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2009 CfAO Summer School
Dr. Matthew Britton
Adaptive Optics Modelingand Simulation
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2009 CfAO Summer School
Lecture Topics
Introductory remarks on adaptive optics
Wave optics simulations
Rytov theory
Turbulence monitoring
PSF estimation
Differential Astrometry
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2009 CfAO Summer School
Introductory Remarks
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2009 CfAO Summer School
Wave Propagation throughTurbulence
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2009 CfAO Summer School
The Adaptive Optics Backend
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2009 CfAO Summer School
Wave Optics Simulations
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2009 CfAO Summer School
Atmospheric TurbulenceWavefront
Phase
1.25 µm PSF
1.65 µm PSF
2.2 µm PSF
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2009 CfAO Summer School
Elements of a Wave OpticsSimulation
Generation of random phase screens to form aturbulence profile.
Wave propagation through turbulence. Simulation of the wavefront sensor Reconstruction and control Simulation of the deformable mirror Formation of the science image
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2009 CfAO Summer School
Turbulence Power Spectra
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2009 CfAO Summer School
Phase ScreensTwo dimensional phasescreens are generated in thespatial frequency domain byfiltering gaussian random noisewith the power spectrum.The resulting two dimensionalarrays are Fourier transformedto produce a physical phasescreen.Periodic or aperiodic phasescreens may be generated.The choice depends on theapplication.
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2009 CfAO Summer School
Phase Screen Dynamics
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2009 CfAO Summer School
Split-Step Wave PropagationThrough Turbulence
PropagateAdd phase errors
PropagateAdd phase errors
Add phase errors
WavefrontAmplitude
WavefrontPhase
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2009 CfAO Summer School
An Example: Anisoplanatism
Differential wavefront phase vs.field location
Propagation Geometry
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2009 CfAO Summer School
An Example: Scintillation
5 meters
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2009 CfAO Summer School
Pupilscircle.g if circle.g if
Annulus SpideredAnnulus Tiled Hexagons
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2009 CfAO Summer School
Pupil and Focal Plane Imagery inthe Absence of Turbulence
circle.g if circle.g if
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2009 CfAO Summer School
Shack-Hartmann WavefrontSensors
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2009 CfAO Summer School
Shack-Hartmann WavefrontSensors Imaging Turbulence
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2009 CfAO Summer School
Deformable Mirrors
The simplest modelconsists of pyramidactuator influencefunctions
More realistic models forinfluence functions may bepursued using the plateequation.
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2009 CfAO Summer School
Deformable Mirror SurfaceSimulation
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2009 CfAO Summer School
Wavefront Reconstruction
X and Y CentroidsActuatorCommands
FriedGeometry
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Actuator+!="
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ActuatorCentroids
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2009 CfAO Summer School
The Resulting DM Surface
Distorted wavefrontentering the ShackHartmann sensor
The correction assensed by the ShackHartmann,reconstructed andplaced onto the DM
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2009 CfAO Summer School
An AO SimulationAberrated Phase TTM Surface DM Surface
Residual Phase PSF DL PSF
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2009 CfAO Summer School
Simulation Parameters Turbulence profile and power spectra
Guide star/science target geometry
Diffractive/geometric propagation
Sensing and detection wavelengths
Phase screen sampling
Aperture diameter/geometry
Style of wavefront sensing and order of correction
Pupil / lenslet array / deformable mirror registration
Reconstruction and control algorithms
Focal plane resolution
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2009 CfAO Summer School
Adaptive Optics PSF for A 30mSegmented Telescope
Diffraction Limited PSF AO Compensated PSF
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2009 CfAO Summer School
Misregistration Tolerance
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2009 CfAO Summer School
Control Law Analysis
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2009 CfAO Summer School
Compute Time Requirementson a 2.4 GHz Pentium
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2009 CfAO Summer School
Parallelization of Wave OpticsSimulations
Memory and CPU resources Duration of simulation Number of guide stars and science targets Level of fidelity
Computational architecture Symmetric multiprocessor Cluster
Simulation Output Data load Results
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2009 CfAO Summer School
Rytov Theroy and TurbulenceMonitoring
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2009 CfAO Summer School
Rytov Approximation
The Rytov approximation providesexpressions for the two-point phase and logamplitude correlation function between raystraveling in different directions throughturbulence.
)()( 21 rrba
rr!!
)()( 21 rrba
rr!!
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2009 CfAO Summer School
Aperture Averaged Rytov Expressionsfor Plane Waves
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2009 CfAO Summer School
Multi-Aperture Scintillation Sensor
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2009 CfAO Summer School
MASS Integrands in the RytovApproximation
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2009 CfAO Summer School
MASS Weighting Functions
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2009 CfAO Summer School
The Inverse Problem
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2009 CfAO Summer School
The DIMM/MASS Instrument
Pupil plane mask
DIMMapertures Kornilov et al., MNRAS 2007
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2009 CfAO Summer School
DIMM/MASS Results
http://odata1.palomar.caltech.edu/massdimm/
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2009 CfAO Summer School
Seeing and r0 Variability
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2009 CfAO Summer School
Seeing and r0 Statistics
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2009 CfAO Summer School
Pupil-Dependent Rytov Expressionsfor Plane Waves
There are many circumstances in which we would like toemploy expressions that are not aperture averaged.
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2009 CfAO Summer School
The Two-Point Correlation Functionfor Plane Waves
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2009 CfAO Summer School
Phase Structure Function forUncompensated Turbulence
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2009 CfAO Summer School
Tilt Removed Phase Variance on anAperture
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2009 CfAO Summer School
Name Lang Methods Availability
CAOS IDL Wave Optics www-luan.unice.fr/caos/
ESO’s sim C++ Wave Optics Contact: Miska Le Louarn
Paola IDL Wave Optics cfao.ucolick.org/software/paola.php
YAO Yorick Wave Optics www.maumae.net/yao/
LAOS Matlab Wave Optics Contact: Luc Gilles
Cibola Matlab Rytov cfao.ucolick.org/software/cibola.php
Arroyo C++ Wave Optics andRytov
eraserhead.caltech.edu/arroyo.html
SciAO Scios Wave Optics sciao.sourceforge.net/
Readily Available Simulations
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2009 CfAO Summer School
PSF Estimation
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2009 CfAO Summer School
The Structure Function for NGS AO
)(ra
r! )(r
b
r!
rr
( ) ( ) ( ) ( )rrrraab
rrrr!!!"~
+#=
Residual phase
( ) ( ) ( )21
~2121
,,, rrDrrDrrD apl
rrrrrr
!" +=
Structure function of the residual phase
( )( ))()()()( 2211 rrrrabab
rrrr!!!! ""
May be computed using the 2 pt correlationfunction:
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2009 CfAO Summer School
The NGS AO Structure Function
( ) ( )
{ }3/521
3/5
21
3/5
21
3/5
0
223/8
21
22
2,
ababab
napl
zrrzrrrrz
zCdzkrrD
!!!rrrrrrrrr
rr
"""+"""+
#= $%
( ) ( ) ( )21
~2121
,,, rrDrrDrrD apl
rrrrrr
!" +=
A bonus! The NGSstructure function isstationary
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2009 CfAO Summer School
The NGS OTF and PSF
( )
!"#
$%&
+'!"#
$%&
'=
!"#
$%&
+'=
((
((
),(2
1exp)(
2
1exp
),(2
1exp
~ srsDsdrD
srsDsdrOTF
apl
rrrrr
rrrrr
)
*
Example OTF and PSF for a perfect AO system
AnisoplanaticTransferFunction
Guide StarOTF
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2009 CfAO Summer School
Long Exposure PSF of a Perfect AOSystem
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2009 CfAO Summer School
Experimental ValidationA 21” binary
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2009 CfAO Summer School
Differential Astrometry
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2009 CfAO Summer School
Differential Tilt Jitter
K+!"
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-1
367.2
23/1
2
2
2
||
DD
.µ
/
/
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2009 CfAO Summer School
Differential Astrometry
[ ] [ ]!"#
$%&
'('(
='
dddddPd
T
d
1
2
1exp
det2
1)(
)
5d
r
2d
r1d
r
4d
r
3d
r
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2009 CfAO Summer School
Grid AstrometryWe would like to establish the location of the target starwith respect to the grid of reference stars that lie in thefield of view. To do so, we would like to average over themeasured separation vectors between the target andreference stars. The question is how to do so in a waythat minimizes the uncertainty in the resultingmeasurement.
1d
r
2d
r
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2009 CfAO Summer School
Optimal Estimation
WWd
T!=!"
Wd=!
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2009 CfAO Summer School
Improvement With Number ofReference Stars
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2009 CfAO Summer School
Three Observational Epochs of M5
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2009 CfAO Summer School
Scaling Laws With TelescopeDiameter
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2009 CfAO Summer School
Lecture Summary
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2009 CfAO Summer School
Topical Review Wave optics simulations and their role in the design
and tolerancing of adaptive optics systems. Use of the Rytov theory in estimation of the
turbulence profile from variance measurements. Use of the Rytov theory in modeling variances,
structure functions and optical transfer functions onthe aperture.
Modeling anisoplanatic errors for PSF estimation anddifferential astrometry.