Wavefront Sensor Design - Center for Adaptive Optics

Wavefront sensing for 

adaptive optics 

Richard Dekany 

Caltech Optical Observatories 

2009

• Thanks to: 

Acknowledgments 

– Marcos van Dam – original screenplay 

– Brian Bauman – adapted screenplay 

– Contributors 

• Richard Lane, Lisa Poyneer, Gary Chanan, 

Jerry Nelson, and others; 

Elements of this presentation were prepared under the auspices of the U.S. Department of Energy 

by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

Outline 

• Wavefront sensing 

– Shack-Hartmann 

• Hartmann test 

• History of Shack-Hartmann WFS 

• Centroid estimation 

• SH WFS design 

– Pyramid 

– Curvature 

• Not covered here 

– Shearing interferometers 

– Direct phase/interferometric measurements 

– Phase retrieval 

– Double curvature sensing 

– PIGS, SPLASH, L-O PWFS, etc. 

• Topics are covered with a bit of the Optical 

Engineer’s point of view 

– “the Boss : Springsteen :: the Optical Engineer : Bauman”

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)). 


z 

Δy 

Slope= Δy/z 


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)). 


z 

Δy 

slope= Δy/z 


Shack-Hartmann test 

• Before there was Shack, there was 

Hartmann (1900, 1904 (in German)). 

z 

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

Lenslets-to-CCD: The “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 to f/50), 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 to a few 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. 

– If subapertures are too small, then spot size increases due 

to diffraction, degrades spot centroid estimate (proportional 

to spot size) 

• In example above, 5% spot-size displacement => 0.1 

μrad => 0.1 μrad * 0.4 m = 40 nm tilt across 

subaperture

Plate scale 

• “Plate scale” refers to the size in arcsec (on the 

sky) of a pixel 

– For SH WFS, this is often ~1-2 arcsec/pixel. 

• Plate scale trade-off 

– Bigger pixels (in arcsec) 

• More dynamic range w/o “crosstalk” between subapertures, 

but lets in more sky background photons (noise) 

– More pixels per subaperture 

• Increases linearity, at cost of more read noise and dark 

current in slope measurement

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 

Low- or zero-noise detectors are starting to 

change astronomical WFS thinking (more pixels)

Centroiding 

• Once you have generated spots, how do you 

determine their positions? 

– The performance of the Shack-Hartmann 

sensor (the quality of the wavefront estimate) 

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.

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: 

• 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

Correlation (matched filtering) 

• Find the displacement of the image 

that gives the maximum correlation: 

= 

!

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 

• Computationally more expensive 

– This used to matter more.

Quad cells 

• 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 

• 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 variance 

• Centroid error can be reduced by 

using the average value of the 

denominator

Laser guide elongation 

• Shack-Hartmann subapertures see a line not a spot 

– Length Θ ≈ t*s / h 2 ; where t is Na layer or range gate thickness 

• Depends on projector offset, not viewing direction

LGS elongation at Keck II 

Laser projected from right

A possible mitigation for LGS elongation 

• Radial format CCD 

– A specially oriented 

array of CCDs on one 

chip 

• Arrange pixels to be 

at same orientation 

as spots 

• Currently hardware 

testing this design 

for TMT 

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 

• 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

Pyramids… 

• … are naturally quite small 

– Size of pyramid ~ n * (λ*f#), where n is # of subapertures 

(natural spatial filter) 

• … 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. 

– It is difficult to make sharp edges 

• … can be based on lenslets (coming up) 

– 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 

• … can transmogrify 

– 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 

•Bauman (2003 (in English)) 

•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 

Image of PWFS 

Johnson, et al., 2006

Why is a PWFS/Foucault test a 

slope sensor? 

• Use Fourier optics! 

focal plane 

mask 

W (x,y) 

p 

CCD 

pupil 

relay 

lenses 

pupil

1/x is poor man’s approximation 

to δ (1) (x)

How to convert SHWFS to PWFS 

• This works only 

when the # of 

subapertures is 

approximately 

equal to the # of 

pixels per 

subaperture 

• Otherwise, other 

optical changes 

need to be made 

image 

plane 

"pupilet plane" 

lenslet 

array 

collimating 

lens 

demagnified 

"pupilet plane" 

relay 

lenses 

WFS 

"dot plane" 

lenslet 

array 

demagnified 

"dot plane" 

relay 

lenses 

Figure 3- 17: Conversion of a SH WFS to a PWFS. Top figure: Converging 

light from the left comes to a focus and is then collimated by a collimating 

lens. The collimating lens creates a pupil downstream, where the lenslet 

array is placed. The lenslets produce a series of images or dots at the focal 

plane of the lenslets or “dot plane”. Subsequent relay optics scale the dots 

as appropriate for the WFS CCD. For clarity, light from only one dot is 

shown after the dot plane. Bottom figure: to convert SH WFS to PWFS, 

remove collimating lens and translate lenslet array, relay optics, and WFS 

CCD upstream until the lenslet array is at the focus of the incoming beam. 

The lenslets now produce “pupilets” at the lenslet focal plane, i.e., where the 

dots were in the top figure. Thus, the relay optics will relay the pupilets to 

the WFS CCD. 

WFS

Curvature sensing 

-z 

Image 2 

Aperture 

Wave-front at aperture 

z 

Image 1


• 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 1 error 

propagation 

1 

Still an area of active research


• Using the irradiance transport 

equation, 

where I 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.

Solution at the boundary 

If the intensity is constant at the aperture, 

I 

I 

1 

1 

! 

+ 

I 

I 

2 

2 

= 

H 

H 

( x 

( x 

! 

! 

R 

R 

! 

! 

zW 

zW 

x 

x 

) 

) 

! 

+ 

H 

H 

( x 

( x 

! 

! 

R 

R 

+ 

+ 

zW 

zW 

x 

x 

) 

) 

I 1 

I 2 

I 1 - I 2

Solution inside the boundary 

I 

I 

1 

1 

! 

+ 

I 

I 

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


As the propagation distance, 

z, increases, sensitivity 

increases. 

• Spatial resolution 

decreases. 

• Diffraction effects 

increase. 

• The relationship between 

the signal, (I 1 - I 2 )/(I 1 + I 2 ) 

and the curvature, W xx + 

W yy , becomes non-linear


• 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 Sensor Design - Center for Adaptive Optics

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