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.

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Adaptive Optics in the VLT and ELT era Atmospheric Turbulence

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Adaptive optics in the vlt and elt era atmospheric turbulence

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

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

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

Turbulence arises in several places

tropopause

10-12 km

wind flow around dome

boundary layer

~ 1 km

stratosphere

Heat sources w/in dome


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

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 flow pattern around a telescope dome

Local “Seeing” -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

Boundary 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 helmholtz instability measured by radar

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

Kolmogorov turbulence, cartoon

solar

Outer scale L0

Inner scalel0

h

Wind shear

convection

h

ground


Kolmogorov turbulence in words

Kolmogorov 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 theory

Assumptions of Kolmogorov turbulence theory

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

Concept 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 seeing monitor measurements

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

Atmospheric structure functions

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 n 2

Typical values of CN2

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

Turbulence profiles from SCIDAR

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

Starfire Optical Range, Albuquerque NM

Siding Spring, Australia


Atmospheric turbulence main points

Atmospheric 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 spatial coherence and r 0

Phase structure function, spatial coherence and r0


Definitions structure function and correlation function

Definitions - Structure Function and Correlation Function

  • Structure function: Mean square difference

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


Relation between structure function and covariance function

Relation between structure function and covariance function

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


Definitions spatial coherence function

Definitions - 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 f


Now evaluate spatial coherence function c r

Now evaluate spatial coherence function C (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 r in terms of the turbulence strength c n 2

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

  • We want to evaluate

  • Remember that


Solve for d r in terms of the turbulence strength c n 2 continued

Solve for D ( 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 r in terms of the turbulence strength c n 2 continued1

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

  • Change variables:

  • Then

Algebra…


Solve for d r in terms of the turbulence strength c n 2 continued2

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

  • Now we can evaluate D ( r )

Algebra…


Solve for d r in terms of the turbulence strength c n 2 completed

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

  • But


Finally we can evaluate the spatial coherence function c r

Finally 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

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

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 optical transfer functions

Examples of PSF’s and their Optical Transfer Functions

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

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 0

Derivation of r0

  • 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 0 continued

Derivation of r0 , 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 0 continued1

Derivation of r0 , 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 0 concluded

Derivation of r0 , concluded


Scaling of r 0

Scaling of r0

  • 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 0

Typical values of r0

  • 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 parameter

Phase PSD, another 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


Adaptive optics in the vlt and elt era atmospheric turbulence

Tip-tilt is single biggest contributor

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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