Wavefront Sensing I

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Wavefront Sensing I

• Richard Lane
• Department of Electrical and Computer Engineering
• University of Canterbury
• Christchurch
• New Zealand

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Location
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Astronomical Imaging Grouppast and present

• Dr Richard Lane
• Professor Peter Gough
• Associate Professor P. J. Bones
• Associate Professor Peter Cottrell
• Professor Richard Bates
• Dr Bonnie Law Dr Roy Irwan
• Dr Rachel Johnston Dr Marcos van Dam
• Dr Valerie Leung Richard Clare
• Yong Chew Judy Mohr

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Contents

• Session 1 Principles
• Session 2 Performances
• Session 3 Wavefront Reconstruction for 3D

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Principles of wavefront sensing

• Introduction
• Closed against open loop wavefront sensing
• Nonlinear wavefront sensing
• Shack-Hartmann
• Curvature
• Geometric
• Conclusions

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Imaging a star
 
 
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The effect of turbulence
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Adaptive Optics system
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Closed loop system
Reduces the effects of disturbances such as
telescope vibration, modelling errors by the
loop gain
Design limited by stability constraints
Does not inherently improve the noise
performance unless the closed loop measurements
are easier to make
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Postprocessing systemfeedforward compensation
Distorted incoming wavefront
Detector plane
telescope
Computer
Image

• Fixed
• mirror

Wavefront sensor
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Open loop system (SPID)
Sensitive to modelling errors No stability
issues with computer post processing Problem is
not noise but errors in modelling the system
T
time
Temporal coherence of the atmosphere
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Modelling the problem (step1)

• The relationship between the measured data and
  the object and the point spread function is
  linear
• Data object convolution point spread function
  noise
• (psf)
• A linear relationship would mean that if we
  multiply the input by a we multiply the output by
  a. The output doesnt change form

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Modelling the problem (step 2)

• The relationship between the phase and the psf is
  non linear
• psf Fourier magnitude phase correlation
  transform

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Phase retrieval
Wrapped ambiguity
Correct MAP estimate
ML estimation
MAP estimation

• Nonlinearity caused by 2p wrapping interacting
  with smoothing

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Role of typical wavefront sensor

• To produce a linear relationship between the
  measurements and the phase
• Speeds up reconstruction
• Guarantees a solution
• Degrades the ultimate performance
• phase weighting basis function

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Solution is by linear equations

• Measurement Interaction Basis function
• vector matrix
  Coefficents
• ith column of T corresponds to the measurement
  that would occur if the phase was the ith basis
  function
• Three main issues
• What has been lost in linearising?
• How well you can solve the system of equations?
• Is it the right equations?

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The effect of turbulence
There is a linear relationship between the mean
slope of the phase in a direction and the
displacement of the image in that direction.
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Trivial example

• There is a linear relationship between the mean
  slope and the displacement of the centroid
• Measurements are the centroids of the data
• Interaction matrix is the scaled identity
• Reconstruct the coefficients of the tip and tilt

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Quality of the reconstruction

• The centroid proportional to the mean slope
  (Primot el al, Welsh et al).
• The best Strehl requires estimating the least
  mean square (LMS) phase (Glindemann).
• To distinguish the mean and LMS slope you need to
  estimate the coma and higher order terms

LMS slope
Mean slope
Phase
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Difference between the lms and mean tilt
Ideal image
Coma distortion

• Peak value is better than the centroid for
  optimising the Strehl
• Impractical for low light data

Detected image
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Where to from here

• The real problem is how to estimate higher
  aberration orders.
• Wavefront sensor can be divided into
• pupil plane techniques, that measure slopes
  (curvatures) in the divided pupil plane,
• Shack-Hartmann
• Curvature (Roddier), Pyramid (Ragazonni)
• Lateral Shearing Interferometers
• Image plane techniques that go directly from data
  in the image plane to the phase (nonlinear)
• Phase diversity (Paxman)
• Phase retrieval

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Geometric wavefront sensing

• Pyramid, Shack-Hartmann and Curvature sensors are
  all essentially geometric wavefront sensors
• Rely on the fact that light propagates
  perpindicularly to the wavefront.
• A linear relationship between the displacement
  and the slope
• Essentially achromatic

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Geometric optics model

• A slope in the wave-front causes an incoming
  photon to be displaced by
• Model is independent of wavelength and spatial
  coherence.

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Generalized wave-front sensor

• This is the basis of the two most common
  wave-front sensors.

Converging lens
Aberration
Focal plane
Shack-Hartmann
Curvature sensor
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Trade-off

• For fixed photon count, you trade off the number
  of modes you can estimate in the phase screen
  against the accuracy with which you can estimate
  them
• To estimate a high number of modes you need good
  resolution in the pupil plane
• To make the estimate accurately you need good
  resolution in the image plane

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Properties of a wave-front sensor

• Linearization want a linear relationship between
  the wave-front and the measurements.
• Localization the measurements must relate to a
  region of the aperture.
• Broadband the sensor should operate over a wide
  range of wavelengths.
• ? Geometric Optics regime

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Explicit division of the pupil
Direct image
Shack-Hartmann
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Shack-Hartmann sensor

• Subdivide the aperture and converge each
  subdivision to a different point on the focal
  plane.
• A wave-front slope, Wx, causes a displacement of
  each image by zWx.

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Fundamental problem

• Resolution in the pupil plane is inversely
  proportional to the resolution in the image plane
• You can have good resolution in one but not both
  (Uncertainty principle)

Pupil
D
w
Image
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Loss of information due to subdivision

• Cannot measure the average phase difference
  between the apertures
• Can only determine the mean phase slope within an
  aperture
• As the apertures become smaller the light per
  aperture drops
• As the aperture size drops below r0 (Fried
  parameter) the spot centroid becomes harder to
  measure

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• Subdivided aperture

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Implicit subdivision

• If you dont image in the focal plane then the
  image looks like a blurred version of the
  aperture
• If it looks like the aperture then you can
  localise in the aperture

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Explanation of the underlying principle

• If there is a deviation from the average
  curvature in the wavefront then on one side the
  image will be brighter than the other

If there is no curvature from the atmosphere
then it is equally bright on both sides of focus.
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Slope based analysis of the curvature sensor
The displacement of light from one pixel to its
neighbour is s determined by the slope of the
wavefront
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Slope based analysis of the curvature sensor

• The signal is the difference between two slope
  signals
• ?Curvature

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Phase information localisation in the curvature
sensor

• Diffraction blurring geometric expansion

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Curvature sensing

• Localization comes from the short effective
  propagation distance,
• Linear relationship between the curvature in the
  aperture and the normalized intensity difference
• Broadband light helps reduce diffraction effects.

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Curvature sensing signal
Simulated intensity measurement
Curvature sensing estimate

• The intensity signal gives an approximate
  estimate of the curvature.
• Two planes help remove scintillation effects

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Irradiance transport equation

• Linear approximation gives

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Solution inside the boundary

• There is a linear relationship between the signal
  and the curvature.
• The sensor is more sensitive for large effective
  propagation distances.

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Solution at the boundary (mean slope)

• If the intensity is constant at the aperture,
• H(z) Heaviside function

I1 I2 I1- I2
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The wavefront also changes

• As the wave propagates, the wave-front changes
  according to

• As the measurement approaches the focal plane the
  distortion of the wavefront becomes more
  important, and needs to be incorpoarated (van Dam
  and Lane)

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Non-linearity due to the wavefront changing

• As a consequence the intensity also changes!
• So, to second order
• The sensor is non-linear!

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Origin of terms

• Due to the difference in the curvature in the x-
  and y- directions (astigmatism).
• Due to the local wave-front
• slope, displacing the curvature
• measurement.

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Consequences of the analysis

• As z increases, the curvature sensor is limited
  by nonlinearities K and T.
• A third-order diffraction term limits the spatial
  resolution to

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Analysis of the curvature sensor

• As the propagation distance, z, increases,
• Sensitivity increases.
• Spatial resolution decreases.
• The relationship between the signal and the
  curvature becomes non-linear.

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Tradeoff in the curvature sensor

• Fundamental conflict between
• Sensitivity which dictates moving the detection
  planes toward the focal plane
• Aperture resolution which dictates that the
  planes should be closer to the aperture

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Geometric optics model

• Slopes in the wave-front causes the intensity
  distribution to be stretched like a rubber sheet
• Wavefront sensing maps the distribution backto
  uniform

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Intensity distribution as a PDF

• The intensity can be viewed as a probability
  density function (PDF) for photon arrival.
• As the wave propagates, the PDF evolves.
• The cumulative distribution function (CDF) also
  changes.

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• Take two propagated images of the aperture.
• D1 m, r00.1 m and ?589 nm.

Intensity at -z
Intensity at z
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• Can prove using the irradiance transport equation
  and the wave-front transport equation that
• This relationship is exact for geometric optics,
  even when there is scintillation.
• Can be thought of as the light intensity being a
  rubber sheet being stretched unevenly

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• Use the cumulative distribution function to match
  points in the two intensity distributions.
• The slope is given by

x1 x2
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Results in one dimension
Actual (black) and reconstructed (red) derivative
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Simulation results
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Comparison with Shack-Hartmann
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Conclusions

• Fundamentally geometric wavefront sensors are all
  based on the same linear relationship between
  slope and displaced light
• All sensors trade off the number of modes you can
  estimate against the quality of the estimate
• The main difference between the curvature and
  Shack-Hartmann is how they divide the aperture
• Question is how to make this tradeoff optimally.