27 April 2005 1

Wavefront Sensing

Alan Greenaway

27 April 2005 2

Credit to:• Heather Campbell PhD student• Clare Dillon EngD student• David Faichnie EngD student• Sijiong Zhang RA• Frank Spaan RA• Anne Marie Johnson RA

Funded by PPARC/dstl through Smart Optics Faraday Partnershipin collaboration with UCL, UK ATC, Zeeko, Scalar Techologies, BAE

SYSTEMS, NPL

27 April 2005 3

Adaptive Optics - what is it?Adaptive = feedback controlAdaptive Optics

3 ComponentsWavefront Modulator (WFM)Wavefront Sensor (WFS)Control loop

Active opticsNo WFSNo on-line control loopControl signal pre-computed off-line (e.g. gravity sag, thermally-induced aberrations, ...)

27 April 2005 4

Background• Adaptive optics

• Metrology

Null wavefront sensor is most suitable

fasterflexiblecheaper (?)

Require wavefront reconstruction

reference to national standardsspeed/cost less important (?)

27 April 2005 5

Metrology• Metrology is a

demanding application

• Speed also required...

• …as is robustness

NumbersEven worse, absolute numbers!Reference to standards!

Production lines for thin plastic films run at 5ms-1

Discontinuous surfacesRough surfacesMultiply-connected surfaces

27 April 2005 6

Err...• For industrial surfaces

• For thin films

• For exo-earth imaging

Sub nm resolution is generally sufficient

Sub Å resolution on wavefront aberrations is required

Better than 0.1 Å is required

An industrialist insisted that he needed a resolution of λ/40000

27 April 2005 7

How to achieve these...Unlikely to give accuracy or spatial format neededData reduction is time consumingEfficient use of data (not of detector)Can be reduced to a very simple algorithmCan be very accurateCan be large format

• Average over lenslet(Shack Hartmann)?

• Shearing interferometer?

• Wavefront curvature (phase diversity)?

27 April 2005 8

Phase-diverse wavefront sensing

(wavefront curvature sensing)

• Solution of ITE gives wavefrontconverging

converging

diverging

diverging

Plane 1 Plane 2

( )( ) ( , )R

I rr k dr G r rz′∂′ ′Ψ =−

∂∫

Plane 1 Plane 2

1 2

I I Iz z z− ∂− ∂

27 April 2005 9

Green’s function solution (GFS)

1 22 ( )I I GI z

π φλ δ

−⋅ ⋅ =-

1I 2I

10

20

30

40

10

20

30

40

-1

0

1

10

20

30

40

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-0.1

-0.05

0

0.05

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

=

27 April 2005 10

How to get the data?

∆x

yx

d

dm2Shift Phase Local x

m∆

π=φ

Phase shift is of opposite sign in orders of opposite sign m.

• Use a diffraction grating as a beamsplitter• Use ‘Detour Phase’ to get different level of

defocus in each diffraction order.• Simultaneous multi-plane imaging on 1 CCD

27 April 2005 11

How to collect data?

)yx(R

dW)y,x( 22

220 +

λ=∆

φα r2phase change

φα − r2

r

+ 1

- 1

0

+1

0

-1

Parallel Beam

Converging Beam

Diverging Beam

Parallel BeamIncident

rR

DistortionW20 defines level of distortion. Equivalent to extra path length introduced at edge of grating aperture

W20 defines level of distortion. Equivalent to extra path length introduced at edge of grating aperture

r is radius from mask centre

r is radius from mask centre

27 April 2005 12

Diffractive Optics• Phase-diversity scheme needs wavefront intensity pattern

on two separate planes: Scheme adopted uses IMP®s

Undistorted Grating Undistorted Grating - identical images of a single object layer in each order

Distorted Grating Distorted Grating - images of different object layers on a single image plane

Diffraction Order-1 0 +1

IMAGE

IMP®

Image-1

0

+1-1 0 +1

Diffraction Order-1 0 +1IMP® is a DERA Trademark

DERA is now QinetiQ

27 April 2005 13

Examples of Data

Blanchard, P.M., et al., Phase-diversity wave-front sensing with a distorted diffraction grating. Applied Optics, 2000. 39(35): p. 6649-6655.

•Some examples of the data seen at the focal plane.

•Easy to see the aberrations present in the data just by eye.

•Defocus•Astigmatism•Coma•Trefoil•Spherical Aberration

27 April 2005 14

Phase Diverse WFS

• Solution of ITE gives wavefront

Plane 1 Plane 2

1 2

I I Iz z z− ∂− ∂

( )( ) ( , )R

I rr k dr G r rz′∂′ ′Ψ =−

∂∫Figure 1: Two intensity planes either side of the wavefront

• DoE used to image Planes 1 & 2

27 April 2005 15

RestrictionsPhase solution and its derivative must be continuous Wavefront must be simply connectedIntensity must be constant (or small variations)Deviation lead to low-pass filtered solutions

• Green’s function approach using Neumann boundary conditions

27 April 2005 16

Image Plane System

( )rψ

( ). .idr e pupilϕψ

1 { . . }i idI Ae e pupilϕ ϕ− ++ = ℑ

0{ } ( )th

iI Ae ϕ ξ= ℑ = Ψ

1 { . . }i idI Ae e pupilϕ ϕ− −− = ℑ

27 April 2005 17

Pupil Plane System

( )rψ( )ξΨ

1 { ( ). . }idI e pupilϕξ ++ = ℑ Ψ

0{ ( )} ( )thI rξ ψ= ℑ Ψ =

1 { ( ). . }idI e pupilϕξ −− = ℑ Ψ

( ). .ide pupilϕξΨ

27 April 2005 18

Metrology requirementsMetrology of rough surfacesMetrology of printed circuits, integrated optics, ...Secondary with support structure

• Wavefront may be scintillated

• Wavefront may be discontinuous

• Wavefront may be multiply connected

All of these are a problem for the GFS approach

27 April 2005 19

Hang on...• Isn’t another way of looking at this as the

wavefront convolved with a defocused psf?• With a defocus symmetric wrt the plane in

which the wavefront is to be determined• So what, if anything, is special about

defocus?• Is the geometric picture just a prop?• What other convolutions might be used?

27 April 2005 20

Convolution model• Wavefront is convolved

with a filter function• In conventional phase

diversity this represents propagation

• Convolution represents propagation equal distances either side of measurement plane

wavefront

Filter psf+

Filter psf-

wavefront

27 April 2005 21

wavefront

Filter psf+ Filter psf+

wavefront

27 April 2005 22

wavefront

Filter psf+ Filter psf+

wavefront

27 April 2005 23

wavefront

Filter psf+ Filter psf+

wavefront

27 April 2005 24

Null sensing• The basic requirement

for AO is a null sensorIt is sufficient, but not necessary, to have a metric that is zero if the wavefront is flatIt is desirable, but not necessary, to have a wfs that indicates the sense of any deviation from flatness

27 April 2005 25

Symmetry conditions• Required properties

for aberration functions are†:

† Campbell, Zhang, Greenaway and Restaino, Opt Lett, 29(2004)2707-2709

Filter function must be complexReal and imaginary parts must have same symmetryTwo filter function with opposite sense can be used to provide signal after subtraction

27 April 2005 26

Metrology• Metrology applications

need numbers…

• Model test wavefront– Hermitian– Anti-Hermitian

parts

Need data inversionIdeally analytic

If not, iterations must converge quickly

How to invert data for generalised phase diversity?

. * . . * .

. * . . * .

d( ) H( ) I( ) H ( ) R( ) H( ) R( ) H ( ) I( )2

A( ) I( ) A ( ) R( ) A( ) R( ) A ( ) I( )

ir ir ir ir

ir ir ir ir

r d e d e d e d ei

d e d e d e d e

′ ′− −

′ ′− −

′ ′ ′ ′ ′ ′= −

′ ′ ′ ′ ′ ′+ −

∫ ∫ ∫ ∫∫ ∫ ∫ ∫

ξ ξ ξ ξ

ξ ξ ξ ξ

ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ

ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ

27 April 2005 27

Diffractive optics• Such filters are

suitable for DOEs• Equal and opposite

errors automatically produced in + orders

• Consistent with use of defocus error

Then:null signal flat wavefrontreversing sign of error reverses sign of signalPosition of signal may show error location

. * . . *

. * . . *

d( ) H( ) I( ) H ( ) R( ) H( ) R( ) H ( ) I( )2

A( ) I( ) A ( ) R( ) A( ) R( ) A ( ) I(

ir ir ir

ir ir ir

r d e d e d e di

d e d e d e d

′ .

.)

ir

ir

e

e

′− −

′ ′− −

′ ′ ′ ′ ′ ′= −

′ ′ ′ ′ ′ ′+ −

∫ ∫ ∫ ∫∫ ∫ ∫ ∫

ξ ξ ξ ξ

ξ ξ ξ

ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ

ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ

27 April 2005 28

Computer simulation• Same wavefront aberration with defocus and with

spherical aberration kernel• Can produce greater signal amplitude for same

signal input and same amplitude in kernel function

27 April 2005 29

Generalised phase diversityMany suitable filters, best signal wavefront dependentTried Gerchberg-Saxton - often hangsSuccessful with small signalsNow using SAE with iterative refinement

• Initially tested signal amplitude

• Generalised analytic solution hard

• Small-angle approximation

27 April 2005 30

Continuous phase...

• Explore the phase reconstruction systematically

• Continuous phase distribution ‘illuminated’ with a ‘laser beam waist’(flat, Gaussian profile)

27 April 2005 31

Continuous phase...

27 April 2005 32

Continuous phase...• Reconstructions seem

good up to a limit, then we have sudden failure

• Note that a section fails - after this the reconstruction shape is good again

• Failure due to slope + phase swing?

27 April 2005 33

Limitations...• We are still trying to

determine how slope & amplitude combine to cause problem

• Hope to modify algorithm - but may need to modify diversity filter

27 April 2005 34

SAE

PV error of inputWavefront = 0.27λ

Defocus diversityUsed.

This figure shows the original input phase, and the retrieved phase calculated by the SAE algorithm. New data is computed from the solution and compared to the ‘measured’ data.An ‘error’ phase is calculated from the difference and used to iteratively refine the solution.

27 April 2005 35

Direct application of SAE(section through 2-d sample)

0 50 100 150 200 250 300-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

original phaseretrieved phaseerror phase

27 April 2005 36

0 50 100 150 200 250 300-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

original phaseretrieved phaseerror phase

After second iteration...

27 April 2005 37

Discontinuous wavefrontsInitially a surprise, but verified by simulations‘No problem’ if step is <<π/2

• Theory suggests signal is null for π phase jump

• Small discontinuities reconstruct easily

27 April 2005 38

Problems with larger steps...

27 April 2005 39

Failure with large steps

27 April 2005 40

SummarySignal can be increased wrt defocusPossibility to optimise filter (using statistical model of zernike modes?)

Rapid convergence, cope with small discontinuities, speckle, multiple-connectivity?

• Phase diversity can be generalised

• Data inversion can be fast

27 April 2005 41

Summary• Completely analytic

solution not yet found• Is speed a problem

• Is increased sensitivity useful, or is need to optimise restricting?

• Is boundary-value problem solved?

Still considered desirable and sought forConvergence is fast and computation scales as NlogN, not N4 (GFS)If restricting, is defocus the best general choice?

Indication that this still requires more work