Tomographic algorithm for multiconjugate adaptive optics systems
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Transcript of Tomographic algorithm for multiconjugate adaptive optics systems
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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
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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,
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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
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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
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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
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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
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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
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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
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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
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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 ,~
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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
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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)
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Plane wave
129 nm rms 155 nm rms
Infinite aperture Finite aperture
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Spherical Wave
421 nm rms388 nm rms
155 nm rms
Infinite Aperture Finite Aperture
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Movie
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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)