Part 1: Image Motion Part 2: AO System Optimization Lecture 8

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Part 1 Image MotionPart 2 AO System
OptimizationLecture 8

• Claire Max
• Astro 289C, UC Santa Cruz
• January 31, 2008

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Outline of lecture

• Image motion and its effect on AO system
  performance
• AO system optimization

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Part One Image motion and its effects on Strehl
ratio

• Sources of image motion
• Wind buffeting of telescope (hard to model a
  priori)
• Atmospheric turbulence
• Image motion due to turbulence
• Sensitive to inhomogenities telescope diam. D
• Hence reduced if outer scale of turbulence is
  D

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Scaling of image motion due to turbulence

• Mean squared angular tilt independent of l and
  D-1/3
• But relative to Airy disk (diffraction limit),
  image motion gets worse for larger D and smaller
  wavelengths

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Typical values of image motion

• Keck Telescope D 10 m, r0 0.2 m, l 2
  microns
• So in theory at least, rms image motion is 10
  times larger than diffraction limit, for these
  numbers.
• Measurements on Keck I (Gary Chanan) suggested
  that actual image motion was smaller than this.
• But Keck II seems to have larger image motion
  than Keck I. Wind shake? Something else?

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What maximum tilt must the tip-tilt mirror be
able to correct?

• For a Gaussian distribution, probability is 99.4
  that the value will be within 2.5 standard
  deviations of the mean.
• For this condition, the peak excursion of the
  angle of arrival is
• Note that peak angle is independent of wavelength

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Use Gaussians to model the effects of image motion

• Model the diffraction limited core as a Gaussian
• G(x) exp (-x2 / 2s2) / s (2p)1/2
• For Gaussian, s 0.425 x FWHM
• A Gaussian profile with standard deviation
  sA 0.44 l / D has same width as an Airy
  function

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Tilt errors spread out the core

• Effect of a random tilt error sa is to spread
  each point of the image into a Gaussian profile
  with standard deviation sa
• If initial profile has width sA then the profile
  with tilt has width sT ( sa2 s?2 )1/2

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Image motion reduces peak intensity

• Conserve flux
• Integral under a Gaussian profile with peak
  amplitude A0 is equal to 2pA0sA2
• Image motion keeps total energy the same, but
  puts it in a new Gaussian with variance sT2
  s?2 sa2
• Peak intensity is reduced by the ratio

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Tilt effects on point spread function, continued

• Since sA 0.44 l / D, the peak intensity of the
  previously diffraction-limited core is reduced by
• Diameter of core is increased by FT-1/2
• Similar calculations for the halo replace D by
  r0
• Since D r0 for cases of interest, effect on
  halo is modest but effect on core can be large

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Typical values for Keck Telescope, if tip-tilt
is not corrected

• Core is strongly affected at a wavelength of 1
  micron
• Core diameter is increased by factor of FT-1/2
  23
• Halo is much less affected than core
• Halo peak intensity is only reduced by factor of
  0.93
• Halo diameter is only increased by factor of 1.04

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Effect of tip-tilt on Strehl ratio

• Define Sc as the peak intensity ratio of the core
  alone
• Image motion relative to Airy disk size 1.22 l /
  D
• Example To obtain Strehl of 0.8 from tip-tilt
  only (no phase error at all, so sp 0), sa
  0.18 (1.22 l / D )
• Residual tilt has to be w/in 18 of Airy disk
  diameter

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Image motion main points

• Image motion can be large, if not compensated
• Keck, l 1 micron, sa 0.5 arc sec
• Enters computation of overall Strehl ratio
  differently than higher order wavefront errors
• Lowers peak intensity of core by Fc-1 1 / 0.002
  500 x
• Halo is much less affected
• Peak intensity decreased by 0.93
• Halo diameter increased by 1.04

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Part Two Optimization of AO systems

• If you are designing a new AO system
• How many actuators?
• What kind of deformable mirror?
• What type of wavefront sensor?
• How fast a sampling rate and control bandwidth
  (peak capacity)?
• If you are using an existing AO system
• How long should you integrate on the wavefront
  sensor? How fast should the control loop run?
• Is it better to use a bright guide star far away,
  or a dimmer star close by?
• What wavelength should you use to observe?

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Issues for designer of AO systems

• Performance goals
• Sky coverage fraction, observing wavelength,
  degree of compensation needed for science program
• Parameters of the observatory
• Turbulence characteristics (mean and
  variability), telescope and instrument optical
  errors, availability of laser guide stars
• AO parameters chosen in the design phase
• Number of actuators, wavefront sensor type and
  sample rate, servo bandwidth, laser
  characteristics
• AO parameters adjusted by user integration time
  on wavefront sensor, wavelength, guide star mag.
  offset

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Factors affecting performance estimates
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Example Keck Observatory AO Blue Book

• Made scientific case for Keck adaptive optics
  system
• Laid out the technical tradeoffs
• Presented performance estimates for realistic
  conditions
• First draft of design requirements

The basis for obtaining funding commitment from
the user community and observatory
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What is in the Blue Book?

• Chapter titles
• 1. Introduction
• 2. Scientific Rationale and Objectives
• 3. Characteristics of Sky, Atmosphere, and
  Telescope
• 4. Limitations and Expected Performance of
  Adaptive Optics at Keck
• 5. Facility Design Requirements
• Appendices Technical details and overall error
  budget

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Other telescope projects have similar Books

• Keck Telescope (10 m)
• Had a Blue Book for the telescope concept
  itself
• CELT Telescope (30 m)
• Green Book
• Includes a chapter on adaptive optics

These Books are the kick-off point for work on
the Preliminary Design
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First, look at individual terms in error budget
one by one
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Dependence of Strehl on l and number of DM
degrees of freedom

• Assume bright natural guide star
• No meast error or iso-planatism or bandwidth
  error

Deformable mirror fitting error only
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Dependence of Strehl on l and number of DM
degrees of freedom

• Assume bright natural guide star
• No meast error or iso-planatism or bandwidth
  error

Deformable mirror fitting error only
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Strehl vs l and seeing (r0)

• Assume bright natural guide star
• No meast error or iso-planatism or bandwidth
  error

Decreasing fitting error
Deformable mirror fitting error only
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Strehl vs l and guide star magnitude (measurement
error)
Assumes no fitting error or other error terms
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Strehl vs l and guide star magnitude (measurement
error)
Assumes no fitting error or other error terms
bright star
Decreasing measurement error
dim star
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Strehl vs l and guide star angular separation
(anisoplanatism)
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Strehl vs l and guide star angular separation
(anisoplanatism)
0 arcsec
4 arcsec
10 arcsec
20 arcsec
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Strehl degradation due to anisoplanatism as a
function of l
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Sky coverage accounting for guide star densities
LGS coverage 80
Tip/tilt sensor magnitude limit
Hartmann sensor magnitude limit
Galactic latitude
NGS coverage 0.1
Isokinetic angle qk
Isoplanatic angle q0
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Point-spread function for different no. of DM
degrees of freedom
PSF, l2.2 mm, D/r0 8.5
1
218 DOF
50 DOF
0.1
24 DOF
12 DOF
2 DOF
0.01
Peak intensity relative to diffraction limit
uncorrected
0.001
0.0001
0
0.1
0.2
0.3
0.4
0.5
Radius (arcsec)
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Encircled energy curves for various l
2.2 m
1.65 m
1.25 m
uncorrected
0.88 m
0.7 m
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Background-limited spectroscopy optimal slit
width
As Strehl is increased (by increasing DOF), the
optimal slit width becomes smaller
Question give a qualitative explanation of why
this is so
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Overall system optimization
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Error model mean square wavefront error is sum
of squares of component errors

• Mean square error in wavefront phase

Measurement error
How to evaluate signal to noise ratio SNR?
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(Long) Digression on signal to noise ratio

• Based on a lecture by Bruce Macintosh and on Ian
  McLeans book Electronic Imaging in Astronomy
• Detector technology
• Basic detector concepts
• Modern detectors CCDs and IR arrays
• Signal-to-Noise Ratio (SNR)
• Introduction to noise sources
• Expressions for signal-to-noise
• Terminology is not standardized
• Two Keys 1) Write out what youre measuring.
  2) Be careful about units!
• Work directly in photo-electrons where possible

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References for detectors and signal to noise ratio

• Excerpt from Electronic imaging in astronomy,
  Ian. S. McLean (1997 Wiley/Praxis)
• Excerpt from Astronomy Methods, Hale Bradt
  (Cambridge University Press)
• Both are in the Reader and on the web

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Primary properties of detectors

• Quantum Efficiency QE Probability of detecting a
  single photon incident on the detector
• Spectral range (QE as a function of wavelength)
• Dark Current Detector signal in the absence of
  light
• Read noise Random variations in output signal
  when you read out a detector
• Gain G Conversion factor between internal
  voltages and computer Data Numbers DNs or
  Analog-to-Digital Units ADUs

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Secondary detector characteristics

• Pixel size (e.g. in microns)
• Total detector size (e.g. 1024 x 1024 pixels)
• Readout rate (in either frames per sec or pixels
  per sec)
• Well depth (the maximum number of photons that a
  pixel can record without saturating or going
  nonlinear)
• Cosmetic quality Uniformity of response across
  pixels, dead pixels
• Stability does the pixel response vary with time?

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Early detectors Eyes, photographic plates, and
photomultipliers

• Eyes
• Photographic plates
• very low QE (1-4)
• non-linear response
• very large areas, very small pixels (grains of
  silver compounds)
• hard to digitize
• Photomultiplier tubes
• low QE (10)
• no noise each photon produces cascade
• linear at low signal rates
• easily coupled to digital outputs

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Modern detectors are based on semiconductors

• Semiconductors have partially-filled valence
  band and empty conduction band energy levels
• Energetic photon can create an electron (and
  corresponding hole) and kick it into
  conduction band

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Bandgap energies for commonly used detectors
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Charge-Coupled-Devices, or CCDs

• CCD consists of an array of storage wells
  defining pixels
• Charge-shifting readout architecture allows large
  numbers of pixels to couple to single output
• Large arrays can be manufactured
• CCDs provided an order of magnitude increase in
  sensitivity relative to photmultipliers
• The most significant development in astronomy in
  the 1980s

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CCD readout process charge transfer

• Adjusting voltages on electrodes connects wells
  and allow charge to move
• Charge shuffles up columns of the CCD and then is
  read out along the top
• Charge on output amplifier (capacitor) produces
  voltage

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CCD phase space

• CCDs dominate inside and outside astronomy
• Even used for x-rays
• Large formats available (4096x4096)
• High quantum efficiency 80
• Dark current from thermal processes
• Long-exposure astronomy CCDs are cooled to reduce
  dark current
• Readout noise can be several electrons per pixel
  each time a CCD is read out
• Trade high readout speed vs added noise

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Main sources of detector noise for wavefront
sensors in common use

• Poisson noise or photon statistics
• Noise due to statistics of the detected photons
  themselves
• Read-noise
• Electronic noise (from amplifiers) each time CCD
  is read out
• Other noise sources (less important for wavefront
  sensors, but important for other imaging
  applications)
• Sky background
• Dark current

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Photon statistics Poisson distribution

• CCDs are sensitive enough that they care about
  individual photons
• Light is quantum in nature. There is a natural
  variability in how many photons will arrive in a
  specific time interval T , even when the average
  flux F (photons/sec) is fixed.
• We cant assume that in a given pixel, for two
  consecutive observations of length T, the same
  number of photons will be counted.
• The probability distribution for N photons to be
  counted in an observation time T is

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Properties of Poisson distribution

• Average value FT
• Standard deviation (FT)1/2
• Approaches a Gaussian distribution as N becomes
  large

Horizontal axis FT
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Properties of Poisson distribution
Horizontal axis FT
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Properties of Poisson distribution

• When is large, Poisson distribution
  approaches Gaussian
• Standard deviations of independent Poisson and
  Gaussian processes can be added in quadrature

Horizontal axis FT
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Look at all the various noise sources

• Wisest to calculate SNR in electrons rather than
  ADU or magnitudes
• Noise comes from Poisson noise in the object,
  Gaussian-like readout noise RN per pixel, Poisson
  noise in the sky background, and dark current
  noise D
• Readout noise
• where npix is the number of pixels and RN is the
  readout noise
• Photon noise
• Sky background for RSky e-/pix/sec from the sky,
  
• Dark current noise for dark current D
  (e-/pix/sec)

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Total signal to noise ratio
where F is the average photon flux, T is the
time interval of the measurement, RSky is the
electrons per pixel per sec from the sky
background, D is the electrons per pixel per sec
due to dark current, and RN is the readout noise
per pixel.
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Some special cases

• Poisson statistics If detector has very low read
  noise, sky background is low, dark current is
  low, SNR is
• Read-noise dominated If there are lots of
  photons but read noise is high, SNR is
• If you add multiple images, SNR ( Nimages
  )1/2

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Now, back to calculating measurement error for
Shack-Hartmann sensor

• Assume we are read-noise limited (usually the
  case with todays CCDs). Then

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Error model mean square wavefront error is sum
of squares of component errors

• where Tcontrol is the closed-loop control
  timescale, typically 10 times the integration
  time Tint

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Question

• For a fixed subaperture diameter d, approximately
  how does each term scale with r0?
• What are the implications for designing a new AO
  system to work at a shorter wavelength?

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Integration time trades temporal error against
measurement error

• Modified from Hardy, Figure 9.23

Measurement error r0 0.1 m
Temporal error 1 / t0 39 Hz
Optimum integration time
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First exercise in optimization

• Minimize the sum of read-noise and temporal
  errors by finding optimal integration time
• Sanity check optimum Tint larger for long t0,
  large read noise RN, and lower photon flux F

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Similarly, subaperture size d trades fitting
error against measurement error

• Hardy, Figure 9.25
• Smaller d
• better fitting error, worse measurement error

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Keck AO error budget (as of a couple of years
ago)
Assumptions NGS is 10 arcsec off-axis NGS is
very bright 30 degree zenith angle Closed-loop
bandwidth 90Hz Chose to have relatively large
fitting error, in order to be able to use fainter
guide stars (lower measurement error) Note that
uncorrectable errors in telescope itself are
significant
Just an example
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Summary What can you really optimize?
Once telescope is built on a particular site,
dont have control over t0, ?0 , r0 But when you
build your AO system, you CAN optimize choice of
subaperture size d , maximum speed of AO system,
field of view, etc. Even when you are observing
with an existing AO system, you can optimize
things wavelength of observations (changes
fitting error) integration time of wavefront
sensor Tint tip-tilt bandwidth brightness of
guide star
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Summary, part two

• Image motion
• Broadens core of AO PSF
• Contributes to Strehl degradation differently
  than high-order aberrations
• Crucial to correct tip-tilt