Transcript ao2004 4626

Tomographic algorithm for multiconjugate
adaptive optics systems
Donald Gavel
Center for Adaptive Optics
UC Santa Cruz
UCO Lick Observatory
NSF Center for Adaptive Optics
Laboratory for Adaptive Optics
Multi-conjugate AO Tomography
using Tokovinin’s Fourier domain approach1
Measurements from guide stars:
~
~
sk  f   M  f   f ,  k   ~k  f 
k  1, ngs
Problem as posed: Find a linear combination of guide
star data that best predicts the wavefront in a given
science direction, 
~ f ,   
N
T~
~
~  f ,  ~


g
s
f

g
s
 k
k
k 1
1Tokovinin,
A., Viard, E., “Limiting precision tomographic phase estimation,” JOSA-A,
18, 4, Apr. 2001, pp873-882.
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Least-squares solution


*
1
~
g f   Af    f I  ~
c f 
c~k f   M  Cn2 h  exp  2if θk  θ h dh
a~kk  f   M
2
c0
2
C
 n h exp  2if θk  θk hdh
c0
c0   Cn2 h  dh
 f   W f  W0 f c0 
A-posteriori error covariance:


W f   W0 f c0 1  cT A   I c *
1
W0 f   9.69  103 2   f 11/ 3
2
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Re-interpret the meaning of the c vector
Solution wavefront
T
1
~ f ,    c~   f  A   f   I   f  ~s  f 
  exp 2ihf  ~ jk  f C n2 h  exp 2i j hf ~
sk  f dh
N
N
j 1 k 1
c0
Filtered sensor data vector:
~  f 




~s  f   A  f   I   f  1 ~s  f 


The solution again, in the spatial domain and in terms of the filtered sensor data:
N
1
2
 x,     Cn h  sk x  h   k dh
c0
k 1
Define the volumetric
estimate of turbulence as
nx, h  
which is the sum of back

2c0
N
 s x  h C h 
k 1
k
k
2
n
projections of the filtered wavefront measurements.
The wavefront estimate in the science direction is then
  x,   
2

 nx  h , h dh
which is the forward propagation along the science direction through
the estimated turbulence volume.
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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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)
N
nest x, h  

z

 2

s
x

h
θ

k
k Cn h 
2c0 k 1  z  h



~s  f   A s  f   I   f  1 ~s  f 
zh

θk  θk  h dh
a~ksk  f    Cn2 h  exp  2if
z


c0
c0   Cn2 h dh
 f   W f  W0 f c0 
•
•
An approximate solution for finite apertures is obtained by mimicking the back
propagation implied by the infinite aperture solutions
~
skFA  f   ~
sk  f   ~
p f ; D
An approximate solution for finite aperture spherical waves (cone beams from laser
guide stars) is obtained by mimicking the spherical wave back propagations
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Spherical Wave Solution
Spatial domain
Forward propagation
Backward propagation
Turbulence at
position x at
altitude h
appears at position
 zh
x
  h
 z 
at the pupil
So back-propagate
position x in pupil
to position
 z 
x
  h
z

h


at altitude h
Frequency domain
Frequencies f at altitude h
scale down to frequencies
Frequencies f at the
pupil scale up to frequencies
zh
f

 z 
 z 
f

zh
at the pupil
at altitude h
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Another algorithm2 projects the volume
estimates onto a finite number of
deformable mirrors
~
d m  f    g~mDM  f , h n~ f , h dh
m  1,, nDM
1
~
~
g DM  f , h   A DM  f b DM  f , h 
~
bmDM
 J 0 2fH m  h 

~ DM  J 2fH  H 
a
mm 
0
m
m
2Tokovinin,
A., Le Louarn, M., Sarazin, M., “Isoplanatism in a multiconjugate adaptive optics
system,” JOSA-A, 17, 10, Oct. 2000, pp1819-1827.
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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MCAO tomography algorithm summary
k=angle of guidestar k
Guide star
angles k
Wavefront slope
measurements
from each
guidestar
sk x    x,k    k x 
x = position on pupil (spatial domain)
f = spatial frequency (frequency domain)
h = altitude
Hm = altitude of DM m
n~ f , h  
Along guidestar directions
~
~
sk  f     f ,k   ~k f 
a~kk   f  
Project
onto DMs
Filter
1
Cn2 h e 2if k k  h dh

c0


~s  f   A  f   I   f  1 ~s  f 
k


 N 2
 n x, h  
Cn h s x  h k 

2c0 k 1


Back-project
Convert slope to
phase (Poyneer’s
algorithm)
 N 2 ~
 Cn h s  f e2ifh
2c0 k 1
1
~
~
g DM  f , h   A DM  f b DM  f , h 
~
bmDM
 J 0 2fH m  h 

DM
~
a  J 2fH  H 
mm 
0
m
m
DM conjugate
heights H m
Field of view 
~
d m  f    g~mDM  f , h n~ f , h dh
Actuator commands
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.
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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The MCAO reconstruction process
a pictoral representation of what’s happening
Measure light from
guidestars
1
BackProject* to volume
2
Combine onto DMs
Propagate light from
Science target
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
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For implementation purposes, combine steps
2 and 3 to create a reconstruction matrix
~
d m  f    g~mDM  f , h n~  f , h dh

  A DM
1
k
 f   Cn2 h b DM  f , h e2ifh dh  ~sk f 
k
~
d  f   P DM  f ~s  f 
~
1
DM
 f   P

d
f  A  f   Iv  f  ~
s f 


  
M vector
of DM
commands
M  K projector
matrix
K  K filter
matrix
K vector of
WFS data
A simple approximation, or clarifying example: assume atmospheric layers
(Cn2) occur only at the DM conjugate altitudes.
g~m  f , h    h  H m 

~
d m  f    Cn2 H m e 2ifHmk ~
sk f 
k
Weighted by Cn2
Filtered measurements from
guide star k
Shifted during back projection
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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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) KM 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
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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
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Plane wave
Finite aperture
Infinite aperture
129 nm rms
155 nm rms
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Spherical Wave
Finite Aperture
Infinite Aperture
388 nm rms
421 nm rms
155 nm rms
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Movie
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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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)
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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