wavefront sensing for adaptive optics
TRANSCRIPT
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Wavefront sensing for adaptive
optics
MARCOS VAN DAM & RICHARD CLARE
W.M. Keck Observatory
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Wilson Mizner : "If you steal from one author it's
plagiarism; if you steal from many it's research."
Thanks to: Richard Lane, Lisa Poyneer, Gary Chanan,
Jerry Nelson
Acknowledgments
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Wavefront sensing
Shack-Hartmann
Pyramid
Curvature
Phase retrievalGerchberg-Saxton algorithm
Phase diversity
Outline
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Properties of a wave-front sensor
Localization: the measurements should relate to aregion of the aperture.
Linearization: want a linear relationship between thewave-front and the intensity measurements.
Broadband: the sensor should operate over a widerange of wavelengths.
=> Geometric Optics regime
BUT: Very suboptimal (see talk by GUYON on Friday)
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Effect of the wave-front slope
A slope in the wave-front causes an incoming photon
to be displaced by
There is a linear relationship between the mean slope
of the wavefront and the displacement of an image
Wavelength-independent
xzWx =
x
z
W(x)
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Shack-Hartmann
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 spots
45-degree astigmatism
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Typical vision science WFS
Lenslets CCD
Many pixels per subaperture
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Typical Astronomy WFS
lensletsrelay lens
CCD
200 μ
2 mm
3.15 reduction
21 pixels
3x3 pixels/subap
Former Keck AO WFS sensor
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The performance of the Shack-Hartmann sensordepends on how well the displacement of the spot isestimated.
The displacement is usually estimated using thecentroid (center-of-mass) estimator.
This is the optimal estimator for the case where thespot is Gaussian distributed and the noise is Poisson.
=),(
),(
yxI
yxIxsx
Centroiding
=),(
),(
yxI
yxIysy
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G-tilt vs Z-tilt
The centroid gives the mean slope of the wavefront
(G-tilt).
However, we usually want the least-mean-squares
slope (Z-tilt).
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Due to read noise and dark current, all pixels arenoisy.
Pixels far from the center of the subaperture aremultiplied by a large number:
The more pixels you have, the noisier the centroidestimate!
= ),( yxIxsx
Centroiding noise
},3,2,1,0,1,2,3,{ LL=x
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The noise can be reduced by windowing the centroid:
Weighted centroid
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Can use a square window, a circular window:
Or better still, a tapered window, w(x,y)
Weighted centroid
= ),(),( yxIyxxwsx
= ),(),( yxIyxywsy
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Find the displacement of the image that gives themaximum correlation:
Use FFT or quadratic interpolation to find thesubpixel maximum correlation
Correlation (matched filtering)
)),(),(max(arg),( yxIyxwss xx =
=
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Noise is independent of number of pixels
Much better noise performance for many pixels
Estimate is independent of uniform backgrounderrors
Estimate is relatively insensitive to assumed image.
Correlation (matched filtering)
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In astronomy, wavefront slope measurements are
often made using a quad cell (2x2 pixels)
Quad cells are faster to read and to compute the
centroid and less sensitive to noise
Quad cells
4321
4321
IIII
IIIIs
x
+++
+=
4321
4321
IIII
IIIIsy
+++
+=
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These centroid is only linear with displacement over a
small region (small dynamic range)
Centroid is proportional to spot size
Quad cells
Displacement
Centroid
Centroid vs. displacement for different spot sizes
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When the photon flux is very low, noise in the
denominator increases the centroid error
Centroid error can be reduced by using the average
value of the denominator
Denominator-free centroiding
][ 4321
4321
IIIIE
IIIIs
x
+++
+=
][ 4321
4321
IIIIE
IIIIsy
+++
+=
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Shack-Hartmann subapertures see a line not a
spot
Laser guide elongation
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LGS elongation at Keck
Laser projected from right
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A possible solution for LGS elongation
Radial format CCD
Arrange pixels to be at
same angle as spots
Currently testing this
design for TMT
laser
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Pyramid wave-front sensor
Focal plane
Images of the aperture
(conjugate aperture plane)
Aperture plane
Pyramid (glass prism)
Lens to image the aperture
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Similar to the Shack-Hartmann using quad cells: it
measures the average slope over a subaperture.
The subdivision occurs at the image plane, not the
pupil plane.
Local slope determines which image receives the light
Pyramid wave-front sensor
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When the aberrations are large, the pyramid sensor is
very non-linear.
Pyramid wave-front sensor non-linearity
4 pupil images x- and y-slopes estimates.
Large focus aberration
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Modulation of pyramid sensor
Without modulation:
Linear over spot width
With modulation:
Linear over modulation width
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+
Pyramid + lens = 2x2 lenslet array
Pyramid
Relay lens
lenslets
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Duality between Shack-Hartmann and pyramid
Shack-Hartmann Pyramid
Low resolution
images of the object
Object
Low resolution
images of the aperture
Aperture
ApertureHigh resolution
image of the
object
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Duality between Shack-Hartmann and pyramid
Shack-Hartmann Pyramid
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Duality between Shack-Hartmann and pyramid
Shack-Hartmann
Aperture Focal Plane
Pyramid
Pixels in Shack-Hartmann = lenslets in PyramidLenslets in pyramid = pixels in Shack-Hartmann
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Multi-sided prisms
Pyramid uses 4-sided glass prism at focal plane
to generate 4 aperture images
Can use any N-sided prism to produce N aperture
images
Limit as N tends to Infinity gives the “cone” sensor
Cone
Relay lens
Aperture image
Aperture
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Wave-front at aperture
Aperture
Image 1
z
-z
Image 2
Curvature sensing
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Localization comes from the shorteffective propagation distance,
Linear relationship between thecurvature in the aperture and thenormalized intensity difference:
Broadband light helps reducediffraction effects.
Curvature sensing
Aperture
Defocused
image I1
Defocused
image I2
l
f l
lffz
)(=
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Curvature sensing
WIWIz
I= .
2
I
IWzWz
II
II+=
+.
2
12
12
Where I is the intensity, W is the wave-front
and z is the direction of propagation, we obtain
a linear, first-order approximation,
Using the irradiance transport equation,
which is a Poisson equation with Neumann
boundary conditions.
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Solution at the boundary
)()(
)()(
21
21
xx
xx
zWRxHzWRxH
zWRxHzWRxH
II
II
++
+=
+
I1
I2
I1- I2
If the intensity is constant at the aperture,
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Solution inside the boundary
)(21
21yyxx WWz
II
II+=
+
There is a linear relationship between the signal and
the curvature
The sensor is more sensitive for large effective
propagation distances
Curvature
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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
)(21
21yyxx WWz
II
II+=
+
Subaru AO system will use two different propagation distancesA large distance for high sensitivityA short distance for high spatial resolution
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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 sensor subapertures
Measure intensity in each subaperture with an
avalanche photo-diode (APD)
Detect individual photons – no read noise
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Wavefront sensing from defocused images
There are more accurate, non-linear, algorithms to
reconstruct the wavefront using defocused images
with many pixels
Defocused images True and reconstructed wavefronts
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Suppose we have an image and knowledge about the
pupil.
Can we find the phase, , that resulted in this image?
Phase retrieval
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Image is insensitive to:
Addition of a constant to (x).Piston does not affect the image
Addition of a multiple of 2 to any point on (x)Phase wrapping
Replacing (x) by - (-x) if amplitude is symmetricale.g., positive and negative defocused images look identical
Called the phase ambiguity problem
Phase retrieval
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Gerchberg-Saxton algorithm
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Phase retrieval suffers from phase ambiguity, slow
convergence, algorithm stagnation and sensitivity to
noise
These problems can be overcome by taking two or more
images with a phase difference between them
In AO, introduce defocus by moving a calibration source.
Phase diversity
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Phase diversity
+2 mm
-2 mm
-4 mm
Defocus
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Phase diversity
Poked actuators Minus poke phase Plus poke phase Difference
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Phase diversity
Theoretical diffraction-limited image Measured image
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Mahalo!