Transcript Presentation (PowerPoint File)

Fundamentals of adaptive optics
and wavefront reconstruction
Marcos van Dam
Institute for Geophysics and Planetary Physics,
Lawrence Livermore National Laboratory
Outline
1. Introduction to adaptive optics
2. Wavefront sensors
• Shack-Hartmann sensors
• Pyramid sensors
• Curvature sensors
3. Wavefront reconstructors
• Least-squares
• Modal reconstructors
4. Dynamic control problem
Uranus and Titan
Courtesy: De Pater
Courtesy: Team Keck.
Adaptive optics
Effect of the wave-front slope
• A slope in the wave-front causes an incoming photon
to be displaced by x  zWx
W(x)
z
x
Shack-Hartmann wave-front sensor
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The aperture is subdivided using a lenslet array.
Spots are formed underneath each lenslet.
The displacement of the spot is proportional to the
wave-front slope.
Shack-Hartmann wave-front sensor
• The centroid (center-of-mass) is proportional to the mean
slope across the subaperture.
• Centroid estimate diverges with increasing detector area
due to diffraction and with increasing pixels due to
measurement noise.
• Correlation or maximum-likelihood methods can be used.
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Quad cells
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Wave-front x- and y-slope measurements are usually
made in each subaperture using a quad cell (2 by 2).
Quad cells are faster to read and to compute the
centroid.
Quad cells
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These centroid is only linear with displacement over a
small region.
Centroid is proportional to spot size.
Centroid vs. displacement for different spot sizes
Centroid
Displacement
Pyramid wave-front sensor
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Similar to the Shack-Hartmann, it measures the
average slope over a subaperture.
The subdivision occurs at the image plane, not the
pupil plane.
Less affected by diffraction.
Curvature sensing
Image 2
-z
Aperture
Wave-front at aperture
z
Image 1
Curvature sensing
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Practical implementation uses a variable curvature
mirror (to obtain images below and above the
aperture) and a single detector.
Curvature sensing
•
Using the irradiance transport equation,
I
  I2W  I .W
z
Where I is the intensity, W is the wave-front and z is
the direction of propagation, we obtain a linear, firstorder approximation,
I 2  I1
I
2
 z W  zW .
I 2  I1
I
which is a Poisson equation with Neumann boundary
conditions.
Curvature sensing
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Solution inside the boundary,
I1  I 2
  z (Wxx  W yy )
I1  I 2
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Solution at the boundary,
I1  I 2 H ( x  R  zWx )  H ( x  R  zWx )

I1  I 2 H ( x  R  zWx )  H ( x  R  zWx )
I1
I2
I1- I2
Curvature sensing
As the propagation distance, z, increases,
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Sensitivity increases.
Spatial resolution decreases.
Diffraction effects increase.
The relationship between the signal, (I1- I2)/(I1+ I2)
and the curvature, Wxx + Wyy, becomes non-linear.
Faint companions
Wave-front reconstruction
• There is a linear relationship between wave-front derivative
and a measurement.
• Don’t want to know the wave-front derivative, but the wavefront or, better, the actuator commands.
• Need to know the relationship between actuator commands
and measurement.
Actuators: Shack-Hartmann
• The lenslets are usually located such that the actuators of
the deformable mirror are at the corners of the lenslets.
• Piston mode, where all the actuators are pushed up, is
invisible to the wave-front as there is no overall slope.
• Waffle mode, where the actuators are pushed up and down
in a checkerboard pattern, is also invisible.
System matrix
• The system matrix, H, describes how pushing each
actuator, a, affects the centroid measurements, s
s  Ha .
• It is created by pushing one actuator at a time and
measuring the change in centroids.
Centroids
Actuators
System matrix
• Alternatively, the system matrix can be computed
theoretically using finite differences to approximate the
derivatives:
six, j  six1, j ai 1, j  ai , j

2
d
d
siy, j  siy, j 1 ai , j 1  ai , j

2
d
• Another formulation is using Fourier transforms (faster
than matrix multiplication).
Actuators: Curvature
• Bimorph mirrors are usually used, which respond to an
applied voltage with a surface curvature.
• The electrodes have the same radial geometry as the
subapertures.
• Curvature sensors tend to be low order.
Reconstruction matrix
s  Ha
• We have the system matrix:
• We need a reconstruction matrix to convert from centroid
measurements into actuator voltages:
a  Rs
• Need to invert the 2N (centroids) by N (actuators) H matrix.
• For well-conditioned H matrices a least-squares algorithm
suffices: unsensed modes, such as overall piston, p, and
waffle, w, are thrown out.
R  ( H T H  pT p  wT w)1 H T
p  [1,1,1,1,1,1,...]
w  [1,1,1,1,1,...]
• Equivalently, use singular value decomposition.
Reconstruction matrix
• Most modes have local waffle but no global waffle.
• Hence, must regularize before inverting.
Reconstruction matrix
1. Penalize waffle in the inversion, e.g., using the inverse
covariance matrix of Kolmogorov turbulence, C and a
noise-to-signal parameter,  (Bayesian reconstructor).
1
R  ( H T H  C )1 H T
SVD
Bayesian
Reconstruction matrix
• Comparison of reconstruction matrices
SVD
Bayesian
Reconstruction matrix
• Comparison of reconstruction matrices
SVD
Bayesian
Reconstruction matrix
2. Only reconstruct certain modes, zi, (modal reconstruction).
1
R  Z [( HZ ) ( HZ )] ( HZ )
T
Z  [ z1 , z 2 , z3, ...]
T
Keck II AO + NIRSPEC (June/00)
Active galaxy
NGC6240:
Max et al.
Keck: K-band
1.6 arc sec
R = 12.5 NGS,
35” separation
2.00 m
2.08 m
2.17 m
HST: J,H,K composite
Faint nucleus
Gas in between
x
Bright nucleus

H2 emission line from shocked or ionized gas (v=580 km/s)
Control problem
• Wave-front sensing in adaptive optics is not only an
estimation problem, it is a control problem.
• There are inherent delays in the loop due to
• Integration time of the camera
• Computation delays
• The AO system should minimize bandwidth errors while
maintaining loop stability.
• The propagation of measurement noise through the loop
also needs to be minimized.
Modeling the system dynamics
• Model the dynamic behavior of the AO system using the
transfer function of each block.
Deformable mirror
Centroid measurements
Modeling the system dynamics
• The turbulence rejection curve can be calculated from a
model of the AO system.
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Rejection
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Frequency (Hz)
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Modeling the system dynamics
• We can calculate the bandwidth and noise terms from a
combination of data from the telescope and modeling the
system.
Bandwidth errors
Noise
Laser guide stars
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Shine a 589 nm 10-20 W laser in the direction of
the atmosphere.
Sodium atoms at an altitude of 90 km are excited
by this light and re-emit.
The return can be used as a guide star.
Laser guide stars
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The laser is equally deflected on the way up and
down, so can’t be used to measure tilt.
The guide star is not at infinity, so the focus is
different.
Hence, need a natural guide star as well (but can
be much fainter).
Acknowledgements
• This work was performed under the auspices of the US
Department of Energy by the University of California,
Lawrence Livermore National Laboratory, under contract W7405-Eng-48.
• The work has been supported by the National Science
Foundation Science and Technology Center for Adaptive
Optics, managed by the University of California at Santa
Cruz under cooperative agreement No. AST-9876783.
• W. M. Keck Observatory has supported
this work.
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Thank you!