Lecture 4: Propagation of light through atmospheric turbulence

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Lecture 4Propagation of light through
atmospheric turbulence

• Claire Max
• Astro 289C, UC Santa Cruz
• January 17, 2008

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Outline for today

• Kolmogorov turbulence
• Physics
• Assumptions
• Derived spectrum
• Effect of Kolmogorov turbulence on the
  propagation of light

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Temperature profile in atmosphere

• Strong temperature gradient in troposphere, at
  low altitudes

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Wind shear mixes layers with different
temperatures

• Wind shear ? Kelvin Helmholtz instability
• If two regions have different temperatures,
  turbulent temperature fluctuations ?T will result

Computer simulation
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Real Kelvin-Helmholtz instability measured by
FM-CW radar

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

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Turbulence strength vs. time of day
Day
Night
Day
Night
Night
Day
Night

• Theory for different heights Theory vs.
  data
• Balance heat fluxes in surface layer
• Credit Wesely and Alcaraz, JGR (1973)

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Leonardo da Vincis view of turbulence
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Kolmogorov turbulence in a nutshell

• Big whorls have little whorls,
• Which feed on their velocity
• Little whorls have smaller whorls,
• And so on unto viscosity.

L. F. Richardson (1881-1953)
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Kolmogorov turbulence, cartoon
ground
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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 10s to 100s of meters l0 is a
  few mm

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Example Breakup of Kelvin-Helmholtz vortex

• Computer simulation
• Start with large coherent vortex structure, as is
  formed in Kelvin-Helmholtz instability
• Watch it develop smaller and smaller substructure
• Analogous to Kolmogorov cascade from large eddies
  to small ones
• From small k to large k

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How large is the Outer Scale?

• Dedicated instrument, the Generalized Seeing
  Monitor (GSM), built by Dept. of Astrophysics,
  Nice Univ.)

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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

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Concept Question

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

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The Kolmogorov turbulence model, derived from
dimensional analysis

• u velocity, energy dissipation rate per
  unit mass, ? viscosity, l0 inner scale, l
  local spatial scale
• Energy/mass u2/2 ? u2
• Energy dissipation rate per unit mass

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What is the inner scale l0 ?

• Navier Stokes Equations
• n viscosity, u velocity, ppressure, r
  mass density
• Take product of 1st equation with u, keep only
  the viscosity term on the right side
  (hypothesized to be dominant at l0 )

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Inner scale l0 , continued

• Two equations to solve for l0
• Solve for l0
• where the Reynolds number is

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Reynolds number

• Reynolds number
• Here L is a typical length scale, u is a typical
  velocity, n is the viscosity
• Re inertial stresses / viscous stresses
• 0 - 1 Creeping flow
• 1 - 100 Laminar flow, strong Re dependence
• 100 - 1,000 Boundary layer
• 1,000 - 10,000 Transition to slightly turbulent
• 104 - 106 Turbulent, moderate Re dependence
• gt 106 Strong turbulence, small Re dependence

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Derive the Kolmogorov Power Spectrum using
dimensional analysis

• 1-D power spectrum of velocity fluctuations k
  2p / l
• For dimensional analysis, divide by k
• 3-D power spectrum divide by k3
• For rigorous calculation see V. I. Tatarski,
  1961, Wave Propagation in a Turbulent Medium,
  McGraw-Hill, NY

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Lab experiments agree

• Air jet, 10 cm diameter (Champagne, 1978)
• Assumptions turbulence is homogeneous,
  isotropic, stationary in time

Credit Gary Chanan, UCI
Slope -5/3
Power (arbitrary units)
Slope -5/3
l0
L0
k (cm-1)
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Structure functions are used a lot in AO. What
are they?

• Mean values of meteorological variables change
  over minutes to hours. Examples T, p, humidity
• If f(t) is a non-stationary random variable,
• Ft(t) f ( t t) - f ( t) is a difference
  function that is stationary for small t .
• Structure function is measure of intensity of
  fluctuations of f (t) over a time
  scale ? t
• Df(t) lt Ft(t) 2gt lt f (t t) - f ( t)
  2 gt

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Structure function for atmospheric fluctuations

• Scaling law we derived earlier for Kolmogorov
  turbulence
• u2 e 2/3 l2/3 r 2/3
• Heuristic derivation Velocity structure function
  u2
• Du ( r ) lt u (x ) - u ( x r ) 2 gt
  Cu2 r 2/3
• here Cu2 constant to clean up the look of
  the equation

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Derivation of Du from dimensional analysis

• If turbulence is homogenous, isotropic,
  stationary
• where f is a dimensionless function of a
  dimensionless argument.
• Dimensions of a are u2, dimensions of b are
  length, and they must depend only on and n
  (the only free parameters in the problem).
• n cm2 s-1 erg s-1 gm-1 cm2
  s-3

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Du, continued

• The only combinations of and n with the right
  dimensions are
• So

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What about temperature and index of refraction
fluctuations?

• Temperature fluctuations are carried around
  passively by the velocity field (for
  incompressible fluids).
• So T and N have structure functions similar to u
  
• DT ( r ) lt T (x ) - T ( x r ) 2 gt CT2
  r 2/3
• DN ( r ) lt N (x ) - N ( x r ) 2 gt CN2
  r 2/3

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How do you measure index of refraction
fluctuations in situ?

• Refractivity
• Index fluctuations
• So measure ?T , p, and T calculate CN2

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Simplest way to measure CN2 is to use
fast-response thermometers

• DT ( r ) lt T (x ) - T ( x r ) 2 gt CT2 r
  2/3
• Example mount fast-response temperature probes
  at different locations along a bar
• X X X X X X
• Form spatial correlations of each time-series T(t)

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Assumptions of Kolmogorov turbulence theory

• Medium is incompressible
• External energy is input on largest scales
  (only), dissipated on smallest scales (only)
• Smooth cascade
• Valid only in inertial range between scales L0
  and l0
• Turbulence is
• Homogeneous
• Isotropic
• In practice, Kolmogorov model works surprisingly
  well!

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Typical values of CN2

• Index of refraction structure function
• DN ( r ) lt N (x ) - N ( x r ) 2
  gt CN2 r 2/3
• Night-time boundary layer CN2 10-13 - 10-15
  m-2/3

Paranal, Chile, VLT
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Turbulence profiles from SCIDAR
Eight minute time period (C. Dainty, Imperial
College)
Starfire Optical Range, Albuquerque NM
Siding Spring, Australia
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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 proportional to r2/3
• All else will follow from these points!

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Part 2 Effect of turbulence on spatial coherence
function of light
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Definitions - Structure Function and Correlation
Function

• Structure function Mean square difference
• Covariance function Spatial correlation of a
  function with itself

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Relation between structure function and
covariance function

• A problem on the homework for next week
• Derive this relationship
• Hint expand the product in the definition of D?
  ( r ) and assume homogeneity to take the averages

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Definitions - Spatial Coherence Function

• Spatial coherence function of field is defined as
• Covariance for complex fns
• Note that Quirrenbach and Hardy call this
  function but Ive called it
  C? (r) in order to avoid confusion with the
  correlation function B? ( r ) . C? (r) is a
  measure of how related the field ? is at one
  position (e.g. x) to its values at neighboring
  positions (say x r ).

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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 ) !

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

• We want to evaluate
• Recall that
• So next we need to evaluate

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

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

• Change variables
• Then

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

• Now we can evaluate D? ( r )

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

• But

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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.
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Given the spatial coherence function, calculate
effect 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

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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( x - r ) PSF ( x )
• Take Fourier Transform F ( I ) F (O ) F ( PSF
  )
• Optical Transfer Function is Fourier Transform of
  PSF
• OTF F ( PSF )

convolved with
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Examples of PSFs and their Optical Transfer
Functions
Seeing limited OTF
Seeing limited PSF
Intensity
?-1
?
l / r0
l / D
r0 / l
D / l
Diffraction limited PSF
Diffraction limited OTF
Intensity
?-1
?
l / r0
l / D
D / l
r0 / l
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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 ? df S ( f ) ? df B ( f ) ? T ( f )

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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
• ? df B ( f ) ? df T ( f ) ? ( p / 4 ) ( r0
  / l )2

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Derivation of r0 , continued

• Now we have to evaluate the contribution of the
  atmospheres OTF ? df B ( f )
• B ( f ) C? ( l f ) (to go from cycles per
  meter to cycles per wavelength)

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Derivation of r0 , continued

• Now we need to do the integral in order to solve
  for r0
• ( p / 4 ) ( r0 / l )2 ? df B ( 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

Replace by r
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Derivation of r0 , concluded
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Scaling of r0

• 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 ? )

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Typical values of r0

• Usually r0 is given at a 0.5 micron wavelength
  for reference purposes. Its up to you to scale
  it by ?-1.2 to evaluate r0 at your favorite
  wavelength.
• At excellent sites such as Mauna Kea in Hawaii,
  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.

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Next time All sorts of good things come from
knowing r0

• Timescales of turbulence
• Isoplanatic angle AO performance degrades as
  astronomical targets get farther from guide star

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Assignment (see web page for details)

• Reading for Tuesday anisoplanatism, tip-tilt
  errors, other contributors to total wavefront
  error of an AO system (see web)
• Homework for next Thursday (see web)