M. P. Oxley, L. J. Allen, W. McBride and N. L. O’Leary School of Physics, The University of Melbourne Iterative wave function reconstruction.

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

W. McBride and N. L. O’LearySchool of Physics, The University of Melbourne

Iterative wave function reconstruction

• 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.

• 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.

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.

rrr T 0The 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.

qqq T0

For coherent aberrations the image may be formed by the convolution of the exit surface wave function withthe transfer function of the imaging system Tr.

r0

qT i[exp ]

max

max

0

1

q

qA

q

qq

A(q) (q)

q4(q) q2 0.53Csf

sample

lens

objective aperture

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).

qI 0q´+qTq´+q0* q´T *q´

In the presence of incoherence the diffractogram (the Fourier transform of the real space intensity) is propagated via the convolution1,2:

[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.

Esq´q, q´E(q´+q,q´)dq´

Envelope function describing finite source size i.e. Spatial coherence

Envelope function describing defocus spread due to variation in the incident wavelength.i.e. Temporal coherence

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).

The general form of the envelope function for spatial coherence, based on a first order Taylor expansion of the phase transfer function q is given by:

2

2

2

s

)(

)(

)(

4exp

),(

q

q

qq

qq

qqq

E

is the is the semi-angle subtended by the finite source size.

For modern HRTEM, using a field emission gun (FEG), 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:

)()(),( cohs

cohss qqqqqq EEE

where

2

2

2cohs

)(

4exp)(

q

qq

E

The general form of the envelope function for temporal coherence, based on a first order Taylor expansion of the phase transfer function q is given by:

22 )()(

4exp

),(

ff

E

qqq

qqq

is the 1/e value of the Gaussian distribution of the defocus spread due to variations in the incident wavelength .

For a FEG, even though the spread in incident wavelengths is quite small,there can still be a substantial defocus spread . We will use the separableapproximation:

)()(),( cohcoh qqqqqq EEE

where

22coh )(

4exp)(

fE

qq

Assuming the form of and presented, the momentum space wave function qmay be calculated using

qqqqq cohcohs0 EET

qcohsE qcoh

E

• 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).

• 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 BECCryptography

rexp1I

rexp2I

rexp3I

rj1

rj3

rj2

rjn

We start with N experimental images at different defocus values.A phase is guessed for each plane: usually for j 1

rexpnI

0rjn

Construct initial wave function at each plane n

rrr jnn

jn iI expexp

rj1

rj2

rj3

rj1

rj2

rj3

qj1

qj2

qj3

Propagate the wave function at each plane to the exit surface 1cohcoh

s0,

nnn EET qqqqq

qj0

~

Construct estimate of exit surface wave function for the jth iteration

N

n

jn

j

N 1 0,0

1~ qqPropagate the estimated exit surface wave function back to each plane

njjn EET qqqqq cohcoh

s0~~

qj1

~

qj2

~

qj3

~

qj1

~

qj2

~

qj3

~

iI jj exp~~11 rr

iI jj exp~~22 rr

iI jj exp~~33 rr

rj1

~

rj3

~

rj2

~

Calculate the sum squared error in the wave function amplitude at each plane

pixels

exp2

pixels

exp ~SSE rrr n

jnn

jn III

j1SSE

j2SSE

j3SSE

Construct average SSE

N

n

jn

j

N 1avg. SSE1

SSEIf j 1, or then j j 1 1avg.avg. SSESSE jj

rj0

~

If output 1avg.avg. SSESSE jj rj

0~

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.

• 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.

A. Ziegler, C. Kisielowski, R.O. Ritchie, Acta

B. Materialia 50 (2002) 565-574.

Case 1: 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.

Silicon

Nitrogen Unit cell

• Si3N4 is a light, hard engineering ceramic with many industrial applications due to its strength.

• [0001] Zone axis orientation.

• It has a hexagonal structure.

f = -2831.7 Å

f = -2812.4 Å

f = -2793.1 Å

f = -2773.8 Å

f = -2754.5 Å

f = -2735.2 Å

f = -2715.9 Å

f = -2696.6 Å

f = -2677.3 Å

f = -2658.0 Å

f = -2638.7 Å

f = -2619.4 Å

f = -2600.1 Å

f = -2580.8 Å

f = -2561.5 Å

f = -2542.2 Å

f = -2522.9 Å

f = -2503.6 Å

f = -2484.3 Å

f = -2465.0 Å

• 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).

Standard Deviation

IWFR 0.275

MAL 0.304

IWFR 0.202

MAL 0.201

• 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.

Unit cell Silicon

2.75ÅNitrogen

0.8Å

• 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.

Defocus step

19.3 Å

f = -2580.8 Å

f = -2561.5 Å

f = -2542.2 Å

Iteration

0 2 4 6 8 10

SS

Ej n(

x10-2

)

0.0

0.5

1.52.02.5

-2831.7 Å-2793.1 Å -2658 Å -2561.5 ÅAverage

f = -2677.3 Å

f = -2658.0 Å

f = -2638.7 Å

• 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.

Number of Input Images

0 5 10 15 20

SS

Eav

g( x

10-2

)

0.05

0.10

0.15

0.20

0.25

0.30• 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 5 the value of the average SSE has stabilized.

f = -2831.7 Å

f = -2812.4 Å

f = -2793.1 Å

f = -2773.8 Å

f = -2754.5 Å

f = -2735.2 Å

f = -2715.9 Å

f = -2696.6 Å

f = -2677.3 Å

f = -2658.0 Å

f = -2638.7 Å

f = -2619.4 Å

f = -2600.1 Å

f = -2580.8 Å

f = -2561.5 Å

f = -2542.2 Å

f = -2522.9 Å

f = -2503.6 Å

f = -2484.3 Å

f = -2465.0 Å

Series A

Series B

Series C

Series D

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

Standard deviations about average value

Image Phase

-217 Å -361 Å -506 Å -651 Å -795 Å

Intensity Image Phase Map

(Data provided by Lian Mao Peng)

B.E. Allman et al.

J Opt. Soc. Am. A17

(2000) 1732-1743

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.