wavefront sensing for adaptive optics
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Wavefront sensing for adaptive
optics
MARCOS VAN DAM & RICHARD CLARE
W.M. Keck Observatory
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
Wavefront sensing
Shack-Hartmann
Pyramid
Curvature
Phase retrievalGerchberg-Saxton algorithm
Phase diversity
Outline
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)
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)
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.
Shack-Hartmann spots
45-degree astigmatism
Typical vision science WFS
Lenslets CCD
Many pixels per subaperture
Typical Astronomy WFS
lensletsrelay lens
CCD
200 μ
2 mm
3.15 reduction
21 pixels
3x3 pixels/subap
Former Keck AO WFS sensor
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
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).
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
The noise can be reduced by windowing the centroid:
Weighted centroid
Can use a square window, a circular window:
Or better still, a tapered window, w(x,y)
Weighted centroid
= ),(),( yxIyxxwsx
= ),(),( yxIyxywsy
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 =
=
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)
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
+++
+=
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
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
+++
+=
Shack-Hartmann subapertures see a line not a
spot
Laser guide elongation
LGS elongation at Keck
Laser projected from right
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
Pyramid wave-front sensor
Focal plane
Images of the aperture
(conjugate aperture plane)
Aperture plane
Pyramid (glass prism)
Lens to image the aperture
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
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
Modulation of pyramid sensor
Without modulation:
Linear over spot width
With modulation:
Linear over modulation width
+
Pyramid + lens = 2x2 lenslet array
Pyramid
Relay lens
lenslets
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
Duality between Shack-Hartmann and pyramid
Shack-Hartmann Pyramid
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
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
Wave-front at aperture
Aperture
Image 1
z
-z
Image 2
Curvature sensing
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
)(=
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.
Solution at the boundary
)()(
)()(
21
21
xx
xx
zWRxHzWRxH
zWRxHzWRxH
II
II
++
+=
+
I1
I2
I1- I2
If the intensity is constant at the aperture,
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
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
Curvature sensing
Practical implementation uses a variable curvature
mirror (to obtain images below and above the
aperture) and a single detector.
Curvature sensor subapertures
Measure intensity in each subaperture with an
avalanche photo-diode (APD)
Detect individual photons – no read noise
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
Suppose we have an image and knowledge about the
pupil.
Can we find the phase, , that resulted in this image?
Phase retrieval
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
Gerchberg-Saxton algorithm
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
Phase diversity
+2 mm
-2 mm
-4 mm
Defocus
Phase diversity
Poked actuators Minus poke phase Plus poke phase Difference
Phase diversity
Theoretical diffraction-limited image Measured image
Mahalo!
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