performance of adaptive optics systems...performance of adaptive optics systems don gavel ucsc...

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


Performance measures

•  Wavefront error

•  Strehl Ratio


Strehl ratio

Lick 3m Telescope Keck 10m Telescope


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 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


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


The SNR-optimal slit-width transitions to λ/d when the Strehl gets > 0.1

Energy in a spectrograph slit

D.L.

unc


Additional measures

•  Field performance

•  PSF stability


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


•  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


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


F kd

(

) ( )

z x

DM

DM fitting error

Influence Function Spatial Frequency Response


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:


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


Wavefront measurement error

Spot-size factor (units: angle on the sky)

Control loop averaging factor

Reconstructor noise propagator


Isoplanatic error

h

Turbulent layer

Light from science object

Light from guide star

•  If the guide star is not the science object...

Isoplanatic angle:


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

19 Gavel, AO Performance, Aug. 2009

Laser guidestar specific errors

•  Cone effect

Z

h

e.g…. h=4 km, r0=10cm => d0=4.5m

Laser Guidestar at finite altitude


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


Laser guide star

Natural star


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)


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


Guide star magnitude, mv

Opt

imum

con

tolle

r ba

ndw

idth

, fc


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


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


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


Error Budget Summary – Key Terms in an Astronomical AO Error Budget


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


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


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


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 =============================================


============================================= 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 =============================================


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).


Lick AO System: performance statistics


Lick AO System: performance statistics 2001-2002


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.


39Gavel, AO Performance, Aug. 2009

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



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



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)



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.



Measured Point Spread Function

Variation in NGS PSF quality from the Lick AO system (all at 2 microns)

Ideal PSF



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.



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"




σ CrB µ Cas ι Cas

70 Oph γ Del WDS 00310+2809

Lick NGS Data

12" 5"

1"

9"

7" 0.5"-7"



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




•  γ Del (sep = 9.22 arcseconds) – ratio = 0.76 ± 0.04 θo= 20.1" ± 2.1"




•  70 Oph (sep = 4.79 arcseconds) – ratio = 0.84 ± 0.04 θo = 14.3" ± 2.5"



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!





•  Analysis Techniques - Iterative Blind (myopic) deconvolution (Christou-CfAO) - Parametric Blind Deconvolution (PSF Modelling) (Drummond-AFRL)

•  Astrometry and Photometry (on following pages)




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.



•  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



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



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%



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 …?



Lick NGS Strehl Stability (10000 frames 22-57ms/frame)

Christou (UCSC), Gladysz (NUI) CfAO Summer School on Adaptive Optics


•  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



PDF Models

x = S x = 100 / (100 - S) x = ln(100 - S) CfAO Summer School on Adaptive Optics


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


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



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


63

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

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

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


performance of adaptive optics systems...performance of adaptive optics systems don gavel ucsc...

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