation between structure function and Relation between structure function and covariance functioncovariance function

To derive this relationship, expand the product in the definition of D ( r ) and assume homogeneity to take the averages

D (r ) 2 B (0) B (

r )

Definitions - Spatial Coherence FunctionDefinitions - Spatial Coherence Function

• Spatial coherence function of field is defined as Covariance for

complex fn’s

C (r) measures how “related” the field is at one position x to its values at neighboring positions x + r .

Do not confuse the complex field with its phase

•

For light wave exp[i(x )], phase is (

x ) kz t

C (rr ) (rx)*(rx rr )

Now evaluate spatial coherence Now evaluate spatial coherence function Cfunction C (r) (r)

• For a Gaussian random variable with zero mean,•

• So

• So finding spatial coherence function C (r) amounts to evaluating the structure function for phase D ( r ) !

expi exp 2 / 2

C (rr ) exp i[(rx) (rx rr )]

exp (rx) (rx rr ) 2 / 2

exp D (rr ) / 2

Next solve for DNext solve for D ( r ) ( r ) in terms of the in terms of the turbulence strength Cturbulence strength CNN

22

• We want to evaluate

• Remember that

D (rr ) 2 B (0) B (rr )

C (r ) exp D (

r ) /2

Solve for DSolve for D ( r ) ( r ) in terms of the in terms of the turbulence strength Cturbulence strength CNN

22, continued, continued

• But for a wave propagating

vertically (in z direction) from height h to height h + h. This means that the phase is the product of the wave vector k (k= [radian/m]) x the Optical path

Here n(x, z) is the index of refraction.

• Hence

B (r ) k 2 d z

h

hh

d z h

hh

n(x , z )n(

x

r , z )

(rx) k dz n(vx, z)

h

hh



• Change variables:

• Then

z z z

B (r ) k 2 d z

h

hh

dz n(x , z )n(

x

r , z z)

h z

hh z

k 2 d z h

hh

dz BNh z

hh z

(r ,z)

B (r ) k 2h dzBN

h z

hh z

(r ,z) k 2h dzBN

(r ,z)

Algebra…



• Now we can evaluate D ( r )

D (r ) 2 B (0) B (

r ) 2k 2h dz BN (0,z) BN (

r ,z)

D (r ) 2k 2h dz BN (0,0) BN (

r ,z) BN (0,0) BN (0,z)

D (r ) k 2h dz

DN (r ,z) DN (0,z) Algebra…


22, completed, completed

• But

•

25(1/ 2)(1/ 6)

(2 / 3)

r 5 / 3 2.914 r5 / 3

DN (rr ) CN2 rr 2 /3 CN

2 r2 z2 1/3 so

D (rr ) k2hCN2 dz r2 z2 1/3

z2 /3

D (rr ) 2.914 k2r5 /3CN

2h 2.914 k2r5 /3 dh0

CN2 (h)

FinallyFinally we can evaluate the spatial we can evaluate the spatial coherence function Ccoherence function C (r) (r)

For a slant path you can add factor ( sec )5/3 to

account for dependence on zenith angle

Concept Question: Note the scaling of the coherence function with separation, wavelength, turbulence strength. Think of a physical reason for each.

C (rr ) exp D (rr ) / 2 exp

12

2.914 k2r5/3 dhCN2 (h)

0

Given the spatial coherence function, calculate Given the spatial coherence function, calculate effect on telescope resolutioneffect 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

Define optical transfer function (OTF)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 = O PSF dx O(r - x) PSF ( x )

• Take Fourier Transform: F ( I ) = F (O ) F ( PSF )

• Optical Transfer Function is Fourier Transform of PSF:

OTF = F ( PSF )

convolved with

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

Seeing limited PSF

Diffraction limited PSF

Inte

nsity

Inte

nsity

Seeing limited OTF

Diffraction limited OTF

/ r0

/ r0 / D

/ D r0 / D /

r0 / D /

-1

-1

Now describe optical transfer function of the Now describe optical transfer function of the telescope in the presence of turbulencetelescope 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 ( / D).

• Measure the resolving power of the imaging system by

R = df S ( f ) = df B ( f ) T ( f )

Derivation of rDerivation of r00

• R of a perfect telescope with a purely circular aperture of (small) diameter d is

R = df T ( f ) = ( / 4 ) ( d / )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 ) ( / 4 ) ( r0 / )2

Derivation of rDerivation of r0 0 , continued, continued

• Now we have to evaluate the contribution of the atmosphere’s OTF: df B ( f )

• B ( f ) = C ( f ) (to go from cycles per meter to cycles per wavelength)

B( f ) C (f ) exp Kf 5/ 3

C (rr ) exp

12

2.914 k2r5 /3 dhCN2 (h)

0

Algebra…

Derivation of rDerivation of r0 0 , continued, continued

• Now we need to do the integral in order to solve for r0 :

( / 4 ) ( r0 / )2 = df B ( f ) = df exp (- K f 5/3)

• Now solve for K:

K = 3.44 (r0 / )-5/3 B ( f ) = exp - 3.44 ( f / r0 )5/3 = exp - 3.44 ( / r0 )5/3

(6 / 5) (6/5) K-6/5

Replace by r

Algebra…

Derivation of rDerivation of r0 0 , concluded, concluded

3.44rr0

5/ 3

12

2.914k 2r5 / 3 sec dhCN2 (h)

r0 0.423k 2 sec dhCN2 (h) 3 / 5

D (r) 6.88(r / ro )5/ 3

Scaling of rScaling of r00

• 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 )

r0 0.423k2 sec CN2 (z)dz

0

H

3 / 5

6 / 5 sec 3 / 5 CN2 (z)dz

-3/5

Typical values of rTypical values of r00

• Usually r0 is given at a 0.5 micron wavelength for reference purposes. It’s up to you to scale it by -1.2 to evaluate r0 at your favorite wavelength.

• At excellent sites such as Paranal, 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.

• r0 changes its value with a typical time constant of 5-10 minutes

Phase PSD, another important parameterPhase PSD, another important parameter

• Using the Kolmogorov turbulence hypothesis, the atmospheric phase PSD can be derived and is

3/11

3/50

023.0)( kr

k

• This expression can be used to compute the amount of phase error over an uncorrected pupil

3/5

0

2 03.1

rD

Units: Radians of phase / (D / r0)5/6

Reference: Noll76

Tip-tilt is single biggest contributor

Focus, astigmatism, coma also big

High-order terms go on and on….

adaptive optics in the vlt and elt era atmospheric turbulence

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