Wavefront sensing for adaptive optics Brian Bauman, LLNL This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.
Acknowledgments Wilson Mizner : "f 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; now add Marcos van Dam
Outline Wavefront sensing Shack-Hartmann Hartmann test History of Shack-Hartmann WFS Centroid estimation SH WFS design Pyramid Curvature Not covered direct phase/interferometric measurements Phase retrieval Topics are covered with a bit of the optical engineer s point of view
Hartmann test Before there was Shack, there was Hartmann (1900, 1904 (in German)). Used for testing figure of optics reference spots wavefront screen Detector (film/ccd)
Hartmann test Before there was Shack, there was Hartmann (1900, 1904 (in German)). Used for testing figure of optics z W( x, y) Δy Slope= Δy/z wavefront screen Detector (film/ccd)
Hartmann masks Originally, polar array of holes to sample aperture; suffered from sparse sampling at outer edge (or over-dense sampling near center), radial patterns hard to see Holes sized according to power, diffraction size Helical pattern for testing Lick 3-meter mirror (Mayall & Vasilevskis, 1960) Square grid was introduced in early 70 s Malacara
Hartmann test Before there was Shack, there was Hartmann (1900, 1904 (in German)). Used for testing figure of optics z W( x, y) Δy slope= Δy/z wavefront screen Detector (film/ccd)
Hartmann test Before there was Shack, there was Hartmann (1900, 1904 (in German)). z W( x, y) Δy Slope= Δy/z wavefront lenslets Detector (film/ccd)
History of Shack s/platt s modifications Original application was for measuring atmospheric distortions to deconvolve images of satellites Replaced holes with lenslets to maximize throughput (application was measuring atmosphere-distorted wavefronts) and to reduce spot size Made lenslets by polishing glass with 150-mm-long cylindrical nylon mandrel sliding on steel shaft until cylindrical divot was desired width (1 mm), then shifted the mandrel by the lenslet pitch Cylinders polished to λ/20 Used glass cylinders as master in molding process Plexiglass was molded between crossed cylindrical sets to form spherical lenslets on square grid Molds formed in Platt s kitchen oven, softening plexiglass until it slumped between masters; trimmed plexiglass with electric kitchen knife Platt and Shack, J. Refractive Surgery, vol. 17, p. S573 S577 (2001)
Shack-Hartmann spots
Shack-Hartmann spots 45-degree astigmatism
Dot relay Lenslets available generally only in fixed sizes; CCD pixels available in fixed sizes; but can adapt lenslet pitch to CCD pixel pitch via relay, often two-lens 4-f telescope for low aberrations/geometric distortion; Relay often necessary anyway because of short lenslet focal lengths and clearance issues Modeled as separate imaging system; dots are objects; entrance pupil is at infinity (telecentric) There is rarely any optical design advantage in modeling the lenslet array as such. Divide design into before-lenslets and after-lenslets. Dot plane Good place for filters CCD plane
Dot relay design considerations Once wavefront is sampled by lenslets, the game is over; the wavefront measurement has been made. The relay need only to not blur spot too much, and not introduce unacceptable distortion, which is interpreted as a wavefront error by the WFS. f# is generally slow (e.g., ~ f/20), so re-imaging dots is not difficult. For quad-cell systems, spacing between lenses needs to be perturbed from 4f otherwise there is no dot magnification adjustment possible There is a pupil wrt imaging the dots this is a good place for filters as it after the measurement of the wavefront and affects all subapertures equally. Dot plane Good place for filters CCD plane
Spot size/subaperture size Spot size ~λ/d, where d is the subaperture size. Typically, d is on the order of the actuator pitch (often exactly the actuator pitch Fried geometry), and is on the order of r 0 at the science wavelength. For λ 0.8μ and d=40 cm, the spot size is approximately 0.8 μ/0.4m=2 μrad=0.4 arcsec Spot size trade-off bigger subapertures => more light, better SNR in centroid measurement, but poorer fit to wavefront. f subapertures are too small, then spot size increases due to diffraction, degrades spot centroid estimate (proportional to spot size) n example above, 5% spot-size displacement => 0.1 μrad => 0.1 μrad * 0.4 m = 40 nm nm tilt across subaperture
Plate scale Plate scale refers to the size in arcsec of a pixel on the CCD of the SH WFS, often ~1-2 arcsec/pixel. Plate scale trade-off Bigger pixels (in arcsec) => more range without crossover between subapertures, but lets in more sky background Could use more pixels per subaperture, but that increases noise in estimate due to read noise/dark current
Typical vision science WFS Lenslets CCD Many pixels per subaperture
Typical Astronomy WFS Former Keck AO WFS sensor 2 mm 21 μ pixels 3x3 pixels/subap 200 μ lenslets relay lens CCD 3.15 reduction
Centroiding Once you have generated spots, how do you determine their positions? The performance of the Shack-Hartmann sensor depends on how well the displacement of the spot is estimated. The displacement is usually estimated using the centroid (center-of-mass) estimator. This is the optimal estimator for the case where the spot is Gaussian distributed and the noise is Poisson. s x = x ( x, y) ( x, y) s y = y ( x, y) ( x, y)
Centroiding noise Due to read noise and dark current, all pixels are noisy. Pixels far from the center of the subaperture are multiplied by a large number: s x = x ( x, y) x = { L, 3, 2, 1,0,1,2,3, L} The pixels with the most leverage on the centroid estimate are the dimmest (therefore, the pixels with the least information), and there are lots of dim pixels The more pixels you have, the noisier the centroid estimate!
Weighted centroid 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 s x = xw( x, y) ( x, y) s y = yw( x, y) ( x, y)
Correlation (matched filtering) Find the displacement of the image that gives the maximum correlation: ( s, s ) = argmax( w( x, y) ( x, y)) x y =
Correlation (matched filtering) Noise is independent of number of pixels Much better noise performance for many pixels Estimate is independent of uniform background errors Estimate is relatively insensitive to assumed image.
n 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 4 3 2 1 4 3 2 1 s x + + + + = 4 3 2 1 4 3 2 1 s y + + + + =
Quad cells The estimated centroid position is linear with displacement only over a small region (small dynamic range) Sensitivity is proportional to spot size Estimated centroid position vs. displacement for different spot sizes Centroid estimated position Displacement
Denominator-free centroiding 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 s 1 + 2 3 4 = x E[ + + + ] 4 1 2 3 s 1 2 + 3 4 = y E[ + + + ] 4 1 2 3
Laser guide elongation Shack-Hartmann subapertures see a line not a spot
LGS elongation at Keck Laser projected from right
A possible solution for LGS Radial format CCD Arrange pixels to be at same orientation as spots Currently testing this design for TMT elongation laser
Dynamic refocusing for pulsed lasers Powered mirror on mechanical resonator (U of A) Segmented MEMS, one segment per subaperture (Bauman; Baranec) Rotating phase plates (e.g., Alvarez lens) (Bauman)
Problems with SH WFS Spot size is large (~ λ/d) Crucial measurement is made at junction between pixel boundaries, which are indistinct (has been reported at ~1/3 pixel charge diffusion) Worst of all worlds: photons near knife-edge generate all the noise and none of the signal!
Foucault knife-edge test Foucault (1858, 1859 (in French)) Knife-edge test for perfect lens (top), and one with spherical aberration (bottom). At right are observer views of pupil in each case. An irregular mirror tested with knifeedge test
Foucault test with mirror
Pyramid WFS Pyramid is simultaneous implementation of 4 Foucault knife-edge measurements SH WFS divides aperture into subapertures (via lenslets), then field into quadrants (via pixels) PWFS does in reverse order: pyramid divides field into quadrants (via pyramid) then aperture into subapertures (via pixels) pyramid field lens pupils with CCD pixels demarking subapertures incoming beam CCD at pupil plane image plane
PWFS details Pyramids are naturally quite small: Size of pyramid ~ n * (λ*f#), where n is # of subapertures (natural spatial filter) Pyramids have tight fabrication tolerances: Edge precision is a fraction of the full-aperture diffraction spot size (e.g., λ=1μ, f/15 sub-micron precision required. Can make beam slower to relax edge requirements, but at cost of length. Can be made of glass, using cemented facets. Difficult to make sharp edges Can use lenslet-based PWFS (coming up) Note advantage: if edges of pyramid can be sharp, then centroid measurement can be quite precise; indistinct CCD pixel boundaries relegated to subaperture division not crucial Also interesting: as wavefront slopes becomes small, the PWFS becomes a direct phase measuring device
SH WFS vs. PWFS Geometrically, identical just remapping of pixels. Diffractive advantage appears in high- Strehl regime.
Pyramid wave-front sensor non-linearity When the aberrations are large (e.g., defocus below), the pyramid sensor is very non-linear (reaches saturation). 4 pupil images x- and y-slopes estimates.
Modulation of pyramid sensor Without modulation: Linear over spot width With modulation: Linear over modulation width
Another pyramid implementation: Pyramid + lens = 2x2 lenslet array Lenslets are inexpensive and easily replicated. The right manufacturing technique produces sharp boundaries between lenslets (where all the action is). pyramid field lens lenslets Bauman dissertation
Brightening of rim is real effect PWFS is not quite a slope detector, but a derivative detector (effect also seen in knife-edge tests) There is a large derivative (in amplitude) at the edge of an aperture Pupils should not be too close to avoid contamination between pupil images mage of PWFS Johnson, et al., 2006
Curvature sensing -z mage 2 Aperture Wave-front at aperture z mage 1
Curvature sensing Developed by Roddier for AO in 1988. Linear relationship between the curvature in the aperture and the normalized intensity difference: Broadband light helps reduce diffraction effects. Tends to be used in lowerorder systems (i.e., fewer subapertures/actuators, because of higher error propagation z = f f l ( f l) l Aperture Defocused image 1 Defocused image 2
Curvature sensing W W z =. 2 where is the intensity, W is the wavefront and z is the direction of propagation, we obtain a linear, first-order approximation, which is a Poisson equation with Neumann boundary conditions. Using the irradiance transport equation, W z W z + = +. 2 1 2 1 2
Solution at the boundary ) ( ) ( ) ( ) ( 2 1 2 1 x x x x zw R x H zw R x H zw R x H zw R x H + + + = + + - 1 2 1-2 f the intensity is constant at the aperture,
Solution inside the boundary 1 1 + 2 2 = z( W xx + W yy ) Curvature There is a linear relationship between the signal and the curvature The sensor is more sensitive for large effective propagation distances
Curvature sensing As the propagation distance, z, increases, sensitivity increases. Spatial resolution decreases. Diffraction effects increase. 1 1 + 2 2 = z( W xx + W yy ) The relationship between the signal, ( 1-2 )/( 1 + 2 ) and the curvature, W xx + W yy, becomes non-linear
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