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NSF Center for Adaptive OpticsUCO Lick Observatory Laboratory for Adaptive Optics

Tomographic algorithm for multiconjugate adaptive optics systems

Donald GavelCenter for Adaptive Optics

UC Santa Cruz

IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 2

Multi-conjugate AO Tomographyusing Tokovinin’s Fourier domain approach1

fffMfs kkk ~,~~ gsnk ,1

sg ~~~,~,~1

TN

kkk fsfgf

1Tokovinin, A., Viard, E., “Limiting precision tomographic phase estimation,” JOSA-A, 18, 4, Apr. 2001, pp873-882.

Measurements from guide stars:

Problem as posed: Find a linear combination of guide star data that best predicts the wavefront in a given science direction,

IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 3

Least-squares solution

A-posteriori error covariance:

00

20

022

02

2exp~

2exp~

cWW

dhhCc

cdhhihCMa

cdhhihCMc

n

kknkk

knk

fff

θθff

θθff

*~~ 1 fcIffAfg

*1 100 cIAcff

TcWW

3/11230 21069.9 fW f

IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 5

Re-interpret the meaning of the c vector

01 1

2

1

~2exp~2exp

~~,~

cdhfshfihCfhfi

fsfffcfN

j

N

kkjnjk

T

IA

Filtered sensor data vector:

f

f

fff sIAs ~

~

~ 1

The solution again, in the spatial domain and in terms of the filtered sensor data:

dhhxshCc

xN

kkkn

1

2

0

1,

Define the volumetric estimate of turbulence as

N

knkk hChxs

chxn

1

2

02,

which is the sum of back projections of the filtered wavefront measurements.

The wavefront estimate in the science direction is then

dhhhxnx ,2

,

which is the forward propagation along the science direction through the estimated turbulence volume.

Solution wavefront

IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 6

The new interpretation allows us to extend the approach into useful domains

• Solution is independent of science direction (other than the final forward projection, which is accomplished by light waves in the MCAO optical system)

• The following is a least-squares solution for spherical waves (guidestars at finite altitude)

• An approximate solution for finite apertures is obtained by mimicking the back propagation implied by the infinite aperture solutions

• An approximate solution for finite aperture spherical waves (cone beams from laser guide stars) is obtained by mimicking the spherical wave back propagations

ffff s sIAs ~~ 1

Dfpfsfs kFA

k ;~~~

N

knkkest hCh

hz

zs

chn

1

2

02, θxx

00

20

02 2exp~

cWW

dhhCc

cdhhz

hzihCa

n

kknskk

fff

θθff

IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 7

Spherical Wave Solution

Turbulence at position x at altitude happears at position

at the pupil

hz

hzx

So back-propagateposition x in pupilto position

at altitude h

hhz

zx

Frequencies f at altitude hscale down to frequencies

at the pupil

z

hzf

Frequencies f at thepupil scale up to frequencies

at altitude h

hz

zf

Forward propagation Backward propagation

Spatial domain

Frequency domain

IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 8

Another algorithm2 projects the volume estimates onto a finite number of

deformable mirrors

2Tokovinin, A., Le Louarn, M., Sarazin, M., “Isoplanatism in a multiconjugate adaptive optics system,” JOSA-A, 17, 10, Oct. 2000, pp1819-1827.

dhhfnhfgfd DMmm ,~,~~

mm

DMmm

mDM

m

DMDMDM

HHfJa

hHfJb

hffhf

2~2

~,

~,~

0

0

1bAg

DMnm ,,1

IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 9

MCAO tomography algorithm summary

Wavefront slope measurements

from each guidestar

Filter

Back-projectAlong guidestar directions

Projectonto DMs

Actuator commands

xxxs kkk ,

Convert slope to phase (Poyneer’s

algorithm)

fffs kkk ~,~~

ffff sIAs ~~ 1

dhehCc

fa

hifn

kk

kk 22

0

1

~

N

kkn

N

k

ifhn

hxshCc

hxn

efshCc

hfn k

1

2

0

1

22

0

2,

~2

,~

dhhfnhfgfd DMmm ,~,~~

mm

DMmm

mDM

m

DMDMDM

HHfJa

hHfJb

hffhf

2~2

~,

~,~

0

0

1bAg

Guide star angles k

DM conjugate heights

Field of view

mH

References:Tokovinin, A., Viard, E., “Limiting precision tomographic phase estimation,” JOSA-A, 18, 4, Apr. 2001, pp873-882.Tokovinin, A., Le Louarn, M., Sarazin, M., “Isoplanatism in a multiconjugate adaptive optics system,” JOSA-A, 17, 10, Oct. 2000, pp1819-1827.Poyneer, L., Gavel, D., and Brase, J., “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,”

JOSA-A, 19, 10, October, 2002, pp2100-2111.Gavel, D., “Tomography for multiconjugate adaptive optics systems using laser guide stars,” work in progress.

k=angle of guidestar kx = position on pupil (spatial domain)f = spatial frequency (frequency domain)h = altitudeHm = altitude of DM m

IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 10

The MCAO reconstruction processa pictoral representation of what’s happening

Propagate light fromScience target

Measure light fromguidestars

Back-Project* to volume

Combine onto DMs

1 2 3 4

*after the all-important filtering step, which makes the back projections consistent with all the data

IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 11

For implementation purposes, combine steps 2 and 3 to create a reconstruction matrix

data WFSof vector

matrixfilter

1

matrixprojector

commands DMof vector

221

~~

~~

~,

,~,~~

KKKKM

DM

M

DM

kk

ifhDMn

DM

DMmm

ffvfff

fff

fsdhehfhCf

dhhfnhfgfd

k

sIAPd

sPd

bA

A simple approximation, or clarifying example: assume atmospheric layers (Cn

2) occur only at the DM conjugate altitudes.

k

kifH

mnm fseHCfd km ~~ 22

Filtered measurements from guide star kShifted during back projection

Weighted by Cn2

mm Hhhfg ,~

IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 12

It’s a “fast” algorithm

• The real-time part of the algorithm requires– O(N log(N))K computations to transform the guidestar measurements

– O(N) KM computations to filter and back-propagate to M DM’s

– O(N log(N))M computations to transform commands to the DM’s

– where N = number of samples on the aperture, K = number of guidestars, M = number of DMs.

• Two sets of filter matrices, A(f)+Iv(f) and PDM(f), must be pre-computed– One KxK for each of N spatial frequencies (to filter measurements)-- these

matrices depend on guide star configuration

– One MxK for each of N spatial frequencies (to compact volume to DMs)-- these matrices depend on DM conjugate altitudes and desired FOV

• Deformable mirror “commands”, dm(x) are actually the desired phase on the DM

– One needs to fit to DM response functions accordingly

– If the DM response functions can be represented as a spatial filter, simply divide by the filter in the frequency domain

IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 13

Simulations

• Parameters– D=30 m

– du = 20 cm

– 9 guidestars (8 in circle, one on axis)

– zLGS = 90 km

– Constellation of guidestars on 40 arcsecond radius

– r0 = 20 cm, CP Cn2 profile (7 layer)

– = 10 arcsec off axis (example science direction)

• Cases– Infinite aperture, plane wave

– Finite aperture, plane wave

– Infinite aperture, spherical wave

– Finite aperture, spherical wave (cone beam)

IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 14

Plane wave

129 nm rms 155 nm rms

Infinite aperture Finite aperture

IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 15

Spherical Wave

421 nm rms388 nm rms

155 nm rms

Infinite Aperture Finite Aperture

IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 17

Conclusions

• MCAO Fourier domain tomography analyses can be extended to spherical waves and finite apertures, and suggest practical real-time reconstructors

• Finite aperture algorithms “mimic” their infinite aperture equivalents

• Fourier domain reconstructors are fast– Useful for fast exploration of parameter space

– Could be good pre-conditioners for iterative methods – if they aren’t sufficiently accurate on their own

• Difficulties– Sampling 30m aperture finely enough (on my PC)

– Numerical singularity of filter matrices at some spatial frequencies

– Spherical wave tomographic error appears to be high in simulations, but this may be due to the numerics of rescaling/resampling (we’re working on this)

– Not clear how to extend the infinite aperture spherical wave solution to frequency domain covariance analysis (it mixes and thus cross-correlates different frequencies)