performance of adaptive optics systems s. severson, j. lloyd, recent science and engineering results...
TRANSCRIPT
Performance of Adaptive Optics
Systems
Don Gavel
UCSC
Center for Adaptive Optics
Summer School
August, 2008
Outline
• Performance Measures
• The construction of error budgets
• AO error contributors
• AO system simulation
• Gathering performance data on real AO systems
• Performance results: Lick AO
The Strehl is related to the wavefront variance
through Marechal’s approximation
SPSF
PSF Nx e d x
D x
P
0 0
0 0
1
0
0
1 2 2
2
,
,
exp
~
~
• Valid approximation for small
~
300 dof
80 dof
Exp - 2~
~2 2 (radians)
Str
ehl R
atio
• Extended region of validity
for AO-corrected wavefronts
Resolution
The Rayleigh criterion: in a diffraction-limited optical system, two
point sources are separately distinguishable at a separation ~l/d
In AO systems with a Strehl
>~0.15, the FWHM of the
corrected image is ~l/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 l=2m
Distance from the primary star, arcseconds
Co
ntr
as
t R
ati
o
The SNR-optimal slit-width transitions to l/d
when the Strehl gets > 0.1
Energy in a spectrograph slit
D.L.
unc
7 layer model atmosphere with r0 =
15.6 cm and 0 = 3.1 arcsec
Optimize on disk
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
~2 2 2 2 2 2 2 DM BW SNR iso cal cone
To some approximation, we can add these terms in quadrature
DM fitting error
DM DM
P
S k F kd d k2 2
S k r k 0 023 0
5 3 11 3.
The DM corrects the wavefront up to
a spatial frequency of 1/(actuator
spacing)
Example spatial filtering function
d
Kolmogorov turbulence
mDM d r2
0
5 3
F kd( )
kd
The fitting error coefficient, m, depends on the
type of deformable mirror
Segmented mirror
Square segment, m=0.174
Hexagonal segment, m=0.116
d
d
Continuous face sheet DM: m=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:
mDM Na aD N r2
0
5 3
mNa
0 355
0 339
0 221
.
.
.
square
hex
contin
Control bandwidth error
Example temporal filtering function
f fc
F f fc
BW cS f F f f df2
S f r v f 2 6 0
5 3 8 3. /
BW g cf f25 3
f f
f f
c
c
2
21
The control loop corrects the wavefront up to a temporal
frequency of f fc s 10
“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
l
SNRx SNR
2
2 0
1d
I d d
I d
x y x y
y y
( , )
( , )
Spot-size factor
(units: angle on the sky)
Control loop
averaging factor
Reconstructor
noise propagator
SNR SNR SNRx y
2 2 2
Isoplanatic error
h
00
r
h
iso
2
0
5 3
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” = (/m )5/3
(Tokovinin&LeLouarn, 2000)
Laser guidestar specific errors
• Cone effect
Z
h
dZ
hr0 02
e.g…. h=4 km, r0=10cm > d0=4.5m
Laser Guidestar at finite altitude
cone d d2
0
5 3
The laser guide star has a larger apparent size than a
natural star
• The wavefront measurement error is increased accordingly
Spot size (arcsec)
DL (d=25cm) 0.4
star (r0=11cm) 0.94
LGS 2.16
Laser
StarStar
StarStar Laser
LGS
Star
Radius, arcsec
Enci
rcle
d e
ner
gy
Lick laser data, from Nov. 1999
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: l:r0(l)=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
Su
ba
pe
rture
siz
e,
d
increasing
brightness
Simultaneous
Solution
Gu
ide
sta
r m
agn
itu
de, m
vSubaperture size, d
Rms wavefront error, nm
Optimizing the error budget
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...
Multiply by the aperture function
Subtract the DM’s phase
Run through the WFS model
Run through the controller model
Apply the DM actuator response
model
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
r0 (Fried seeing parameter) Histogram
0
10
20
30
40
50
0 5 10 15 20 25 30
r0, cm
# o
f o
cc
ura
nc
es
median r0 = 10 cm
Seasonal Variation of Seeing
02468
101214161820222426
1 2 3 4 5 6 7 8 9 10 11 12
Month
r0,
cm
(2000-2002)
Histogram of Wind Speeds
0
50
100
150
200
0 5 10 15 20 25 30 35 40 45 50
Wind Speed, m/sec
# o
f o
ccu
ran
ces
Greenwood Frequency Histogram
0
5
10
15
20
0 5 10 15 20 25
fg, Hz
# o
f o
ccu
ran
ces
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).
Performance vs Guide Star Brightness
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
10 100 1000 10000
Brightness, ph/subap/ms
Str
eh
l BrG
Ks
H
Performance vs Greenwood Frequency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 10 20 30 40
fg, Hz
Str
eh
l
BrG
Ks
Ks-dim
H
Performance vs Seeing
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 5 10 15 20 25
r0, cm
Str
eh
l
BrG
Ks
Ks-dim
H
Lick AO System: performance statistics
LGS Performance
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 5 10 15 20 25
r0
Str
eh
l Ks Nov01
BrG Oct00
Ks Oct00
Strehl Histogram BrG Filter
0
1
2
3
4
5
6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Strehl
# o
f o
ccu
ran
ces
BrG
Strehl Histogram Ks Filter
0
1
2
3
4
5
6
7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Strehl
# o
f o
ccu
ran
ces
Ks-Dim
Ks
Lick AO System: performance statistics 2001-2002
Lick AO System: On-line Performance Analysis
Fill in the seeing and other system parameters in the green boxes and read the Strehl in the blue box
Lick Error Budget r0 0.15 m
[email protected] nm Strehl@lambdaObs v-wind 10 m/s
counts 100 photo-electrons Fitting 2.405180443 210.538 0.70 tau0 0.015 s
read noise 6 electrons Bandwidth 0.838952777 73.43791 0.96 fg 9 Hz
spot FWHM 2 arcsec SNR 1.223671374 107.1143 0.91 mu 1
spot sigma 1.442695 arcsec Calibration 1.583226455 138.5881 0.85 d 0.43 m
pixel size 2 arcsec Aniso 0 0 1.00 fs 100 Hz
crosstalk 0.2 arcsec Strehl 283.5481 0.52 fc 10 Hz
centroider quad FWHM open loop 0.75630429 arcsec at lambdaObs lambda 550 nm
SNR 6.401844 note: need to load math package in Excel (erf) to connect this to calculations belowObserved 0.35 lambdaObs 2200 nm
theta 0 arcsec Unaccounted 2.510864718 219.7891 0.67 calibration 0.85 Strehl (BrG)
Quad Cell SNR 4.125565 theta0 6 sec
• 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
spectraAccount 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
correctionCalculate fitting
error
SNR
BW
DM
Control
matrix
Actuator spacing
Compute the
compensator
function
Determine
sensor noise
Measure Hartmann spot
size of internal source
Compute noise
averaging
factor
Correcting the closed loop residual phase
spectrum for the effects of noise
f
fDM
en
e
e H H ncor cl f
H fH f
H fcl
ol
ol
1
H f H fH f
cor cl
ol
11
1
Closed-loop transfer function: low-pass “Correction” transfer function: high-pass
e H ncor f
f n 0
S H S H S
S H S H S
e cor cl n
e cl cl n
2 2
2 2
f
f
S S H He e cor cl
2 2
=============================================
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
=============================================
• 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
2
2
1
i
i
h
hS
pk
3
ih
hS
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.
)(
)(
3
3pkpk
pS
hS
p
p
h
hSR
ii
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.
Ideal PSF
• 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)
Relationship 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.
• 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.
• Science Targets- Basic Astronomy; stellar classification; stellar motion – orbits
• AO Performance - Isoplanatic Issues – on-axis vs. off-axis performance
- Isoplanatic angle - o
• Analysis Performance
- Measurement of Photometry and Astrometry
• Lick Observatory Data- NGS
- 0.5" Separations 12"
• 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 is the binary separation
oaxis-on
axis-off exp
SR
SR
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" o 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.
- CrB (J = 0.5)
- m Cas (J = 0.4; Br = 0.2)
- Cas Aa (J = 0.2; Ks = 0.2)
- Cas Ac (H = 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.
Astronomical AO System Data
Analysis
Julian Christou (UCSC)
Szymon Gladysz (NUI)
Gladysz, S., Christou, J., Redfern, M., Characterization of the Lick adaptive optics point
spread function, SPIE Proc., 6272, June, 2006
• 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 2005
Lick 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 …?
• 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
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.
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
AMOS 9-9-05
Separation of the components of 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% l/D)
o Separation dispersion ~ 50 mas
Julian C. Christou, Austin Roorda, and David R. Williams, Deconvolution of adaptive
optics retinal images, J. Opt. Soc. Am. A 21, 1393-1401 (2004)
xxxxxx ndfhg
Deconvolution of final images, using data from the
wavefront sensing
Object NoisePSFImage
ffff NFHG Fourier Transform:
framemulti
*2
framemulti
* NHFHGH
Then (in the Fourier domain):
0
= +
Then solve for object F …
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, 2008