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Adaptive Optics in the VLT and ELT era Atmospheric TurbulencePowerPoint Presentation

Adaptive Optics in the VLT and ELT era Atmospheric Turbulence

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### Adaptive Optics in the VLT and ELT era Atmospheric Turbulence

François Wildi

Observatoire de Genève

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

Atmospheric 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 due to temperature fluctuations

- Refractivity of air
where P = pressure in millibars, T = temp. in K, in microns

n = 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)

Turbulence arises in several places fluctuations

tropopause

10-12 km

wind flow around dome

boundary layer

~ 1 km

stratosphere

Heat sources w/in dome

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

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

Within dome: “mirror seeing”credit: M. Sarazin

credit: M. Sarazin

convective cells are bad

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

Local “Seeing” - equilibrium is reached. Flow 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 night equilibrium is reached.

- 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-Helmholtz instability) measured by radar

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

Kolmogorov turbulence, cartoon instability) measured by radar

solar

Outer scale L0

Inner scalel0

h

Wind shear

convection

h

ground

Kolmogorov turbulence, in words instability) measured by radar

- 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 theory instability) measured by radar

Questionable

- 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!

Concept Question instability) measured by radar

- What do you think really determines the outer scale in the boundary layer? At the tropopause?
- Hints:

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

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

Df(t) = < [ f (t + 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 C measurementsN2

10-14

- 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

Paranal, Chile, VLT

Turbulence profiles from SCIDAR measurements

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

Starfire Optical Range, Albuquerque NM

Siding Spring, Australia

Atmospheric Turbulence: Main Points measurements

- 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, spatial coherence and r measurements0

Definitions - Structure Function and Correlation Function measurements

- Structure function: Mean square difference
- Covariance function: Spatial correlation of a random variable with itself

Relation between structure function and covariance function measurements

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

Definitions - Spatial Coherence Function measurements

- 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 f

Now evaluate spatial coherence function C measurements (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 ) !

Next solve for D measurements ( r )in terms of the turbulence strength CN2

- We want to evaluate
- Remember that

Solve for D measurements ( r )in terms of the turbulence strength CN2, 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=2p/l [radian/m]) x the Optical path

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

- Hence

Solve for D measurements ( r )in terms of the turbulence strength CN2, continued

- Change variables:
- Then

Algebra…

Solve for D measurements ( r )in terms of the turbulence strength CN2, continued

- Now we can evaluate D ( r )

Algebra…

Solve for D measurements ( r )in terms of the turbulence strength CN2, completed

- But

Finally measurements we can evaluate the spatial coherence function C (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.

Given the spatial coherence function, calculate effect on telescope resolution

- Define optical transfer functions of telescope, atmosphere
- Definer0 as the telescope diameter where the two optical transfer functions are equal
- Calculate expression for r0

Define optical transfer function (OTF) telescope resolution

- 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 Optical Transfer Functions telescope resolution

Seeing limited OTF

Seeing limited PSF

Intensity

-1

l / D

l / r0

r0 / l

D / l

Diffraction limited PSF

Diffraction limited OTF

Intensity

-1

l / r0

l / D

D / l

r0 / l

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 = dfS ( f ) = dfB ( f ) T ( f )

Derivation of r the presence of turbulence0

- 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:
dfB ( f ) = dfT ( f ) ( p / 4 ) ( r0/ l )2

Derivation of r the presence of turbulence0 , continued

- Now we have to evaluate the contribution of the atmosphere’s OTF: dfB ( f )
- B ( f ) = C ( l f ) (to go from cycles per meter to cycles per wavelength)

Algebra…

Derivation of r the presence of turbulence0 , continued

(6p / 5) G(6/5) K-6/5

- Now we need to do the integral in order to solve for r0 :
( p / 4 ) ( r0/ l )2 =dfB ( 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

Algebra…

Replace by r

Derivation of r the presence of turbulence0 , concluded

Scaling of r the presence of turbulence0

- r0is size of subaperture, sets scale of all AO correction
- r0gets smaller when turbulence is strong (CN2 large)
- r0gets bigger at longer wavelengths: AO is easier in the IR than with visible light
- r0gets smaller quickly as telescope looks toward the horizon (larger zenith angles )

Typical values of r the presence of turbulence0

- 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, the presence of turbulenceanother important parameter

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

- This expression canbeused to compute the amount of phase error over an uncorrectedpupil

Tip-tilt is single biggest contributor the presence of turbulence

Focus, astigmatism, coma also big

High-order terms go on and on….

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

Reference: Noll76

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