The University of Melbourne Astigmatic diffraction – A unique solution to the non-crystallographic phase problem Keith A. Nugent School of Physics The University of Melbourne Australia

Why do we recover wavefields Recover as much information about the wavefield as possible Compensate for the experimental conditions Relate measurement to structure

What Characterizes Wavefields? For Gaussian statistics, a wavefield is fully characterised by the mutual coherence function. This is a complex four-diimensional function describing the phase and visibility of fringes created by Young’s experiments as a function of the two-dimensional positions of the pinholes.

The coherence function 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 Phase Refraction of light passing through water is a phase effect. The twinkling of a star is an analogous phenomenon

The conventional view

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

Sensing Phase Phase gradients are reflected in the flow of energy

Neutron Imaging P.J.McMahon et al, Contrast mechanisms in neutron radiography, Appl.Phys.Letts, 78, 1011-1013 (2001)

One approach to solution Assume the paraxial approximation: 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, Partially coherent fields, the transport of intensity equation, and phase uniqueness. J.Opt.Soc.Am.A., 12, 1942-1946 (1995).

Intensity profile Phase structure Optical Doughnuts Intensity profile Phase structure

X-ray Vortices A 9keV photon carrying 1ħ of orbital angular momentum A.G.Peele et al, Observation of a X-ray vortex, Opt.Letts., 27, 1752-1754 (2002).

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

X-ray Vortex from Simple Three-Molecule Diffraction

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

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

X-Ray Complex Phase Tomography

Modern Diffraction Physics <70nm Rayleigh point

Far-Field Diffraction with Curved Incident Beam Far-field : Detected field has negligible curvature Fraunhofer: Detected field AND incident field have negligible curvature

Far-Field Diffraction with Curved Incident Beam Incident field has parabolic curvature

Change in measured intensity is formally identical to the ToI equation! Vortices are ubiquitous in the far-field and so this equation cannot be solved uniquely, except under very special conditions.

Far-Field Diffraction with Cylindrical Incident Beam Written in this way, we see that the intensity difference is proportional to the divergence of the Poynting vector in the far-field.

Far-Field Diffraction with Cylindrical Incident Beam Now consider cylindrical curvature incident. An identical argument gives: This may be directly integrated to obtain:

Boundary Conditions – Neuman Problem G

Full Phase Recovery In this way, we are able to obtain both components of the Poynting vector. The Poynting vector completely specifies the field. This may be integrated to recover the phase but is not so easy as care needs to be taken in the presence of vortices.

Phase guess for object structure Apply planar diffraction data constraints to intensity Apply weak support constraint and x-cylinder curvature Apply x-cylinder diffraction data constraints to intensity Apply weak support constraint and y-cylinder curvature Apply y-cylinder diffraction data constraints to intensity Apply weak support constraint and zero curvature Apply planar diffraction data constraints to intensity Check for convergence FINISH

“Homometric” Structures These are finite structures that produce identical diffraction patterns and have identical autocorrelation functions – they cannot be resolved using oversampling techniques.

Summary Can view phase as a rather geometric property of light. This yields methods that are very simple to implement. Phase dislocations are important. Can work with radiation of all sorts. Can do tomographic measurements.

Collaborators David Paterson (now @ APS) Ian McNulty (APS) 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)