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This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.
Wavefront Reconstruction
Lisa A. PoyneerLawrence Livermore National Laboratory
Center for Adaptive Optics 2008 Summer SchoolUniversity of California, Santa Cruz
August 11, 2009
LLNL-PRES-405137
What is wavefront reconstruction?
• Most* wavefront sensors do not measure the wavefront phase directly, but instead measure the average derivative
• Most* wavefront correctors are used to conjugate that phase on the mirror’s surface
• We must reconstruct the phase from the WFS slopes, achieving the most accurate, lowest noise estimate possible in the least amount of computation
• We’ll start with just one sensor and the phase as a two-dimensional surface
2
* I’m ignoring direct phase sensors as well as curvature sensors. I’m also ignoring AO systems which operate without a WFS or that do not conjugate the phase
Method 1: zonal matrix reconstruction
• The slope vector contains x- and y-slopes for all valid subapertures in the pupil
• The phase vector contains all controllable actuators
3
s
φ
Mapping subapertures and actuators
4
Phase in pupil
Slo
pe
vect
or: b
oth
x- a
nd y
-slo
pes
Act
uato
r ve
ctor
s φ
Mapping subapertures and actuators
4
Phase in pupil
Slo
pe
vect
or: b
oth
x- a
nd y
-slo
pes
Act
uato
r ve
ctor
s φ
Mapping subapertures and actuators
4
Phase in pupil
Slo
pe
vect
or: b
oth
x- a
nd y
-slo
pes
Act
uato
r ve
ctor
s φ
Mapping subapertures and actuators
4
Phase in pupil
Slo
pe
vect
or: b
oth
x- a
nd y
-slo
pes
Act
uato
r ve
ctor
s φ
Mapping subapertures and actuators
4
Phase in pupil
Slo
pe
vect
or: b
oth
x- a
nd y
-slo
pes
Act
uato
r ve
ctor
s φ
Mapping subapertures and actuators
4
Phase in pupil
Slo
pe
vect
or: b
oth
x- a
nd y
-slo
pes
Act
uato
r ve
ctor
s φ
Method 1: zonal matrix reconstruction
• The slope vector contains x- and y-slopes for all valid subapertures in the pupil
• The phase vector contains all controllable actuators
• The basis set for reconstruction is the actuators• We model the WFS measurement process as
• With the matrix pseudo-inverse , the reconstruction is obtained by a matrix-vector multiplication
5
s = Wφ
s
φ
φ = Es
E = W+
Method 2: modal matrix reconstruction
• We first define a orthogonal modal basis set to represent the actuators
6
〈mk,ml〉 = 0, for k #= l
Examples of modal basis sets
7
• Actuators
Examples of modal basis sets
7
• Actuators
• Zernike modes
Examples of modal basis sets
7
• Actuators
• Zernike modes
• Fourier modes
Examples of modal basis sets
7
Method 2: modal matrix reconstruction
• We first define a orthogonal modal basis set to represent the actuators
• We can analyze the phase in terms of modal coefficients with the inner product
• The phase is synthesized from the modal coefficients as
• We can put the modes into rows or columns to produce modal analysis and synthesis matrices
8
ck = !!,mk"
! =
n!1!
k=0
ck
mk
!mk,mk".
M M−1
〈mk,ml〉 = 0, for k #= l
Modal reconstruction, continued
• Now the slope measurement process is
• And the modal reconstruction is
• With modes, we can think about weighting or manipulating them. For example, If we choose the Zernike modes for a basis set, we can easily remove piston, tip and tilt (or other Zernikes) by zeroing the correct coefficients before converting back to actuators, using matrix
9
s = WM!1
c
c = MW+s
E = M!1
GMW+
G
! = Es , where
Suppressing local waffle via matrix
• The regular error criterion, which produces SVD, is
• We add a term which penalizes certain actuator patterns
• Using the weighting matrix to penalize local waffle
10
Figures from D. T. Gavel, “Suppressing anomalous localized waffle behavior in least squares wavefront reconstruction,” Proc. SPIE 4839, pp. 972–980 (2002).
J = (s ! W!)T (s ! W!)
J = (s ! W!)T (s ! W!) + !T
V!
VGavel, Phase Reconstruction Page 9
Figure 3. Zoomed in view of other modes 15-18 (top row) and 33-36 (bottom row) in the SVD mode set.
Every mode shows waffle behavior.
Figure 4. The mode set resulting from the actuator-penalty method (20), ranked in reverse order of
magnitude of the singular value of the reconstruction matrix K (largest is upper left). Waffle behavior is
relegated to modes with the lowest singular values. Modes with high singular values look like low-order
Zernikes, with no waffle. Piston is the lowest non-zero singular value mode (by design – we added piston-
mode actuator penalty). The “zero-visiblity” (zero singular value) modes are associated with actuators
outside the aperture and so are last in this set.
Gavel, Phase Reconstruction Page 10
Figure 5. Zoomed in view of modes 1-4 (top row) and 19-22 (bottom row) from the actuator-penalty
method.
Figure 6. Zoomed in view of modes 15-18 (top row) and 33-36 (bottom row) from the actuator-penalty
method. Although high-order modes have high spatial frequencies, they have no localized waffle behavior.
a b
Figure 7. Long-exposure point-spread-functions from a closed loop simulation using a) a standard SVD
pseudo-inverse matrix, with modes rejected if the singular value falls below 0.15 times the maximum
singular value, and b) a reconstruction matrix generated using the actuator-penalty method with no modes
rejected. The deformable mirror consists of a circular aperture with 276 actuators arranged on a rectilinear
grid (within the circle inscribing a square 18x18 grid). The input wavefront presented to the controller was
flat (zero phase over the aperture for the duration of the simulation) while the Hartmann slope measurement
error was 1 radian per subaperture-width rms white noise, spatially and temporally uncorrelated. The
SVD modes New modes
Method 3: Fourier reconstruction
• The zonal perspective: the slopes and phase are spatial signals. If we describe the WFS process with a filter, we can simply inverse filter to obtain the phase
• The modal perspective: the Fourier modes (sines and cosines) form a basis set. The matrices and are simply the DFT matrices if we embed the aperture in a square grid
• How do we deal with this circular aperture/square grid problem? (What to do with slopes that equal zero?)
11
M M−1
True phase Reconstruction
Do nothing Edge CorrectionExtension
Essential to solve boundary problem
• Without slope management, region outside pupil will be forced flat, making phase estimate incorrect
• Two methods for fixing this: Extension and Edge Correction
12
True phase Reconstruction
Do nothing Edge CorrectionExtension
Essential to solve boundary problem
• Without slope management, region outside pupil will be forced flat, making phase estimate incorrect
• Two methods for fixing this: Extension and Edge Correction
12
True phase Reconstruction
Do nothing Edge CorrectionExtension
True phase Reconstruction
Do nothing Edge CorrectionExtension
Essential to solve boundary problem
• Without slope management, region outside pupil will be forced flat, making phase estimate incorrect
• Two methods for fixing this: Extension and Edge Correction
12
True phase Reconstruction
Do nothing Edge CorrectionExtension
True phase Reconstruction
Do nothing Edge CorrectionExtension
Fourier reconstruction, continued
• The spatial domain / frequency domain pair is
• Fourier modes are eigenfunctions of LSI systems - for each mode the filter is simply multiplication by a complex number
13
x[m, n] !" X[k, l]
+
+
+
ReconstructionWFS
Gwx[k, l]
Gwy[k, l]
Qx[k, l]
Qy[k, l]
![k, l]
Vx[k, l]
Vy[k, l]
![k, l]
• “Modified-Hudgin model” is accurate, but has less noise, no problems with waffle or local waffle
![m,n] ![m + 1, n]
![m + 1, n + 1]![m,n + 1]
![m + 0.5, n]
φ[m + 0.5, n + 1]
![m,n + 0.5] ![m + 1, n + 0.5]
![m,n] ![m + 1, n]
![m + 1, n + 1]![m,n + 1]
x[m,n]
y[m,n]
x[m,n]
y[m,n]
Modified-Hudgin modelFried model
Shack-Hartmann Shack-Hartmann
A better point-based model
14
x[m, n] = ![m + 1, n + 0.5] ! ![m, n + 0.5]
Gwx[k, l] = exp
!
j!l
N
" #
exp
!
j2!k
N
"
! 1
$
A complete Fourier Optics model
• Based on a continuous-time model of average derivative
• When sampled with correct grid spacing and converted into a filter, it becomes
15
x[m, n] =
! m+1
x=m
! n+1
y=n
"
d
dx!(x, y)
#
dx dy
Gwx[k, l] =N
2!l
!"
cos
#
2!l
N
$
! 1
%
sin
#
2!k
N
$
+
"
cos
#
2!k
N
$
! 1
%
sin
#
2!l
N
$&
+
jN
2!l
!"
cos
#
2!k
N
$
! 1
% "
cos
#
2!l
N
$
! 1
%
! sin
#
2!k
N
$
sin
#
2!l
N
$&
Other filters
Fourier Transform Reconstruction
164
WFS x-slopes
FFT-1
WFS y-slopes
FFT FFT
Desired phase (actuators)
Solve boundary problem
Complex-valued Fourier coefficients
! =G!
wxX + G!
wyY
|Gwx|2 + |Gwy|2Recon. filter
Other filters
Fourier Transform Reconstruction
164
WFS x-slopes
FFT-1
WFS y-slopes
FFT FFT
Desired phase (actuators)
Solve boundary problem
Complex-valued Fourier coefficients
! =G!
wxX + G!
wyY
|Gwx|2 + |Gwy|2
Recon. filter
Reconstruction: summary
• Matrix reconstruction is widely used and familiar• It allows easy measurement of arbitrary system alignments
and geometries (e.g. mis-matched WFS-actuator grids as in many vision AO systems)
• Many well-established mathematics techniques can be used to formulate more sophisticated control
• Fourier transform reconstruction is a computationally efficient method which uses the Fourier basis set
• Many well-established signal processing techniques can be used to formulate more sophisticated control
17
Brief break - any questions?
18
Fundamental design decisions
• The number of actuators in the pupil affects performance and system design
• Incorporation of statistical information about the signal and noise for better reconstruction performance
19
More actuators = a better fit20
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
Phase
Fitting a phase shape
More actuators = a better fit20
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
Phase
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=8
Fitting a phase shape
More actuators = a better fit20
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
Phase
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=8
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=12
Fitting a phase shape
More actuators = a better fit20
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
Phase
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=8
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=12
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=16
Fitting a phase shape
More actuators = a better fit20
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
Phase
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=8
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=12
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=16
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=24
Fitting a phase shape
More actuators = a better fit20
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
Phase
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=8
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=12
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=16
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=24
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=32
Fitting a phase shape
More actuators = a better fit20
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
Phase
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=8
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=12
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=16
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=24
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=32
Fitting a phase shape
0 10 20 30 40
Actuator (N=48)
-30
-20
-10
0
10
20
Ph
ase
(A
U)
PhaseN=48
Fitting a phase shape
More actuators = more noise
• More actuators require more WFS measurements in order to properly sample the phase
• These smaller subapertures receive less light each, resulting in more noisy measurements
• Where is the flux from the guide star, the number of electrons received per subaperture is
• Following Guyon, the intensity at a PSF location is
21
e !
Fd2
fAO
F
I !
N2
f2F
Shack-Hartmann noise halo22
0.1 1
Arcsec
1x10-7
1x10-6
1x10-5
1x10-4
1x10-3In
ten
sity
24324048
PSF intensity with system size, noise only
• The computational cost of implementing a full matrix-vector multiplication is prohibitive for systems with thousands of actuators: for actuators
• We could just rely on Moore’s Law...• Original Keck AO computer (~1997): 16 Intel i860 processors, 1.35 ms latency• Keck NextGen WC (~2007): 3 TigerSharc DSPs, 0.081 ms latency
• Determine an efficient algorithm to solve the matrix equation:
• Fourier reconstruction, which uses FFTs:
More actuators = more computation
23
O(n2) n
O(n log n) O(n)
O(n log n)
Fast reconstruction algorithms
• Pseudo open-loop minimum variance unbiased (MVU) formulation can be solved with several different techniques• Sparse matrix techniques [Ellerbroek, 2002]:• Conjugate gradient algorithm [Gilles, 2002] and variations by Vogel, Yang:
• Open-loop reconstructions• MV conjugate gradient with multi-grid [Gilles, 2003]: • Multi-grid for least-squares [Vogel , 2006]:
• Many other proposals• Fractal iterative preconditioning for MV (FRIM) [Béchet, 2006]: • Local reconstruction [MacMartin, 2003]: or • Sparse reconstruction [Shi, 2002]:• Fourier demodulation from WFS CCD spots [Ribak, 2006], [Glazer, 2007]:• Wavelet reconstruction [Hampton, 2008]:
24
O(n log n)
O(n3/2)
O(n)O(n)
O(n)O(n3/2) O(n4/3)
O(n2 log n)O(n log n)
O(n)
Actual FLOPs counts for algorithms
MGCG from Gilles, 2003. FTR from Poyneer, 2007.
25
10 100
N x N system size
0.01
0.1
1
10
100
1x103M
FL
OP
s p
er
tim
e s
tep
VMM
Computational costs - reconstruction
GPI
Keck
TMT PFI
Actual FLOPs counts for algorithms
MGCG from Gilles, 2003. FTR from Poyneer, 2007.
25
10 100
N x N system size
0.01
0.1
1
10
100
1x103M
FL
OP
s p
er
tim
e s
tep
VMM
Computational costs - reconstruction
GPI
Keck
TMT PFI
10 100
N x N system size
0.01
0.1
1
10
100
1x103M
FL
OP
s p
er
tim
e s
tep
VMMFTR
Computational costs - reconstruction
GPI
Keck
TMT PFI
Actual FLOPs counts for algorithms
MGCG from Gilles, 2003. FTR from Poyneer, 2007.
25
10 100
N x N system size
0.01
0.1
1
10
100
1x103M
FL
OP
s p
er
tim
e s
tep
VMM
Computational costs - reconstruction
GPI
Keck
TMT PFI
10 100
N x N system size
0.01
0.1
1
10
100
1x103M
FL
OP
s p
er
tim
e s
tep
VMMFTR
Computational costs - reconstruction
GPI
Keck
TMT PFI
10 100
N x N system size
0.01
0.1
1
10
100
1x103M
FL
OP
s p
er
tim
e s
tep
VMMFTRMGCG
Computational costs - reconstruction
GPI
Keck
TMT PFI
Advanced matrix techniques: Keck AO
• Instead of using the pseudo inverse, the Keck AO control matrix is generated using prior information in a Bayesian formulation
• The covariance matrix of Kolmogorov turbulence is• The relative noise matrix for the subapertures is • The control matrix is
• This is based on the open-loop statistics of the wavefront, but achieves good results in Keck AO closed-loop
26
C!
N
E = (WTN
!1W + !C
!1
! )!1W
TN
!1
See M. A. van Dam, D. L. Mignant, and B. A. Macintosh, “Performance of the Keck Observatory adaptive-optics system,” Appl. Opt. 43, 5458–5467 (2004).
Minimum Variance Unbiased model
• The slopes now have noise:• The error that we wish to minimize is set by the difference
between the phase (given viewing angle) and the actuator commands (given DM response):
• The best actuator commands are obtained from the slopes
• Where the control matrix is given by , where
27
s = W! + v
! = H!" ! Haa
a = Cs
C = FE
E = (WTP
!1
v W + P!1
! )!1W
TP
!1
v
F = (HTa BHa + NwN
Tw + kI)!1
HTa BH!
See B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. A 19, 1803–1816 (2002).
MVU details
• MVU incorporates statistical information and other knowledge:• the atmospheric covariance • the WFS noise covariance• the DM response• the target angle (to deal with anisoplanatism)• and error weighting matrix• a constraint matrix, based on DM and weighting • a regularization parameter
• This model is based on open-loop statistics. As such, to be used in closed-loop, the pseudo-open loop measurements are first generated from the slopes by adding in the DM shape
• The structure of MVU can be solved efficiently
28
P!
Pv
Ha
H!
B
Nw
k
Conjugate gradient (CG) methods
• CG and variants (FD-PCG, MG-PCG, etc) are fast ways to iteratively solve a large AO matrix equation, provided is SPD
• CG for AO uses the MVU model (or variants) to generate the AO equation. Instead of a matrix-multiplication, the unknown phase/actuator vector is solved for
• The fundamental concept of CG, whether used for multivariate optimization or the solution of linear systems, is that at each iteration you search/update your estimate in a conjugate (perpendicular) direction.
29
Ax = b
A
How CG works - simple example
30
x0
• We’ll use CG to solve the equation
• To produce our estimate , we “searched” along
How CG works - simple example
30
!
3 1
1 1
"
x =
#
8
2
$
x1 =
!
7/3
1
" !
4
0
"
x0
• We’ll use CG to solve the equation
• To produce our estimate , we “searched” along
How CG works - simple example
30
!
3 1
1 1
"
x =
#
8
2
$
x1 =
!
7/3
1
" !
4
0
"
x0
x1
• We’ll use CG to solve the equation
• To produce our estimate , we “searched” along
• The error of this estimate is
How CG works - simple example
30
!
3 1
1 1
"
x =
#
8
2
$
x1 =
!
7/3
1
" !
4
0
"
!
8
2
"
!
#
3 1
1 1
$ !
7/3
1
"
=
!
0
!4/3
"
x1
• We’ll use CG to solve the equation
• To produce our estimate , we “searched” along
• The error of this estimate is
• For the next iteration, we’ll “search” along
How CG works - simple example
30
!
3 1
1 1
"
x =
#
8
2
$
x1 =
!
7/3
1
" !
4
0
"
!
8
2
"
!
#
3 1
1 1
$ !
7/3
1
"
=
!
0
!4/3
"
!
0
!4/3
"
x1
• We’ll use CG to solve the equation
• To produce our estimate , we “searched” along
• The error of this estimate is
• For the next iteration, we’ll “search” along
How CG works - simple example
30
!
3 1
1 1
"
x =
#
8
2
$
x1 =
!
7/3
1
" !
4
0
"
!
8
2
"
!
#
3 1
1 1
$ !
7/3
1
"
=
!
0
!4/3
"
!
0
!4/3
"
x1
x2
Most AO CG algorithms are preconditioned
• Convergence of CG depends on condition number of matrix• Convergence can be sped up by preconditioning
• Preconditioning is accomplished with a matrix • If , then we solve • Performance now depends on condition not of , but of
• What might be a good ? How about the DFT matrix?
31
P
P!1
P = I P!1
AP!1
Px = P!1
b
A
P!1
AP
P
Using statistical information in FTR
• If we know nothing about signal and noise, the filter is
• If we know the signal power and noise power for all Fourier modes, we have the Wiener filter
• If you use Fourier modes in MVU, the answer is the same32
! =G!
wxX + G!
wyY
|Gwx|2 + |Gwy|2
σ2
φ !2
v
NSR =!2
v
!2
!
1
|Gwx|2 + |Gwy|2
! =
!
1
1 + NSR
"
G!
wxX + G!
wyY
|Gwx|2 + |Gwy|2
Gain and prediction filters
• The Wiener gain is implemented simply as a real-valued gain filter after reconstruction
• A real-valued gain filter, based on temporal optimization, not just NSR, is used in Optimized-gain Fourier Control
• Another easy filter to implement is a shift filter. A linear-phase complex exponential shifts the phase estimate
• This concept forms the basis of Predictive Fourier Control, where the Kalman filter to predict a multi-layer atmosphere uses shift filters
33
System design: summary
• More actuators means a better fit to the phase, but...• More actuators result in more noise (more scattered light)• More actuators require more (too much) computation
• Many computationally efficient methods for solving a matrix equation, which may involve statistical priors• MVU formulation and CG solution
• Fourier reconstruction filtering is also efficient, and deals with statistical priors via filtering
34
Brief break - any questions?
35
What happens in a real AO system?
• How do we obtain the control matrix?• How do we get the filters and use FTR?• How do we adjust for the response of the DM?
36
* Al Gore is not affiliated with the CfAO
Computer simulation Real-world AO system
Images from Time Magazine and Gemini Observatory
Starting from aaaaaaa
• The slopes are arrayed• The actuators are arrayed • A theoretical model of the WFS and DM are used to generate
the slopes that are measured when each actuator is poked
37
x0, y0, x1, y1, · · ·
!0, !1, !2, · · ·
ordering
s = Wφ
Starting from aaaaaaa
• The slopes are arrayed• The actuators are arrayed • A theoretical model of the WFS and DM are used to generate
the slopes that are measured when each actuator is poked
37
x0, y0, x1, y1, · · ·
!0, !1, !2, · · ·
ordering
• x-slopes go down-up, down-up• y-slopes go down-down, up-up
s = Wφ
Altair interaction matrix W
• “Poke” an actuator and “measure” the slopes
• Those slopes become a specific column in the interaction matrix
• Note that due to influence function model, only close-by subapertures measure a poke
38
W
1 column per actuator
Eac
h ro
w c
orre
spon
ds
to a
n x-
or
y- s
lop
e
Altair control matrix aaaaaaaaaaE
• Altair matrix is obtained via the modal method, with some modes suppressed
• Note that the control matrix is not particularly sparse
39
1 column per x- or y-slope
Eac
h ro
w c
orre
spon
ds
to
an a
ctua
tor
E = M!1
GMW+
Examples of non-matching systems
• Though many AO systems have matched subapertures and actuators, some do not. A control matrix can deal with this.
40
Sample Arm
M9 M8
MiC
M5
M3
!
!
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)*)+$,!/)!13>?/.$.!=>44>7$.!5*!2!.$+2/4$.!.$)63/1+/>0!>=!+#$!26#3>,2+/@/0A!4$0)!20.!4/A#+!)>-36$)!-)$.!=>3!/,2A/0A<!
!
!"0"+&1234563789:+;<;836+
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M1426$.!/0!=3>0+!>=!+#$!)2,14$!23,!=/5$3!6>44/,2+>3P<!F>+#!]^_)!23$!>1+/6244*!6>0U-A2+$.!7/+#!+#$!)-5U$6+_)!$*$!1-1/4J!
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20.!12+#!4$0A+#!>=!+#$!)2,14$!23,<!
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%&1+/D!20.!2!9::!26+-2+>3!^b^Y!]$=>3,254$!^/33>3!M]^CP!,20-=26+-3$.!5*!F>)+>0!^/63>,26#/0$)<!"#$!%&!)*)+$,!
-)$)!+#$!5/,>31#!]^!=>3!4>7'>3.$3!25$332+/>0!6>33$6+/>0!M.$=>6-)!20.!2)+/A,2+/),P!$4/,/02+/0A!+#$!0$$.!=>3!+3/24!4$0)$)!
+>!6>33$6+!/0./?/.-24!)-5U$6+_)!3$=326+/?$!$33>3)!-1!+>!c!`!./>1+$3)!.$=>6-)<!"#$!)2,$!,/33>3!/)!24)>!-)$.!+>!)#/=+!2D/244*!
+#$!=>6-)!>=!+#$!%&'&("!5$2,!+>!3$+/024!)+3-6+-3$)!>=!/0+$3$)+<!B/A-3$!C!)#>7)!5>+#!]^)!-)$.!/0!>-3!)*)+$,<!
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]^!M3/A#+P<!"#$!)#2.$.!6/364$)!.$1/6+!+#$!)/@$!>=!+#$!)-5U$6+_)!1-1/4!M26+-24!./2,$+$3!Q<aN!,,PJ!7#/6#!/)!,2A0/=/$.!+>!
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$=>4?6985@57A+
)37;+
!"#$%&#'&(!)*&+#,%&-./0&&-./0102/
UC Davis AO OCT
Figure from Zawadzki, SPIE 6429 (2007)
Examples of non-matching systems
• Though many AO systems have matched subapertures and actuators, some do not. A control matrix can deal with this.
40
Sample Arm
M9 M8
MiC
M5
M3
!
!
!"! #$%&'($)*+$,-+#&%./-*+
"#$!%&'&("!)*)+$,!-)$.! /0! +#/)!121$3!#2)!243$2.*!5$$0!.$)63/5$.! /0!.$+2/4! $4)$7#$3$! 89:;<!%!53/$=!>?$3?/$7!>=! +#$!
)*)+$,!/)!13>?/.$.!=>44>7$.!5*!2!.$+2/4$.!.$)63/1+/>0!>=!+#$!26#3>,2+/@/0A!4$0)!20.!4/A#+!)>-36$)!-)$.!=>3!/,2A/0A<!
!
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2! 6-)+>,'.$)/A0! )1$6+3>,$+$3! 244>7/0A! C! ,,! 2D/24! )6200/0A! .$1+#! /0! +#$! $*$<! B.'&("! .2+2! 26E-/)/+/>0! )>=+723$!
.$?$4>1$.!5*!F/>1+/A$0! G06<! #2)! 5$$0!-)$.! +>! 26E-/3$! 20.!./)142*!F')620)! /0! 3$24! +/,$!7/+#! +#$! 26E-/)/+/>0! )1$$.!>=!
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%&'&("!/,2A/0A<!G0!+#$!>1+/624!.$)/A0J!7$!-)$.!2!62)62.$!>=!=>624!+$4$)6>1$)!M63$2+$.!5*!12/3)!>=!)1#$3/624!,/33>3)P!+>!
13>.-6$!6>0U-A2+$!1420$)!>=! +#$!$*$!1-1/4!7/+#!244!V$*!>1+/624!6>,1>0$0+)J! /064-./0A!W!20.!X!)6200/0A!,/33>3)J! +7>!
72?$=3>0+!6>33$6+>3)!20.!2!S23+,200'Y#26V!MS'YP!72?$=3>0+!)$0)>3!M7#/6#!-)$)!+#$!&("!/,2A/0A!4/A#+!=>3!72?$=3>0+!
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M1426$.!/0!=3>0+!>=!+#$!)2,14$!23,!=/5$3!6>44/,2+>3P<!F>+#!]^_)!23$!>1+/6244*!6>0U-A2+$.!7/+#!+#$!)-5U$6+_)!$*$!1-1/4J!
W'X!)6200$3)!20.!+#$!S'Y!72?$=3>0+!)$0)>3<!"#$!3$=$3$06$!23,!>=!+#$!%&'&("!)*)+$,!/)!.$)/A0$.!+>!,2+6#!./)1$3)/>0!
20.!12+#!4$0A+#!>=!+#$!)2,14$!23,<!
!
G0!>-3!%&!)-5')*)+$,!7$!-)$.!+7>!72?$=3>0+!6>33$6+>3)[!2!`a!26+-2+>3!F/,>31#!]$=>3,254$!^/33>3!M]^9P!=3>,!
%&1+/D!20.!2!9::!26+-2+>3!^b^Y!]$=>3,254$!^/33>3!M]^CP!,20-=26+-3$.!5*!F>)+>0!^/63>,26#/0$)<!"#$!%&!)*)+$,!
-)$)!+#$!5/,>31#!]^!=>3!4>7'>3.$3!25$332+/>0!6>33$6+/>0!M.$=>6-)!20.!2)+/A,2+/),P!$4/,/02+/0A!+#$!0$$.!=>3!+3/24!4$0)$)!
+>!6>33$6+!/0./?/.-24!)-5U$6+_)!3$=326+/?$!$33>3)!-1!+>!c!`!./>1+$3)!.$=>6-)<!"#$!)2,$!,/33>3!/)!24)>!-)$.!+>!)#/=+!2D/244*!
+#$!=>6-)!>=!+#$!%&'&("!5$2,!+>!3$+/024!)+3-6+-3$)!>=!/0+$3$)+<!B/A-3$!C!)#>7)!5>+#!]^)!-)$.!/0!>-3!)*)+$,<!
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]^!M3/A#+P<!"#$!)#2.$.!6/364$)!.$1/6+!+#$!)/@$!>=!+#$!)-5U$6+_)!1-1/4!M26+-24!./2,$+$3!Q<aN!,,PJ!7#/6#!/)!,2A0/=/$.!+>!
H9K!,,!2+!+#$!F/,>31#!20.!,/0/=/$.!+>!H`<9!,,!2+!+#$!^b^Y<!!
$=>4?6985@57A+
)37;+
!"#$%&#'&(!)*&+#,%&-./0&&-./0102/
UC Davis AO OCT
Figure from Zawadzki, SPIE 6429 (2007)
Moving beyond on-axis, single conjugate
• Given multiple WFS measurements, the phase can be reconstructed either in layers or in a volume
41Figure from Tokovinin, 2001
A. Tokovinin et al.: Optimized modal tomography in adaptive optics 711
GS1GS2
s
c
Command
Signal
Telescope
WFS 1 WFS 2
DM2
DM1
!
Command matrix
M
Fig. 1. Scheme of an MCAO with modal control. Atmosphericlayers matching the conjugation altitudes of the 2 DMs areshown. Both the WFS signal vector s and the DM commandvector c are specified as the coe!cients of wavefront expansionon Zernike modes. For DM2, this expansion is defined on ameta-pupil of the diameter D+2!H2, where D is the telescopediameter, ! is the radius of the FOV, H2 is the conjugationaltitude of DM2.
some command matrix M to control the shape of the DMs.This shape is also specified in terms of Zernike modes, andfor this reason such an MCAO system can be called modal,as opposed to zonal systems where both WFS and DM sig-nals are specified as local parameters (wavefront slopes oractuator voltages).
Although Zernike modes are slightly sub-optimal forturbulence correction (Roddier 1999), the Zernike basisplays an important role in theoretical studies of AO andgives reasonable performance predictions. We extend thesestudies to 3-dimensional turbulence correction and presentan analitycal tool for fast performance estimates whichpermits to explore rapidly the vast space of system pa-rameters in search of best configurations. Further refinedanalysis of selected configurations must be done by Monte-Carlo simulation which will take into account many addi-tional details.
In an MCAO system the shape of DMs is driven insuch a way as to obtain the best possible correction in aclosed loop; the WFSs then measure the remaining resid-ual wave-front aberrations. Here we consider an open-loopMCAO system, where the WFSs measure the uncorrectedperturbations, not residual aberrations. The DMs correctthe turbulence in another (scientific) beam. As shown byEllerbroek (1994), this approximation can describe a real(closed-loop) AO system under certain assumptions. Weneglect the temporal aspects of MCAO operation by sup-posing that all measurements and corrections are done
instantaneously. Thus, our attention is focused on thespatial aspects of turbulence tomography and correction,which are indeed new and specific to MCAO.
The command matrix plays a role of a “magic” blackbox that receives the WFS signals and produces the re-constructed wave-fronts at the output. How should thecommand matrix be selected to obtain the best possiblecorrection? What is “the best possible”? How good is thecorrection finally? Which parameters of MCAO need to beoptimized, and how? These are the questions addressed inthe present work.
In Sect. 2 we briefly outline possible correction strate-gies and the method to estimate the first-order MCAOperformance based on modal covariances (mathematicaldetails of the derivation are given in Appendix B). Ournumerical code is presented in Sect. 3. In Sect. 4 the re-sults on tomography are given, and in Sect. 5 a low-orderMCAO system is considered. The conclusions are summa-rized in Sect. 6.
2. Principles of optimized modal tomography
2.1. Inverse control
The concept of modal MCAO was discussed in the paper ofRagazzoni et al. (1999, hereafter RMR99). It was shownthat the deformations of several DMs can be measured(and hence controlled) by measuring a certain number ofZernike modes on natural or laser GSs.
In a classical AO there is a one-to-one correspondencebetween the wavefront and DM modes, but in MCAO thisrelation is not so simple. As can be seen in Fig. 1, fora finite Field of View (FOV) of a radius θ, the diame-ter of the second DM (called meta-pupil) must be largerthan the telescope pupil D by at least 2θH2, where H2
is the conjugation height of the second DM. A beam ofsome GS illuminates only a portion of the meta-pupil: thebeam footprint diameter is smaller than D for an LGS(as shown in the figure) or equal to D for an NGS. Inaddition, the footprints are displaced from the center ofthe meta-pupil as soon as the GSs are not on-axis. Thewavefront is decomposed into Zernike modes on a smallcircle (footprint), and DM commands are decomposed ona larger circle (meta-pupil). The relation between thosetwo sets of modes is called mode projection, and is givenby a mode projection matrix. It is discussed in RMR99and in Appendix A.
Using mode projections, it is possible to express by amatrix the reaction of all WFSs to a given Zernike modeon a meta-pupil. This interaction matrix A relates the vec-tor of WFS signals s to the control signals (commands) c,as in any AO system, and its specific form (projections)is only a consequence of the modal representation of bothsignals that we choose. As noted in RMR99, the inter-action matrix can be inverted, to give a command ma-trix Minv:
s = Ac and c = Minvs, where Minv = A!1". (1)
Moving beyond on-axis, single conjugate
• This enables the AO system to use one or multiple mirrors to improve correction across a field of view
42
Strehl maps from MAD on-sky test of Omega Centauri.SCAO - 1 NGS, MCAO - 3 NGS.
Figure from ESO: 19d/07
New concepts and algorithms
• MAD: 3 NGSs, 2 DMs, one altitude conjugated. Matrix formulation to optimize a uniform Strehl across FoV.
• Proposed NFIRAOS for TMT: 6 LGS + NGS, 2 DMs, correction over a 1-2 arcmin FoV• candidate algorithm is FD-PCG in MCAO MVU formulation [Yang, 2006] and [Vogel
& Yang, 2006]
• LAO’s MCAO test bench• 3 simulated LGSs, 3 DMs, Fourier-domain tomography [Gavel, 2004]
• Ground layer AO: using one mirror and many WFSs, correct the common ground layer across a very wide field• ESO’s GRAAL: 4 LGS, 1 DM, 7.5 arcmin FoV in the NIR• ESO’s GALACSI: 4 LGS, 1 DM, 1 arcmin FoV in the visible
43
Reconstruction summary
• The WFS does not measure the phase. So we need to reconstruct it.
• This is usually done with a matrix, though other methods such as Fourier reconstruction also work
• While many actuators improve phase fitting, they require computationally efficient algorithms
• Both matrix and Fourier methods can incorporate statistical information to improve performance
• Covered the process of developing a control matrix of reconstruction filter for a specific AO system and DM.
44
(Non-exhaustive) bibliography
[1] C. Bechet, et al, “Frim: minimum-variance reconstructor with a fractal iterative method,” in “Advances in Adaptive Optics II,” , B. L. Ellerbroek and D. B. Calia, eds. (2006), Proc. SPIE 6272, p. 62722U.
[2] B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. A 19, 1803–1816 (2002).[3] K. Freischlad and C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete fourier transform,” J. Opt. Soc. Am. A 3, 1852–
1861 (1986).[4] D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).[5] D. T. Gavel, “Suppressing anomalous localized waffle behavior in least squares wavefront reconstruction,” in “Adaptive Optical System Technologies II,” , P. L.
Wizinowich and D. Bonaccini, eds. (2002), Proc. SPIE 4839, pp. 972–980.[6] D. T. Gavel, “Tomography for multiconjugate adaptive optics systems using laser guide stars,” in “Advancements in Adaptive Optics,” , D. B. Calia, B. L. Ellerbroek, and
R. Ragazzoni, eds. (2004), Proc. SPIE 5490, pp. 1356–1373.[7] L. Gilles, et al, “Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction,” J. Opt. Soc. Am. A 19, 1817–1822 (2002).[8] L. Gilles, “Order-N sparse minimum-variance open-loop reconstructor for extreme adaptive optics,” Opt. Lett. 28, 1927–1929 (2003).[9] O. Glazer, et al, “Adaptive optics implementation with a fourier reconstructor,” Appl. Opt. 46, 574–580 (2007).[10] J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).[11] R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).[12] D. G. MacMartin, “Local, hierarchic and iterative reconstructors for adaptive optics,” J. Opt. Soc. Am. A 20, 1084–1093 (2003).[13] L. A. Poyneer, et al, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A 24, 2645–2660 (2007).[14] L. A. Poyneer, et al, “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. A 19, 2100–2111 (2002).[15] L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wave-front control,” J. Opt. Soc. Am. A 22, 1515–1526 (2005).[16] E. N. Ribak, et al, “Full wave front reconstruction in the fourier domain,” in “Advances in Adaptive Optics II,” , B. L. Ellerbroek and D. B. Calia, eds. (2006), Proc. SPIE
6272, p. 627254.[17] F. Shi, et al, “Sparse-matrix wavefront reconstruction: simulations and experiments,” in “Adaptive Optical System Technologies II,” , P. L. Wizinowich and D. Bonaccini,
eds. (2002), Proc. SPIE 4839, pp. 981–988.[18] A. Tokovinin, et al, “Isoplanatism in a multiconjugate adaptive optics system,” J. Opt. Soc. Am. A 17, 1819–1827 (2000).[19] A. Tokovinin and E. Viard, “Limiting precision of tomographic phase estimation,” J. Opt. Soc. Am. A 18, 873–882 (2001).[20] M. A. van Dam, et al, “Performance of the Keck Observatory adaptive-optics system,” Appl. Opt. 43, 5458–5467 (2004).[21] C. R. Vogel and Q. Yang, “Multigrid algorithm for least-squares wavefront reconstruction,” Appl. Opt. 45, 705–715 (2006).[22] C. R. Vogel and Q. Yang, “Fast optimal wavefront reconstruction for multi-conjugate adaptive optics using the fourier domain preconditioned conjugate gradient
algorithm,” Opt. Exp. 14, 7487–7498 (2006).[23] Q. Yang, et al, “Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography,” Appl. Opt. 45, 5281–5293 (2006).
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