ELECTRON INTERFEROMETRY USING AN AMPLITUDE DIVIDING
GRATING BEAMSPLITTER: DEVELOPMENT AND APPLICATION
by
FEHMI SAMI YASIN
A DISSERTATION
Presented to the Department of Physics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2019
DISSERTATION APPROVAL PAGE
Student: Fehmi Sami Yasin
Title: Electron Interferometry Using an Amplitude Dividing Grating Beamsplitter:
Development and Application
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Physics by:
Benjamı́n Alemán Chair
Benjamin J. McMorrran Advisor
Michael Raymer Core Member
Brad Nolen Institutional Representative
and
Janet Woodruff-Borden Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded June 2019
ii
©c 2019 Fehmi Sami Yasin
This work is licensed under a Creative Commons
Attribution (United States) License.
cb
iii
DISSERTATION ABSTRACT
Fehmi Sami Yasin
Doctor of Philosophy
Department of Physics
June 2019
Title: Electron Interferometry Using an Amplitude Dividing Grating Beamsplitter:
Development and Application
Electron microscopes can be used for atomic resolution imaging myriad materials
including semiconductor nanomaterials integral to modern technology, biomolecular
materials that compose all of life and the carbon energy cycle, and 2-D materials
with various potential applications. Due to the high electron dose required to form
contrast in conventional electron microscopy imaging techniques, many biomolecular
and low atomic number materials are destroyed before an image can be formed.
Electron interferometry shows promise as a lower dose imaging technique due to
the difference in how contrast is formed. Electron holography, for example, uses an
electrostatic charged wire as an electron biprism in order to overlap the interaction
and vacuum electron waves to form interference fringes. These fringes can be imaged
directly and processed to measure the object transmission function, providing both
spatial information such as atomic locations, and quantitative phase and amplitude
information, necessary for thickness and electric and magnetic field measurement
within the specimen. In this work, we combine electron holography with Scanning
Transmission Electron Microscopy (STEM), which forms a focussed probe beam at the
iv
specimen and raster scans the beam over a field of view. We’ve experimentally realized
a path-separated electron interferometer in this mode, called STEM holography
(STEMH), and apply it to image gold nanoparticles on thin amorphous carbon
at subnanometer resolution. STEMH simultaneously forms efficient, interpretable
contrast for both of these materials, allowing us to confirm the presence of string-like
structure within the amorphous carbon, previously thought to be randomly bonded
and oriented. Additionally, we’ve devised and implemented a multi-biprism design
that enables tuning of the path separation between the arms of the interferometer
at the specimen plane, and we demonstrate the largest path-separated amplitude
division electron interferometer to date. This flexible STEMH enables large geometry
experiments and a means to precisely place the probes in the specimen plane,
enabling imaging around beam-sensitive materials and probing fundamental physical
phenomena in or around materials. On its own, STEMH can probe fundamental fields
with atomic resolution, and advances in detector technology may allow STEMH to
image beam-sensitive materials without destroying them in the process.
This dissertation includes previously published co-authored material.
v
CURRICULUM VITAE
NAME OF AUTHOR: Fehmi Sami Yasin
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene
Westminster College, Salt Lake City, UT
DEGREES AWARDED:
Doctor of Philosophy, Physics, 2019, University of Oregon
Bachelor of Science, Physics, 2013, Westminster College
AREAS OF SPECIAL INTEREST:
Electron Interferometry
Phase-Contrast Scanning Transmission Electron Microscopy
Scanning Transmission Electron Microscope Holography
PROFESSIONAL EXPERIENCE:
NSF Graduate Research Opportunities Worldwide Fellow, Center for Exploratory
Research, Hitachi, Ltd., Hatoyama, Saitama, JAPAN, 2017-2018
NSF Graduate Research Fellow, University of Oregon, 2015-2018
Science Literacy Program Fellow, University of Oregon, 2015
Graduate Research Assistant, University of Oregon, 2014-2015
Graduate Teaching Fellow, University of Oregon, 2013-2015
GRANTS, AWARDS AND HONORS:
Microscopy Society of America Travel Scholarship Award, International Congress
of Microscopy 19, 2018
Special ”opps” Travel and Research Award, International Congress of Microscopy
19, 2018
National Science Foundation Graduate Research Opportunities Worldwide
Fellowship, 2017
vi
Microscopy and Microanalysis Meeting Student Scholar Award, Microscopy and
Microanalysis Meeting, 2016
Graduate Teaching Fellowship, UO Science Literacy Program, Spring 2015
Graduate Teaching Fellowship, UO Science Literacy Program, Winter 2015
National Science Foundation Graduate Research Fellowship, 2015
Weiser Senior Teaching Assistant Award, UO Department of Physics, 2015
PUBLICATIONS:
TR Harvey, FS Yasin, JJ Chess, JS Pierce, RMS dos Reis, VB Özdöl, P Ercius,
J Ciston, W Feng, NA Kotov, BJ McMorran, C Ophus. Phys. Rev. Appl.,
2018, 10, 6, 061001
FS Yasin, K Harada, D Shindo, H Shinada, BJ McMorran, T Tanigaki. Appl.
Phys. Lett., 2018, 113, 233102
FS Yasin, TR Harvey, J Chess, JS Pierce, C Ophus, P Ercius, BJ McMorran.
Nano Lett. 2018, 18, 11, 7118-7123
BJ McMorran, TR Harvey, C Ophus, J Pierce, FS Yasin. Microsc. Microanal.
2018, 24, S1, 200-201
TR Harvey, V Grillo, F Venturi, JS Pierce, FS Yasin, JJ Chess, S Frabboni, E
Karimi, BJ McMorran. Microsc. Microanal. 2018, 24, S1, 938-939
FS Yasin, TR Harvey, J Chess, JS Pierce, BJ McMorran. J. Phys. D: Appl.
Phys., 2018, 51, 205104
J Ziegler, A Blaikie, A Fathalizadeh, D Miller, FS Yasin, K Williams, J
Mohrhardt, BJ McMorran, A Zettl, B Alemán. Nano Lett. 2018, 18, 4,
2683-2688
FS Yasin, TR Harvey, JJ Chess, JS Pierce, BJ McMorran. Microsc. Microanal.
2016, 22, 506
CW Warren, DW Miller, FS Yasin, JT Heath. 2013 IEEE 39th Photovoltaic
Specialists Conference (PVSC). 2013, 0170-0173
vii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. DEVELOPMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Note on ‘Path-separated electron interferometry in a
scanning transmission electron microscope’ . . . . . . . . . 6
Introduction to Path-separated electron interferometry in
a scanning transmission electron microscope . . . . . . . . 7
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Treatment of the second order diffraction probes . . . . . . . . . . 24
Supplemental Figure . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 26
III. APPLICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Note on ‘Probing Light Atoms at Subnanometer
Resolution: Realization of Scanning Transmission
Electron Microscope Holography’ . . . . . . . . . . . . . . 27
Introduction to Probing Light Atoms at Subnanometer
Resolution: Realization of Scanning Transmission
Electron Microscope Holography . . . . . . . . . . . . . . 28
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Chapter Page
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Theory and Reconstruction . . . . . . . . . . . . . . . . . . . . . . 35
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 42
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Full form of equation 3.4 . . . . . . . . . . . . . . . . . . . . . . . 46
Derivation of transfer function reconstruction . . . . . . . . . . . . 46
Numerical calculation of σφ . . . . . . . . . . . . . . . . . . . . . 48th
Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 50
IV. MEASUREMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Note on ‘A tunable path-separated electron interferometer
with an amplitude-dividing grating beamsplitter.’ . . . . . 51
Introduction to A tunable path-separated electron
interferometer with an amplitude-dividing grating
beamsplitter . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
V versus area enclosed by interferometer . . . . . . . . . . . . . . . 65
Additional figures . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 72
V. CONCLUSION AND FUTURE DIRECTIONS . . . . . . . . . . . . . 73
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Chapter Page
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
x
LIST OF FIGURES
Figure Page
1. (a) Mach-Zehnder Interferometry . . . . . . . . . . . . . . . . . . . . . 10
1. (b) Experimental setup for an analagous electron interferometer
that uses a grating beamsplitter and magnetic lenses as mirrors.
The electron beam is split into diffracted probes by a diffraction
grating. The +1 order probe interacts with the phase-object
while the 0 and -1 order probes pass through vacuum. The
TEM’s imaging system recombines the paths, magnifies the
resulting interference pattern and projects it onto the CCD
camera. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2. TEM image of the 50 µm diameter, 200 nm pitch sine profile
diffraction grating within the condenser lens aperture. A
6.5 µm×6.5 µm slice of the grating is shown in the green cutout.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3. (a) A CCD acquisition of the diffracted probes at the specimen
plane with the microscope configured in STEM mode. The
asymmetry of the probes’ shape is due to aberrations in the
projection lenses. An inset of the 0-, +1 and +2 order diffraction
probes with a 1D summed profile for comparison of probe
intensities. These images are normalized log profiles of the
probe. (b) Overlapping STEM images of a test phase object (C
membrane with Au nanoparticles for alignment purposes) using
multiple diffracted probes. The image is formed by scanning the
diffracted probes across the phase object while measuring the
amount of scattered electrons using a high angle annular dark
field (HAADF) detector. Each of the diffracted probes creates
an HAADF image. From these images, we measure the spatial
separation by measuring the distance between the bright 0th
order image and the lower contrast 1st order image (constructed
from the interaction of the 1st order diffraction probe and the
phase object). These images were acquired using conventional
probe-forming lens settings (convergence angle 6 mrad), and
shows a separation of 30 nm between beams. . . . . . . . . . . . . . . . 15
xi
Figure Page
4. (a) A conventional STEM-HAADF image of the graphitic
carbon. A line is drawn to mark the position of the +1-order
probe as it scanned across the edge of the carbon. The line is
in an orientation consistent with the diffraction axis, so that the
probes followed one after the other. The inset plot is a smoothed
slice of the image along the line scan, or the 1D HAADF profile
of the scan. (b) A 1-D slice of the interference fringes (image
of the diffraction grating) for various locations along the line
scan shown in (a). Note that the fringes translate as the probes
scan across the phase object, imparting a different phase value
along each location. Note xp = 40 nm, where the lowest spatial
frequency disappears because of destructive interference, leaving
fringes with double frequency. (c) Same as (b) but predicted from
theory using the HAADF profile as a simulated phase object.The
HAADF values were scaled to correspond to a maximum phase
shift of 2.3 π rad. with maximum height . . . . . . . . . . . . . . . . . 16
5. Simulated interference profile predicted from Equation 2.5 as
a function of the phase φ introduced to one path. Here, the
horizontal axis is wave number k, the vertical axis is phase φ
and the color represents I(k). Each row represents a 1D profile
of the image of the grating (interference fringes). . . . . . . . . . . . . . 19
6. (a) Measured interference fringes using STEM Holography.
Along the y-axis are 1D interference fringes, with each line
corresponding to the mean pattern at that location in the line
scan. Line 0 corresponds to xp = 0 nm, etc. The change in
contrast around position 15 nm is due to the +1-order diffraction
probe initiating illumination of the phase object. (b) Predicted
interference fringes from the HAADF profile in Fig. 4a. The
HAADF values were scaled to correspond to a maximum phase
shift of 2.3π rad. Note the agreement with the measured fringes
in (a), as well as the blurred fringes over small length scales, due
to noise in the HAADF profile. . . . . . . . . . . . . . . . . . . . . . . 22
7. Image of interference fringes used to calculate the fringe visibility.
Inset shows the 1D average of 100 images along the fringes
‘vertical’ axis. The color fill represents one standard deviation
from the mean for each point. The electron microscope was
configured in STEM mode with a spot size of 8 and a convergence
angle of 6 mrad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
8. STEM holography electron optical setup. . . . . . . . . . . . . . . . . . 31
xii
Figure Page
9. (a-d) Build-up of single electron events resulting in interference
fringes after (a) 0.0025 s, (b) 0.045 s, (c) 0.1425 s, and (d)
0.2875 s. The FFT of each frame is shown in the inset image.
Note that frames with 0.0025 s exposure were used to reconstruct
the phase image shown in Figure 11. . . . . . . . . . . . . . . . . . . . 34
10. Mean interference fringes averaged over the scan with the
background subtracted. The inset shows a 1D profile of the
sum of fringes along the direction perpendicular to the line trace
shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
11. (a,e) Conventional annular dark field images of two different Au
nanoparticles on thin C support. (b,f) Phase reconstruction of
the same regions using STEMH. (c-d) Corresponding Fourier
Transforms of (a) and (b). (g-h) Selected line traces from (e-f),
highlighting the low atomic number material contrast seen using
STEMH. The profiles are normalized to the maximum value of
each image after offsetting to a mean value of zero in vacuum.
(i) Selected line trace from (b) along just the carbon substrate,
from which the thickness is calculated. (g-i) are plots of the
mean along three line traces with the root mean square of the
the deviations shaded. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
12. Inset from center of Figure 11b in a carbon-only region to
enhance contrast. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
13. (a) The fringe visibility and (b) root-mean-squared uncertainty
as a function of phase imparted onto probe+1 for both m = 1
(red) and m = 2 (green). . . . . . . . . . . . . . . . . . . . . . . . . . . 49
14. Experimental setup for a tunable path-separated interferometer.
The right-hand-side illustrates the change in path with the
biprisms engaged. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
15. Interference fringes at the detector. Inset is the mean of 400 1D
slices of the interference fringe pattern. . . . . . . . . . . . . . . . . . . 56
16. fSTEMH image of a fabricated Si phase ramp that increases
(linearly f)rom vacuum with a gradient of 25
rad on one side
µm
and decreases linearly with twice the gradient on the other side
−50 rad . (a,b) The reconstructed unwrapped phase image as
µm
well as the amplitude of the object wave using conventional off-
xiii
Figure Page
axis electron holography. (c,d) Same as (a,b), but using fSTEMH
with ∆x = 5 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
17. (a-c) 1D profile of p+1 and p−1 in vacuum with a two slit window
aperture inserted in the sample position to block all higher
diffraction orders. (d-f) Interference fringes acquired over a 10 s
exposure for the corresponding path separations shown in (a-c).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
18. Interference fringe visibility versus path separation length
between the electron probes at the specimen plane. . . . . . . . . . . . 61
19. fSTEMH setup for different path separations. Notice the change
in the enclosed areas A1, A2, A3 and A4. As the path separation
increases initially, the total area enclosed decreases, but then
increases monotonically, leading to a loss in fringe visibility for
very large path separations. . . . . . . . . . . . . . . . . . . . . . . . . 63
20. Experimentally measured interference fringe visibility versus
the biprism 1 voltage applied, the absolute value of which is
proportional to the area enclosed by the interferometer. . . . . . . . . . 67
21. Simulated V(A) for ω̄ = 5 kHz and a) σ 5B0 = B̄ , σ = 50 ω ω̄, b)10 10
σ = 7.5B0 B̄0, σ =
7.5
ω ω̄, c) σB = B̄0, σω = ω̄, d-f) same as a-c)10 10
except with ω̄ = 5 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . 68
22. Simulated V(A) for increasing A1 and decreasing An>1, which
corresponds to an increasing path separation at the sample plane.
For a-c) ω̄ = 5 kHz. a) σ = 5B0 B̄ , σ
5 7.5
10 0 ω
= ω̄, b) σ
10 B0
= B̄0,10
σ = 7.5ω ω̄, c) σB = B̄0, σω = ω̄, d-f) same as a-c) except with10
ω̄ = 5 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
23. Scanning electron microscopy micrograph of the two-slit
aperture used to block the higher order diffraction probes
while performing the fringe visibility versus path separation
experiment described in the main text. . . . . . . . . . . . . . . . . . . 70
24. a) An image of the diffraction probes at the specimen plane with
∆x = 25 µm. b) An image of the Fourier transform of the
interference fringes with the peaks corresponding to the fringe
spacing circled in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
xiv
CHAPTER I
INTRODUCTION
In 1887, Michelson demonstrated a practical light interferometer, paving the way
for applications ranging from imaging of transparent phase objects [1], to fundamental
quantum measurements [2], to the recent gravitational wave detections [3]. Since
Dennis Gabor proposed an “electron interference microscope” in 1948 [4], electron
interferometry has also been utilized for nanometer to atomic resolution imaging
applications and to probe fundamental physics. Because electrons and photons
both exist as particle-wave phenomena, they may be used in such interferometric
setups to great effect. Electrons, however, offer several differences from their
optical counterparts. Electrons are massive, charged and have a shorter De Broglie
wavelength than photons. Yet they can still accelerate to relativistic speeds, and their
charge enables them to interact strongly with electromagnetic fields. Their picometer
wavelength provides electron interferometers with higher resolving power, making
them important tools for exploring fundamental physics and materials research.
Electron interferometers were first built in 1953 by Marton et al. using
amplitude-dividing polycrystalline epitaxially grown copper membrane diffraction
gratings [5]. Möllenstedt and Düker developed the more versatile wavefront-dividing
electron biprisms in 1955 [6], enabling the development of electron holography, now
a trusted technique for quantitative high precision imaging [7, 8, 9, 10, 11] and
probing basic physics [12, 13, 14]. While electron biprisms have proven a dependable
beam-splitting technology, they have a few drawbacks when compared to amplitude-
dividing beamsplitters. These include strict demands on the illuminating electron
1
beam coherence width, assymetry in the output beams’ profile and Fresnel diffraction
phenomena associated with the edges of the biprism [15].
Another consideration is whether to configure the beam into a wide, transmission
electron microscopy (TEM) mode, or the focussed-probe, scanning TEM (STEM)
mode within the microscope. In conventional off-axis TEM holography, an electron
biprism is placed in an image plane of the specimen. It must be placed in a vacuum
region parallel to the edge of the specimen and engaged to overlap a vacuum reference
wave with the interaction wave to form interference fringes. These fringes are then
imaged directly on a charge-coupled device (CCD) camera. This setup limits sample
geometries to those with an easily available vacuum region near the edge of the
specimen large enough to contain a biprism within the field of view. Alternatively,
STEM holography employs a pre-specimen beamsplitter, and the optics are configured
to focus the multiple, spatially separated beams down to probes at the specimen. Such
a setup was theorized and experimentally attempted in three groups simultaneously
in the late 1980s/ early 1990s [16, 17, 18, 19]. Ultimately, detector and biprism
technology at the time limited the practicality of such an interferometer, and it was
set aside.
In 2014, conversations with my advisor Benjamin McMorran and colleagues Tyler
Harvey, Jordan Chess and Jordan Pierce lead to a push for me to develop a path-
separated STEM interferometer using amplitude-dividing nanofabricated diffraction
gratings that have been used as phase plates within the McMorran lab since
its creation in 2011 [20, 21]. After achieving the promising preliminary results
presented in Chapter I of this dissertation, we discovered the aforementioned literature
and Tyler, Jordan and I pushed to change the name of the technique to STEM
holography (STEMH) as an homage to its history. My advisor still argues that
2
the word ‘interferometry’ is technically more accurate. While presenting the results
at a conference, Toshiaki Tanigaki of Hitachi, Ltd. Research and Development
Group suggested that we combine wavefront-dividing electron biprisms with STEMH.
Together we discovered that this would enable a tunable path-separated electron
interferometer with all of the coherence-width advantages of the STEMH experimental
setup mentioned above but with the added flexibility of pushing the interferometer’s
paths further apart or closer together, enabling more flexibility in experimental
setups. We therefore coapplied for the National Science Foundation Graduate
Research Opportunities Worldwide (GROW) fellowship in order to build such an
interferometer.
Simultaneously, realizing that the detector technology severely limited STEMH
in the past, we employed the technique in the TEAM I at the National Center
for Electron Microscopy (NCEM) at Lawrence Berkeley National Lab in Berkeley,
California, a Thermo Fisher Titan 80/300 kV (S)TEM equipped with both probe
and image correction as well as a Gatan Summit K2 fast-readout detector. The K2
is a direct electron detector, meaning that it is sensitive to single electron events
and therefore requires less dose to acquire a high signal-to-noise interference fringe
pattern. This collaboration resulted in the first demonstration of STEMH at atomic
resolution, presented in Chapter III.
Following this demonstration, we were notified that I received the GROW
fellowship and was to be named a Japanese Society for the Promotion of Science
(JSPS) international research fellow beginning in the Fall of 2017. Upon arrival at
Hitachi, Ltd. Center for Exploratory Research in Hatoyama, Saitama, Japan, Dr.
Tanigaki and I installed a diffraction grating into a Hitachi TEM containing five
electron biprisms. I built a scanning mode for the microscope and by Christmas, we
3
acheived our first STEMH images. The following year we finished developing flexible
STEMH, presented in Chapter IV.
This dissertation consists of work from five previously published papers. I have
chosen to include three of the previously published papers that I am first author of,
organized into three chapters. These chapters describe our historical development
of STEMH, starting with our initial proof-of-principle demonstration, a model of
the interferomter output, and an experimental comparison to the model. The next
chapter describes a technique to extract measured phase shifts from the interference
fringe patterns and demonstrates an application of this to an atomic resolution
quantitative phase and amplitude imaging technique. The final chapter describes
our development of an electron interferometer with tunable path separation. Each
chapter suggests applications of this new tool, such as capabilities for probing physics
within nanomaterials and measuring electric and magnetic nano-fields.
The following manuscripts are included in this work:
Chapter II. Initial Development and Theory
Fehmi S. Yasin, Tyler R. Harvey, Jordan J. Chess, Jordan S. Pierce, and
Benjamin J. Mcmorran. “Path-separated electron interferometry in a scanning
transmission electron microscope.” Journal of Physics D: Applied Physics 51 205104
(2018).
Chapter III. Application
Fehmi S. Yasin, Tyler R. Harvey, Jordan J. Chess, Jordan S. Pierce, Colin Ophus,
Peter Ercius, and Benjamin J. Mcmorran. “Probing Light Atoms at Subnanometer
Resolution: Realization of Scanning Transmission Electron Microscope Holography.”
Nano Letters 18 (11) 7118-7123 (2018).
Chapter IV. Further Development
4
Fehmi S. Yasin, Ken Harada, Daisuke Shindo, Hiroyuki Shinada, Benjamin J.
Mcmorran, and Toshiaki Tanigaki. “A tunable path-separated electron interferometer
with an amplitude-dividing grating beamsplitter.” Applied Physics Letters 113
233102 (2018).
This work was partially supported by the U.S. Department of Energy, Office
of Science, Basic Energy Sciences, under Award DE-SC0010466 and by the National
Science Foundation Graduate Research Fellowship Program under grant No. 1309047.
F.S.Y. is a JSPS International Research Fellow.
5
CHAPTER II
DEVELOPMENT
Note on ‘Path-separated electron interferometry in a scanning
transmission electron microscope’
Reprinted (adapted) with permission from Fehmi S Yasin et al. Journal of
Physics D: Applied Physics 51, 205104 (2018). Copyright 2018 IOP PUBLISHING,
LTD.
Benjamin McMorran conceived of the idea, and he and I developed the
experimental plan. Tyler Harvey showed me how to use the electron microscope and
we performed the first experimental attempt together. I performed the experiment
from which the published data was collected, and Tyler and I developed the theory
independently, making different approximations but meeting regularly to discuss and
integrate our ideas. Tyler developed the final generalized theory inspired by [22]
that we used to develop the phase reconstruction software. Tyler and I developed
the software in python, meeting regularly with Jordan Chess to optimize the code.
Jordan Chess and Jordan Pierce both contributed to the theoretical development
as well through regular meetings where they offered advice and experience integral
to developing a final working product. Jordan Pierce fabricated the diffraction
gratings used in the experiment. I performed the STEMH simulations with the advice
and guidance of Tyler and Jordan C.. I recorded experimental data, analyzed the
experimental and simulated data, produced all figures, and wrote the supplemental
material and manuscript with input from co-authors.
6
Introduction to Path-separated electron interferometry in a scanning
transmission electron microscope
Since Michelson’s practical demonstrations of interferometry in 1887, interferometric
techniques have been used in optics for a great variety of applications, from
fundamental quantum measurements [2], to gravitational wave sensing [3], to imaging
of transparent objects [1]. Using the wave-like characteristics of light, interferometers
split a beam of light into two or more paths that then recombine to form an
interference pattern. If one path acquires a phase shift relative to the other, for
example by passing through an object, then the resulting phase can be measured by
a modification in the interference pattern. In a Mach-Zehnder interferometer, shown
schematically in Fig. 1a, the interference pattern is typically measured using a second
beamsplitter to analyze the interference that occurs where the two paths overlap, but
in principle the interference fringes could be imaged directly using a camera.
In 1948, Dennis Gabor proposed a variation of this setup within an electron
microscope. Later called holography, Gabor hypothesized an “electron interference
microscope” and demonstrated the principle within an optical setup [4]. Originally
meant to overcome the difficulties posed by lens aberrations within a TEM, two new
issues emerged when Haine and Mulvey [23] experimentally realized Gabor’s idea:
the ‘twin-image problem’ of in-line holography, which arises because the object and
reference wave forming the interference pattern propagate in the same direction in
space [11, 24, 25], and the limited extension of spatial coherence in the hologram
plane. Much work has been done to remedy these issues [24, 25, 26], including the
creation of ‘off-axis’ holography by Leith and Upatnieks [27]. Once adapted to the
electron microscope, the beam was split using a Möllenstedt electron biprism [28]
after the sample, rather than before. This solved the ‘twin-image problem’ because
7
the reference wave interferes with the object wave at an angle, allowing them to be
separated in Fourier space [11]. Additionally, H. Wahl overcame the limited coherence
problem by recording the interference pattern (hologram) in an image plane [26]. Off-
axis electron holography has since been developed and applied[9, 10, 29, 30, 31, 32] to
push the boundaries of electron microscopy through feats such as the atomic resolution
electrostatic potential mapping of graphene sheets [33]. Although it remains a state
of the art imaging technique, off-axis electron holography uses a wavefront-division
beamsplitter, which requires a wide, coherent beam, which in turn requires a highly
coherent source. As stated before, this requirement can be met, but remains a non-
trivial task that requires expertise. Additionally, the phase resolution is limited by
the number of fringes within the field of view. Finally, the biprism has a non-trivial
installation in commercial electron microscopes and creates unwanted artifacts in the
interference pattern due to diffraction from the biprism’s edge.
To overcome these restraints, multiple amplitude-division crystal lattice
beamsplitters have been proposed and utilized by Matteucci et al.[34], Ru et al.[35],
Zhou [36], Agarwal et al.[37] and others since Marton et al. first demonstrated
them in 1952 [5, 38, 39]. Additionally, diffraction gratings have also been developed
as amplitude-dividing beamsplitters for electrons [40, 41] and applied to electron
interferometers [42, 43, 44]. Gronniger et al. had a three-grating setup which required
and external laser source for alignment, while Cronin and McMorran used two gratings
for a Lau [44] and Talbot [43] interferometer setup. Holographic diffraction gratings
for electrons have been developed by several groups [20, 21, 45, 46, 47] to carefully
design the intensity and phase structure of the probe beams. As diffraction grating
fabrication techniques have improved, so has the spatial separation range of the
diffracted probes.
8
Here we demonstrate that a single nanoscale grating can be used as a coherent
beamsplitter to form a separated-path interferometer in an electron microscope.
These gratings diffract the electron wavefront into probes with spatial separation of
tens of nanometers under standard lens settings in commercial TEMs. This separation
is sufficient to study nano-materials using classic interferometric techniques. Such
a technique within an electron microscope was initially theorized and explored
experimentally in the late 1980’s and early 1990’s [16, 18, 22]. The initial experimental
attempts proved too challenging for the reasons mentioned above, and so the
technique was ultimately abandoned, although Cowley expanded the theory in 2003
[48]. In this paper we experimentally explore and discuss the feasibility of his idea of
“STEM holography.”
9
FIGURE 1. (a) Mach-Zehnder Interferometry
10
FIGURE 1. (b) Experimental setup for an analagous electron interferometer that
uses a grating beamsplitter and magnetic lenses as mirrors. The electron beam is
split into diffracted probes by a diffraction grating. The +1 order probe interacts
with the phase-object while the 0 and -1 order probes pass through vacuum. The
TEM’s imaging system recombines the paths, magnifies the resulting interference
pattern and projects it onto the CCD camera.
11
FIGURE 2. TEM image of the 50 µm diameter, 200 nm pitch sine profile diffraction
grating within the condenser lens aperture. A 6.5 µm× 6.5 µm slice of the grating is
shown in the green cutout.
12
Experimental Setup
We performed this experiment on an FEI 80-300 Titan operated at 300 keV in
STEM mode. Using a Focused Ion Beam (FIB), we fabricated a 50 µm diameter,
200 nm pitch diffraction grating (shown in Fig. 2) in a Ti and Pt coated silicon nitride
membrane [21]. The grating bars were milled using a 30 kV accelerating voltage, 24
pA beam of Ga ions. Once milled, the grating was positioned in the Condenser 2
aperture plane.
As illustrated in Fig. 8b, the input plane wave electron beam travels down
the microscope column to the C2 aperture, where it diffracts through the grating
and forms isolated electron probes at the specimen plane with tens of nanometers
spatial separation (shown in Fig. 3). In this experiment, we position a phase object
such that the three most intense diffraction probes initially pass through vacuum.
The phase object we used was a Ted Pella Combined TEM Test Specimen on a 3
mm grid, consisting of graphitic carbon and gold nanoparticles on nearly electron-
transparent amorphous carbon. The nanoparticles provide recognizable features that
can be used for alignment of the diffracted probe beams, and they have a known mean
inner potential that can be used to predict the phase shift imparted to transmitted
electrons. The probes are then scanned across the field of view so that the +1 order
probe interacts with the phase object while the 0 and -1 order probes pass through
vacuum. They are then recombined through the post-specimen optics and interfere
in the image plane on a Gatan Ultrascan 1000 CCD camera. An image of the grating
(interference pattern) is acquired by the CCD camera at each location in the scan.
The phase information at each location on the phase object may be extracted by a
post process described in the theory section below. We specifically illuminated the
13
graphitic carbon, as it provides an easily recognizable change in thickness (4z) when
the probe scans from vacuum onto the phase object.
The phase object is shown in Fig. 4a using conventional STEM High Angle
Anular Dark Field (HAADF) imaging. A regular focused probe is rastered across
the phase object. At each probe position, the electrons are scattered by the atoms
they pass. The more atoms the beam of electrons interact with, the higher number
of counts on the HAADF detector. The projection lens system is set to a camera
length of 130 mm. At this camera length, a 6 mrad convergence angle corresponds
to a beam radius of 0.78 mm at the phosphor screen, three times smaller than the
HAADF detector’s inner radius. Therefore, the HAADF signal is proportional to the
thickness of the phase object, as well as the atomic number, Z [49]. Since the phase
object we’re considering is homogeneous, we approximate the signal to be proportional
to thickness.
With the diffraction grating inserted, the electron probes were scanned from
vacuum onto the edge of the carbon, along the drawn line indicated in Fig. 4a. Along
each point in the scan, an image of the diffraction grating (interference fringes) was
recorded and analyzed. Some of these fringes are shown in Fig. 4b, which confirms the
translation of the fringes shift across the scan as expected. In addition to translation,
however, we observed a doubling of the fringe frequency and a loss of contrast of the
lowest spatial frequency. We explore why this occurs in the next section.
14
(a)
(b)
FIGURE 3. (a) A CCD acquisition of the diffracted probes at the specimen plane
with the microscope configured in STEM mode. The asymmetry of the probes’ shape
is due to aberrations in the projection lenses. An inset of the 0-, +1 and +2 order
diffraction probes with a 1D summed profile for comparison of probe intensities. These
images are normalized log profiles of the probe. (b) Overlapping STEM images of a
test phase object (C membrane with Au nanoparticles for alignment purposes) using
multiple diffracted probes. The image is formed by scanning the diffracted probes
across the phase object while measuring the amount of scattered electrons using
a high angle annular dark field (HAADF) detector. Each of the diffracted probes
creates an HAADF image. From these images, we measure the spatial separation by
measuring the distance between the bright 0th order image and the lower contrast
1st order image (constructed from the interaction of the 1st order diffraction probe
and the phase object). These images were acquired using conventional probe-forming
lens settings (convergence angle 6 mrad), and shows a separation of 30 nm between
beams.
15
(b)
(a) (c)
FIGURE 4. (a) A conventional STEM-HAADF image of the graphitic carbon. A
line is drawn to mark the position of the +1-order probe as it scanned across the
edge of the carbon. The line is in an orientation consistent with the diffraction
axis, so that the probes followed one after the other. The inset plot is a smoothed
slice of the image along the line scan, or the 1D HAADF profile of the scan. (b)
A 1-D slice of the interference fringes (image of the diffraction grating) for various
locations along the line scan shown in (a). Note that the fringes translate as the
probes scan across the phase object, imparting a different phase value along each
location. Note xp = 40 nm, where the lowest spatial frequency disappears because
of destructive interference, leaving fringes with double frequency. (c) Same as (b)
but predicted from theory using the HAADF profile as a simulated phase object.The
HAADF values were scaled to correspond to a maximum phase shift of 2.3 π rad.
with maximum height
16
Theory
Here we develop a theory to describe the output of a multiple-beam
interferometer transmitting through a phase object, which is then scanned. Similar
to Cowley’s 2-beam electron interferometer proposal [48], we model the wave function
of an electron that has been coherently diffracted by a grating into several spots that
are focused in the plane of the sample:
∑+1
ψ0(x) = sn (x− nx0) . (2.1)
n=−1
where the complex amplitude unique to the nth diffraction order is described
by sn(x). The spatial separation between diffracted probes can be described by
x = λf10 x̂, where λ is the de Broglie wavelength of the electron, d the spatial frequencyd
of the beam-splitting grating, and f1 the focal length of the probe-forming lens system.
Usually, a sinusoidal diffraction grating diffracts the electron beam into many
diffraction orders. Here, however, we consider a diffraction grating in the aperture
plane, fabricated such that all but three of the diffracted spots of varying amplitude
cn are negligible, and they are all sharply peaked with very small beam radii relative
to the phase gradient of the phase object transfer function. As such, they can be
approximated delta functions:
∑
ψ0(x) = cnδ (x− nx0) . (2.2)
n
The +1 diffraction spot transmits through an electron-transparent phase object
described by a complex transmission function
t(x) = Θ(−x + xedge) + Θ(x− xedge)A(x)eıφ(x) (2.3)
17
such that the complex amplitude of the electron wave function immediately after
the phase object is
ψ′0(x) = ψ0(x)t(x). (2.4)
Note that 1− |A(x)|2 is the probability for an electron to be elastically scattered out
of the optical system and φ(x) is the phase profile of the specimen. For electron-
transparent objects, |A(x)|2 ≈ 1. The Heavyside theta functions represent the edge
of the phase object.
In the experimental setup, the post-specimen optical system of the electron
microscope is configured to record a far field diffraction pattern of the sample, which
has the effect of spreading and overlapping the separated electron paths to form an
interference pattern at the imaging detector (a CCD camera). Thus, this interference
pattern can be described by the modulus squared of the Fourier transform of Equation
2.4:
I(k) =|c + c e2πıx0·k + c e−2πıx0·keıφ(x0) 20 −1 +1 | . (2.5)
Fig. 5 shows the behavior of this interference pattern described in Equation 2.5
as a function of the phase φ introduced to the +1 diffracted path. Note that the
fringes shift laterally for small relative phase shifts as with a 2-path interferometer,
but the inclusion of a third path returns a double-spatial-frequency fringe pattern,
most noticeable in Fig. 5 when φ is an odd multiple of π rad. This can be interpreted
as the output of three parallel 2-beam interferometers - one formed by the -1- and
+1- order diffracted probes, one by the -1- and 0- order diffracted probes and the
18
FIGURE 5. Simulated interference profile predicted from Equation 2.5 as a function
of the phase φ introduced to one path. Here, the horizontal axis is wave number k,
the vertical axis is phase φ and the color represents I(k). Each row represents a 1D
profile of the image of the grating (interference fringes).
other formed by the 0- and +1-order probes. As the phase increases, the 0/+1 and
-1/+1 interference fringes translate to the right, while the -1/0 interference fringes
stay stationary. The -1/+1 interference between the diffracted probes contributes the
double frequency structure seen in Figure 5. Of course, there also exists interference
between the higher order diffraction probes, but the intensity of such beams is so low
so as to contribute little to the overall interference pattern; i.e., for a typical grating
used for these experiments, |c+2||c±1|  |c0||c±1|. This is illustrated in Fig. 3a. The
higher order probe interference effects are seen more in the higher frequency fringes
structure seen in Fig. 6.
19
This interferometer can be applied to image phase objects to realize the STEM
holography proposed by Cowley [22, 48]. Eq. 2.5 can be expanded to model the
behavior of the interference pattern resulting from a nonuniform object that induces
a position-dependent phase φ(xp) onto electrons. In principle, the object can be
scanned in the vicinity of the +1 diffracted probe, but in practice TEM optics can be
used to scan the diffraction probes in the sample plane and de-scan the interference
pattern. For non-magnetic materials, the position-dependent phase imparted onto an
electron by a phase object is given by
∫
φ(xp) = CE V (xp) dz, (2.6)
where CE is a constant that is determined by the kinetic energy of the incident
electron and V (xp) is the electric potential of the sample at scan position xp [50];
at low resolution and for amorphous materials, we can consider only the mean inner
potential. Specimens with a uniform mean inner potential V0 can be described by a
more simplified transmission function,
φ(xp) = CE V0 T (xp), (2.7)
where T (xp) is the thickness of the phase object [50].
Results
We collected interference fringes for a 60 nm line scan with a 1 nm step size. We
report fringe contrast of up to 39.7% ± 1.4. We include the image and a 1D profile
of the mean along the vertical fringes from which this value was calculated. The
fringes from the experiment are summarized in the visualization in Fig. 6a. In this
20
plot, each row corresponds to a mean 1D plot of the interference fringes detected at
a single position in the line scan. We compared this to simulated interference fringes
from the HAADF profile (Fig. 4a) in Fig. 6a and found the experimental fringe shifts
to be consistent with a phase shift of up to 2.3π rad.
Illumination of the phase object with the 0- order diffraction probe
In practice, a scan could be performed where one allows the 0-order diffraction
probe to follow the +1-order probe onto the phase object. Mathematically, this
would be represented by an eıφ(x) factor in the c0 term of Equation 2.5. The phase
retrieved from the previous scan positions could in principle be utilized to subtract
the phase imparted onto the 0-order probe. This analysis is integrated along with
our coefficients into the interference pattern shown in Fig. 6b, and we provide
the full equation (including the +2 order diffraction probe interaction term) in the
supplemental materials.
Recalling Equation 2.7, an electron probe wave function experiences a phase
modulation that is directly proportional to the thickness of the phase object T (x).
Therefore, a wide variety of TEM samples can be probed using this experimental
setup. Additionally, an electron probe experiences a similar phase shift due to local
magnetization of specimens, and so this experiment can be expanded to provide nano-
magnetic field profiles of magnetic samples, while also providing a path separated
interferometric method to explore the Aharonov-Bohm effect.
21
(a)
(b)
FIGURE 6. (a) Measured interference fringes using STEM Holography. Along the
y-axis are 1D interference fringes, with each line corresponding to the mean pattern
at that location in the line scan. Line 0 corresponds to xp = 0 nm, etc. The change
in contrast around position 15 nm is due to the +1-order diffraction probe initiating
illumination of the phase object. (b) Predicted interference fringes from the HAADF
profile in Fig. 4a. The HAADF values were scaled to correspond to a maximum
phase shift of 2.3π rad. Note the agreement with the measured fringes in (a), as well
as the blurred fringes over small length scales, due to noise in the HAADF profile.
22
Conclusion
We have demonstrated a path-separated electron interferometer using a single
grating within a widely available, commercial TEM. The nanofabricated phase grating
sits within a conventional aperture. We attained spatial separation between electron
probes of up to 30 nm, and an interference fringe contrast of up to 39.7± 1.4%. We
developed a theoretical model of the interference pattern due to three separate paths
under the influence of a phase induced to just one path. We performed an experiment
in which the interferometer was scanned across a phase object. An independently
measured thickness of the test phase object was used as an input to our model, and
closely matches the experimental interference pattern.
Our use of a single grating in a modified, user-replaceable condenser aperture in
a TEM instrument opens a path to STEM Holography, in which the interferometer
is scanned across an electron-transparent phase object in a TEM to form an image of
the induced phase. To this end, we are working to invert our theoretical model
to quantitatively measure an unknown phase from an experimental scan of the
interferometer. STEM Holography has the potential to probe both electrostatic
potentials of phase specimens and magnetic fields at sub-nanometer scales. As a
path-separated, charged particle-wave interferometer, future work may also include
an exploration of the coherence length of electron wavepackets. To be certain, the
progress in nano-fabrication techniques for diffraction gratings as well as their easy
installation in conventional aperture holders has enabled the use of the electron
microscope as an electron optics bench.
23
Treatment of the second order diffraction probes
One might ask what the second order diffraction probe contribution to the
interference pattern is in this multi-armed interferometer. Beginning from Equation
(5), we can add the second order terms, while also including the specimen interaction
term for the 0-, +1, and +2 terms, since they all illuminate the specimen at some
point in the scan. Here, as in Equation 2.5, we only consider one point in the line
scan, and so we drop our scan variable xp. We also choose our origin to be located
at the 0-order diffraction probe position.
I(k) =|c eıφ(0) + c e2πıx0·k + c e−2πıx0·keıφ(x0)0 −1 +1 + c e4πıx0·k−2 + c+2e−4πıx0·keıφ(2x0)|2.
(2.8)
Expanding this produces a sum of periodic functions. Here we will only keep
the first and second frequency terms that include a phase modulation due to the
specimen. We’ll also consider the probe coefficients to have both amplitude and
phase, cn = |c |eıθnn
∑+2
I(k) = |cn|2+2|c−2||c0| cos (4πk · x0 − φ(0) + θ−2 − θ0) + 2|c−1||c0| cos (2πk · x0 − φ(0) + θ−1 − θ0) +
n=−2
2|c−1||c+1| cos (4πk · x0 − φ (x0) + θ−1 − θ+1) +
2|c0||c+1| cos (2πk · x0 + φ(0)− φ (x0) + θ0 − θ+1) +
2|c0||c+2| cos (4πk · x0 + φ(0)− φ (2x0) + θ0 − θ+2) +
2|c+1||c+2| cos (2πk · x0 + φ(x0)− φ (2x0) + θ+1 − θ+2)
(2.9)
24
Supplemental Figure
FIGURE 7. Image of interference fringes used to calculate the fringe visibility. Inset
shows the 1D average of 100 images along the fringes ‘vertical’ axis. The color
fill represents one standard deviation from the mean for each point. The electron
microscope was configured in STEM mode with a spot size of 8 and a convergence
angle of 6 mrad.
25
Chapter Conclusion
In this chapter we presented a path-separated electron interferometer using an
amplitude-dividing grating beamsplitter placed in the condenser lens system within a
commercial FEI Titan 80-300 TEM. We demonstrated that interference fringes could
be recorded and a shift in the fringes could be detected when introducing a phase onto
one of the interferometer’s arms. We developed a theoretical model of the interference
pattern produced by three separate electron beam paths with a phase imparted onto
one of the paths. We performed an experiment in which we scanned the focused probe
beams from vacuum onto a weak phase object, here thin graphitic carbon, so that
only one of the beams would illuminate it. Using an independent measurement of the
sample’s thickness as an input to our theoretical model, we found that the shift in the
simulated interference fringe pattern closely matches our experimental data. Finally,
we discussed how such an interferometer may have quantitative imaging applications,
and suggested a path forward to STEMH.
26
CHAPTER III
APPLICATION
Note on ‘Probing Light Atoms at Subnanometer Resolution: Realization
of Scanning Transmission Electron Microscope Holography’
Reprinted (adapted) with permission from Fehmi S Yasin et al. Nano Letters
18, (11) 7118-23 (2018). Copyright 2018 American Chemical Society.
Benjamin McMorran, Tyler Harvey and I conceived of the experimental setup.
Peter Ercius and Colin Ophus provided the specimen. Jordan Pierce fabricated the
diffraction grating beamsplitter. Peter Ercius trained me on the microscope, and
explained how to use the Gatan K2 Summit camera software. Colin and Peter helped
explain how the data structure was saved and provided working examples for how
to open and read .dm4 files 300 GB. Jordan Chess provided long hours helping me
with code development and optimization. Tyler and I developed theories for STEMH
using a weak phase approximation, and then Tyler wrote a general theory of STEMH,
resulting in this manuscripts ‘sister’ article, [51]. Colin Ophus contributed a new phase
reconstruction idea which Tyler integrated into his theory and I integrated into the
final software. Jordan Pierce generated one figure. I recorded all data, generated all
but one of the figures, and wrote the supplemental material manuscript with input
from coauthors.
27
Introduction to Probing Light Atoms at Subnanometer Resolution:
Realization of Scanning Transmission Electron Microscope Holography
Phase contrast for low-atomic-number, beam-sensitive materials has long been
pursued in electron microscopy, seeing the advent of multiple transmission electron
microscopy (TEM) and scanning TEM (STEM) techniques over the past 60+
years, including electron holography or interferometry using both wavefront-dividing
beamsplitters [4, 9, 23, 31, 32, 33, 52] and amplitude-dividing beamsplitters [5, 34, 35,
36, 38, 39, 53], ptychography [54, 55, 56, 57, 58], cryo-electron microscopy [58, 59, 60],
matched illumination and detector interferometry [57, 61], differential phase contrast
[62, 63], and more. These techniques have benefited from the development of
technologies such as fast readout detectors and aberration correctors that have driven
imaging resolution of STEM below 0.41 Å[64] and TEM below 0.43 Å[65].
Several decades ago, an interferometric technique called STEM holography
(STEMH) was initially developed as a phase contrast electron imaging technique
[16, 17, 18, 19]. These arrangements used a charged biprism wire to split an electron
beam into two probes focused at the specimen. With one beam transmitted through
the specimen, the interference between the two was recorded. Due to the slow
throughput and limited geometries of detectors at the time, STEMH was never
widely implemented. The recent advent of fast-readout direct electron detectors
enables STEMH as a practical imaging technique. Additionally, advances in FIB
fabrication technologies allowed us to expand on this technique with the addition of a
static, nanofabricated, amplitude-dividing diffraction grating for use as probe-forming
aperture and beam splitter in a multiple-path-separated interferometer [53]. In this
article, we provide such a demonstration.
28
Amplitude-dividing beamsplitters in the form of nanofabricated electron
diffraction gratings have been developed by multiple groups. In contrast to wavefront-
dividing beamsplitters such as electrostatic biprisms, these diffraction gratings lower
the coherence width requirements of the beam, while also allowing for careful shaping
of the electron wave fronts phase and amplitude structure [20, 21, 45, 46, 47]. They
form symmetric profile probes at the specimen plane (grating’s diffraction plane) that
are absent of any unwanted edge-diffraction artifacts, and have one passive working
part equal in size and shape to conventional apertures, making them easily installable
into commercial electron microscopes. Additionally, although many diffraction order
probes are generated from the grating, the diffraction efficiency of the grating can be
tuned to decrease the intensity in the higher orders [21, 53].
Another technological advance that enables STEMH is the advent of fast-readout
direct electron detectors. These detectors are capable of acquiring thousands of
images in seconds and are sensitive to individual electrons. Such a fast readout
is necessary for any high resolution 4D-STEM imaging technique. In addition to a
fast readout of 102 fps, the high detective quantum efficiency of such detectors should
allow for a decrease in the electron dose seen by the specimen by at least two orders
of magnitude [66]. STEMH combines the aforementioned direct electron detector,
amplitude-division diffraction gratings, interference fringe phase reconstruction, and
aberration correction to provide quantitative phase contrast, including the dc-
component with respect to vacuum.
In this article, we provide the theoretical framework for a three-beam, path-
separated electron interferometer with a phase imparted onto one or more paths.
We then provide two proof-of-principle STEMH images of Au on C, with high-angle
annular dark field (HAADF) images for comparison. In HAADF STEM, the beam
29
current is focused to a sub-nanometer width and is scanned across a field of view,
dwelling at each location until a sufficient number of high-angle scattering events have
illuminated an annulus detector, forming contrast. In STEMH, we extract the phase
contrast in these images from the interference fringe patterns using the aformentioned
model. STEMH reveals a string-like phase structure in the amorphous carbon region,
which is consistent with the thick-bonding theoretical model proposed by Ricolleau
et al. [67].
30
d
C3 Aperture
Grating
Plane
Pre-Specimen
Optics
Diffraction −1 0 1
Orders
Specimen
Sample Plane
Post-Specimen
Optics
Image
Plane
FIGURE 8. STEM holography electron optical setup.
31
Experimental Setup
As illustrated in Figure 8, the input plane wave electron beam travels down
the microscope column to the probe-forming aperture, where a diffraction grating
coherently splits the electron beam into multiple diffraction orders that are sharply
peaked at the specimen plane, with tens of nanometers spatial separation. The
specimen is positioned such that all three diffraction probes, which we’ll call probe+1,
probe0, and probe−1 in the text, initially pass through vacuum. These probes are
then rastered across the field of view along the same line as the diffraction pattern’s
orientation using the scanning (deflection) coils in the microscope. probe+1 interacts
with the specimen while probe0 and probe−1 pass through vacuum, acting as reference
beams in three parallel interferometers. An interference pattern is focused onto the
detector, and the fringes shift as the phase imparted onto the interacting probe varies.
Note that as long as the path separation and scan step size are well known, the
diffraction probes may be scanned further into the specimen, such that the reference
beam also acquires a phase shift. In principle, this reference phase shift, having
already been determined at the previous data points in the scan, can be subtracted
through post-processing.
As shown in Figure 9a, it is hard to make out interference fringes from a single
frame exposed to a beam current of 0.041 nA. Increasing the detector’s exposure
time to acquire a greater number of events results in fringes discernible to the human
eye, as in Figures 9b-9d and 10. Using a computer, however, we can resolve the
fringes in a single frame via a Fourier Transform, and so direct electron detectors
have decreased STEM convergent beam electron diffraction (CBED) recording time,
therefore decreasing the electron dose seen by the specimen.
32
We inserted a selected area aperture in an image plane of the diffraction probes
in order to reduce noise due to unwanted high-angle scattering. This large aperture
only blocks high order diffraction probes (> 4th order) which are assumed to be
negligible. The passed probes are then recombined through the post specimen optics
and interfere in the image plane on the detector. The phase information of each
location on the specimen is extracted by a post process described in the Theory and
Reconstruction section below.
We performed this experiment on TEAM I, an FEI Titan 80-300 operated at
300 KeV in STEM mode with both probe and image aberration correction and a
semi-angle of 30 mrad. A 50 um diameter, 200 nm pitch sinusoidal phase grating is
positioned in the Condenser 3 aperture plane. We imaged a specimen consisting of
Au nanoparticles on a thin, amorphous carbon support. The images shown in Figures
11b and 11f are reconstructed from a 128 x 296 and 115 x 300 2D scan of 1920 x 1792
images, forming two 4D data sets with a field of view of 11.1 nm x 25.8 nm and 8.6
nm x 22.5 nm, respectively.
33
(a) (b)
(c) (d)
FIGURE 9. (a-d) Build-up of single electron events resulting in interference fringes
after (a) 0.0025 s, (b) 0.045 s, (c) 0.1425 s, and (d) 0.2875 s. The FFT of each frame
is shown in the inset image. Note that frames with 0.0025 s exposure were used to
reconstruct the phase image shown in Figure 11.
34
Theory and Reconstruction
The pre-specimen probe wavefunction is defined to be
∑
ψi(x) = a(x− xp) = cnan (x− xp − nx0) (3.1)
where xp is the offset-position of our probe, an is the phase and intensity distribution
of the nth diffraction order, cn is the complex amplitude of the n
th diffraction order
probe, and x0 is the real-space path separation of any one diffraction order probe
from it’s nearest neighbor. Note that the grating could, in principle, incorporate
holographic designs [21] that produce different phase and intensity distributions in
each diffraction order, such as vortex beams [20] or aberration-corrected beams [68,
69]. In these experiments, we used a large, straight grating within the aperture,
encoding flat phase structure in the probes such that each term in a(x−xp) describes
a sharply-peaked, symmetric function that only differs by a linear phase, or an = a0.
Recall that the probes are scanning through space at the specimen plane, which
is why an offset-position of the probe is needed here. We’ll use a specimen transfer
function t(x) resulting in a post-specimen wavefunction
ψf (x) = a(x− xp) · t(x), (3.2)
where t(x) is the object transmission function. The far field interference pattern at
the detector at probe position xp is then
35
Ip(k) = |(Ψf (k)|
2
p
= A∗
)
p(k)⊗ T ∗(k) (Ap(k)⊗ T (k)) , (3.3)
where ⊗ represents convolution and ∗ represents complex conjugate. We use
lower-case and capitilized letters to denote real versus reciprocal space variables,
respectively.
Now lets make the assumption that there are only three beams, or c|n|>1 = 0;
n ∈ [−1, 0, 1], and that only probe+1 interacts with the specimen with the other two
being reference beams passing through vacuum. Taking the Fourier Transform of 3.3
results in five sharp peaks, which are visible in insets to Figures 9a, 9b, 9c and 9d.
Ip(x) =I−2 (xp,x) +I−1 (xp,x) +
I0 (xp,x) +I+1 (xp,x) +I+2 (xp,x) (3.4)
Equation 3.4 is expanded into its full form in the supplemental materials. We
can extract the specimen’s transfer function by integrating around one of the sharp
peaks, along the variable x, which would leave us with the transfer function of the
scan position variable xp. We could do this for each peak in Ip(x), which would
give us redundant information for peaks that include a signal from more than one of
the interferometers that includes the scanning probe interacting with the specimen.
For example, if probe+1 is the interaction scanning probe, the object transmission
function information probed by the interaction scanning probe is encoded in fringes
with spacing k0 =
1
| | due to interference between the probe+1 and probe0. This periodx0
36
corresponds to the −1- and +1- order peaks in Ip(x), from which the transmission
function can be extracted.
This information is also encoded in fringes with spacing k 10 = | | due to2 x0
interference between probe+1 and probe−1, and can therefore be extracted from the
−2- and +2-order peaks in Ip(x). In summary, for a three beam interferometer
in which one first order diffraction probe interacts with the specimen, the object
transmission function information is stored in both the the first and second orders,
respectively, of the Fourier transform of the interference fringe image.
A non-negligible +2-order diffraction probe probe+2 complicates this picture,
and the −2 and +2 peaks in Ip(x) also contain that information via interference with
probe0. Because the nanofabricated gratings are designed such that cn>1 should be
weak, we assume that it is negligible.
We can also make the assertion that the specimen function in vacuum is just 1,
simplifying equation 3.4 even further. Integrating around I+1 (xp,x) in equation 3.4,
using a0 (x) as a kernel, and noting that A0 (k) is a circular aperture, we arrive at
the solution.
∫
a0 (x)I+1 (xp,x) dx
Ω(+x0)
= c∗c h (x )⊗ t∗0 +1 p (x0 + xp) (3.5)
where h (xp) = |a 20 (xp)| . The full derivation is provided in both the supplemental
materials and another manuscript that provides a full treatment of the general theory
of STEMH [70].
To summarize the numerical object wave reconstruction procedure:
37
1. At each probe position, take the Fourier transform of the interference fringe
pattern, resulting in equation (3.4).
2. Isolate a small (we used < 10× 10 pix2) region around a peak that contains the
desired object wave information, I+1 (xp,x).
3. Define a kernel a0(x) by taking the Fourier transform of a reference image of the
interference fringes, i.e. an image when all three probes pass through vacuum,
and isolate a small region around the center peak.
4. Multiply these two peaks and integrate, taking the complex conjugate, equation
(3.11).
5. Repeat for each pattern in the scan, i.e. each xp value.
Phase-thickness relation
The specimen transfer function contains an amplitude and phase, which can
be used to calculate the thickness of a specimen. For a non-magnetic specimen,
the phase imparted onto an electron wave-front is proportional to the electrostatic
potential projected through the bulk of the specimen [53]. For amorphous materials,
we may consider only the mean inner potential, Vi. Thus,
φ = CEViT (xp) , (3.6)
where T (xp) is the thickness of the specimen for each location in the scan, xp, CE =
2π e E0+E , λ is the relativistic wavelength of the electron, 1.97 pm for E = 300 keV,
λ E 2E0+E
where E is the kinetic energy of the electron, E0 is the rest energy of the electron,
and e is the electron unit charge.
38
Phase uncertainty
The theory of phase detection uncertainty in electron holography has been
worked out in detail by Lichte et al. and de Ruijter et al. [71, 72], whose work
was experimentally supported by Harscher and Lichte [73]. If we only consider the
counting statistics for the number of electrons per unit area of the detector at any
time (shot noise), the standard deviation for detection of the phase from interference
fringes with visibility V = Imax−Imin is
Imax+Imin
√
2
σφ = V , (3.7)th 2N
where N is the number of electrons in the measurement area.
Detectors will also contribute to the phase uncertainty, and their contribution is
typically characterized by a detective quantum efficiency,
(SNR)2
DQE = out
(u)
, (3.8)
(SNR)2in(u)
where (SNR)out(u) and (SNR)in(u) are signal-to-noise ratios at the output and input
of the detector as a function of spatial frequency, u [73]. The DQE modifies equation
3.7 to be
√
− 1 2σ 4φ = (DQE)th V . (3.9)2N
For our experiment, the number of electrons per frame was estimated by summing
the intensity values in a frame to be N ≈ 105 and we measured our fringe visibility
39
8
5.0 7
2.5 6
5
4
3
2
1
0
FIGURE 10. Mean interference fringes averaged over the scan with the background
subtracted. The inset shows a 1D profile of the sum of fringes along the direction
perpendicular to the line trace shown.
from Figure 10 to be V = 42.7% ± 4.8. The predicted fringe visibility from an ideal
three beam interferometer depends on the phase imparted onto probe+1. The fringe
spacing at the camera was ≈ 0.38 × fN , where fN is the Nyquist frequency. At this
spatial frequency, the Gatan K2 Summit camera has a DQE ≈ 0.56 [66]. Using these
values, we plot the numerically calculated V and σφ in the supplemental materials.th
The mean theoretical uncertainty in phase measurement is σφ < 15 mrad whenth
probe+1 transmits through a weak phase object.
40
Intensity [a.u.]
 HAADF Image of Au on C STEMH Phase Image of Au on C
1.2
5
1.0
4
0.8
3
i 0.62 0.4
1 0.2
10 nm 10 nm
0 0.0
(a) (b)
0.258 nm
0.244 nm
2 nm 1 2 nm 1
(c) (d)
 HAADF Image of Au on C STEMH Phase Image of Au on C
8
7 1.75
6 1.50
5 g g 1.25
4 1.00
3 0.75
2 0.50h h
1 0.25
0 5 nm 5 nm 0.00
(e) (f)
0.3 6
1.0 STEMH 1.0 STEMH 0.2 4
0.5 HAADF 0.5 HAADF 0.1 2
0.0 0
0.0 0.0
0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 0.1 0 2 4 6 8
Line Trace [nm] Line Trace [nm] Line trace [nm]
(g) (h) (i)
FIGURE 11. (a,e) Conventional annular dark field images of two different Au
nanoparticles on thin C support. (b,f) Phase reconstruction of the same regions using
STEMH. (c-d) Corresponding Fourier Transforms of (a) and (b). (g-h) Selected line
traces from (e-f), highlighting the low atomic number material contrast seen using
STEMH. The profiles are normalized to the maximum value of each image after
offsetting to a mean value of zero in vacuum. (i) Selected line trace from (b) along
just the carbon substrate, from which the thickness is calculated. (g-i) are plots of the
mean along three line traces with the root mean square of the the deviations shaded.
41
Intensity [a.u.]
e  Counts x104 e  counts x10
4
Intensity [a.u.]
Phase [rad]
rad rad
Thickness [nm]
Results and Discussion
STEMH Phase Contrast
The STEMH phase reconstructions and HAADF images of two randomly
oriented Au nanoparticles embedded on a thin amorphous carbon film are shown
in Figure 11. Compared to the HAADF image, STEMH allows for a much higher
contrast of the thin amorphous carbon. Additionally, the dc-component of the phase
is reconstructed using STEMH, resulting in a comparable signal with the HAADF,
but with additional amorphous carbon signal barely visible in the HAADF. Figure
11d shows that under the experimental conditions we used, STEMH has 0.24 nm
resolution of the Au atomic lattice, comparable to the HAADF resolution shown
in 11c. Notice how the high frequency information between the two techniques are
comparable, whereas the STEMH reconstruction contains much more low frequency
information because of the higher contrast on the carbon substrate. Note that these
scans were under-sampled in order to achieve a large field of view and decrease both
the scan time and file size. The achievable probe size for STEM is sub-angstrom as
discussed in the introduction, suggesting that STEMH should be able to achieve even
higher resolution than we report.
Figures 11g - 11h shows selected line traces along the carbon film and Au
nanoparticles for both the STEMH and HAADF signals. For comparison, the signals
are normalized to the maximum value of each image after offsetting to a mean value of
zero in vacuum. For 11h the STEMH signal begins to rise earlier than the HAADF due
to the amorphous carbon preceding the Au nanoparticles. In 11g, the two signals rise
simultaneously because the nanoparticle hangs off of the edge of the carbon. However,
the STEMH signal continues to rise around 7 nm, because unlike HAADF STEM the
42
STEMH signal is sensitive to the carbon film lying beneath the Au nanoparticle. This
is due to STEMH’s phase contrast, resulting in a gap between the two signals after
7 nm.
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
2 nm 0.05
FIGURE 12. Inset from center of Figure 11b in a carbon-only region to enhance
contrast.
The line trace in Figure 11b follows a path along the amorphous carbon film and
is plotted in Figure 11i. The thickness calculated from equation (3.6) is shown on
the right vertical axis. Interestingly, the carbon in Figures 11b and 11f, isolated in
Figure 12, shows a string-like topography, which is consistent with a thick-bonding
model detailed by Ricolleau, et al. in 2013 [67].
43
rad
We previously assumed that probe+2 was weak, or c+2 <
c+1 . This results in a
10
phase signal < 1 the probe+1 signal. Such a weak signal is present in Figures 11b10
and 11f in the form of a ‘shadow’ image of the nanoparticle in vacuum, although it
is weak enough to be barely identifiable above the noise from the primary signal on
carbon.
As shown in the previous section, however, this signal is on the same order or
larger than our theoretical uncertainty, which is confirmed by our measurement of
uncertainty in the phase. Here, we measured the deviation from the mean for both a
single line and an area of 50 lines within the vacuum region of Figure 11b. We found
that for a single line, σφexp = 30 mrad and σφexp = 35 mrad for an area of 50 lines.
This difference can be attributed to scan noise. The increase in noise between theory
and experiment is consistent with contributions due to higher order probes, and so
future consideration should be taken when designing gratings so as to optimize the
output SNR. Alternatively, a smaller selected area aperture could be used to block
higher orders.
Conclusions
In this article, we demonstrated sub-nanometer resolution electron phase imaging
using STEMH, a multiple-arm, path-separated interferometer with a phase imparted
onto one or more paths. We measured a fringe visibility of V = 42.7% experimental
uncertainty in phase measurement to be σφexp ≈ 0.03 rad. We then provided two
0.24 nm resolution phase-contrast images of Au nanoparticles on a thin carbon
substrate, with conventional HAADF images for comparison.
STEMH provides quantitative phase contrast, including the dc-component,
which we utilized to analyze the thickness of the carbon support. In addition to
44
thickness mapping due to the electrostatic potential within nanomaterials, STEMH
enables magnetic field mapping at subnanometer resolution. With an aperture
inserted to block higher order probes, the electron scanning coils can be utilized
to image electric and magnetic fields surrounding a material without illuminating
the material at all. Recall that we used a straight grating in this experiment to
prepare sharply-peaked, symmetric probes at the sample plane. Note, however, that
different phase-structured diffraction grating designs can be used to holographically
vary the complex amplitude cn of the diffraction orders [21, 46, 47, 61], potentially
enabling more complicated electron-specimen interactions with signals extractable
via STEMH. Finally, future additions of faster readout detectors and different grating
designs would further reduce the electron dose, potentially allowing STEMH to image
beam-sensitive, bio-molecular materials, electronics and magnetic materials at atomic
resolution.
45
Full form of equation 3.4
∑+1
Ip(x) = |cnt (nx0 + xp)|2
n=−1 [ ]
+c∗ c ∗−1 0 a−1 (x + 1x0) t
∗ (x + xp) ⊗ [a0 (x + 1x0) t (x + 1x0 − xp)]
+c∗0c
∗ ∗
−1 [a[0 (x− 1x0) t (x− 1x0 + xp)]⊗][a−1 (x− 1x0) t (x− xp)]
+c∗−1c+1 [a
∗ ∗
−1 (x + 2x0) t (x + 1x0 + xp)]⊗ [a+1 (x + 2x0) t (x + 1x0 − xp)]
+c∗+1c
∗
−1 a+1 (x− 2x0) t∗ (x− 1x0 + xp) ⊗ [a−1 (x− 2x0) t (x− 1x0 − xp)]
+c∗c [a∗0 +1 0 (x + 1x0) t
∗ (x + 1x0 + xp)]⊗ [a+1 (x + 1x0) t (x− xp)]
+c∗
[
c a∗
]
+1 0 +1 (x− 1x0) t∗ (x + xp) ⊗ [a0 (x− 1x0) t (x− 1x0 − xp)] (3.10)
Collecting the `th peak terms a (x + `x0), we can write this in the simpler form
seen in equation 3.4.
Derivation of transfer function reconstruction
Let us integrate out the x variable around the +1 order peak in Ip(x), using a0
as a kernel.
∫ ∫
a0 (x)I+1 (xp,x) dx = c
∗ ∗
0c+1a0 (x) [a0 (x) t
∗ (x + 1x0 + xp)]⊗ a+1 (x) dx
Ω(+x0)
∫ ∫ Using the commutivity of convolutions:
= c∗0c+1 a0 (x) a (x− x′) [a∗+1 0 (x′) t∗ (x′ + 1x0 + xp)] dxdx′
(3.11)
46
∫
Because an (x) is a symmetric function, a0 (x) a+1 (x− x′) dx = a∗ (x′0 ) ⊗
a ′+1 (−x ) = a0 (−x′) ⊗ a ′ ′+1 (x ) = a0 (x ) ⊗ a+1 (x′). We can simplify this further
using the convolution theorem, and noting that the circular aperture A0 (k) is a top
hat function:
 1 |k| ≤ KπK2Am (k) = A0 (k) =  (3.12)0 |k| > K
∫
a0 (x)I+1 (x∫p,x∫) dxΩ(+x0)
′
=c∗0c+1 ∫ ∫ e−2πik·x A0 (k)A+1 (k) [a∗ (x′) t∗ (x′0 + 1x0 + x ′p)] dkdx
=c∗c ∫ ∫ e−2πik·x
′ |A (k)|2 ∗ ′ ∗ ′ ′0 +1 0 [a0 (x ) t (x + 1x0 + xp)] dkdx
=c∗
′
0c ∫ e−2πik·x+1 A0 (k) [a∗ (x′0 ) t∗ (x′ + 1x0 + xp)] dkdx′
=c∗c ∫ a (x′) [a∗ (x′) t∗0 +1 0 0 (x′ + 1x0 + xp)] dx′
=c∗0c+1 ( |a (x
′
0 )|2t∗) (x
′ + 1x0 + xp) dx
′
∗
=c∗0c+1 |a0 (xp)|2 ? t∗ (x0 + xp)
=c∗c 2 ∗0 +1|a0 (−xp)| ⊗ t (x0 + xp)
=c∗0c+1h (xp)⊗ t∗ (x0 + xp) (3.13)
Since a0 is symmetric, h (xp) = |a0 (x )|2p . For the case where an (x) is
asymmetric, refer to the supporting information of our other article[70].
47
Numerical calculation of σφth
For an ideal three beam interferometer, the three probes are of equal amplitude
(cn ≈ √1 ). In the following calculation, we simulated a phase grating with the3
following transmission function:
( ( ( )))
1 + cos 2πk
G (k) = exp ∆φ i d × A0(k), (3.14)
2
where ∆φ is the phase depth, a complex coefficient that determines the diffraction
grating efficiency and wavefunction amplitude loss, while d is the grating pitch.
For the simulation, we used ∆φ = 2.869, which corresponds to diffraction probe
amplitudes of cn = 0.299, for n ∈ [−1, 0, 1]. The grating pitch was d = 160 nm and the
diameter was 50 µm. We then calculated the probe wavefunction and applied a phase
to probe+1. We calculated the fringe visibility from equation (3.15), which utilizes
the fast Fourier transform of the fringe pattern. There are two fringe spacings, and so
equation (3.15) calculates the fringe visibility of the mth FFT peak, corresponding to
the probe+1/probe−1 interferometer (m = 2) and the probe+1/probe0 interferometer
(m = 1).
V Im +Im ∆I= = 〈 〉 (3.15)I0 2 I
As shown in Figure 13a, the fringe visibility V varies between 0 % and 91 %.
This of course means that for a pure phase grating, the phase uncertainty diverges
at φ = `2π, where ` is an integer value. Realistically, these gratings are partially
amplitude gratings, and so the visibility is nonzero in vacuum. The corresponding
phase uncertainty is shown in Figure 13b. Since the phase information is measured
48
0.10
1st order FFT peak 1st order FFT peak
1.0 2nd order FFT peak 2nd order FFT peak
0.08
0.8
0.06
0.6
0.4 0.04
0.2 0.02
0.0 0 1 2 3 4 5 6 0 1 2 3 4 5 6
ϕ [rad] ϕ [rad]
(a) (b)
FIGURE 13. (a) The fringe visibility and (b) root-mean-squared uncertainty as a
function of phase imparted onto probe+1 for both m = 1 (red) and m = 2 (green).
in both interferometers, STEMH can utilize both signals to decrease the phase
uncertainty over a range of phase values.
49

σth[rad]
Chapter Conclusion
In this chapter we demonstrated scanning transmission electron microscope
holography at sub-nanometer resolution. We presented a theoretical model to extract
the transmission function of a specimen from directly imaging and analyzing the
interference patterns at each point in an image scan. We perform the technique to
image gold nanoparticles on a thin carbon substrate. In the next chapter, we will
present a further innovation to improve the ability of this electron interferometer
to probe more fundamental physics, image across large fields of view, and lower the
electron dose by increasing the signal to noise ratio of the interference fringe patterns.
50
CHAPTER IV
MEASUREMENT
Note on ‘A tunable path-separated electron interferometer with an
amplitude-dividing grating beamsplitter.’
Reprinted from Fehmi S Yasin et al. Applied Physics Letters 113, 233102 (2018),
with the permission of AIP Publishing.
Benjamin McMorran and Toshiaki Tanigaki first chatted about mixed-type
interferometers using both amplitude- and wavefront-dividing beamsplitters in 2014
at the Tonomura FIRST Meeting. Toshiaki Tanigaki and I concieved the original
idea and developed the experimental setup with hands-on guidance from Toshiaki
and through helpful conversations and meetings with Ken Harada, Daisuke Shindo
Benjamin McMorran and Hiroyuki Shinada. I fabricated the grating beamsplitters
used, built scanning mode into the Hitachi TEM with help from Toshiaki, recorded
all data, generated all figures, and wrote the supplemental material manuscript with
input from coauthors.
51
Introduction to A tunable path-separated electron interferometer with
an amplitude-dividing grating beamsplitter
Electron interferometry has been utilized to probe fundamental physics
and provide object wave imaging since Dennis Gabor hypothesized an ‘electron
interference microscope’ in 1948 [4]. Electrons offer different advantages from their
optical counterparts. Electrons are massive, yet can still be accelerated to relativistic
speeds. Their charge allows for a strong coupling to electromagnetic fields and
their shorter De Broglie wavelength provides electron interferometers with higher
resolving power, making them potent tools for materials research and the exploration
of fundamental physics.
Marton et al. built the first electron interferometer using amplitude-dividing
polycrystalline epitaxially grown copper membrane diffraction gratings in 1953 [74]
and Möllenstedt et al. followed close behind with wavefront-dividing electron biprisms
in 1955 [6]. The latter technology proved to be quite versatile allowing Lichte,
Tonomura, Matteucci, Pozzi and many others to establish electron holography as
a trusted technique for either high precision imaging [8, 9, 10, 11, 75] or probing
basic physics [12, 13, 14]. Tanigaki et al. expanded on this setup through split-
illumination electron holography (SIEH) [76], which boasts an additional two biprisms
for pre-specimen beam splitting. This method utilizes the electron biprism to tune
the path separation at the specimen plane, although with two notable difficulties.
First, SIEH uses plane wave illumination incident on the specimen, similar to normal
off-axis electron holography. This requires custom beam-blocking apertures in order
to measure a field-of-interest surrounding a beam-sensitive specimen [77]. Second,
although it remains a useful technology, Möllenstedt electron biprisms require wide,
52
coherent incident beams when employed as a beamsplitter, thus demanding a highly
coherent beam.
One way to alleviate this coherence requirement is by using electron diffraction
grating beamsplitters. Similar to Marton et al.’s original beamsplitter, diffraction
gratings are advantageous because they are amplitude-dividing beamsplitters, which
create copies of the original wavefront propagating in different directions. They
have the advantage that only a few grating bars, typically tens to hundreds of
nanometers in pitch, must be coherently illuminated in order to maintain fringe
visibility in the interference pattern [7, 78, 79]. This decreases the coherence width
requirements by at least an order-of-magnitude when compared to current biprism
beamsplitters, and could potentially allow implementation of this interferometer
without highly coherent electron source guns such as cold field-emission guns or
Schottky sources, for example. Holographic diffraction gratings have been developed
by several groups [20, 21, 40, 45, 46, 47] and have been employed as beamsplitters
in a couple of path separated electron interferometers [40, 53] and in a proposed
electron interferometer with path separations of 10−2 m [80]. Using current focused
ion beam (FIB) engineering techniques, we fabricated gratings that form electron
diffraction orders (henceforth called pn for the nth order) with a spatial separation
of hundreds of nanometers at the Lorentz sample plane, located in a field-free region
above the objective lens useful for imaging magnetic materials, and tens of nanometers
at the high-resolution sample plane in a commercial transmission electron microscope
(TEM) configured in scanning (STEM) mode [53].
In this article, we combine an amplitude-dividing beamsplitter with the
versatility of electrostatic biprisms to create a tunable path-separated electron
interferometer. Harvey et al. and Yasin et al. previously developed and demonstrated
53
full object wave measurement using STEM holography (STEMH) [81, 82]. Here, we
use this interferometer to perform flexible STEM holography (fSTEMH), where we
have increased flexibility via the tunability of the path separation. This increases the
field of view of STEMH and opens the door to fundamental physics experiments as
well as microscopy applications requiring large path-separations and localized-probe.
Experimental Setup
This interferometer was setup within a Hitachi HF-3000X TEM equipped with
a cold field-emission gun and several positionable electrostatic biprism wires placed
both pre- and post-specimen. As illustrated in Figure 14, the input electron wave
diffracts through a diffraction grating with pitch d = 190 nm and is focused into
electron probes with an estimated convergence semi-angle of 0.2 mrad and hundreds
of nanometers spatial separation at the Lorentz specimen plane using a two-condenser
lens illumination system. Note that the original probe separation at the specimen
plane depends on both the physical pitch of the diffraction grating as well as the setup
of the microscope lens system and biprisms. These gratings can be fabricated so that
the amplitude of the diffraction orders other than p−1, p0 and p+1 are approximately
negligible [21].
Four BPs are positioned along the optical axis and are tuned such that the
diffraction probes straddle the three BPs located further down the microscope column.
The probes are focused onto the first bi-prism (BP1), which blocks p0 entirely.
This increases the signal-to-noise of the desired frequency fringes for two reasons.
First, the probe intensity |p0|2 > |p 2n>0| , so it contains the majority of inelastically
scattered electrons from the grating. Second, when p−1, p0 and p+1 are all utilized,
the amplitude |p0| may be large enough for a non-negligible interference signal with
54
d
Grating Aperture Plane
Condenser 1 Lens
Cross-over 1 -Xo +Xo Biprism (BP) 1 -Xo - +Xo
BP2 +
Condenser 2 Lens
Cross-over 2
Specimen Plane
Objective Lens
Int 1 Lens
Grating Image 1 BP3 -
Int 2 Lens
Cross-over 3 BP4 +
Grating Image 2
Projection Optics
Image Plane
Detector
FIGURE 14. Experimental setup for a tunable path-separated interferometer. The
right-hand-side illustrates the change in path with the biprisms engaged.
p±2, adding noise [82]. Blocking p0 removes these two sources of noise. While the
diffracted probes may contain inelastic scattering, Shiloh et al. has shown that a
200 nm thick SiN membrane has a ratio of elastic to inelastically scattered electrons
of ≈ 0.66, with most of the inelastically scattered electrons contained in the long
tails of the probes [83]. Since our diffraction grating is 75 nm thick, we expect the
proportion of inelastically scattered electrons to be even lower. BP1 changes the
overlap of the interference fringe discs while BP2 tunes the path separation of the
probes in the specimen plane. BP1 and BP2 are tuned such that the remaining
diffraction probes straddle BP2 and spatially separate at the specimen plane to a
desired value. p−1 interacts with the phase-object while p+1 passes through vacuum.
The image of the grating is then focused onto BP3, with two spatially separated
images formed due to the voltage bias engaged in BP1. We tuned this voltage so that
55
300
160
140
250
120
100
200
150
100
50
0
FIGURE 15. Interference fringes at the detector. Inset is the mean of 400 1D slices
of the interference fringe pattern.
the images straddle BP3. We then engage BP3 to decrease the spatial separation of
the diffraction probes focused onto BP4 in the reciprocal plane, affectively increasing
the fringe spacing in the image plane at the detector. Finally, we engage BP4 to
overlap the grating images onto the detector, a Gatan US4000 charge-coupled device
(CCD) that records the interference pattern, shown in Figure 15.
Results
fSTEMH phase reconstructions of Si ramp
For use as an easily-characterizable phase object specimen, we placed Si on
a Mo substrate with W as an adhesion layer using FIB microsampling [84]. We
nanofabricated the Si to form a linear phase ramp, increasing from vacuum to
≈ 30 rad, (≈ 360 nm thick) over ≈ 1.2 µm on one side and decreases linearly with
twice the gradient on the other side, or −50 rad . We performed both fSTEMH with a
µm
56
Intensity [a.u.]
path separation of ∆x = 5 µm and off-axis electron holography on this test specimen.
As shown in Figure 16, fSTEMH accurately measures the phase profile of the Si, as
compared to off-axis electron holography.
The microscope we used wasn’t outfitted with a native STEM mode or aberration
correction, so the probe width at the specimen plane limits our resolution to ≈ 45 nm.
A dedicated STEM can improve this resolution, and STEM holography has been
previously demonstrated at subnanometer resolution [82]. We adjusted our scan
step size to be as large as the probe width to decrease acquisition time, maximize
beam stability and minimize data size. Due to undersampling, the phase image is
pixelated, but contains quantitative amplitude and phase information that compares
well to the off-axis electron holography reconstruction. Since we weren’t using a fast-
readout detector, the scan time was quite large as each pixel in Figures 16c and 16d
corresponds to an ≈ 5 s dwell time. The beam current decreased over time, resulting
in a decrease in the measured amplitude as seen in Figure 16d from the start of the
scan (bottom left) to the end (upper right). This decrease notably does not affect
the phase image in Figure 16c, and isn’t present in previous STEMH data sets that
utilize a fast readout detector [81, 82].
Fringe visibility versus path separation
In order to determine a range of path separations usable by such an interferometer
under these experimental conditions and limit any noise from higher order diffraction
probes, we fabricated an aperture with a series of two-slit windows (Figure S.4. in
the supplemental materials with well defined, varied path separations. We used this
aperture to isolate p+1 and p−1 at the specimen plane and measure the fringe visibility
V of the interference fringes at the detector over a range of path separations ∆x.
57
We adjusted BP2 to set ∆x, BP3 and BP4 to maintain a constant fringe spacing
dI ≈ 147 µm, and held BP1= −100 V and all lens values constant throughout. Images
of the probes for a selection of path separations and their corresponding averaged,
normalized interference fringe profiles are shown in Figure 17.
The results are shown in Figure 18. V (∆x) increases until ∆x ≈ 4 µm, after
which it decreases monotonically. This decrease in V can be explained by stray
magnetic fields passing through the area enclosed by the interferometer. According
to the Aharanov-Bohm effect, the phase difference between two paths of an electron
interferometer depends linearly on both the area enclosed by the two paths, A, and the
time-dependent alternating current (AC) stray magnetic field, B(t). This introduces
a time dependent phase in the interference fringes at the detector, modeled as B(t) =
B0sin(ωt).
Here, the interferometer has four enclosed areas. Referring to Figure 19, A1
encloses BP2, A2 is located at the objective lens, A3 is located at the intermediate 1
lens and A4 encloses BP3. This results in four independent phase terms that affect
the interference fringes recorded for a finite time interval, or 2 s for this experiment.
This time average has a couple of consequences. First, it is a source of noise that
decreases the fringe visibility of the interferometer at the onset. Second, increasing An
increases the amplitude of these phase fluctuations without changing the frequency.
When time averaged over the same 2 s time interval, destructive interference decreases
the fringe visibility to a minimum.
Initially, BP2 and BP3 are not engaged, and so the path separation at the Lorentz
sample position is ∆x = 1.9 µm. As BP2 is engaged and increased in value, A1 begins
to decrease. We measure an increase in V for these first changes in BP2, 3 and 4. This
could be explained by the stochastic fluctuations of the amplitude and frequency of the
58
thermal magnetic field noise, as described and demonstrated previously by Uhlemann
et al [85]. A1 is initially large, suggesting that fluctuations in B1 and ω1 dominate
the phase fluctuations. As A1 decreases, these phase fluctuations decrease, suggesting
that the fringe visibility increases due to more constructive interference over the 2 s
exposure. Simultaneously, An>1 all increase in magnitude, suggesting that there must
also be an eventual decrease in V as the path separation increases. This increase in
An>1 would explain the decrease in V for ∆x > 3.8 µm. We simulated this fringe
visibility experiment with thermal magnetic field fluctuations and present the results
in the supplemental materials in Figure S.3.
To test the largest path separation possible, we increased the path separation
to ∆x = 25 µm. However, due to considerations of the voltage that can be applied
to BP3 safely, this path separation could only be achieved at a much smaller fringe
spacing, dI ≈ 30 µm. This spacing corresponds to the Nyquist frequency of the
detector, which results in an expected decrease in V . We measured V = 0.67%±0.15%
under these conditions as shown in the supplemental materials.
59
35 35
30 30
25 25
20 20
15 15
10 10
5 5
200 nm 0−5 200 nm
0
−5
(a) (c)
1.2 1.2
1.0 1.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
200 nm 0.0 200 nm 0.0
(b) (d)
FIGURE 16. fSTEMH image of a fabricated Si phase ramp that increases linearly
from vacuum with a gradient o(f 25 rad on one side and decreases linearly with twiceµm )
the gradient on the other side −50 rad . (a,b) The reconstructed unwrapped phase
µm
image as well as the amplitude of the object wave using conventional off-axis electron
holography. (c,d) Same as (a,b), but using fSTEMH with ∆x = 5 µm.
60
ϕ [rad]
Amplitude [a.u.]
Amplitude [a.u.] ϕ [rad]
 100000 1.0
 50000
0.5
 0
0 5 10 15 20 25 30
m 0.0
(a) (d)
 100000 1.0
 50000
0.5
 0
0 5 10 15 20 25 30
m 0.0
(b) (e)
 100000 1.0
 50000
0.5
 0
0 5 10 15 20 25 30
m 0.0
(c) (f)
FIGURE 17. (a-c) 1D profile of p+1 and p−1 in vacuum with a two slit window
aperture inserted in the sample position to block all higher diffraction orders. (d-
f) Interference fringes acquired over a 10 s exposure for the corresponding path
separations shown in (a-c).
0.25
0.20
0.15
0.10
0.05
2.5 5.0 7.5 10.0 12.5 15.0 0.00
Δx [μm]
FIGURE 18. Interference fringe visibility versus path separation length between the
electron probes at the specimen plane.
61
Intensity [counts] Intensity [counts] Intensity [counts]
Norm I(x) [a.u.] Norm I(x) [a.u.] Norm I(x) [a.u.]

Discussion
The BPs allow for increased flexibility in multiple ways. The ability to tune
the path separation to arbitrarily large values at the specimen plane enables large-
geometry electron interferometry experiments. For example, studies of forward-
scattering due to atoms or molecules located in an isolated gas cell [86], the
nature of the Aharonov-Bohm effect from an isolated solenoid [87, 88], decoherence
theory and the quantum-classical boundary as the delocalized probe entagles with
the environment, and enclosing the arms of a charged particle interferometer
in a Faraday cage for rotation sensing [89] could all be enabled by this setup.
Additionally, fSTEMH may enable quantitative phase mapping with respect to
vacuum of programmable phase plates such as the ones proposed by Verbeeck et
al. [90]. Each of the above experiments requires either a large spatial separation
between interferometer arms in order to place a physical boundary between the two,
or the ability to tune the path separation over a significant range of values. fSTEMH
provides such an interferometric setup.
Furthermore, fSTEMH independently positions the localised reference beam
anywhere in a small area of the specimen, whereas conventional TEM holography
requires that one wide reference field pass through vacuum outside of the sample,
which places limits on the types of specimen geometries that can be imaged. Finally,
the use of a holographic beamsplitter allows additional control over the electron beam,
such as removing the spherical aberration [68] or introducing phase vortices that can
be used to measure magnetic fields [91].
62
Aperture Plane
Condenser 1 Lens
Biprism (BP) 1 -Xo - +Xo BP2	=	0	V	
BP2 +
A BP2	=	13	V	1	
Condenser 2 Lens BP2	=	99	V	
Specimen Plane Δx=	1.9	μm	
Objective Lens A Δx	=	3.8	μm	2	 Δx	=	15	μm	
Int 1 Lens A3	 BP3	=	0	V	
BP3 - BP3	=	-57V	
Int 2 Lens A BP3	=	-433	V	4	
BP4	=	51	V	
BP4 + BP4	=	55	V	
Projection Optics BP4	=	86	V	
Image Plane
FIGURE 19. fSTEMH setup for different path separations. Notice the change in the
enclosed areas A1, A2, A3 and A4. As the path separation increases initially, the total
area enclosed decreases, but then increases monotonically, leading to a loss in fringe
visibility for very large path separations.
63
Conclusion
We demonstrate a tunable path-separated electron interferometer within a
STEM. We use a nanofabricated grating as an amplitude-dividing beamsplitter
capable of preparing multiple spatially separated, coherent electron probe beams
with 950 nm spatial separation between neighbor diffraction orders at the specimen.
We configure four electrostatic bi-prisms (BPs) down the optical column and tune
the voltage applied to each to achieve path separations between p+1 and p−1 at the
specimen plane of up to 25 µm while maintaining fringe visibility at the detector.
We performed fSTEMH on a Si test specimen with the path separation tuned
to 5.0 µm. We measure highly interpretable, quantitative amplitude and phase
contrast that agrees with an independent measurement using off-axis electron
holography. We measure the fringe visibility of this interferometer over a range of path
separations to establish the interferometer’s utility at large path separations. This
experimental design can potentially be applied to phase imaging and fundamental
physics experiments, such as the Aharonov-Bohm effect, decoherence theory and
electromagnetic field mapping around a specimen’s edge without exposing the
specimen to radiation.
64
V versus area enclosed by interferometer
Theory
As stated in many textbooks and review papers (see [92, 93], for example), the
phase between two arms of an electron interferometer depends on a magnetic field B
normal to the area S enclosed by the interferometer as
∫
− 2πeφ2 φ1 = B · dS, (4.1)
h S
where h is Planck’s constant, e is the elementary electric charge and S denotes the
perimeter of the interferometer. Note that stray alternating current B field depends
on time and points orthogonal to the local area enclosed, B · S = B⊥(t)A. This
reduces equation (4.1) to
2πe
∆φ = − B⊥(t)A, (4.2)
h
where A is the area enclosed by the two paths and B⊥(t) = B0sin(ωt). The amplitude
B0 and frequency ω of the stray fields fluctuate at each time increment dt. To simulate
thermal fluctuations in the magnetic field, we will treat each amplitude and frequency
term as a mean B̄0 = 1 nT, ω̄ = 5 kHz and stochastically vary each independently
from a gaussian distribution.
In this experimental setup, there are four enclosed areas. Due to beam crossovers
between each area, the phase difference as electrons propagate around A1 and A3
accumulates in the opposite direction as when they propagate around A2 and A4.
However, due to rotation in the beam around the optical axis as the beam travels down
the column, the normal vector of each area points in a unique direction. Additionally,
65
we expect thermal magnetic field noise to be a main contributor to the stray fields
penetrating these areas as previously studied and demonstrated by Uhlemann et al
[85]. These fields not only fluctuate in time, but depend on z as well. Therefore, we
treat the magnetic field within each area as uniform, but local to each area.
For a 2-beam interferometer, like the one employed using the 2-slit aperture
shown in Figure 23 to block all higher order diffraction probes, the intensity pattern
at the detector is
∑ ( ) ( )
∗ (i 4
∑
∆φn) −2πi x 4 2πi xI(x) = I0 + c n=1 d
∗
− c+1e e I1 + c−1c+1e
(−i n=1 ∆φn)e dI , (4.3)
where c thm is the complex amplitude of the m electron diffraction probe beam, I0 =
|c 2 2−1| + |c+1| , dI is the fringe spacing, and ∆φn is the magnetic phase difference
acquired around the nth area. Setting cm =
1 for ease, expanding ∆φn and time2
averaging over a tf [s] exposure,
∫ [ ( )]t 4f 1 x 2πe∑
Ī(x, An) = I0 + cos + (−1)nAnB̄nsin(ω̄nt) dt (4.4)
0 2 dI h n=1
We know fringe visibility is defined as
V Imax − Imin= . (4.5)
Imax + Imin
Experiment and simulation
Using this definition, we can simulate multiple time averaged (2 s exposure)
interference patterns as a function of A and calculate V for each. As seen in Figure
66
0.6
0.5
0.4
0.3
0.2
0.1
0.0 0 25 50 75 100 125
|BP1|[V]
FIGURE 20. Experimentally measured interference fringe visibility versus the biprism
1 voltage applied, the absolute value of which is proportional to the area enclosed by
the interferometer.
19 in the main text, all enclosed areas An of the interferometer increase for decreasing
the voltage in BP1, or increasing |BP1|. Therefore, we performed an experiment and
simulation, decreasing the BP1 voltage, which increases An, while increasing BP4 to
maintain an overlap of the interference fringes at the detector. As shown in Figure
20, a decrease in the BP1 voltage results in a decrease in fringe visibility V . Note
that this is not the same as Figure 18, which plots V versus the path separation of
the probes at the specimen plane.
We simulated Gaussian stochastic fluctuations over various standard deviations
in both the amplitude B0 and frequency ω using an average B̄0 = 1 nT and two
mean frequencies ω̄ = 5 kHz and ω̄ = 5 MHz, representing the range of frequencies
expected for the field noise [85]. We included Poissonian shot noise and used fringe
spacing dI = 147 µm. The resulting simulated plots are shown in Figure 21. They
67

1.0 1.0 1.0
0.8 0.8 0.8
0.6 0.6 0.6
0.4 0.4 0.4
0.2 0.2 0.2
10..70e-09 1.1e-06 2.2e-06 3.3e-06 4.4e-06 5.5e-06 10..70e-09 1.1e-06 2.2e-06 3.3e-06 4.4e-06 5.5e-06 10..70e-09 1.1e-06 2.2e-06 3.3e-06 4.4e-06 5.5e-06
A  [m2] A  [m2] A  [m2enclosed enclosed enclosed ]
(a) (b) (c)
1.0 1.0 1.0
0.8 0.8 0.8
0.6 0.6 0.6
0.4 0.4 0.4
0.2 0.2 0.2
10..70e-09 1.1e-06 2.2e-06 3.3e-06 4.4e-06 5.5e-06 10..70e-09 1.1e-06 2.2e-06 3.3e-06 4.4e-06 5.5e-06 10..70e-09 1.1e-06 2.2e-06 3.3e-06 4.4e-06 5.5e-06
A 2 2enclosed [m ] Aenclosed [m ] Aenclosed [m2]
(d) (e) (f)
FIGURE 21. Simulated V(A) for ω̄ = 5 kHz and a) σ 5 5B0 = B̄10 0, σω = ω̄, b)10
σB0 =
7.5B̄0, σ
7.5
ω = ω̄, c) σB = B̄0, σω = ω̄, d-f) same as a-c) except with ω̄ = 5 MHz.10 10
were calculated using simulated interference fringes from equation 4.4. As seen in
Figure 21, V decreases with A for each standard deviation.
Fringe visibility versus path separation simulation
As shown in Figure 18 in the main text, V(An) increases initially before
monotonically decreasing. We explained this phenomena as being due to a decreasing
A1 while A
2
n>1 increases. To simulate this, we set A1 to decrease linearly from 0.83 µm
to 0.083 µm2 while An>1 increase linearly from 0.083 µm
2 to 0.83 µm2. These values
are a rough Fermi estimate of An ≈ ∆x ∗ L, where the length between crossovers
L ≈ 10−1 m The results are shown in Figure 22. The curve follows a similar qualitative
trend as Figure 18.
68
 [a.u.]  [a.u.]
 [a.u.]  [a.u.]
 [a.u.]  [a.u.]
1.0 1.0 1.0
0.8 0.8 0.8
0.6 0.6 0.6
0.4 0.4 0.4
0.2 0.2 0.2
60..30e-07 8.8e-07 1.1e-06 1.4e-06 1.6e-06 1.9e-06 60..30e-07 8.8e-07 1.1e-06 1.4e-06 1.6e-06 1.9e-06 60..30e-07 8.8e-07 1.1e-06 1.4e-06 1.6e-06 1.9e-06
A 2 2 2enclosed [m ] Aenclosed [m ] Aenclosed [m ]
(a) (b) (c)
1.0 1.0 1.0
0.8 0.8 0.8
0.6 0.6 0.6
0.4 0.4 0.4
0.2 0.2 0.2
60..30e-07 8.8e-07 1.1e-06 1.4e-06 1.6e-06 1.9e-06 60..30e-07 8.8e-07 1.1e-06 1.4e-06 1.6e-06 1.9e-06 60..30e-07 8.8e-07 1.1e-06 1.4e-06 1.6e-06 1.9e-06
Aenclosed [m2] A 2 2enclosed [m ] Aenclosed [m ]
(d) (e) (f)
FIGURE 22. Simulated V(A) for increasing A1 and decreasing An>1, which
corresponds to an increasing path separation at the sample plane. For a-c) ω̄ = 5 kHz.
a) σ 5B0 = B̄0, σ =
5 ω̄, b) σ = 7.5B̄ , σ = 7.5ω B0 ω̄, c) σ = B̄ , σ = ω̄, d-f) same10 10 10 0 ω 10 B 0 ω
as a-c) except with ω̄ = 5 MHz.
69
 [a.u.]  [a.u.]
 [a.u.]  [a.u.]
 [a.u.]  [a.u.]
Additional figures
FIGURE 23. Scanning electron microscopy micrograph of the two-slit aperture used
to block the higher order diffraction probes while performing the fringe visibility
versus path separation experiment described in the main text.
70
1.0
FFT 2D
0.9
0.8
0.7
0.6
1.5
1.2 0.5
0.9
0.6 0.4
10 μm 0.3
0.0
0.3
(a) (b)
FIGURE 24. a) An image of the diffraction probes at the specimen plane with
∆x = 25 µm. b) An image of the Fourier transform of the interference fringes with
the peaks corresponding to the fringe spacing circled in red.
71
Intensity x103
Intensity [a.u.]
Chapter Conclusion
In this chapter we demonstrated a tunable path-separated electron interferometer
with the largest demonstrated path separations for amplitude-dividing electron
interferometers. Such large path separations enable large field of view STEMH
imaging as well as large geometry physics experiments such as an exploration of
the Aharonov-Bohm effect or imaging the electric and magnetic near fields around
a beam-sensitive object without exposing the object to radiation. Additionally,
the addition of a direct beam-blocking biprism coupled with a prespecimen two-slit
aperture to block higher diffraction-order beams allows all electrons illuminating the
specimen to be used to form the signal in the interference fringe pattern, potentially
making fSTEMH the most dose-efficient form of electron holography available. This
dose-efficiency would essentially be limited by inelastic scattering to the p+1 and p−1.
72
CHAPTER V
CONCLUSION AND FUTURE DIRECTIONS
We demonstrated that STEMH may be used as both a practical atomic resolution
phase-contrast imaging technique and a tunable electron interferometer capable of
probing basic physics. As an imaging technique, we achieved atomic resolution of
multiple gold nanoparticles on an ultra-thin carbon substrate. We were able to
confirm a unique, non-random carbon bond structure in the thin substrates, while
resolving the gold atomic lattice with the same resolution as the simultaneously
acquired conventional STEM HAADF images. We were also able to expand the
initially proposed setup using four electrostatic biprisms, two pre- and two post-
specimen, into a tunable electron interferometer. We showed that fSTEMH remains
a quantitative imaging technique capable of both phase and amplitude contrast in
this experimental setup, and were able to explore the nature of the magnetic field
noise incident on such an interferometer as predicted and modeled by Uhlemann et
al [85].
We adapted this interferometric setup into multiple electron microscopes in two
different countries including a Thermo Fisher Titan 80-300 KV TEM with STEM
capabilities located within the Center for Advanced Materials Characterization in
Oregon at the University of Oregon, a Thermo Fisher Titan 80-300 kV TEM with
STEM capabilities, probe and image aberration correction and a Gatan K2 Summit
direct electron detector located within the National Center for Electron Microscopy
at Lawrence Berkeley National Laboratory, and finally a Hitachi HF-3000X TEM
equipped with a cold field-emission gun and several positionable electrostatic biprism
wires placed both pre- and post-specimen. These rapid installations are a testament to
73
the utility and flexibility that comes with using nanofabricated diffraction gratings as
the beamsplitter within the experimental setup. The ease-of-installation, scalability
and low cost of these grating beamsplitters could enable scanning transmission
electron microscopes around the world to have a working, quantitative imaging
technique for electron transparent “phase” materials and nanomaterials of interest
within the semiconductor industry, especially those with lesser known electro-
magnetic field distributions.
In addition to the benefits of having such an imaging technique as STEMH, we are
excited for the potential of developing STEMH as a quantum electron interferometric
imaging technique, which could potentially lower the dose incident on specimens by a
couple order of magnitudes, enabling quatitative, non-destructive biological imaging
applications. While such imaging may not be currently possible, future technologies
such as faster readout cameras and higher efficiency amplitude-dividing beamsplitters
might allow for such a low dose while maintaining fringe visibility at the detector.
The addition of a path separation tuning knob within fSTEMH enables the probing of
new sample geometries and electro-magnetic fields in or around specimens of interest.
It also enables interesting quantum “which way” experiments, and may lead to
interesting insights and a greater understanding of an electron quantum wavefunction.
74
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