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Performance of Adaptive Optics Systems
Don Gavel UCSC
Center for Adaptive Optics Summer School
August, 2009
CfAO Summer School on Adaptive Optics 1 Gavel, AO Performance, Aug. 2009
Outline
• Performance Measures
• Design / error budgeting
• AO error contributors
• AO system simulation
• Performance analysis
• Examples from real systems
CfAO Summer School on Adaptive Optics 2 Gavel, AO Performance, Aug. 2009
Performance measures
• Wavefront error
• Strehl Ratio
CfAO Summer School on Adaptive Optics 3 Gavel, AO Performance, Aug. 2009
Strehl ratio
Lick 3m Telescope Keck 10m Telescope
CfAO Summer School on Adaptive Optics 4 Gavel, AO Performance, Aug. 2009
The Strehl is related to the wavefront variance through Marechal’s approximation
€
S =PSF 0,0( )
PSF 0,0( )˜ ϕ = 0
≅ exp −σ ˜ ϕ 2{ }
• Valid approximation for small
300 dof
80 dof
Stre
hl R
atio
• Extended region of validity for AO-corrected wavefronts
• Caveat on using the Marechal approximation (next slide)
CfAO Summer School on Adaptive Optics 5 Gavel, AO Performance, Aug. 2009
CfAO Summer School on Adaptive Optics Gavel, AO Performance, Aug. 2009 6
Marechal’s condition
• If wavefront phase is contained within confocal spheres λ/2 apart everywhere where the intensity is significant
• The waves will add up at the focus • Consequence of Fermat’s principle
Δx < λ/2
wavefront surface
focus
Resolution
The Rayleigh criterion: in a diffraction-limited optical system, two point sources are separately distinguishable at a separation ~λ/d
In AO systems with a Strehl >∼0.15, the FWHM of the corrected image is ~λ/d
F. Rodier introduced the concept of “Strehl-resolution” = width you have to enclose to get the same energy as in the FHWM of the ideal PSF
CfAO Summer School on Adaptive Optics 7 Gavel, AO Performance, Aug. 2009
Image contrast
• Contrast = ratio of halo to core “surface” brightness
• Integration time required to detect a faint object in the halo is proportional to (contrast)-2
Keck AO example at λ=2µ
Distance from the primary star, arcseconds
Con
tras
t Rat
io
uncorrected
CfAO Summer School on Adaptive Optics 8 Gavel, AO Performance, Aug. 2009
The SNR-optimal slit-width transitions to λ/d when the Strehl gets > 0.1
Energy in a spectrograph slit
D.L.
unc
CfAO Summer School on Adaptive Optics 9 Gavel, AO Performance, Aug. 2009
Additional measures
• Field performance
• PSF stability
CfAO Summer School on Adaptive Optics 10 Gavel, AO Performance, Aug. 2009
7 layer model atmosphere with r0 = 15.6 cm and θ0 = 3.1 arcsec
DM at 0 km DMs at 0,10 km
DMs at 0,5,10 km
Field Performance of Multi-conjugate AO
CfAO Summer School on Adaptive Optics 11 Gavel, AO Performance, Aug. 2009
• Fitting error (DM)
• Control error (sample rate)
• Measurement error (Hartmann sensor)
• Isoplanatic error (field angle)
• Calibration error
• Laser guide-star specfic errors: cone effect, guide-star elongation
AO system error contributors
To some approximation, we can add these terms in quadrature
CfAO Summer School on Adaptive Optics 12 Gavel, AO Performance, Aug. 2009
DM fitting error
€
σDM2 = Sϕ k( )FDM kd( )d2k
P∫∫
The DM corrects the wavefront up to a spatial frequency of 1/(actuator spacing)
Example spatial filtering function
d
Kolmogorov turbulence
CfAO Summer School on Adaptive Optics 13 Gavel, AO Performance, Aug. 2009
F kd
(
) ( )
z x
DM
DM fitting error
Influence Function Spatial Frequency Response
CfAO Summer School on Adaptive Optics 14 Gavel, AO Performance, Aug. 2009
The fitting error coefficient, µ, depends on the type of deformable mirror
Segmented mirror
Square segment, µ=0.174
Hexagonal segment, µ=0.116
d
d
Continuous face sheet DM: µ=0.3
• Segmented mirrors requre 3 (piston, tip, tilt) actuators per segment
• Rewriting the fitting error in terms of number of actuators, Na shows its more economical to use a continuous mirror:
CfAO Summer School on Adaptive Optics 15 Gavel, AO Performance, Aug. 2009
Control bandwidth error
Example temporal filtering function
The control loop corrects the wavefront up to a temporal frequency of
“Greenwood frequency” - depends on wind velocity, r0, etc., but simply defined here as the control frequency where the bandwidth term=1 radian2
CfAO Summer School on Adaptive Optics 16 Gavel, AO Performance, Aug. 2009
Wavefront measurement error
Spot-size factor (units: angle on the sky)
Control loop averaging factor
Reconstructor noise propagator
CfAO Summer School on Adaptive Optics 17 Gavel, AO Performance, Aug. 2009
Isoplanatic error
h
Turbulent layer
Light from science object
Light from guide star
• If the guide star is not the science object...
Isoplanatic angle:
CfAO Summer School on Adaptive Optics 18 Gavel, AO Performance, Aug. 2009
DM at 0 km DMs at 0,10 km
DMs at 0,5,10 km
Anisoplanatic error can be controlled by MCAO
Residual error is the “generalized anisoplanatism” = (θ/θm )5/3
(Tokovinin&LeLouarn, 2000)
CfAO Summer School on Adaptive Optics
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Laser guidestar specific errors
• Cone effect
Z
h
e.g…. h=4 km, r0=10cm => d0=4.5m
Laser Guidestar at finite altitude
CfAO Summer School on Adaptive Optics 20 Gavel, AO Performance, Aug. 2009
The laser guide star has a larger apparent size than a natural star
• The wavefront measurement error is increased accordingly
Lick laser data, from Nov. 1999
CfAO Summer School on Adaptive Optics 21 Gavel, AO Performance, Aug. 2009
Laser guide star
Natural star
CfAO Summer School on Adaptive Optics 22 Gavel, AO Performance, Aug. 2009
Optimizing the error budget
• In the design, select d (subaperture size =~ DM actuator spacing) to trade between DM fitting term and measurement term. This will set the NGS limiting magnitude, or “sky coverage”. It will also set the “optimized wavelength” of the AO system: λ:r0(λ)=d.
• For a laser guide star system, trade measurement error for laser power. Select the optimum d for the predicted LGS brightness. Brighter lasers (and more actuators) get to shorter wavelengths.
• On-line tuning:
• Select a frame rate that will best trade off measurement and bandwidth terms
• Select a natural guide star to trade off brightness (measuement error) for field angle (isoplanatic error)
CfAO Summer School on Adaptive Optics 23 Gavel, AO Performance, Aug. 2009
Contoller bandwidth, fc
Sub
aper
ture
siz
e, d
∂ σ ∂ ϕ d = 0
increasing brightness
Simultaneous Solution
Gui
de s
tar m
agni
tude
, mv
Subaperture size, d
40
50
60
70
30
Rms wavefront error, nm
80
90
Optimizing the error budget
CfAO Summer School on Adaptive Optics 24 Gavel, AO Performance, Aug. 2009
Guide star magnitude, mv
Opt
imum
con
tolle
r ba
ndw
idth
, fc
CfAO Summer School on Adaptive Optics 25 Gavel, AO Performance, Aug. 2009
Simulating an AO system
• Heirarchy of modeling
• Scaling laws • “Analytic” models (usually working in transform space) • Monte-carlo wave-optic simulation
• Tools: • Kolmogorov screen generator
• Wavefront propagation code • DM model, WFS model • Imaging model
CfAO Summer School on Adaptive Optics 26 Gavel, AO Performance, Aug. 2009
Monte-carlo Simulation of an AO system
Generate a guide star
Near-field propagation
Generate a phase screen, add to wavefront’s phase
Continue to propagate
Generate another phase screen, add to wavefront’s phase...
Telescope Multiply by the aperture function
Deformable Mirror Subtract the DM’s phase
Wavefront Sensor Run through the WFS model Controller
Run through the controller model
Actuators
Apply the DM actuator response model
Science Camera
Image residual wavefront
wind
CfAO Summer School on Adaptive Optics 27 Gavel, AO Performance, Aug. 2009
Gathering performance data on a real AO system
• Telemetry:
• Wavefront sensor data (slopes, intensities) -> controller’s rejection curve, bandwidth error term, measurement error term
• DM actuator commands -> simutaneous r0
• Image data: • Open loop -> r0 • Closed loop -> Strehl
CfAO Summer School on Adaptive Optics 28 Gavel, AO Performance, Aug. 2009
Error Budget Summary – Key Terms in an Astronomical AO Error Budget
CfAO Summer School on Adaptive Optics 29 Gavel, AO Performance, Aug. 2009
Lick AO System: On-line Performance Analysis
• The spreadsheet errorbudget.xls can help diagnose the sources of Strel loss and aid with on-line AO system parameter adjustments
• Other on-line metrics at the operator interface, based on AO system telemetry data analysis:
• Seeing r0 • Wind velocity • Temporal power spectrum of turbulence • Control loop rejection curves
k-8/3 spectrum
wind clearing time scale
noise floor
CfAO Summer School on Adaptive Optics 30 Gavel, AO Performance, Aug. 2009
Lick AO Telemetry Data Analysis Pipeline
Hartmann slopes
Actuator voltages
Subaperture intensities
Raw
Hartmann images
Average over illuminated subaps
Frame rate
Determine guidestar intensity
Verify proper background subtraction & photometry
Determine SNR
Generate phase spectra
Frame rate Generate controller rejection curve
Fit effective loop gain
Derive Hartmann spot size
Electronic loop gain
Determine wavefront measurement error
Control params
Generate tilt spectra Account for tilt in phase spectra
Account for sensor noise in phase spectra
Calculate Greenwood frequency
Calculate integrated temporal power rejection
Compare to Greenwood model
Open-loop images
Pre-calibrate rms actuator voltage to micron ratio
Calculate rms phase correction by DM
Determine r0 from rms phase correction
Calculate fitting error
σSNR
σBW
σDM
Control matrix
Actuator spacing
Compute the
compensator function
Determine sensor noise
Measure Hartmann spot size of internal source
Compute noise averaging factor
CfAO Summer School on Adaptive Optics 31 Gavel, AO Performance, Aug. 2009
noise, in WFS units
atmosphere ++
Wavefront sensor
H K
To science image
+ Hol(f)
KTelemtery post-analysis
Reconstructed phase residual-estimates
atmosphere +
equivalent noise, in phase units
+ H K
To science image
+ Hol(f)HK = I
≡Wavefront sensor / reconstructor
Reconstructed phase residual-estimates
Modeling the effect of noise in closed loop
CfAO Summer School on Adaptive Optics 32 Gavel, AO Performance, Aug. 2009
Correcting the closed loop residual phase spectrum for the effects of noise
+++ HOL
φ
φDM
e n
Closed-loop transfer function: low-pass “Correction” transfer function: high-pass
⇒
€
Se = S ˆ e − Hcor2− Hcl
2[ ]CfAO Summer School on Adaptive Optics 33 Gavel, AO Performance, Aug. 2009
============================================= Lick 3m error budget /duck5/lickdata/sep00/lgs6data/sep08/cent_07 Saturday 09/09/00 23:03:44 PDT --------------------------------------------- Fitting Error (sigmaDM) 117.827 nm d = 42.8571 cm r0Hv = 13.6763 cm --------------------------------------------- Servo Error (sigma_BW) 85.8510 nm fc = 45.9980 Hz fgHv = 28.5525 Hz fs = 500 Hz --------------------------------------------- Measurement Error (sigma2phase) 81.9109 nm SNR = 45.7691 control loop averaging factor = 0.452526 spotSizeFactor = 0.882759 arcsec --------------------------------------------- TOTAL: 167.221 nm =============================================
CfAO Summer School on Adaptive Optics 34 Gavel, AO Performance, Aug. 2009
============================================= Lick 3m error budget /duck5/lickdata/may00/lgs6/may21/cent_03 5/22/00, 5:09 UT --------------------------------------------- Fitting Error (sigmaDM) 122.912 nm d = 42.8571 cm r0Hv = 13.0001 cm --------------------------------------------- Servo Error (sigma_BW) 174.682 nm fc = 30.5027 Hz fgHv = 40.2416 Hz fs = 500.000 Hz --------------------------------------------- Measurement Error (sigma2phase) 15.2976 nm SNR = 100.543 control loop averaging factor = 0.257468 spotSizeFactor = 1.23077 arcsec --------------------------------------------- TOTAL: 214.138 nm =============================================
CfAO Summer School on Adaptive Optics 35 Gavel, AO Performance, Aug. 2009
Lick seeing statistics
D. Gavel, E. Gates, C. Max, S. Olivier, B. Bauman, D. Pennington, B. Macintosh, J. Patience, C. Brown, P. Danforth, R. Hurd, S. Severson, J. Lloyd, Recent Science and Engineering Results with the Laser Guidestar Adaptive Optics System at Lick Observatory, Proc SPIE, 4839, pp. 354-359 (2003).
CfAO Summer School on Adaptive Optics 36 Gavel, AO Performance, Aug. 2009
Lick AO System: performance statistics
CfAO Summer School on Adaptive Optics 37 Gavel, AO Performance, Aug. 2009
Lick AO System: performance statistics 2001-2002
CfAO Summer School on Adaptive Optics 38 Gavel, AO Performance, Aug. 2009
Adaptive Optics Performance
How to measure it from focal plane images?• Conventional approach is using the Strehl Ratio.
where both are normalised to the same volume• Exactly how best to measure Strehl is currently being
investigated.
This depends upon generating the perfect PSF; the presence of additive noise (detector and photon); image plane sampling; the effects of incorrect bias subtraction and flat-fielding, finding the actual peak-location etc.
CfAO Summer School on Adaptive Optics
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Measuring Image Quality
• Other Approaches besides Strehl Ratio • Image Sharpness (originally described by Muller and Buffington,
1974) S1 - Size of PSF
S3 - Normalised peak value – directly related to Strehl Ratio
Advantage – independent of knowing peak location and value. - Can be applied to extended sources.
Disadvantage – The numerator is contaminated by an additive noise term ≈ n2.
Disadvantage – sensitive to measurement of peak location and value. Advantage – No noise bias
CfAO Summer School on Adaptive Optics
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1. Palomar pupil geometry: primary mirror diameter of 4.88m and a central obscuration of 1.8m. No secondary supports modelled.
2. H-band (1.65 microns) with different levels of AO correction.
Synthetic Data
Ideal PSF
CfAO Summer School on Adaptive Optics
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Adaptive Optics Performance - Sharpness• Sharpness criteria compared with residual wavefront error from the simulations.
• S1 has a steeper slope for smaller rms phases.
S1 – -0.45 nm-1 S3 – -0.30 nm-1
(nm)
CfAO Summer School on Adaptive Optics
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Adaptive Optics Performance - SharpnessRelationship between S1, S3 and the Strehl Ratio.
S1 and S3 values generated from noise-free simulations as part of the CfAO Strehl study.
Both S1 and S3 are normalised to those of the ideal PSF.
The effect of constant noise is shown on S1.
CfAO Summer School on Adaptive Optics
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Measured Point Spread Function
Variation in NGS PSF quality from the Lick AO system (all at 2 microns)
Ideal PSF
CfAO Summer School on Adaptive Optics
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Adaptive Optics Performance - Sharpness
• Sharpness (normalised S1) compared with Strehl ratio for NGS Lick AO data.
• Data obtained with different SNR, observing conditions, nights.
• Dashed line obtained hueristically from the noiseless simulations. .
Departure from simulations could be due to either overestimating S1 (e.g. presence of noise) or underestimating Strehl ratio (not accurately locating the peak). Further analysis on noisy simulations needed.
Accuracy of system performance measurements can be obtained from SR and S1.
CfAO Summer School on Adaptive Optics
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Binary Star Measurements
• Science Targets - Basic Astronomy; stellar classification; stellar motion – orbits
• AO Performance - Isoplanatic Issues – on-axis vs. off-axis performance - Isoplanatic angle - θo
• Analysis Performance - Measurement of Photometry and Astrometry
• Lick Observatory Data - NGS - 0.5" ≤ Separations ≤ 12"
CfAO Summer School on Adaptive Optics
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Binary Star Measurements
σ CrB µ Cas ι Cas
70 Oph γ Del WDS 00310+2809
Lick NGS Data
12" 5"
1"
9"
7" 0.5"-7"
CfAO Summer School on Adaptive Optics
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Anisoplanatism via Strehl Ratio
• Binary stars permit direct measurement of anisoplanatism by comparing the PSFs.
• An effective measure of anisoplanatism is the fall off of the Strehl ratio of the off-axis source compared to the on-axis source.
where θ is the binary separation
CfAO Summer School on Adaptive Optics
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Anisoplanatism via Strehl Ratio
• γ Del (sep = 9.22 arcseconds) – ratio = 0.76 ± 0.04 θo= 20.1" ± 2.1"
CfAO Summer School on Adaptive Optics
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Anisoplanatism via Strehl Ratio
• 70 Oph (sep = 4.79 arcseconds) – ratio = 0.84 ± 0.04 θo = 14.3" ± 2.5"
CfAO Summer School on Adaptive Optics
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Summary of Binary Strehl Ratio Measurements
• Strehl ratio changes vary similarly for both components.
• Strehl ratio is quite variable for a set of observations (≈ seconds - minutes) up to changes of 20%.
• Differential Strehl ratio also varies – relative position on the detector?
• Isoplanatic angle (as determined from differential Strehl ratio) also varies with 15" ≤ θo ≤ 30" with some results implying minutes!
Anisoplanatism via Strehl Ratio
CfAO Summer School on Adaptive Optics
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Binary Star Measurements
• Analysis Techniques - Iterative Blind (myopic) deconvolution (Christou-CfAO) - Parametric Blind Deconvolution (PSF Modelling) (Drummond-AFRL)
• Astrometry and Photometry (on following pages)
CfAO Summer School on Adaptive Optics
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Binary Star Measurements
Summary of Astrometry and Photometry
• Astrometry between the two techniques shows good agreement (≈ 0.001")
• Differential Photometry is in general good agreement (≈ 0.02 mag) with a few exceptions.
- σ CrB (ΔJ = 0.5) - µ Cas (ΔJ = 0.4; ΔBrγ = 0.2) - ι Cas Aa (ΔJ = 0.2; ΔKs = 0.2) - ι Cas Ac (ΔH = 0.15)
Christou, J.C., Drummond, J.D., Measurements of Binary Stars, Including Two New Discoveries, with the Lick Observatory Adaptive Optics System, The Astronomical Journal, Volume 131, Issue 6, pp. 3100-3108.
CfAO Summer School on Adaptive Optics
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• Data sets obtained at Lick almost monthly between July 2005 and Feb 2006.
• IRCAL fastsub mode (“freeze” images) - texp = 22ms and 57ms - Duty cycle ~ texp + 30ms
• field size of 4.864 × 4.864 arcseconds (64 × 64 pixels) • Target objects: mv~ 6-8 • Typically 10 sets of data each of 1000 frames - 104 total frames
High Speed PSF Measurements
CfAO Summer School on Adaptive Optics
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Long Term PSF Stability Ideal PSF Fiber 1 (Sep-2005) Fiber 2 (Oct-2005) 12-Oct-2005
Reference Change
18-Aug-2005 18-Aug-2005 25-Jul-2005 25-Jul-2005
28-Aug-2004 29-Aug-2004 27-Aug-2004
CfAO Summer School on Adaptive Optics
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78%
20 Nov 2005
80%
13 Oct 2005
75%
17 Sep 2005Lick AO Fiber Source
• Stable structure in atmospheric-free PSF • Strehl Ratios typically 75% -- 82%
CfAO Summer School on Adaptive Optics
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PSF Structure • Fiber Source no better than ~ 80% Strehl ratio.
– What’s the best we can do - 90-95%? • Strong high-order Residual Aberration limiting performance.
– Relatively stable over minutes → hours → days → months → years! – No significant change with change of DM references
– Where is this from? • DM flatness • Unsensed aberrations in main path • Non-common path errors • Incorrect SH References • Obtain Wavefront map from Phase Retrieval/Diversity measurements.
– Typically the image is “sharpened” on the sky • Relative peak value metric - other metrics e.g. S1 • First 10 Zernike terms and increasing to 20.
– Use mirror modes?
• Important to understand for PSF Reconstruction algorithms. – We can deal with the atmosphere but can we deal with the system …?
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Lick NGS Strehl Stability (10000 frames 22-57ms/frame)
Christou (UCSC), Gladysz (NUI) CfAO Summer School on Adaptive Optics
58 Gavel, AO Performance, Aug. 2009
• Distribution of Strehl ratios (for relative stable performance) all show a similar non-gaussian behaviour. • Similar distributions seen in data from Palomar, Keck and AEOS
Strehl Ratio Distributions
CfAO Summer School on Adaptive Optics
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PDF Models
x = S x = 100 / (100 - S) x = ln(100 - S) CfAO Summer School on Adaptive Optics
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PDF Models
Implication is that the instantaneous Strehl ratio has an underlying Gaussian distribution: of r0 !
• Using Hudgin and Marachel approximations produces a distribution of Strehl ratios similar to that measured, i.e. skewed to a low Strehl ratio tail. • Need to obtain simultaneous r0 and S measurements.
• Speckle noise dominating. CfAO Summer School on Adaptive Optics
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PSF Calibration and Quantitative Analysis • The complicated nature of the AO PSF makes quantitative analysis
problematic.
– How well does deconvolution preserve astrometry and photometry?
i Cas
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Gavel, AO Performance, Aug. 2009
Separation of the components of σ CrB Sub-pixel peaks located by Fourier interpolation
o Six separate measurements of a binary star on different days on different positions on the IRCAL detector. o Separation depends upon location on detector o Precision for each location ~ 2 mas (= 0.03 pixels = 1.5% λ/D) o Separation dispersion ~ 50 mas
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Julian C. Christou, Austin Roorda, and David R. Williams, Deconvolution of adaptive optics retinal images, J. Opt. Soc. Am. A 21, 1393-1401 (2004) CfAO Summer School on Adaptive Optics 64 Gavel, AO Performance, Aug. 2009
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Deconvolution of final images, using data from the wavefront sensing
Object Noise PSF Image
Fourier Transform:
Then (in the Fourier domain):
0
= +
Then solve for object F … CfAO Summer School on Adaptive Optics 74 Gavel, AO Performance, Aug. 2009
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AO Performance Measurement
• AO performance-hitters (intro to error budget) • AO modeling and simulation • AO performance metrics
– Sharpness and anisoplanatism measures from the AO corrected science image
– Spectral analysis of telemetry from the AO system (wavefront sensor and deformable mirror signals)
• Astronomical AO data analysis • Vision science AO data analysis
Summary conclusion
CfAO Summer School on Adaptive Optics 86 Gavel, AO Performance, Aug. 2009