Astigmatic diffraction – A unique Astigmatic diffraction – A unique solution to the non-crystallographic solution to the non-crystallographic

phase problemphase problem

Keith A. NugentKeith A. NugentSchool of PhysicsSchool of Physics

The University of MelbourneThe University of Melbourne

AustraliaAustralia

Why do we recover wavefieldsWhy do we recover wavefieldsRecover as much Recover as much

information about the information about the wavefield as possiblewavefield as possible

Compensate for the Compensate for the experimental conditionsexperimental conditions

Relate measurement to Relate measurement to structurestructure

What Characterizes Wavefields?What Characterizes Wavefields?

For Gaussian statistics, For Gaussian statistics, a wavefield is fully a wavefield is fully characterised by the characterised by the mutual coherence mutual coherence function.function.

This is a complex four-diimensional function describing the This is a complex four-diimensional function describing the phase and visibility of fringes created by Young’s phase and visibility of fringes created by Young’s experiments as a function of the two-dimensional positions experiments as a function of the two-dimensional positions of the pinholes.of the pinholes.

2*2, xrExrExrJ

The coherence The coherence functionfunction

We have recently measured the correlations for 7.9keV x-rays from the 2-ID-D beamline at the APS

J.Lin, D.Paterson, A.G.Peele, P.J.McMahon, C.T.Chantler, K.A.Nugent, B.Lai, N.Moldovan, Z.Cai, D.C.Mancini and I.McNulty, Measurement of the spatial coherence function of undulator radiation using a phase mask, Phys.Rev.Letts, in press.

Seeing PhaseSeeing Phase

Refraction of Refraction of light passing light passing through water is through water is a phase effect.a phase effect.

The twinkling of a The twinkling of a star is an star is an analogous analogous phenomenonphenomenon

The conventional viewThe conventional view

An alternative perspectiveAn alternative perspective

dx

d

D.Paganin and K.A.Nugent, D.Paganin and K.A.Nugent, Non-Interferometric Phase Imaging with Partially-Coherent LightNon-Interferometric Phase Imaging with Partially-Coherent Light, Physical Review Letters, 80, 2586-2589 (1998), Physical Review Letters, 80, 2586-2589 (1998)

Sensing PhaseSensing Phase

Phase Phase gradients are gradients are reflected in reflected in the flow of the flow of energyenergy

Neutron ImagingNeutron Imaging

P.J.McMahon et al, P.J.McMahon et al, Contrast mechanisms in neutron radiographyContrast mechanisms in neutron radiography, Appl.Phys.Letts, , Appl.Phys.Letts, 7878, 1011-1013 (2001), 1011-1013 (2001)

One approach to solutionOne approach to solution

Assume the paraxial approximation:

rrIzrI

2

This equation has a unique solution in the case where the phase front is not discontinuous.

A measurement of the probability (intensity) and its longitudinal derivative specifies the complete wave (function) over all space!

T.E.Gureyev, A.Roberts and K.A.Nugent, T.E.Gureyev, A.Roberts and K.A.Nugent, Partially coherent fields, the transport of intensity equation, and phase uniqueness. Partially coherent fields, the transport of intensity equation, and phase uniqueness. J.Opt.Soc.Am.AJ.Opt.Soc.Am.A., ., 12, 1942-1946 (1995).12, 1942-1946 (1995).

Optical DoughnutsOptical Doughnuts

Intensity profileIntensity profile Phase structurePhase structure

X-ray VorticesX-ray Vortices

A.G.Peele et al, A.G.Peele et al, Observation of a X-ray vortex, Observation of a X-ray vortex, Opt.Letts.,Opt.Letts., 27 27, 1752-1754 (2002)., 1752-1754 (2002).

A 9keV photon carrying 1A 9keV photon carrying 1ħ of orbital angular momentumħ of orbital angular momentum

250 m

X-ray Vortex – Charge 4X-ray Vortex – Charge 4A 9keV photon carrying 4A 9keV photon carrying 4ħ of orbital angular momentumħ of orbital angular momentum

X-ray Vortex from Simple Three-Molecule X-ray Vortex from Simple Three-Molecule DiffractionDiffraction

Hard X-ray PhaseHard X-ray Phase

KA Nugent, T.E.Gureyev, D.F.Cookson, D.Paganin and Z.Barnea, KA Nugent, T.E.Gureyev, D.F.Cookson, D.Paganin and Z.Barnea, Quantitative Phase Imaging Using Hard X-Rays, Quantitative Phase Imaging Using Hard X-Rays, Phys.Rev.Letts, 77, Phys.Rev.Letts, 77, 2961-2964 (1996)2961-2964 (1996)

High Resolution X-ray TomographyHigh Resolution X-ray Tomography

P.J.McMahon et al, P.J.McMahon et al, Quantitative Sub-Micron Scale X-ray Phase TomographyQuantitative Sub-Micron Scale X-ray Phase Tomography, , Opt.CommunOpt.Commun., ., in pressin press..

X-Ray Complex Phase TomographyX-Ray Complex Phase Tomography

Modern Diffraction PhysicsModern Diffraction Physics

<70nm<70nm

Rayleigh pointRayleigh point

Far-Field Diffraction with Curved Incident Far-Field Diffraction with Curved Incident BeamBeam

r

ek~krU

~ rik

detf

0

k~cosikk~scatdet

02

Far-fieldFar-field : Detected field has negligible curvature : Detected field has negligible curvature

FraunhoferFraunhofer: Detected field : Detected field ANDAND incident field incident field have negligible curvaturehave negligible curvature

rder k~ rkiobjscat

Far-Field Diffraction with Curved Incident Far-Field Diffraction with Curved Incident BeamBeam

R

rkieer zik

zkR

rki

inc

2010

0

20

Incident field has parabolic curvatureIncident field has parabolic curvature

rdez,rrR

k cosik~ rik

scat

20

k*~k~iRek~cos

kkI scatscatf

2

2

222

0

Change in measured intensity is formally Change in measured intensity is formally identical to the ToI equation!identical to the ToI equation!

kkIkIRk ff

02

1

Vortices are ubiquitous in the far-field and so Vortices are ubiquitous in the far-field and so this equation cannot be solved uniquely, except this equation cannot be solved uniquely, except

under very special conditions.under very special conditions.

Far-Field Diffraction with Cylindrical Far-Field Diffraction with Cylindrical Incident BeamIncident Beam

kSIR f

2

1

Written in this way, we see that the intensity Written in this way, we see that the intensity difference is proportional to the divergence of difference is proportional to the divergence of the Poynting vector in the far-field.the Poynting vector in the far-field.

kkIk

kS

0

1

kkIkIRkxx kfkf

02

1

yxxfx kgdkkIRk

kS

02

1

Far-Field Diffraction with Cylindrical Far-Field Diffraction with Cylindrical Incident BeamIncident Beam

Now consider cylindrical curvature incident. Now consider cylindrical curvature incident. An identical argument gives:An identical argument gives:

This may be directly integrated to obtain:This may be directly integrated to obtain:

Boundary Conditions – Neuman ProblemBoundary Conditions – Neuman Problem

xn

0

kS x

Full Phase RecoveryFull Phase Recovery

In this way, we are able to obtain both components of In this way, we are able to obtain both components of the Poynting vector. The Poynting vector completely the Poynting vector. The Poynting vector completely specifies the field.specifies the field.

This may be This may be integrated to recover integrated to recover the phase but is not so the phase but is not so easy as care needs to easy as care needs to be taken in the be taken in the presence of vortices.presence of vortices.

Check for convergence

FINISH

Phase guess for

object structure

Apply planar diffraction data constraints to

intensityApply weak support

constraint and x-cylinder curvature

Apply x-cylinder diffraction data

constraints to intensity

Apply weak support constraint and y-cylinder

curvatureApply y-cylinder diffraction data

constraints to intensity

Apply weak support constraint and zero

curvature

Apply planar diffraction data

constraints to intensity

““Homometric” StructuresHomometric” Structures

These are finite These are finite structures that structures that produce identical produce identical diffraction patterns diffraction patterns and have identical and have identical autocorrelation autocorrelation functions – they functions – they cannotcannot be resolved be resolved using oversampling using oversampling techniques.techniques.

SummarySummary

• Can view phase as a Can view phase as a rather geometric rather geometric property of light.property of light.

• This yields methods that This yields methods that are very simple to are very simple to implement.implement.

• Phase dislocations are Phase dislocations are important.important.

• Can work with radiation Can work with radiation of all sorts.of all sorts.

• Can do tomographic Can do tomographic measurements.measurements.

CollaboratorsCollaborators

• David Paganin (now @ Monash U)

• Anton Barty (now @ LLNL)• Justine Tiller (now a

Management Consultant)• Eroia Barone-Nugent (UM –

Botany)• Phil McMahon (now @

DSTO)• Brendan Allman (now with

IATIA Ltd)• Andrew Peele (UM)• Ann Roberts (UM)• Chanh Tran (ASRP Fellow)

•David Paterson (now @ APS)

•Ian McNulty (APS)

•Barry Lai (APS)

•Sasa Bajt (LLNL)

•Henry Chapman (LLNL)

•Anatoly Snigirev (ESRF)