M. P. Oxley, L. J. Allen,

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Iterative wave function reconstruction

• M. P. Oxley, L. J. Allen,
• W. McBride and N. L. OLeary
• School of Physics, The University of Melbourne

2

• C. Kisielowski, National Center for Electron
  Microscopy (NCEM), Lawrence Berkeley National
  Laboratory Si3N4 data and related results from
  the MAL algorithm.
• J. Ayache, National Center for Electron
  Microscopy SrTiO3 bi-crystal data.
• Lian Mao Peng, Peking University, Beijing
  Potassium titanate nanowire data.
• P. J. McMahon, The University of Melbourne AFM
  tip X-ray data.
• D Paganin, Melbourne/Monash University for many
  useful discussions.

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• Output microscope images are often not directly
  interpretable due to incoherence effects and
  aberrations on top of phase modulation.
• Many objects in electron microscopy fall in to
  the category of phase objects, i.e. intensity
  measurements contain minimal information.
• Wave function reconstruction allows
• Removal of coherent aberrations,
• Correction for partial coherence to the extent it
  is present in modern FEG TEM,
• Provides structural information for phase
  objects.

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• Why iterative wave function reconstruction?
• Iterative methods are simply understood and
  straight forward to implement.
• They are applicable to many experimental
  circumstances.
• a) Resolutions from atomic level to nano level.
• b) Periodic and non periodic structures.
• The global nature of the method presented here
  makes the method robust in the presence of noise.
  
• The method is robust in the presence of phase
  discontinuities.

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For coherent aberrations the image may be formed
by the convolution of the exit surface wave
function with the transfer function of
the imaging system T(r).
The vector r is perpendicular to the direction of
propagation. This is most conveniently done in
momentum space. i.e. by Fourier transforming both
sides. The vector q is that conjugate to r.
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A(q)
c(q)
sample
lens
objective aperture
q4
c(q) p
l
q2 0.5p
l
3
Cs
Df
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• Advantages of coherent propagation
• Image formation is based on the propagation of
  the whole wave function, i.e. Intensity and
  Phase.
• Propagation is numerically efficient using fast
  Fourier transforms.
• Problems with coherent propagation
• Many sources are NOT strictly coherent.
• In particular high resolution transmission
  electron microscopy (HRTEM) requires careful
  treatment of
• a) Finite source size (Spatial coherence),
• b) Defocus spread (Temporal coherence).

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In the presence of incoherence the diffractogram
(the Fourier transform of the real space
intensity) is propagated via the convolution1,2
Y0(qq)T(qq) Y0 (q)T (q)
Es(qq, q)
ED(qq,q)
dq
Envelope function describing defocus spread due
to variation in the incident wavelength. i.e.
Temporal coherence
Envelope function describing finite source size
i.e. Spatial coherence
1 K. Ishizuka, Ultramicroscopy 5 (1980)
55-65. 2 W.M.J. Coene, A. Thust, M. Op de
Beeck, D. Van Dyck, Ultramicroscopy 64 (1996)
109-135.
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• Advantages of partially coherent propagation
• The intensities at each defocus plane are
  calculated from the incoherent addition of
  propagated intensities as is appropriate.
• Accounts for the blurring of images due to
  incoherence.
• Problems with partially coherent propagation
• Only the intensity is propagated. There is no
  estimate of the quantum mechanical phase other
  than at the exit surface.
• Propagation of intensity is based on the
  evaluation of a two dimensional convolution
  integral as the envelope functions are not in
  general separable Numerically intensive,
  especially for large numbers of measured pixels
  (e.g. 1024 by 1024).

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The general form of the envelope function for
spatial coherence, based on a first order Taylor
expansion of the phase transfer function c(q),
is given by
b is the is the semi-angle subtended by the
finite source size.
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For modern HRTEM, using a field emission gun
(FEG), b is small. In particular, for focused
beams (e.g. CBED or STEM), beam convergence is
due to the coherent focussing of the beam by the
probe forming optics. Treating total beam
convergence incoherently may lead to an
overcorrection. We hence approximate the spatial
coherence envelope, in the separable form, as
where
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The general form of the envelope function for
temporal coherence, based on a first order
Taylor expansion of the phase transfer function
c(q), is given by
D is the 1/e value of the Gaussian distribution
of the defocus spread due to variations in the
incident wavelength l.
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For a FEG, even though the spread in incident
wavelengths is quite small, there can still be a
substantial defocus spread D. We will use the
separable approximation
where
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Assuming the form of and
presented, the momentum space wave function Y(q)
may be calculated using

• This allows rapid calculation wave function
  propagation using fast Fourier transforms.
• The extension to propagation between planes other
  than the exit surface is obvious.
• The ability to rapidly calculate propagation of
  the wave function makes this approximation
  amenable to methods based upon iterative wave
  function reconstruction (IWFR).

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• The method is generally based on the measurement
  of a through focal series (TFS) of images in real
  space.
• May in principle use information from other than
  variation in defocus, for example diffraction
  patterns.
• Based on the propagation of the entire wave
  function.
• Works in the presence of phase discontinuities.
• Requires that images be aligned.

Over-sampling
Astigmatic fields
Time evolution in BEC
Cryptography
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• The most appealing feature of this method is its
  simplicity.
• In the spirit of the original Gerchberg-Saxton
  algorithm, the intensity is simply updated at
  each iteration
• It can be easily modified to suit a number of
  experimental regimes.
• It is easily understood and simple to implement.

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• The method is global. The exit surface wave
  function is calculated using equally weighted
  information from all images in the TFS.
• Because of its global nature of this method copes
  well with noise.
• Because the SSE is calculated at each plane, for
  each iteration, faulty data planes can be
  removed or re-measured.
• Fast due to the use of fast Fourier transforms.
• Produces consistent results from independent
  image sets.

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• Case 1 b phase Si3N4
• Phillips CM30/FEG/UT microscope at NCEM with a
  resolution
• of less than one Angstrom.
• Sample 100 Å thick with thin amorphous carbon
  layer to allow for
• determination of defocus and spherical
  aberration.
• 0001 zone axis orientation.
• 20 images in total.

1. Ziegler, C. Kisielowski, R.O. Ritchie, Acta
2. Materialia 50 (2002) 565-574.

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Silicon

• Si3N4 is a light, hard engineering ceramic with
  many industrial applications due to its strength.
• 0001 Zone axis orientation.
• It has a hexagonal structure.

Nitrogen
Unit cell
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Df -2831.7 Å
Df -2754.5 Å
Df -2522.9 Å
Df -2677.3 Å
Df -2600.1 Å
Df -2812.4 Å
Df -2735.2 Å
Df -2658.0 Å
Df -2503.6 Å
Df -2580.8 Å
Df -2793.1 Å
Df -2715.9 Å
Df -2638.7 Å
Df -2561.5 Å
Df -2484.3 Å
Df -2773.8 Å
Df -2696.6 Å
Df -2619.4 Å
Df -2542.2 Å
Df -2465.0 Å
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• After alignment the images were reduced in size
  to 902 by 940 pixels and padded back to 1024 by
  1024.
• Only 18 of 20 images were used. This will be
  expanded on later.
• Results are compared to the MAL algorithm using
  the same parameters (all 20 images used in MAL
  reconstruction).

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Standard Deviation IWFR 0.275 MAL 0.304
IWFR 0.202 MAL 0.201
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• Excellent quantitative agreement is achieved
  between the two methods.
• While IFWR uses a coherent treatment of temporal
  coherence, MAL uses a more exact formulation.
• The close agreement between the two methods
  suggests the coherent approximation is good in
  this experimental regime.
• The close agreement shows that damping down of
  the image and phase is due to the nature of the
  data set (Stobbs?) and not an artifact of the
  method.

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• As expected for a nominal weak phase object, the
  projected structure is seen in the phase.
• The hexagonal symmetry is obvious.
• The Si-N pairs are easily seen.
• N locations are not well resolved.

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Defocus step 19.3 Å
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• Wave function reconstruction may be done with as
  few as two images (from experiencethree may be
  required for uniqueness).
• Alignment of images however requires closely
  spaced defocus steps.
• With fewer images the noise level on each image
  has a greater effect on the result.
• For small numbers of images, the presence of
  faulty data will have a greater effect on the
  result.

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• Here we compare the average SSE for differing
  numbers of images.
• For few images the SSE is small due to the weak
  constraint in the presence of noise.
• For N ³ 5 the value of the average SSE has
  stabilized.

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Series A
Df -2831.7 Å
Df -2754.5 Å
Df -2677.3 Å
Df -2600.1 Å
Df -2522.9 Å
Series B
Df -2812.4 Å
Df -2735.2 Å
Df -2658.0 Å
Df -2580.8 Å
Df -2503.6 Å
Series C
Df -2793.1 Å
Df -2715.9 Å
Df -2638.7 Å
Df -2561.5 Å
Df -2484.3 Å
Series D
Df -2773.8 Å
Df -2696.6 Å
Df -2619.4 Å
Df -2542.2 Å
Df -2465.0 Å
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Standard deviations about average value
Image Phase
IFWR 0.275 0.202
A 0.295 0.210
B 0.271 0.169
C 0.230 0.169
D 0.298 0.209
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Image
Phase
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-217 Å
-361 Å
-506 Å
-651 Å
-795 Å
Intensity Image
Phase Map
(Data provided by Lian Mao Peng)
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B.E. Allman et al. J Opt. Soc. Am. A17 (2000)
1732-1743
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• Advantages
• Based on direct propagation of the wave function.
  Both phase
• and intensity are found
• The global nature of the algorithm assures
  robustness in the
• presence of noise and discontinuities.
• Straight forward to implement.
• Applicable to a wide range of experimental
  conditions.
• Suitable for periodic and non-periodic samples.
• Monitoring of convergence provides valuable
  information about
• the data sets.