Lecture 6: TipTilt and Wavefront Sensing

1
Lecture 6Tip-Tilt and Wavefront Sensing

• Claire Max
• Astro 289C, UCSC
• January 24, 2008

2
Topics

• Image motion
• Finish discussion from the last lecture
• Wavefront sensing
• General principles
• Shack-Hartmann sensors
• Curvature sensors
• Pyramid sensors
• Summary

3
Image motion or tip-tilt also contributes to
total wavefront error

• Turbulence both blurs an image and makes it move
  around on the sky (image motion).
• Due to overall wavefront tilt component of the
  turbulence across the telescope aperture
• Can correct this image motion either by taking
  a very short time-exposure, or by using a
  tip-tilt mirror (driven by signals from an image
  motion sensor) to compensate for image motion

(Hardy Eqn 3.59 - one axis)
4
Scaling of tip-tilt with l and D the good news
and the bad news

• In absolute terms, rms image motion in radians is
  independent of l, and decreases slowly as D
  increases
• But you might want to compare image motion to
  diffraction limit at your wavelength
• Now image motion relative to
• diffraction limit is almost D,
• and becomes larger fraction of
• diffraction limit for small l

5
Correcting tip-tilt has relatively large effect,
for seeing-limited images

• For completely uncompensated images
• s2uncomp 1.02 ( D / r0 )5/3
• If image motion (tip-tilt) has been completely
  removed
• s2tiltcomp 0.134 ( D / r0 )5/3
• (Tyson, Principles of AO, eqns 2.61 and 2.62)
• Removing image motion can (in principle) improve
  the wavefront variance of an uncompensated image
  by a factor of 10
• Origin of statement that Tip-tilt is the single
  largest contributor to wavefront error

6
But you have to be careful if you want to apply
this statement to AO correction

• If tip-tilt has been completely removed
• s2tiltcomp 0.134 ( D / r0 )5/3
• But typical values of ( D / r0 ) are 10-50 in
  near-IR
• Keck, D10 m, r0 60 cm, ( D/r0 ) 17
• s2tiltcomp 0.134 ( 17 )5/3 15
• so wavefront phase variance is gtgt 1
• Conclusion if ( D/r0 ) gtgt 1, removing tilt
  alone wont give you anywhere near a diffraction
  limited image

7
Effects of turbulence depend on size of telescope

• Coherence length of turbulence r0 (Frieds
  parameter)
• For telescope diameter D ? (2 - 3) x r0
• Dominant effect is "image wander"
• As D becomes gtgt r0
• Many small "speckles" develop
• Computer simulations by Nick Kaiser image of a
  star, r0 40 cm

D 2 m
D 8 m
D 1 m
8
Effect of atmosphere on long and short exposure
images of a star

• Hardy p. 94
• Vertical axis is image size in units of l/r0

Image motion only
FWHM l/D
9
Error budget concept (sum of s2 s)

• ?stot2 s12 s22 s32 ...
• Individual terms we know so far
• Anisoplanatism sanisop2 (? / ?0 )5/3
• Temporal error stemporal2 28.4 (t / t0 )5/3
• Fitting error sfitting2 m ( d / r0 )5/3

Theres not much to be gained by making any
particular term much smaller than all the others.
Try to keep terms roughly equal in magnitude.
10
Error budget so far

• ?stot2 sfitting2 sanisop2 stemporal2
  smeas2 scalib2

v
v
v
Still need to work on these two
11
We want to relate phase variance to the Strehl
ratio

• Two definitions of Strehl ratio (equivalent)
• Ratio of the maximum intensity of a point spread
  function to what the maximum would be without
  aberrations
• The normalized volume under the optical
  transfer function of an aberrated optical system

12
Relation between variance and Strehl

• Maréchal Approximation
• Strehl exp(- s?2)
• where s?2 is the total wavefront variance
• Valid when Strehl gt 10 or so
• Under-estimate of Strehl for larger values of s?2

13
Relation between Strehl and residual wavefront
variance
Strehl exp(-s?2)
Dashed lines Strehl (r0/D)2 for high wavefront
variance
14
Error Budgets Summary

• Individual contributors to error budget (total
  mean square phase error)
• Anisoplanatism sanisop2 (? / ?0 )5/3
• Temporal error stemporal2 28.4 (t / t0 )5/3
• Fitting error sfitting2 m ( d / r0 )5/3
• Measurement error
• Calibration error, .....
• In a different category
• Image motion lta2gt1/2 2.56 (D/r0)5/6 (l/D)
  radians2
• Try to balance error terms if one is big, no
  point struggling to make the others tiny

15
Wavefront Sensing Topics

• General overview of wavefront sensing
• Types of wavefront sensors (a list)
• Three types in more detail
• Shack-Hartmann wavefront sensors
• Curvature sensing
• Pyramid sensing

16
Overview of wavefront sensing

• Measure phase by measuring intensity variations
• Difference between various wavefront sensor
  schemes is the way in which phase differences are
  turned into intensity differences
• General box diagram

Wavefront sensor
Computer
Transforms aberrations into intensity variations
17
How to use intensity to measure phase?

• Irradiance transport equation A is complex
  field amplitude.
• (Teague, 1982, JOSA 72, 1199)
• Follow I(x,y,z) as it propagates along the z axis
  (paraxial ray approximation small angle w.r.t.
  z)

Wavefront curvature Curvature sensors
Wavefront tilt Hartmann sensors
18
Types of wavefront sensors

• Direct in pupil plane split pupil up into
  subapertures in some way, then use intensity in
  each subaperture to deduce phase of wavefront.
  Sub-categories
• Slope sensing Shack-Hartmann, lateral shear
  interferometer, pyramid sensing
• Curvature sensing
• Indirect in focal plane wavefront properties
  are deduced from whole-aperture intensity
  measurements made at or near the focal plane.
  Iterative methods - take a lot of time.
• Image sharpening, multi-dither
• Phase diversity, phase retrieval, Gerchberg-Saxton

19
How to reconstruct wavefront from measurements of
local tilt
20
Shack-Hartmann wavefront sensor concept - measure
subaperture tilts
CCD
CCD
21
Example Hartmann test of one Keck segment
(static)

• Reference flat wavefront Measured wavefront

Gary Chanan, UCI
22
Resulting displacement of centroids

• Definition of centroid
• Each arrow represents an offset proportional to
  its length

Gary Chanan, UCI
23
Reminder of some optics definitions focal length
and magnification

• Focal length f of a lens or mirror
• Magnification M
  y/y -s/s

f
f
y
y
s
s
24
Displacement of Hartmann Spots
25
Quantitative description of Shack-Hartmann
operation

• Relation between displacement of Hartmann spots
  and slope of wavefront
• where k 2p/l , Dx is the lateral displacement
  of a subaperture image, M is the
  (de)magnification of the system, f is the focal
  length of the lenslets in front of the
  Shack-Hartmann sensor

26
Example Keck adaptive optics system

• Telescope diameter D 10 m, M 2800 ? size of
  whole lenslet array (10/2800) m 3.57 x 10-3 m
• Lenslet array is approx. 18 x 18 lenslets ? each
  lenslet is 200 microns in diameter
• Sanity check size of subaperture on telescope
  mirror lenslet diameter x magnification 200
  microns x 2800 56 cm r0 for wavelength l
  between 1 and 2 microns
• Now look at scale of pixels on CCD detector
• Lenslet array size (200 microns) is larger than
  size of the CCD detector, so must put a focal
  reducer lens between the lenslets and the CCD
  scale factor 3.15

27
Keck AO example, continued

• Each subaperture is then mapped to a size of 200
  microns ? 3.15 63 microns on the CCD detector
• Choose to make this correspond to 3 CCD pixels
  (two to measure spot position, one for guard
  pixel to keep light from spilling over between
  adjacent subapertures)
• So each pixel is 63/3 21 microns across.
• Now calculate angular displacement corresponding
  to one pixel, using

28
Keck AO example, concluded

• Angle corresponding to one pixel Dz/Dx where
  the phase difference D? k Dz.
• Dz / Dx (pixel size x 3.15) ? (2800 x 200 x
  10)
• Pixel size is 21 microns.
• Dz / Dx (21 x 3.15) ? (2800 x 2000) 11.8
  microradians
• Now use factoid 1 arc sec 4.8 microradians
• Dz / Dx 2.4 arc seconds.
• So when a subaperture has 2.4 arc seconds of
  slope across it, the corresponding spot on the
  CCD moves sideways by 1 pixel.

29
How to measure distance a spot has moved on CCD?
Quad cell formula
?
30
Disadvantage gain depends on spot size b which
can vary during the night
31
Question

• What might happen if the displacement of the spot
  is gt radius of spot? Why?

?
?
32
Signal becomes nonlinear and saturates for large
angular deviations
Rollover corresponds to spot being entirely
outside of 2 quadrants
33
Measurement error from Shack-Hartmann sensing

• Measurement error depends on size of spot as seen
  in a subaperture, ?b , wavelength l , subap size
  d, and signal-to-noise ratio SNR
• (Hardy equation 5.16)

34
General expression for signal to noise ratio of a
pixelated detector

• S flux of detected photoelectrons /
  subap npix number of detector
  pixels per subaperture
• R read noise in electrons per
  pixel
• The signal to noise ratio in a subaperture for
  fast CCD cameras is dominated by read noise, and

See McLean, Electronic Imaging in Astronomy,
Wiley, Sect. 10.9
We will discuss SNR in much more detail in a
later lecture
35
Order of magnitude, for r0 d

• Estimate ?b l / r0
• If we want the wavefront error to be lt ?/20, we
  need

36
Trade-off between dynamic range and sensitivity
of Shack-Hartmann WFS

• If spot is diffraction limited in a subaperture
  d, linear range of quad cell (2x2 pixels) is
  limited to ? lref/2d.
• Can increase dynamic range by enlarging the spot
  (e.g. by defocusing it).
• But uncertainty in calculating centroid ?
  width x Nph1/2 so centroid calculation will be
  less accurate.
• Alternative use more than 2x2 pixels per
  subaperture. Decreases SNR if read noise per
  pixel is large (spreading given amount of light
  over more pixels, hence more read noise).

Linear range
37
Curvature wavefront sensing

• F. Roddier, Applied Optics, 27, 1223- 1225, 1998

More intense
Less intense
Normal derivative at boundary
Laplacian (curvature)
38
Curvature sensor lenslet shapes are different for
edge, middle of pupil
Lenslet array

• Example This is what wavefront tilt (which
  produces image motion) looks like on a curvature
  wavefront sensor
• Constant I on inside
• Excess I on right edge
• Deficit I on left edge

39
Simulation of curvature sensor response
Wavefront pure tilt
Curvature sensor signal
G. Chanan
40
Curvature sensor signal for astigmatism
G. Chanan
41
Third order spherical aberration
G. Chanan
42
Practical implementation of curvature sensing
More intense
Less intense

• Use oscillating membrane mirror (2 kHz!) to
  vibrate rapidly between I and I- extrafocal
  positions
• Measure intensity in each subaperture with an
  avalanche photodiode (only need one per
  subaperture!)
• Detects individual photons, no read noise, QE
  60
• Can read out very fast with no noise penalty

43
Measurement error from curvature sensing

• Error of a single set of measurements is
  determined by photon statistics, since detector
  has NO read noise!
• where d subaperture diameter and Nph is no. of
  photoelectrons per subaperture per sample period
• Error propagation when the wavefront is
  reconstructed numerically using a computer scales
  with no. of subapertures N
• (Error)curvature ? N, whereas (Error)Shack-Hartman
  n ? log N

44
Question

• What might be the pros and cons for
• Shack-Hartmann sensing
• Curvature sensing

45
Advantages and disadvantages of curvature sensing

• Advantages
• Lower noise ? can use fainter guide stars than
  S-H
• Fast readout ? can run AO system faster
• Can adjust amplitude of membrane mirror excursion
  as seeing conditions change. Affects
  sensitivity.
• Well matched to bimorph deformable mirror (both
  solve Laplaces equation), so less computation.
• Curvature systems appear to be less expensive.
• Disadvantages
• Avalanche photodiodes can fail if too much light
  falls on them. They are bulky and expensive.
  Hard to use a large number of them.

46
Pyramid sensing

• From Brian Baumans PhD dissertation

47
Schematic of pyramid sensor

• From Esposito et al.

48
Review of Shack-Hartmann geometry
49
Pyramid sensor reverses order of operations in a
Shack-Hartmann sensor
50
Heres what a pyramid-sensor meast looks like

• Courtesy of Jess Johnson

51
(No Transcript)
52
Potential advantages of pyramid wavefront sensors

• Wavefront measurement error can be much lower
• Shack-Hartmann size of spot limited to ? / d,
  where d is size of a sub-aperture and usually d
  r0
• Pyramid size of spot can be as small as ? / D,
  where D is size of whole telescope. So spot can
  be D/r0 20 - 100 times smaller than for
  Shack-Hartmann
• Measurement error (e.g. centroiding) is
  proportional to spot size/SNR. Smaller spot
  lower error.
• Avoids bad effects of charge diffusion in CCD
  detectors
• Fuzzes out edges of pixels. Pyramid doesnt mind
  as much as S-H.

53
Potential pyramid sensor advantages, continued

• Linear response over a larger dynamic range
• Naturally filters out high spatial frequency
  information that you cant correct anyway

54
Detectors for wavefront sensing

• Shack-Hartmann and pyramid usually use fast CCDs
  (charge-coupled devices), sizes up to 128 x 128
  pixels
• Sensitive to visible light (out to 1 micron)
• Can have high quantum efficiency (up to 85)
• Practical frame rates limited to a few kHz
• Read noise currently gt 3 electrons per pixel per
  read
• Recent development infrared detectors (more
  noise)
• Curvature usually use avalanche photodiodes (1
  pixel)
• Sensitive to visible light
• Slightly lower quantum efficiency than CCDs
• NO NOISE
• Very fast
• Discrete components ? maintenance headache if
  there are too many of them

55
Where are the various wavefront sensing methods
used?

• Shack-Hartmann
• Keck, VLT/NACO, Gemini North and South, MMT,
  Palomar, Lick
• Curvature sensing
• Subaru, VLT/MACAO, Gemini South/NICI
• Pyramid sensing
• LBT, WHT?
• Phase Diversity (very low bandwidth)
• James Webb Space Telescope

56
Wavefront Sensing Summary of main points

• Wavefront sensors in common use for astronomy
  measure intensity variations, deduce phase
• Shack-Hartmann
• Curvature sensors
• Curvature systems are cheaper, have fewer degrees
  of freedom, scale more poorly to high no. of
  degrees of freedom, but can use fainter guide
  stars
• Shack-Hartmann systems excel at very large no. of
  degrees of freedom
• Complementary advantages
• New kid on the block pyramid sensors
• Current status testing in lab. Future testing
  on the sky.