Wavefront Sensing I

Richard Lane

Department of Electrical and Computer Engineering

University of Canterbury

Christchurch

New Zealand

Location

Astronomical Imaging Grouppast and present

Dr Richard LaneProfessor Peter GoughAssociate Professor P. J. BonesAssociate Professor Peter CottrellProfessor Richard Bates

Dr Bonnie Law Dr Roy Irwan Dr Rachel Johnston Dr Marcos van DamDr Valerie Leung Richard ClareYong Chew Judy Mohr

Contents

• Session 1 – Principles

• Session 2 – Performances

• Session 3 – Wavefront Reconstruction for 3D

Principles of wavefront sensing

• Introduction

• Closed against open loop wavefront sensing

• Nonlinear wavefront sensing

• Shack-Hartmann

• Curvature

• Geometric

• Conclusions

Imaging a star

  

The effect of turbulence

Adaptive Optics system

Wavefrontsensor

Image plane

Deformablemirror

Distorted incomingwavefront

telescope

Closed loop system

K Compensation system

error

Noise

True output

Measured output

Desired Output (zero phase)

Reduces the effects of disturbances such as telescope vibration, modelling errors by the loop gain

Does not inherently improve the noise performanceunless the closed loop measurements are easier to make

Design limited by stability constraints

Postprocessing systemfeedforward compensation

Wavefrontsensor

Detector plane

Distorted incomingwavefront

telescope

Fixedmirror

Computer

ImageImage

Open loop system (SPID)

Distortion estimation

Estimatedoutput

True image

Wavefront sensor

Compensation Distortion by the atmosphere

Sensitive to modelling errors

No stability issues with computer post processing

Problem is not noise but errors in modelling the system

timeTemporal coherenceof the atmosphere

T

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 α we multiply the output by α. The output doesn’t change form

),(),(),(),( yxnyxhyxfyxd

Modelling the problem (step 2)

• The relationship between the phase and the psf is non linear

psf Fourier magnitude phase correlation transform

iφAiφA h expexp F

Correct MAP estimate Wrapped ambiguity

ML estimation MAP estimation

Phase retrieval

• Nonlinearity caused by 2 wrapping interacting with smoothing

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

1

),(),(n

ii vuavu

Solution is by linear equations

Measurement Interaction Basis functionvector matrix Coefficents• ith column of Θ 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?

am

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.

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

c

c

y

x

k

k

0

0

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

Mean slope

LMS slope

Phase

50 100 150 200 250

50

100

150

200

250

Coma distortion

Detected image

• Peak value is better than the centroid for optimising the Strehl• Impractical for low light data 50 100 150 200 250

50

100

150

200

250

Difference between the lms and mean tilt

Ideal image

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

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

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.

z

W(x)

x

xzWx

Generalized wave-front sensor• This is the basis of the two most common

wave-front sensors.

Converging lens

Aberration

Focal plane

Curvature sensor

Shack-Hartmann

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

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

Explicit division of the pupilDirect image

Shack-Hartmann

20 40 60 80 100 120

20

40

60

80

100

120

20 40 60 80 100 120

20

40

60

80

100

120

Shack-Hartmann sensor

0

5

10

15

20

0

5

10

15

20-10

0

10

20

05

1015

20

0

5

10

15

200

0.02

0.04

0.06

0.08

• 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.

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)

D

w

Pupil

Image

wD

1

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

• Subdivided aperture

5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

Log10

Number of photons

Mea

n sq

uare

d-er

ror D/r

0=1

D/r0=2

D/r0=4

Implicit subdivision

• If you don’t 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

Focal plane

Aperture

Defocused Aperture I1

Defocused Aperture I2

l

lffz

)(

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

Focal Plane Aperture

Phase

Intensity 1 I1

Intensity 2 I2

l

f

If there is no curvaturefrom the atmosphere thenit is equally bright on bothsides of focus.

Slope based analysis of the curvature sensor

Planar Wavefront

Detector Pixels

Distorted Wavefront

Detector Pixels

The displacement of light from one pixel to its neighbour iss determined by the slope of the wavefront

Slope based analysis of the curvature sensor

Distorted Wavefront

Detector Pixels

Distorted Wavefront

Detector Pixels

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

Distorted Wavefront

Detector Pixels

Phase information localisation in the curvature sensor

r0?

Kolmogorov phase screen

Detector Plane

Blurring due to the roughness of the phase screen

Focal plane

Aperture resolution

Detector plane blurring

• Diffraction blurring + geometric expansion

• 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.

Curvature sensing

A p e r t u r e

D e f o c u s e d i m a g e I 1

D e f o c u s e d i m a g e I 2

l

f l

lffz

)(

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

Irradiance transport equation

WIWIz

I 2.

I

IWzWz

II

II

.2

12

12

• Linear approximation gives

WzIWIzI 21 .

WzIWIzI 22 .

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.

Solution at the boundary (mean slope)

)()(

)()(

21

21

xx

xx

zWRxHzWRxH

zWRxHzWRxH

II

II

If the intensity is constant at the aperture,

H(z) = Heaviside function

+ -

I1

I2

I1- I2

The wavefront also changes

222

22

2

22 )(

816)(

2

11 I

II

IWWz

• 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)

Non-linearity due to the wavefront changing

• As a consequence the intensity also changes!

• So, to second order :

• The sensor is non-linear!

)(1

)(2

21

21

TKz

WWz

II

II yyxx

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.

2xyyyxx WWWK

yyyyxxyyxyyxxxxx WWWWWWWWT

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

z

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.

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

• Slopes in the wave-front causes the intensity distribution to be stretched like a rubber sheet

• Wavefront sensing maps the distribution backto uniform

Geometric optics model

z

W(x)

xxzWx

20 40 60 80 100 120

20

40

60

80

100

120

• 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.

Intensity distribution as a PDF

Intensity distribution

Photon arrival p(x)

x

• Take two propagated images of the aperture.

D=1 m, r0=0.1 m and λ=589 nm.

Intensity at -z

Intensity at z

• 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

),(CDF),(CDF)0,(CDF zx

Wzxz

x

Wzxx

• Use the cumulative distribution function to match points in the two intensity distributions.

• The slope is given by

x1 x2

z

xxxxW

x 2

21

2

21

Results in one dimensionActual (black) and reconstructed (red) derivative

Simulation results

Comparison with Shack-Hartmann

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.