Adaptive Optics Modeling and Simulation - … Optics Modeling and Simulation. 2009 CfAO Summer School ... LAOS Matlab Wave Optics Contact: Luc Gilles Cibola Matlab Rytov cfao.ucolick.org/software/cibola.php

2009 CfAO Summer School

Dr. Matthew Britton

Adaptive Optics Modelingand Simulation


Lecture Topics

Introductory remarks on adaptive optics

Wave optics simulations

Rytov theory

Turbulence monitoring

PSF estimation

Differential Astrometry


Introductory Remarks


Wave Propagation throughTurbulence


The Adaptive Optics Backend


Wave Optics Simulations


Atmospheric TurbulenceWavefront

Phase

1.25 µm PSF

1.65 µm PSF

2.2 µm PSF


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


Turbulence Power Spectra


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.


Phase Screen Dynamics


Split-Step Wave PropagationThrough Turbulence

PropagateAdd phase errors

PropagateAdd phase errors

Add phase errors

WavefrontAmplitude

WavefrontPhase


An Example: Anisoplanatism

Differential wavefront phase vs.field location

Propagation Geometry


An Example: Scintillation

5 meters


Pupilscircle.g if circle.g if

Annulus SpideredAnnulus Tiled Hexagons


Pupil and Focal Plane Imagery inthe Absence of Turbulence

circle.g if circle.g if


Shack-Hartmann WavefrontSensors


Shack-Hartmann WavefrontSensors Imaging Turbulence


Deformable Mirrors

The simplest modelconsists of pyramidactuator influencefunctions

More realistic models forinfluence functions may bepursued using the plateequation.


Deformable Mirror SurfaceSimulation


Wavefront Reconstruction

X and Y CentroidsActuatorCommands

FriedGeometry

[ ]CentroidsCommands

Actuator+!="

#

$%&

'

[ ] !"

#$%

&'=Commands

ActuatorCentroids


The Resulting DM Surface

Distorted wavefrontentering the ShackHartmann sensor

The correction assensed by the ShackHartmann,reconstructed andplaced onto the DM


An AO SimulationAberrated Phase TTM Surface DM Surface

Residual Phase PSF DL PSF


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


Adaptive Optics PSF for A 30mSegmented Telescope

Diffraction Limited PSF AO Compensated PSF


Misregistration Tolerance


Control Law Analysis


Compute Time Requirementson a 2.4 GHz Pentium


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


Rytov Theroy and TurbulenceMonitoring


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!!


Aperture Averaged Rytov Expressionsfor Plane Waves

)(

2sin

2cos

)()(2

2

22

2

0

2

0 !!

!

!!"

# rrrF

k

z

k

z

fdzCdzk

o

o

n

L

a

a

$$$

%%%%%

&

'

(((((

)

*

++,

-../

0

++,

-../

0

1=%%&

'

(()

*

Sasiela 2007207379.=!


Multi-Aperture Scintillation Sensor


MASS Integrands in the RytovApproximation


MASS Weighting Functions

! "=

L

nzzCdz

0

22 )()(#


The Inverse Problem

!!!!!!!!

"

#

$$$$$$$$

%

&

=

!!!!!!!!!!!!!!!!!!

"

#

$$$$$$$$$$$$$$$$$$

%

&

dzC

dzC

dzC

dzC

dzC

dzC

n

n

n

n

n

n

CD

BD

BC

AD

AC

AB

D

C

B

A

)km5(.

)km1(

)km2(

)km4(

)km8(

)km16(

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

Ã

'

'

'

'

'

'

'

'

'

'


The DIMM/MASS Instrument

Pupil plane mask

DIMMapertures Kornilov et al., MNRAS 2007


DIMM/MASS Results

http://odata1.palomar.caltech.edu/massdimm/


Seeing and r0 Variability


Seeing and r0 Statistics


Pupil-Dependent Rytov Expressionsfor Plane Waves

There are many circumstances in which we would like toemploy expressions that are not aperture averaged.

[ ]22121 )()(),( rrrrDrrrr

!!! "=

( )!"#

$%&

+'= (( ),(2

1exp srsDsdrOTF

rrrrr)

[ ]21)(rr

!


The Two-Point Correlation Functionfor Plane Waves

( )

( )

!!

"

!!

#

$

%%%%

&

'

((((

)

*

+

,+,

!-

!./

,%%

&

'

((

)

*,+

%%

&

'

((

)

*+

0= 1

D

zG

D

z

D

rr

D

z

D

rG

D

z

D

rG

zzCdDkrr

abab

abab

nab

22

22

33

rrrr

rrrr

rr

222

2222

)()(

2

3/5

21

21

11

2

0

3/52

21

rr

)(ra

r! )(r

b

r!

458986.=!

( ) ( ) ( )( )21

2

21,,,,,, rrzCDkFrr

abnab

rrrrr!"## =

NOTE:


Phase Structure Function forUncompensated Turbulence

[ ] [ ] [ ]221

2

2

2

121 )()(2)()(),( rrrrrrDrrrrrr

!!!!! "+=

( ) ( )zzCdrrkn

2

0

3/5

21

2

!"

#$=rr

( )3/5

2121 ),( !!

"

#$$%

& '(=

or

rrrrD

rrrr

)

!"

#=

$)(2 223/83/5zCdzkr

noo88388.6=!


Tilt Removed Phase Variance on anAperture

( ) ( )[ ]2rrT rr!! "


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


PSF Estimation


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:


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


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


Long Exposure PSF of a Perfect AOSystem


Experimental ValidationA 21” binary




Differential Tilt Jitter

K+!"

#$%

&'(

)*+

,=

!!"

#

$$%

&

-1

367.2

23/1

2

2

2

||

DD

.µ

/

/



[ ] [ ]!"#

$%&

'('(

='

dddddPd

T

d

1

2

1exp

det2

1)(

)

5d

r

2d

r1d

r

4d

r

3d

r


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


Optimal Estimation

WWd

T!=!"

Wd=!


Improvement With Number ofReference Stars


Three Observational Epochs of M5


Scaling Laws With TelescopeDiameter


Lecture Summary


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.

Documents

Adaptive Optics Modeling and Simulation - … Optics Modeling and Simulation. 2009 CfAO Summer School ... LAOS Matlab Wave Optics Contact: Luc Gilles Cibola Matlab Rytov cfao.ucolick.org/software/cibola.php