Coherent Diffractive Imaging Using Phase Front Modifications
I. Johnson,1K. Jefimovs,1,*O. Bunk,1C. David,1M. Dierolf,1J. Gray,2D. Renker,1and F. Pfeiffer1,3
1Paul Scherrer Institut, 5232 Villigen PSI, Switzerland
2Lawrence Livermore National Laboratory, Livermore, California 94550, USA
3Ecole Polytechnique Fe ´de ´rale de Lausanne, 1015 Lausanne, Switzerland
(Received 17 January 2008; published 17 April 2008)
We introduce a coherent diffractive imaging technique that utilizes multiple exposures with modifica-
tions to the phase profile of the transmitted wave front to compensate for the missing phase information.
This is a single spot technique sensitive to both the transmission and phase shift through the sample. Along
with the details of the method, we present results from the first proof of principle experiment. The
experiment was performed with 6.0 keV x rays, in which an estimated spatial resolution of 200 nm was
DOI: 10.1103/PhysRevLett.100.155503 PACS numbers: 61.05.cp, 42.30.Rx, 42.30.Wb
The high degree of coherence of today’s x-ray sources
has promoted the development of various diffractive imag-
ing techniques that are not hindered by imperfections in
optical elements. In general, coherent diffractive imaging
(CDI) uses the intensity distribution of a diffracted wave to
deduce information about the specimen, such as attenu-
ation and phase shift distributions. A common challenge in
CDI is recovering the phase component of the wave field,
which is lost in the measurement. Iterative algorithms are
commonly used to retrieve the phase information [1–4].
Since the first demonstration by J. Miao et al. , CDI has
proven successful in many instances [6–9]. These tech-
niques attempt to reconstruct the image of a specimen from
a single diffraction pattern. This procedure requires a sub-
stantial number of iterations, has difficulties with complex-
valued exit waves, and may not lead to a unique solution.
Furthermore, one of the most stringent limitations to these
methods is the need for isolated objects (the so-called
These limitations may be overcome with complemen-
tary information from multiple diffraction patterns. The
Ptychographical Iterative Engine (PIE) [10–12] has re-
cently demonstrated the power of combining multiple
diffraction data sets to ease the retrieval of the phase
information. In this technique, overlapping illuminations
lead to the complementary information. Here, we report on
a new technique that takes advantage of phase front mod-
ifications [13,14] after the sample to generate different
diffraction patterns. As a result, the altered exit waves
force each exposure to have a unique interference pattern
and compensate for the lack of phase information. Similar
to the PIE method, complementary information directs the
convergence of a numerical algorithm and solves ambigu-
ities in the reconstruction. In contrast to the conventional
techniques, a sharp support is not needed and extended
specimens may be imaged. Additionally, since both the
computational and data acquisition times can be short
(seconds), this new technique provides the opportunity
for realtime image reconstruction. With regards to micros-
copy, this technique is optimal for specimens that fit within
a coherent area of the radiation, in our case a few micro-
meters in size. In this Letter, we report on the technique
and results from the first demonstration experiment.
Coherent radiation is locally attenuated and phase
shifted as it traverses the sample. In the description of
this new technique, we will focus on the wave front in
three distinct locations: the wave front incident on the
sample inc, the wave front just after the sample s, and
the exiting wave front after the phase plate exit. The phase
plate, P translated by ri, is introduced just behind the
sample to further modify the phase profile,
?r? ? s?r?P?r ? ri?:
Thus, the resulting far-field interference pattern is also
altered. Independent diffraction patterns of the same illu-
mination area on the sample ( sis identical for all expo-
sures) are recorded by shifting the phase plate with respect
to the wave front after the sample. These independent
exposures lead to an over-determination of the attenuation
and phase shift through the sample.
The reconstruction algorithm cycles through these ex-
posures. Starting with either an initial guess or the previous
estimate of the wave front following the sample, s, the
reconstruction algorithm calculates the exit wave front for
the ith illumination, Eq. (1). This exit wave front is then
propagated to the far-field with a Fourier transformation. In
the far-field, the phase information of the estimated com-
plex interference pattern is untouched, while the amplitude
distribution is replaced by the measured diffraction pattern.
Back propagating this data corrected interference pattern
with an inverse Fourier transformation generates a new
estimate of the exit wave, exit0
of this new estimate and the previous estimate of the wave
front following the sample produces an improved estimate
of the wave front just after the sample,
. A weighted combination
s0?r? ? ?1 ? F?r?? s?r? ? F?r?P??r ? ri?
jP?r ? ri?j2 exit0
PRL 100, 155503 (2008)
18 APRIL 2008
© 2008 The American Physical Society
This brings us to the next phase plate position or itera-
tion, where P?is the complex conjugate of P, and F is the
relative update factor. The relative update factor biases the
update in regions where the illumination on the sample has
a higher intensity. In fact, it is directly proportional to the
relative intensity of the illumination function across the
j inc?r?P?r ? ri?j
max?j inc?r?P?r ? ri?j?:
After typically 10 to 20 iterations, the algorithm con-
verges to a unique solution for sin the plane of the phase-
shifting aperture. It is the untouched wave front preceding
the phase-shifting aperture and the complementary infor-
mation in the diffraction plane from the various exposures
that directs the reconstruction to a solution in the plane of
the phase-shifting aperture. As in all phase retrieval meth-
ods, the sample illuminating wave front, inc, must be
known in order to deduce the properties of the sample.
Lastly, the attenuation and phase-shifting properties of the
sample, S, are deduced by dividing out the illuminating
S?r? ? s?r?
This technique is applicable to many types of coherent
transmission microscopy: like laser light, electrons, and x
rays. We have performed a demonstration experiment with
x rays at the microXAS beamline of the Swiss Light
Source, Paul Scherrer Institut (PSI). A schematic of the
experiment is shown in Fig. 1. An optimal x-ray energy of
6.0 keV (? ? 2:1?A) was chosen to balance the transmis-
sion power of the x rays and the coherence length of the
beam. A 10 ?m pinhole  35 meters from the source
selected a coherent portion [16,17] of the beam coming
from the double-crystal monochromator. This coherent
radiation illuminated the test sample, a nanofabricated
16 ?m PSI logo, located 0.5 mm downstream from the
pinhole. Half a millimeter further downstream came the
transparent phase-modifying plate.
The ideal phase plate would populate the diffraction
intensity uniformly in reciprocal space and, thus, interferes
with all length scales of the specimen. The chosen circu-
larly symmetric concentric ring structure (shown in Fig. 1)
accomplishes this fairly well and is not complicated to
fabricate with high precision. This structure was nanofab-
ricated by electroplating gold through a polyimide mould.
Details of the fabrication process may be found in
Ref. . It is composed of four concentric phase-shifting
rings constructed from 680 nm of gold on a 3 ?m thick
silicon membrane. At 6 keV, the rings result in about ?=2
phase shift and 44% attenuation in the wave front .
In the far field, 3.2 m down stream, diffraction patterns
were captured with a fiber coupled CCDdetector (Photonic
Science Hystar, effective pixel size 4:5 ?m). Diffraction
patterns of the sample were recorded with the phase-
modifying plate in seven different lateral positions 
with respect to the coherent illumination area on the sam-
ple. Thus, the final interference patterns are independent
and carry identical information about the exit wave of the
At each position, a short (700 ms) exposure was ac-
quired to capture the intense central part of the diffraction
pattern within the dynamic range of the detector.Long (7 s)
exposures were also recorded at each position to obtain
statistically significant data at higher spatial frequencies.
These two images were combined into a high-dynamic
range diffraction pattern. Of the 2048 ? 2096 pixels of
the camera, data from the central 1536 ? 1536 pixels
(6:91 ? 6:91 mm2) were selected. This corresponds to a
maximum spatial frequency of about 3:3 ? 10?2nm?1.
FIG. 1 (color online).Schematic of the experimental setup.
PRL 100, 155503 (2008)
18 APRIL 2008
The short and long exposure images were also used to
estimate the nonlinear response of the detector versus
integrated flux, the illumination function and diffuse scat-
A constant attenuation (a ? 0:5) and zero phase shift
(? ? 0) acrossthe samplewas assumed forthe initial guess
of the wave front following the sample, s. This guess
provided the starting point for the reconstruction. The
amplitude of the diffraction pattern from the exit wave
—thewave front following the phase-modifying plate,
deduced from the product of the guess and the phase-
modifying plate at the first position [Eq. (1)]—was re-
placed by the corresponding measured diffraction pattern
data. Upon Fourier back transformation, the estimate of the
wave front following the sample, s, was updated accord-
ing to Eq. (2). Following further iterations, the algorithm
quickly converged to the correct solution. The resulting
reconstructed transmission and phase shift through the
sample after 20iterations of cycling throughthe 7positions
are shown in Fig. 2.
As the iteration number in the reconstruction increases,
the change in the image becomes smaller and the image
converges on the solution. The difference between images
of adjacent iterations given by the invariant error metrics
introduced by J.R. Fienup, Eq. 2 of , is shown in Fig. 3.
The lower data points in the figure represent the conver-
gence for the uniform starting guess described in the text
above, while the higher points represent the average value
and its standard deviation for 100 random starting guesses.
The random starting guesses were both random in their
attenuation and phase shift. It can be seen that regardless of
the starting point, the multiple exposures quickly guide,
within tens of iterations, the reconstruction to the 1%
update level (level of Fig. 2). All initial guesses are di-
FIG. 2 (color online).
location of the profile transmission plot (c) and phase shift plot (d).
Reconstructed transmission (a) and phase shift (b) images through the sample. The horizontal lines indicate the
102030 405060 7080 90 100
FIG. 3 (color online).
lower circles are the change in the image update for the uniform
attenuation (a ? 0:5) and zero phase shift starting guess; the
upper triangles correspond to the average update progression for
100 random initial guesses.
Object update vs iteration number. The
PRL 100, 155503 (2008)
18 APRIL 2008
rected to the correct solution; furthermore, the speed of Download full-text
convergence is faster for the more realistic, uniform start-
ing guess. This justifies our starting guess of a constant
attenuation and phase shift across the sample.
In the 10 ?m diameter field of view of the reconstructed
image [Fig. 2(a) and 2(b)] are the micrometer size letters
‘‘SI,’’ the trailing strike, and the lower hundred-nanometer
size letters of the PSI Logo (Fig. 1). The large ‘‘SI’’ letters
are clearly readable in both the transmission and phase
images. The boundaries of the letters are 680 nm high gold
structures. They theoretically correspond to a 44% attenu-
ation and a phase shift of 1.58 radians [opposite direction
in the gray scales of Figs. 2(a) and 2(b)] relative to the
surrounding 700 nm polyimide mould. The reconstructed
transmission and phase shift in the gold regions are in good
agreement with these expected values. This agreement is
better seen in Figs. 2(c) and 2(d) where projections of the
reconstructed images along the lines shown in Figs. 2(a)
and 2(b) are plotted. A 400 nm wide gold bar separates the
large ‘‘S’’ and ‘‘I’’ letters of the logo. In this region, the
transmission drops from approximately 90% to 40%, cor-
responding to 44% attenuation, the dip in Fig. 2(c) located
at ?0:12 ?m. The phase is also shifted from ?0:65 to 0.95
radians, a 1.6 rad phase shift, as shown in the same location
in Fig. 2(d). The overall 10% attenuation in the trans-
mission image is attributed to the 3 ?m silicon membrane
that supports the sample.
In principle, the resolution of the technique is limited by
the angular extent of statistically significant data. In this
demonstration experiment, the resolution of both the trans-
mission and phase shift is on the order of the limit posed by
the pixel size of the reconstruction 95 nm, which is fixed by
the extent of the measured diffraction pattern. The transi-
tion—the lateral width between a change from 10% to
90% of the signal—of the sharp boundaries of the logo
occur in less than a few pixel widths. From this, we
estimate the resolution in both the transmission and phase
shift images to be about 200 nm. The imperfect contrast
and noise in the reconstructed images are attributed to the
experimental data. These artifacts are not present in noise-
free computer simulations , thus confirming that the
artifacts are not produced by the method.
In this Letter, we have presented a new technique that
utilizes multiple exposures with modifications to the phase
of the exiting wave front to direct the iterative reconstruc-
tion to the correct solution. It is similar to the common
microscope where a single spot is placed into a focus; then
the sample may be appropriately repositioned for further
investigation. The technique overcomes the loss of the
phase information, which leads to reconstruction ambigu-
ities in other techniques. It is not limited to isolated speci-
mens, and complex objects may be imaged in their natural
environment. Furthermore, the independent, however com-
plementary, diffraction patterns direct the iterative recon-
struction to the correct solution in thewell-defined plane of
the phase plate.
The method is applicable and has a future in laser light,
electron, and x-ray microscopy. In this Letter, the feasibil-
ity of the method has been demonstrated with the first
proof of principle experiment, conducted with x rays.
The fast convergence and data acquisition times make
this technique suitable for both real time imaging and
three-dimensional nanoscale tomography. These points
lead us to believe that this technique with improved data
quality can assist the investigation of small material sci-
ence and biological samples.
We gratefully acknowledge the assistance of C. Borca,
X. Donath, D. Grolimund, and B. Meyer during the experi-
ments. We also thank A. Menzel, D.K. Satapathy, and
P. Thibault for fruitful discussions. This work has been
performed at the Swiss Light Source, Paul Scherrer
Institut, Villigen, Switzerland.
*Current affiliation: EMPA, 8600 Du ¨bendorf, Switzerland
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