Astigmatic diffraction A unique solution to the noncrystallographic phase problem

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Astigmatic diffraction A unique solution to the
non-crystallographic phase problem

• Keith A. Nugent
• School of Physics
• The University of Melbourne
• Australia

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Why do we recover wavefields
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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 Youngs experiments as a function of
the two-dimensional positions of the pinholes.
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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.
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Seeing Phase
Refraction of light passing through water is a
phase effect. The twinkling of a star is an
analogous phenomenon
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The conventional view
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An alternative perspective
D.Paganin and K.A.Nugent, Non-Interferometric
Phase Imaging with Partially-Coherent Light,
Physical Review Letters, 80, 2586-2589 (1998)
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Sensing Phase

• Phase gradients are reflected in the flow of
  energy

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

• Intensity profile Phase structure

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X-ray Vortices
A 9keV photon carrying 1h of orbital angular
momentum
A.G.Peele et al, Observation of a X-ray vortex,
Opt.Letts., 27, 1752-1754 (2002).
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X-ray Vortex Charge 4
A 9keV photon carrying 4h of orbital angular
momentum
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X-ray Vortex from Simple Three-Molecule
Diffraction
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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)
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High Resolution X-ray Tomography
P.J.McMahon et al, Quantitative Sub-Micron Scale
X-ray Phase Tomography, Opt.Commun., in press.
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X-Ray Complex Phase Tomography
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Modern Diffraction Physics
lt70nm
Rayleigh point
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Far-Field Diffraction with Curved Incident Beam
Far-field Detected field has negligible
curvature
Fraunhofer Detected field AND incident field
have negligible curvature
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Far-Field Diffraction with Curved Incident Beam
Incident field has parabolic curvature

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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.
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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.
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Far-Field Diffraction with Cylindrical Incident
Beam
Now consider cylindrical curvature incident. An
identical argument gives
This may be directly integrated to obtain
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Boundary Conditions Neuman Problem
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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.
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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
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Homometric Structures
These are finite structures that produce
identical diffraction patterns and have identical
autocorrelation functions they cannot be
resolved using oversampling techniques.
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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.

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Collaborators

• David Paganin (now _at_ Monash U)
• Anton Barty (now _at_ LLNL)
• Justine Tiller (now a Management Consultant)
• Eroia Barone-Nugent (UM Botany)
• Phil McMahon (now _at_ DSTO)
• Brendan Allman (now with IATIA Ltd)
• Andrew Peele (UM)
• Ann Roberts (UM)
• Chanh Tran (ASRP Fellow)

• David Paterson (now _at_ APS)
• Ian McNulty (APS)
• Barry Lai (APS)
• Sasa Bajt (LLNL)
• Henry Chapman (LLNL)
• Anatoly Snigirev (ESRF)