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Title

Experimental evaluation of spatial resolution inphase maps retrieved by transport of intensity
equation

Author(s)Zhang, Xiaobin; Oshima, Yoshifumi

Citation

Microscopy, 64(6): 395‑400

Issue Date

2015‑08‑12

Type

Journal Article

Text version

author

URL

http://hdl.handle.net/10119/16079

RightsThis is a pre‑copy‑editing, author‑produced PDF
of an article accepted for publication in
MICROSCOPY following peer review. The definitive
publisher‑authenticated version MICROSCOPY (2015)
64(6): 395‑400 is available online at:
http://dx.doi.org/10.1093/jmicro/dfv045

Description

Japan Advanced Institute of Science and Technology

Microscopy Advance Access published August 12, 2015

Microscopy, 2015, 1–6
doi: 10.1093/jmicro/dfv045

ArticleExperimental evaluation of spatial resolution
in phase maps retrieved by transport of intensity
equation
Xiaobin Zhang1,2,* and Yoshifumi Oshima1,2
1*To whom correspondence should be addressed. E-mail: zxiaobin@jaist.ac.jp
Received 4 April 2015; Accepted 24 July 2015

Abstract
The transport of intensity equation (TIE) is a convenient method of obtaining a potential
distribution, as it requires only three transmission electron microscopy images with different amounts of defocus. However, the spatial resolution of the TIE phase map has not
yet been evaluated experimentally. In this study, we investigated the phase distribution of
spherical gold nanoparticles and its dependence on the defocus difference and found thatthe spatial resolution was ﬁner than 2 nm, even for a defocus difference of 4 µm. Theoretical calculations reproduced the experimental results well.
Key words: transport of intensity equation, phase retrieval, mean inner potential, gold nanoparticle, spatial resolution

Introduction
The potential distribution of a two-phase interface such as
the cathode–electrolyte interface of a lithium ion batteryor the carrier distribution in a semiconductor device is key
to the development of these devices. Transmission electron
microscopy (TEM) is a powerful tool for obtaining a twodimensional potential distribution, because the phase of theincident electron wave is shifted not only by an object but
also by the electric and magnetic ﬁelds. However, TEM
images and transmission electron diffraction (TED) patterns
do not show the phase directly. To obtain phase information, electron holography (EH) [1], focal series reconstruction methods [2], diffractive imaging [3], and differential
phase contrast (DPC) imaging [4,5] have been proposed.
Among these methods, EH is the most popular. In EH, thephase is reconstructed from the interference between the object
wave passing through the specimen and the reference wave

passing through vacuum. However, to enable interference, abiprism must be installed in the TEM column, and a vacuum
path for the reference wave is also essential. These requirements restrict the applicability of EH.
One particular focal series reconstruction method, thetransport of intensity equation (TIE), is a convenient noninterferometric approach, which retrieves the phase information at the image plane using wave propagation principles. The most important beneﬁt of the TIE method is that it
requires only three TEM images with different amounts ofdefocus in order to obtain a phase distribution. The
TIE-retrieved phase has been reported to be equivalent to
the phase retrieved by a weak phase object approximation
when obtained under the conditions of a small-angle
approximation [6]. Beleggia et al. reconstructed the π/7phase jump at the interface between an amorphous carbon
ﬁlm and vacuum using both TIE and EH methods [7].

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1

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School of Materials Science, JAIST, 1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan, and 2JST-CREST,
7-gobancho, Chiyoda-ku, Tokyo 102-0075, Japan

2

Experimental methods
TEM observation was carried out using a 50-pm-resolution
electron microscope (R005) equipped with a cold ﬁeld emission gun and double spherical aberration correctors, operated at an acceleration voltage of 300 kV [19]. To remove
the strongly diffracted electrons, colloidal gold nanoparticles that did not show lattice fringes of more than two different planes were chosen for observation. The exposure timefor each image was 5 s, with a current density of 1 × 10−15
A/nm2 at the specimen position. The scale of the TEM
images was 0.19 nm/pixel. The imaging lens aberrations
were corrected to below fourth-order, and the chromaticaberration was 1.65 mm. Through-focus images were takenfrom an under-focus of −2000 nm to an over-focus of2000 nm by changing the objective lens current. The TIE
phase retrieval procedure was carried out using plug-in software QPt (HREM Research Inc.) for Digital Micrograph(Gatan Inc.). In this procedure, three TEM images (underfocused, in-focus and over-focused) were aligned to within
0.05 nm. The high-pass ﬁlter, which was expected to reduce
low-frequency noise [12,18], was determined to be unnecessary, as long as the phase distribution was retrieved not overthe entire area of the TEM image but only from the regions
containing the gold nanoparticles.
In our experiment, the gold nanoparticles had to be
spherical in shape because the radial intensity proﬁle, which
is used to increase the signal-to-noise (S/N) ratio of TEM
images, is obtained by averaging the intensity around concentric circles as a function of distance from the center. The
radial proﬁle represents the two-dimensional intensity distribution of the gold nanoparticle and has already been demonstrated to be effective for noise reduction by Oshima et al.
[20]. Colloidal gold nanoparticles were selected for observation because of their spherical shape. To prepare a specimen,
a colloidal solution with gold nanoparticles averaging 5 nm
in diameter was dispersed in ethanol and then dropped onto
a copper mesh with a thin carbon support ﬁlm. After drying,
the specimen was observed.
We employed two methods to conﬁrm the sphericity of thegold nanoparticles. The ﬁrst method used a high-angle annulardark ﬁeld scanning transmission electron microscope (HAADFSTEM), because HAADF intensity has been reported to be proportional to specimen thickness [21]. HAADF-STEM images
were obtained with a convergence semi-angle of 30 mrad andinner and outer detector semi-angles of 75 and 120 mrad,respectively. In Fig. 1, the normalized HAADF intensity proﬁlecan be seen to ﬁt well with the thickness distribution of a
7-nm-diameter particle, which indicates that the gold nanoparticle was almost spherical in shape. The blur observed near theparticle edge is believed to be due to the accumulation of
amorphous carbon from the support substrate.
The other method involved ﬁnding the centroid andcenter of mass of the gold nanoparticles. A particle can beapproximated as a sphere if these two centers coincide. The

Fig. 1. Graph of the HAADF-STEM intensity proﬁle of the observed
nanoparticle (solid curve), normalized for comparison with the thickness
distribution of a ∼7 nm gold nanoparticle (dashed curve). The inset
shows the corresponding HAADF-STEM image of the gold nanoparticle.

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De Graef et al. separated the electrostatic and magnetic
phase shifts in magnetic materials using TIE [8]. Once theseresults demonstrated the reliability of the method, TIE wasused to investigate p–n junctions in semiconducting devices[9], to measure the mean inner potential (MIP) of MgOnano-cubes [10] and metal nanoparticles [11–13], and toobserve helical spin order [14] or two-dimensional skyrmion crystals [15,16].
Although quantitative TIE-retrieved phase maps havebeen well demonstrated, their spatial resolution requires
further investigation. The spatial resolution depends on the
defocus difference between the under- and over-defocused
TEM images. Van Dyck and Coene [17] calculated that the
spatial resolution of a phase map was 0.14 nm, when the
defocus difference was smaller than 1 nm at an accelerating
voltage of 400 kV. Recently, Ishizuka and Allman [18]experimentally demonstrated that the TIE method canprovide an atomic-resolution phase map for a Si3N4 crystal
sample and estimated theoretically that to obtain a spatial
resolution of 0.14 nm at an acceleration voltage of 400 kV,
a defocus difference smaller than 9.5 nm is sufﬁcient. These
results are inconsistent. Furthermore, the defocus dependence of the spatial resolution of the TIE-retrieved phase map
has not yet been investigated experimentally.
In this study, the phase distributions of individual gold
nanoparticles supported on a thin carbon ﬁlm were retrieved
by TIE. The defocus dependence of the spatial resolution of
the TIE-retrieved phase map was determined experimentally.

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centroid is the central point of the selection, i.e. the averageof the x and y coordinates of all pixels. The center of mass is

Fig. 2. This TEM image was digitalized in order to distinguish the area of
the gold nanoparticle. The centroid coordinates of the distinguishedarea were used as the center when drawing the radial proﬁle. The radial
circle in (b) closely matches the periphery of the demarcated area in (a).

the brightness-weighted average of the x and y coordinatesof all pixels in the selection. Figure 2 shows an in-focus
TEM image of a gold nanoparticle that was digitized inorder to distinguish the area of the gold nanoparticle. Figure 2ashows the periphery of a typical spherical gold particle. In thiscase, the coordinates of the centroid were measured to be(65.3, 64.5), and the center of mass was also (65.3, 64.5); thatis, the two coincided exactly. Moreover, we conﬁrmed thatthe roundness of this particle was 0.984, as calculated via
4 × ð½area=π × ½major axis2 Þ. Figure 2b shows a circle centered at the centroid coordinates. This circle agrees well withthe periphery of the demarcated area in Fig. 2a. Using thesemethods, spherical gold nanoparticles were selected for analysis, and radial proﬁles of the selected particles were drawn
using the centroid coordinates determined above.
Next, a TEM image was reconstructed from the radialintensity proﬁle, and a TIE phase map was obtained fromthese reconstructed TEM images with three different levelsof focus (under-focused, in-focus and over-focused), as
shown in Fig. 3. In this ﬁgure, the original TEM images andthe corresponding phase map (upper row) are so noisy that
the Fresnel fringes cannot be clearly observed in someplaces. However, the reconstructed TEM images (bottomrow) show the Fresnel fringes clearly, and the corresponding
phase map was obtained without noise. The spatial resolution was evaluated by Fourier transformation.

Results and discussion
Figure 4 shows through-focus TEM images of gold nanoparticles, which were reconstructed from the radial intensity

Fig. 4. (a) Through-focus TEM images, reconstructed using radial intensity proﬁles ranging from an under-focus of−2000 nm to an over-focus of 2000 nm, of a 5-nm spherical gold nanoparticle. (b) TIE-retrieved phase maps at
defocus differences of 200, 400, 1000, 2000, 3000 and 4000 nm, respectively. All of the phase maps were obtainedusing three images taken at under-focus, in-focus and over-focus. The under- and over-focused images had
symmetric defocus distances with respect to the in-focus image.

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Fig. 3. Typical TEM images with different amounts of defocus (−500 nm
under-focus, in-focus, and 500 nm over-focus, from left to right) and a
TIE-retrieved phase map. (a) Results of raw TEM images and (b) resultsof TEM images reconstructed from the radial intensity proﬁles.

3

4Ishizuka et al., theoretically calculated the spatial resolution of the TIE-retrieved phase map. On the basis of their
discussion, the spatial resolution can be estimated to be 1.41,
1.00, 0.63, 0.45, 0.36 and 0.31 nm−1 for a defocus differenceof 200, 400, 1000, 2000, 3000 and 4000 nm, respectively.
The experimental spatial resolution was slightly higher than
the theoretical resolution, except for defocus differencesof 200 and 400 nm, at which the experimental value was signiﬁcantly lower than the theoretical value. We believe thatthe experimental spatial resolution may be reduced below∼1 nm−1 by the quantization error, which occurs when converting real analog values into digital values. The Nyquist
frequency of 2.63 nm−1 in the present phase map (pixel size:
0.19 nm) seems to be insufﬁcient to reproduce the phaseFig. 6. Graph of spatial resolution as a function of defocus difference.
The experimental results are indicated by dots. The circular dots
correspond to TIE phase maps obtained with a pixel size of 0.19 nm,
while the square ones had a pixel size of 0.1 nm. The theoretical results
obtained by Ishizuka et al. are shown by the solid curve, the simulation
results by the dashed curve and the dotted line is a curve ﬁt to thesimulated results.

Fig. 5. Fourier transformations of the TIE-retrieved phase maps shown in
Fig. 4b. The left column shows the radial intensity proﬁles of the Fouriertransformation patterns in the right column. In these patterns, the brightrings appear at 0.25, 0.42, 0.58, 0.73 and 0.89 nm−1, as indicated by thebars.

Fig. 7. Radial intensity proﬁles of the Fourier transformation patterns of
the TIE phase maps, obtained with defocus differences of 200 and
400 nm. To verify that TIE can obtain phase maps with higher resolution
when the defocus difference is smaller, TEM observations of gold
nanoparticles were carried out with the pixel size reduced to 0.1 nm.

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proﬁles. The Fresnel fringes around the gold nanoparticles
changed, depending on the amount of defocus. Phase maps
were retrieved for defocus differences (2Δz) of 200, 400,
1000, 2000, 3000 and 4000 nm. Since the noise was reducedin these maps, the phase distribution from the edge of agold nanoparticle to its center became clearer. Moreover,the phase maps gradually became blurred with increasing
defocus difference.
Figure 5 shows fast Fourier transform (FFT) patterns of
the phase maps in Fig. 4 and the corresponding radial proﬁles. In these patterns, rings appear around 0.25, 0.42,
0.58, 0.73 and 0.89 nm−1, but the maximum radius of thevisible ring decreased with increasing defocus difference.
This indicates that a reasonable phase map of the gold
nanoparticle was retrieved, and the spatial resolution in thephase map decreased with increasing defocus difference.
The spatial resolutions were determined to be 0.89, 0.73,
0.73 0.58, 0.42 and 0.42 nm−1 for defocus differences of200, 400, 1000, 2000, 3000 and 4000 nm, respectively.
The defocus dependence of the spatial resolution wasinvestigated for ﬁve gold nanoparticles with different diameters. The particles generated rings at slightly different
positions in the FFT patterns, but all showed a similar
defocus dependence. The average spatial resolutions were
1.02, 0.86, 0.68, 0.55, 0.42 and 0.42 nm−1 for defocus differences of 200, 400, 1000, 2000, 3000 and 4000 nm,
respectively, as shown in Fig. 6.

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5modulation above ∼1 nm−1. Practically, the frequency component above ∼1 nm−1 is lost in the Fourier transformation,
as shown in Fig. 4. To obtain atomic-resolution phase maps
(with a minimum spatial resolution of 0.1 nm), the pixels
should be smaller than 0.02 nm.
To obtain the actual spatial resolutions for defocus differences of 200 and 400 nm, we carried out TEM observationof gold nanoparticles at high magniﬁcation (pixel size:
0.1 nm). The Nyquist frequency of 5.00 nm−1 should be sufﬁcient to reproduce the phase modulation at high frequency.
Figure 7 shows a radial proﬁle of FFT patterns of the
TIE-retrieved phase maps. Clearly, higher spatial resolutionsof 1.48 and 1.20 nm−1 were achieved for defocus differencesof 200 and 400 nm. These coincide well with the theoretical
calculations by Ishizuka et al., as summarized in Fig. 6. In
this study, the actual spatial resolution was determined to be∼0.68, 0.83, 1.5, 1.8, 2.4 and 2.4 nm for defocus differencesof 200, 400, 1000, 2000, 3000 and 4000 nm, respectively.
In this section, we did not use raw TEM images of thegold nanoparticles, but instead utilized TEM images reconstructed from the radial intensity proﬁles of the gold nanoparticles, owing to the improved S/N ratio of the latter. To
conﬁrm that the reconstruction process was reasonable,
TEM images of a gold nanoparticle 5 nm in diameter were
simulated by the multi-slice method, and TIE phase maps
were retrieved from these images (Fig. 8). The gold nanoparticle was tilted to the [1 5 33] direction to avoid diffraction effects and positioned at the center of the supercell(25 × 25 × 5 nm). Each slice was 0.5 nm thick. The microscopy conditions for the simulation were as follows: the spherical aberration coefﬁcient was 0.001 mm, the defocus spreadfor the electron source was 3 nm and the convergence semiangle was 0.5 mrad. In Fig. 8, the simulated defocus

dependence of the Fresnel fringes shows the same trends as
the experimental dependence. Moreover, the simulated phase
map was gradually blurred by increasing the defocus difference, which is similar to the experimental phase map behavior. The FFT analysis results are shown in Fig. 6 as a dashedline. The simulated upper limit of the visible ring was slightly
higher than the experimental and theoretical limits, probably
because of the noise-free and more coherent electron waves
used in the simulation. However, the (dotted) curve ﬁtted tothe simulation data in Fig. 6 shows a similar defocus difference dependence as the experimental and theoretical curves.
These results support the idea that TEM images reconstructedfrom a radial intensity proﬁle can reasonably be used toevaluate TIE phase maps of gold nanoparticles.
This study suggests that the electron phase shift associated with the two-phase interface can principally be recovered by the TIE technique. For example, in the case of a p–njunction, the potential difference between the p- and n-typesemiconductors is ∼1 V high, and the transition region is∼10 nm wide, even when the carrier concentration was relatively high (∼1018 cm−3) (the potential change is more gradualat low carrier concentrations). Therefore, the potential distribution of a p–n junction may be obtained even at a defocus
difference of 4 µm.

Conclusion
Spherical gold nanoparticles ∼5 nm in diameter wereobserved using a 50-pm-resolution electron microscope at anacceleration voltage of 300 kV. Phase maps were retrieved
using the TIE method. We conclude that the spatial resolution
was 0.68, 0.83, 1.5, 1.8, 2.4, and 2.4 nm for defocus differences of 200, 400, 1000, 2000, 3000, and 4000 nm,

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Fig. 8. (a) Simulated through-focus TEM images of a 5-nm gold nanoparticle placed in vacuum, obtained
from an under-focus of −2000 nm to an over-focus of 2000 nm. (b) The corresponding TIE-retrieved phasemaps at defocus differences of 200, 400, 1000, 2000, 3000 and 4000 nm.

6respectively. Theoretical spatial resolution closely reproduced
these experimental results. We believe that atomic-resolution
phase maps can be obtained from TEM images with pixels
smaller than 0.03 nm. We also believe that the TIE method
can provide the potential distribution of a two-phaseinterface.

Acknowledgements
The authors thank Prof Takayanagi and Dr Mitome for their encouragement and useful advice.

Funding
This work was supported by the Japan Science and Technology
Agency (JST) under the CREST project (114209).

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References

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