1. 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

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

3. 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.

4. 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.

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

6. The conventional view

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

8. Sensing Phase
Phase gradients are reflected in the flow of energy

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

10. 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, (1995).

11. Intensity profile Phase structure
Optical Doughnuts
Intensity profile Phase structure

12. 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, (2002).

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

14. X-ray Vortex from Simple Three-Molecule Diffraction

15. 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, (1996)

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

17. X-Ray Complex Phase Tomography

18. Modern Diffraction Physics
<70nm
Rayleigh point

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

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

21. 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.

22. 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.

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

24. Boundary Conditions – Neuman ProblemG

25. 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.

26. 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

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

29. 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.

30. Collaborators David Paterson (now @ APS) Ian McNulty (APS)
David Paganin Monash U)
Anton Barty LLNL)
Justine Tiller (now a Management Consultant)
Eroia Barone-Nugent (UM – Botany)
Phil McMahon DSTO)
Brendan Allman (now with IATIA Ltd)
Andrew Peele (UM)
Ann Roberts (UM)
Chanh Tran (ASRP Fellow)
David Paterson APS)
Ian McNulty (APS)
Barry Lai (APS)
Sasa Bajt (LLNL)
Henry Chapman (LLNL)
Anatoly Snigirev (ESRF)