1. Instrumentation Concepts Ground-based Optical Telescopes
Keith Taylor
(IAG/USP)
Aug-Nov, 2008
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏ 1

2. Atmospheric Turbulence (appreciative thanks to USCS/CfAO)
Adaptive Optics
Atmospheric Turbulence
(appreciative thanks to USCS/CfAO)
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

3. 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!

4. Kolmogorov Turbulence
Atmospheric refractive index inhomogeneities warp the wave front incident on the telescope pupil.
These inhomogeneities appear to be associated with atmospheric turbulence
Thermal gradients
Humidity fluctuations
Wind shear and associated hydrodynamic instabilities
Image quality is directly related to the statistics of the perturbations of the incoming wave front
The theory of seeing combines
Theory of atmospheric turbulence
Optical physics
Predict the modifications to the image caused by refractive index gradients.
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

5. Modeling the atmosphere
Random fluctuations in the motion of the atmosphere occur predominantly due to:
Drag encountered by the air flow at the Earth's surface and resultant wind shear
Differential heating of different portions of the Earth's surface and the resultant thermal convection
Phase changes involving release of heat (condensation/crystallization) and subsequent changes in temperature and velocity fields
Convergence and interaction of air masses with various atmospheric fronts;
Obstruction of air flows by mountain barriers that generate wavelike disturbances and vortex motions on their leeside.
The atmosphere is difficult to study due to the high Reynolds number (Re ~ 106 ), a dimensionless quantity, that characterizes the ratio of inertial to viscous forces.
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

6. Cloud formations Indicative of atmospheric turbulence profiles
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

7. Atmospheric Turbulence: Outline
What determines the index of refraction in air?
Origins of turbulence in Earth’s atmosphere
Kolmogorov turbulence models

8. The Atmosphere Kolmogorov model of turbulence
Kinetic energy in large scale turbulence cascades to smaller scales
Inertial interval
Inner scale l0 ­ 2mm. Outer scale L0­10 to 100 m
Turbulence distributed within discrete layers
The strength of these layers is described by a refractive index structure function:
Strength of turbulence can be described by a single parameter, r0 , Fried’s parameter
Fried parameter is the diameter of a circular aperture over which the wavefront phase variance equals 1 rad2
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏
J. Vernin, Universite de Nice.
Cerro Pachon for Gemini IGPO

9. IAG/USP (Keith Taylor)‏
Cn2 profiles
Median seeing conditions on Mauna Kea are taken to be ro ~ 0.23 meters at 0.55 microns.
The 10% best seeing conditions are taken to be ro ~ 0.40 meters.
A set of SCIDAR data taken by Francois Roddier et al. (1990 SPIE Vol ) on Mauna Kea.
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

10. IAG/USP (Keith Taylor)‏
Wavefront variance   DT r0
5/3
2 =
radians2
This gives the phase variance over a telescope of diameter DT
A phase variance of less than ~ 0.2 gives diffraction limited performance
There are 3 regimes
DT < r Diffraction dominates
DT ~ r0 - 4r Wavefront tilt (image motion) dominates
DT >> r Speckle (multiple tilts across the telescope aperture) dominates
4m telescope:
D/ r0(500nm)=20
D/ r0(2.2 m)=3.5
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

11. IAG/USP (Keith Taylor)‏
Strehl ratio
Corrected 0.20” FWHM
Uncorrected 0.49” FWHM
MARTINI WHT, K-band
There are two components of the PSF for 2 < 2 radians2
So ‘width’ of the image is not a useful parameter, use height of PSF:
Strehl ratio:
For small 2 : R ~ exp (-2)
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

12. Isoplanatic angle, temporal variation
Angle over which wavefront distortions are essentially the same:
It is possible to perform a similar turbulence weighted integral of transverse wind speed in order to derive an effective wind speed and approximate timescale of seeing
τ0 is the characteristic timescale of turbulence
Note the importance of Cn2(h) in both cases
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

13. Atmospheric Seeing - Summary
Dependence on Wavelength
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

14. Turbulence arises in several places
stratosphere
tropopause
10-12 km
wind flow around dome
boundary layer
~ 1 km
Heat sources w/in dome

15. 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.
credit: M. Sarazin
credit: M. Sarazin
convective cells are bad
To control mirror temperature: dome air conditioning (day), blow air on back (night), send electric current through front Al surface-layer to equalize temperature between front and back of mirror

16. Dome seeing (TMT CFD calculations)
Wind direction
Wind velocities
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

17. 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

18. Top View - Flow pattern around a telescope dome
Computational fluid dynamics simulation (D. de Young)

19. Thermal Emission Analysis: mid-IR
A good way to find heat sources that can cause turbulence
VLT UT3, Chile
19 Feb. 1999
0h34 Local Time
Wind summit: 4m/s
Air Temp summit: 13.8C
Here ground is warm, top of dome is cold. Is this a setup for convection inside the dome?

20. 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 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 rising from hot ground. Quite turbulent.
Night-time: boundary layer collapses to a few hundred meters, is stably stratified. Perturbed if winds are high.

21. Boundary layer is much thinner at night
Surface layer: where viscosity is largest effect
Credits: Stull (1988) and Haggagy (2003)‏

22. Convection takes place when temperature gradient is steep
Daytime: ground is warm, air is cooler
If temperature gradient between ground and ~ 1 km is steeper than the “adiabatic gradient,” a warm volume of air that is raised upwards will find itself still with cooler surroundings, so it will keep rising
These warm volumes of air carry thermal energy upwards: convective heat transport

23. Boundary layer profiles from acoustic sounders
Turbulence profile
Vertical wind profile

24. Boundary layer has strongest effect on astronomical seeing
Daytime: Solar astronomers have to work with thick and messy turbulent boundary layer
Big Bear Observatory
Night-time: Less total turbulence, but still the single largest contribution to “seeing”
Neutral times: near dawn and dusk
Smallest temperature difference between ground and air, so wind shear causes smaller temperature fluctuations

25. Big Bear Solar Observatory
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

26. Wind shear mixes layers with different temperatures
Wind shear  Kelvin Helmholtz instability
If two regions have different temperatures, temperature fluctuations T will result
Computer simulation

27. Real Kelvin-Helmholtz instability measured by FM-CW radar
Colors show intensity of radar return signal.
Radio waves are backscattered by the turbulence.

28. Temperature profile in atmosphere
Temperature gradient at low altitudes  wind shear will produce index of refraction fluctuations

29. Turbulence strength vs. time of 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)‏

30. Leonardo da Vinci’s view of turbulence

31. IAG/USP (Keith Taylor)‏
Reynolds number
When the average velocity (va) of a viscous fluid of characteristic size (l) is gradually increased, two distinct states of fluid motion are observed: 
Laminar (regular and smooth in space and time) at very low va
Unstable and random at va greater than some critical value.
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

32. Reynolds number Re = Reynolds number = v L / 
Here L is a typical length scale, v is a typical velocity,  is the viscosity
Re ~ inertial stresses / viscous stresses
Creeping flow
Laminar flow, strong Re dependence
,000 Boundary layer
1, ,000 Transition to slightly turbulent
Turbulent, moderate Re dependence
> Strong turbulence, small Re dependence

33. Effects of variations in Re
When Re exceeds critical value a transition of the flow from laminar to turbulent or chaotic occurs.
Between these two extremes, the flow passes through a series of unstable states.
High Re turbulence is chaotic in both space and time and exhibits considerable spatial structure
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

34. Velocity fluctuations
The velocity fluctuations occur on a wide range of space and time scales formed in a turbulent cascade.
Kolmogorov turbulence model states that energy enters the flow at low frequencies at scale length L0.
The size of large-scale fluctuations, referred to as large eddies, can be characterized by their outer scale length  L0. (spatial frequency  kL0 = 2/L0)
These eddies are not universal with respect to flow geometry; they vary according to the local conditions.
A mean value  L0 = 24m from the data obtained at Cerro Paranal, Chile (Conan et al. 2000).
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

35. Energy transport to smaller scales
The energy is transported to smaller and smaller loss-less eddies until
At  Re ~ 1 the KE of the flow is converted into heat by viscous dissipation
The small-scale fluctuations with sizes l0 < r < L0 is the inertial subrange have scale-invariant behavior, independent of the flow geometry.
Energy is transported from large eddies to small eddies without piling up at any scale.
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

36. Kolmogorov turbulence, cartoonh
convection
solar
Wind shear
Outer scale L0
Inner scale l0
ground

37. 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

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

39. How large is the Outer Scale?
Dedicated instrument, the Generalized Seeing Monitor (GSM), built by Dept. of Astrophysics, Nice Univ.)‏

40. Outer Scale ~ 15 - 30 m, from Generalized Seeing Monitor measurements
F. Martin et al. , Astron. Astrophys. Supp. v.144, p.39, June 2000

41. Credit: Gary Chanan, UCI
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
 (cm-1)‏

42. 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 L0
Turbulence is
Homogeneous
Isotropic
In practice, Kolmogorov model works surprisingly well!
Questionable at best

43. Typical values of CN2 Index of refraction structure function
DN ( r ) = < [ N (x ) - N ( x + r ) ]2 > = CN2 r 2/3
Night-time boundary layer: CN2 ~ m-2/3
10-14
Paranal, Chile, VLT

44. Turbulence profiles from SCIDAR
Eight minute time period (C. Dainty, Imperial College)‏
Starfire Optical Range, Albuquerque NM
Siding Spring, Australia

45. 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!

46. IAG/USP (Keith Taylor)‏
Wavefronts
Zernike polynomials are normally used to describe the actual shape of an incoming wavefront
Any wavefront can be described as a superposition of zernike polynomials
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

47. Atmospheric Wavefront Variance after Removal of Zernike Polynomialsj n m
Zernike Polynomial
Name
Resid. Var. (rad2) 1
Constant
1.030 (D/r0)5/3 2
Tilt
0.582 (D/r0)5/3 3
0.134 (D/r0)5/3 4
Defocus
0.111 (D/r0)5/3 5
Astigmatism
(D/r0)5/3 6
(D/r0)5/3 7
Coma
(D/r0)5/3 8
(D/r0)5/3 9
(D/r0)5/3 10
(D/r0)5/3
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏

48. Gary Chanan, UCI
Piston
Tip-tilt

49. Astigmatism
(3rd order)‏
Defocus

50. Trefoil
Coma

51. “Ashtray”
Spherical
Astigmatism
(5th order)‏

53. Tip-tilt is single biggest contributor
Units: Radians of phase / (D / r0)5/6
Focus, astigmatism, coma also big
High-order terms go on and on….
Reference: Noll

54. What is Diffraction?
Slides based on those of James E. Harvey, Univ. of Central Florida, Fall ‘05

56. Details of diffraction from circular aperture
1) Amplitude
2) Intensity
First zero at
r = 1.22  / D
FWHM
 / D

57. Diffraction pattern from hexagonal Keck telescope
Ghez: Keck laser guide star AO
Stars at Galactic Center

58. Considerations in the optical design of AO systems: pupil relays
Deformable mirror and Shack-Hartmann lenslet array should be “optically conjugate to the telescope pupil.”
What does this mean?

59. Let’s define some terms
“Optically conjugate” = “image of....”
“Aperture stop” = the aperture that limits the bundle of rays accepted by the optical system
“Pupil” = image of aperture stop
optical axis
object space
image space
symbol for aperture stop

60. So now we can translate:
“The deformable mirror should be optically conjugate to the telescope pupil”
means
The surface of the deformable mirror is an image of the telescope pupil
where
The pupil is an image of the aperture stop
In practice, the pupil is usually the primary mirror of the telescope

61. Considerations in the optical design of AO systems: “pupil relays”
‘PRIMARY MIRROR

62. Typical optical design of AO system
telescope primary mirror
Pair of matched off-axis parabola mirrors
Deformable mirror
collimated
Science camera
Wavefront sensor (plus optics)‏
Beamsplitter

63. More about off-axis parabolas
Circular cut-out of a parabola, off optical axis
Frequently used in matched pairs (each cancels out the off-axis aberrations of the other) to first collimate light and then refocus it
SORL

64. High order AO architecture
Wavefront controller
Typically a deformable mirror (DM)
May not be optically conjugate to an image of the primary
Wavefront sensor (WFS)
Shack Hartmann (WFS) or Curvature Sensor (CS)
Beamsplitter
Dichroic, multi-dichroic, intensity, spatial or combination
Controller
Typically multi-processor or multi-DSP
Interfaces
Can be complex and include removal of non-common path errors to science instrumentation (hence an interface to science data path)
Laser beacons
Aug-Nov, 2008
IAG/USP (Keith Taylor)‏