The W. M. Keck Observatory Optical Telescopes

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Fundamentals of adaptive optics and wavefront
reconstruction

• Marcos van Dam
• Institute for Geophysics and Planetary Physics,
  Lawrence Livermore National Laboratory

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Outline

• Introduction to adaptive optics
• Wavefront sensors
• Shack-Hartmann sensors
• Pyramid sensors
• Curvature sensors
• Wavefront reconstructors
• Least-squares
• Modal reconstructors
• Dynamic control problem

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Uranus and Titan
Courtesy De Pater Courtesy
Team Keck.
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Adaptive optics
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Effect of the wave-front slope

• A slope in the wave-front causes an incoming
  photon to be displaced by

?x
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Shack-Hartmann wave-front sensor

• 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.

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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

• 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.

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Quad cells

• 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
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Pyramid wave-front sensor

• 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.

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Curvature sensing
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Curvature sensing

• Practical implementation uses a variable
  curvature mirror (to obtain images below and
  above the aperture) and a single detector.

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Curvature sensing

• Using the irradiance transport equation,

Where I is the intensity, W is the
wave-front and z is the direction of propagation,
we obtain a linear, first-order approximation,
which is a Poisson equation with Neumann
boundary conditions.
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Curvature sensing

• Solution inside the boundary,

• Solution at the boundary,

I1 I2 I1- I2
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Curvature sensing

• As the propagation distance, z, increases,
• 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.

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Faint companions
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Wave-front reconstruction

• There is a linear relationship between wave-front
  derivative and a measurement.
• Dont want to know the wave-front derivative, but
  the wave-front or, better, the actuator commands.
• Need to know the relationship between actuator
  commands and measurement.

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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.

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System matrix

• The system matrix, H, describes how pushing each
  actuator, a, affects the centroid measurements, s
• .
• It is created by pushing one actuator at a time
  and measuring the change in centroids.

Centroids
Actuators
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System matrix

• Alternatively, the system matrix can be computed
  theoretically using finite differences to
  approximate the derivatives
• Another formulation is using Fourier transforms
  (faster than matrix multiplication).

d
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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.

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Reconstruction matrix

• We have the system matrix
• We need a reconstruction matrix to convert from
  centroid measurements into actuator voltages
• 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.
  
• Equivalently, use singular value decomposition.

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Reconstruction matrix

• Most modes have local waffle but no global
  waffle.
• Hence, must regularize before inverting.

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Reconstruction matrix

• Penalize waffle in the inversion, e.g., using the
  inverse covariance matrix of Kolmogorov
  turbulence, and a noise-to-signal parameter,
  (Bayesian reconstructor).

SVD Bayesian
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Reconstruction matrix

• Comparison of reconstruction matrices

SVD Bayesian
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Reconstruction matrix

• Comparison of reconstruction matrices

SVD Bayesian
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Reconstruction matrix
2. Only reconstruct certain modes, zi, (modal
reconstruction).
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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.

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Modeling the system dynamics

• Model the dynamic behavior of the AO system using
  the transfer function of each block.

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Modeling the system dynamics

• The turbulence rejection curve can be calculated
  from a model of the AO system.

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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.

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Laser guide stars

• 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.

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Laser guide stars

• The laser is equally deflected on the way up and
  down, so cant 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).

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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 W-7405-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!