Coherent X-ray diffractive imaging: applications and limitations.

Abstract

The inversion of a diffraction pattern offers aberration-free diffraction-limited 3D images without the resolution and depth-of-field limitations of lens-based tomographic systems, the only limitation being radiation damage. We review our experimental results, discuss the fundamental limits of this technique and future plans.

Full text

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arXiv:physics/0308064v1 [physics.optics] 15 Aug 2003
Coherent X-ray Diffractive Imaging; applications and limitations
S. Marchesini, H. N. Chapman, S. P. Hau-Riege, R. A. London, and A. Sz¨ oke
Lawrence Livermore National Laboratory, 7000 East Ave., Livermore CA 94550 USA.
H. He, M. R. Howells, H. Padmore, and R. Rosen
Advanced Light Source, Lawrence Berkeley Lab, 1 Cyclotron Rd., Berkeley, CA 94720. USA.
J. C. H. Spence and U. Weierstall
Department of Physics and Astronomy, Arizona State University, Tempe AZ 85287-1504, USA.
The inversion of a diffraction pattern offers aberration-free diffraction-limited 3D images without
the resolution and depth-of-field limitations of lens-based tomographic systems, the only limitation
being radiation damage. We review our experimental results, discuss the fundamental limits of this
technique and future plans.
I.INTRODUCTION
Three ideas, developed over half a century, have now
converged to provide a working solution to the non-
crystallographic phase problem. In this paper we outline
some applications and our development of this solution,
which provides a method for lensless, diffraction-limited,
aberration-free X-ray imaging of nano-objects in three-
dimensions at high resolution. We suggest the acronym
CXDI (Coherent X-ray Diffractive Imaging) for this field.
We also discuss the fundamental limitations on resolu-
tion set by radiation damage, and some new approaches
to this problem based on femtosecond diffraction.
The ideas start with Sayre’s 1952 observation that
Bragg diffraction undersamples diffracted intensity rel-
ative to Shannon’s theorem [1]. Secondly, the develop-
ment of iterative algorithms with feedback in the early
nineteen-eighties produced a remarkably successful opti-
mization method capable of extracting phase information
from adequately sampled intensity data [2]. Finally, the
important theoretical insight that these iterations may
be viewed as Bregman Projections in Hilbert space has
allowed theoreticians to analyze and improve on the ba-
sic Fienup algorithm [3]. A parallel development was the
real space algorithm for “phase recovery” in crystallog-
raphy [4] and non-periodic diffraction [5].
Experimental work started for X-rays with the im-
ages of lithographed structures reconstructed from their
soft-X-ray diffraction patterns by Miao et al. in 1999
[6]. More recently we have seen higher resolution X-
ray imaging [7], 3D imaging [7, 8], the introduction of
algorithms which do not require a conventional lower-
resolution image to provide the boundary (support) of
the object [9, 10, 11] and the striking atomic-resolution
image of a single nanotube reconstructed from an experi-
mental electron micro-diffraction pattern (an example of
the more general CDI method) [12].
The rapid growth of nanoscience (described recently
as “the next industrial revolution”) has produced an ur-
gent need for techniques capable of revealing the inter-
nal structure, in three dimensions, of inorganic nanos-
tructures and large molecules which cannot be crystal-
lized (such as the membrane proteins of vital importance
for drug delivery).Scanning probe methods are lim-
ited to surface structures, and the electron microscope
can provide atomic resolution images of projections of
crystalline materials in thicknesses up to about 50nm,
or tomography of macromolecular assemblies and inor-
ganics at lower resolution. No technique at present can
provide three-dimensional imaging at nanometer resolu-
tion of the interior of particles in the micron size range.
The development of such a method would have a deci-
sive impact on several fields of science, and would be a
timely enabling technology for the Molecular Foundary
at Lawrence Berkeley National Laboratory, the Linac
Coherent Light Source (LCLS) at Stanford and other
nanoscience user facility initiatives.
see initial applications for CXDI (in the 0.5–4kV X-ray
range) in materials science as follows: 1. The visualiza-
tion of the internal labyrinth structure of the new meso-
porous framework structures (eg. glassy foams, now find-
ing uses for molecular sieves and hydrogen storage; 2.
Imaging the complex tangles of dislocation lines which
are responsible for work-hardening; 3. Imaging the cav-
ities within duplex steels, responsible for their very high
uniform extension; 4. 3D imaging defect structures in
magnetic multilayers; 5. The tomographic imaging of
misfit dislocations at interfaces, free of the thin-film elas-
tic relaxation processes which distort the images ob-
tained by transmission electron microscopy; 6. Imaging
of the three-dimensional arrangement of Orowan disloca-
tion loops which, by entanglement with particles, provide
the dispersion-hardening of copper alloys; 7. The imag-
ing of precipitates in metal-matrix composite materials;
8. The imaging of electronic device elements for future
computing schemes, such as quantum computing. This
application is particularly promising, since the ability
to prepare the elements lithographically provides the a-
priori information needed to solve the phase problem. In
life sciences such a technique is needed to determine the
bulk (internal) structure of assemblies of macromolecules
(molecular machines), protein complexes, and virus par-
ticles at a resolution sufficient to recognize known pro-
teins and determine their relationships to each other.
Briefly, we fore-

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The apparatus we have developed for CXDI is reviewed
in section 3 and reconstruction methods are described in
section 4. Section 5 summarizes data on the radiation
damage limitations to resolution, and the theoretical ba-
sis for these limits, while Section 6 describes a method for
overcoming those limits by using ultrafast x-ray pulses.
We begin by describing a new algorithm for CXDI which
avoids the need for a priori knowledge of the object sup-
port.
II.THE CXDI TECHNIQUE
A CXDI experiment consists of three steps: (a) the
sample is illuminated by monochromatic coherent x-rays
and a recording is made of a single diffraction pattern (for
2D) or a tilt series (for 3D); (b) the phases of the pattern
are recovered from the measured intensities using estab-
lished phase-retrieval algorithms; (c) the unknown object
is then recovered by Fourier inversion. In the Gerchberg-
Saxton-Fienup scheme one starts with random phases,
which lead to noise when transformed from reciprocal to
real space. One then imposes the “finite support” con-
straint (namely that there must be a blank frame around
the specimen), before transforming back to reciprocal
space. In reciprocal space the phases so generated are
combined with the measured diffraction magnitudes to
start the next iteration. After a large number of itera-
tions, in most cases the object emerges from the noise.
Our novel improvement is that the estimate for the ob-
ject support is continually updated by thresholding the
intensity of the current object reconstruction [11]. We
start from a threshold of the transform of the diffraction
pattern and as the iterations progress the support con-
verges to a tight boundary around the object. This, in
turn, improves the image reconstruction, which gives a
better estimate of the support. An example of the re-
construction of a simulated diffraction pattern produced
by a cluster of gold balls is shown in the movie (Fig. 3)
together with the support.
The algorithm does not require any “atomicity” con-
straint provided by the gold balls as demonstrated by
the reconstruction of a greyscale image. The algorithm
also successfully reconstructs complex objects (those that
cause large variations in the phase of the exit wavefield
in two dimensions), which hitherto have been experimen-
tally difficult to reconstruct. This opens up the possi-
bility of image reconstruction from microdiffraction pat-
terns, where the illumination is tightly focused on the
object.
III.CXDI EXPERIMENTS AT THE ALS
Our experiments in coherent diffraction began with tri-
als using electron [13] and visible-light [14] optics and
continued with CXDI at the Advanced Light Source
(ALS). The experiments used the “pink” beam at beam-
line 9.0.1 [15] which is fed by a 10-cm-period undulator
operating in third harmonic with deflection parameter
(K) equal to 1.2 and delivering 588 eV (2.11 nm) photons.
Features of the beam line include a 0.5 µm-thick, 750 µm-
square Be window to separate the UHV beam line from
the low-vacuum sample environment, a monochromator
consisting of an off-axis segment of a zone plate and the
diffraction experiment itself (Fig. 1).
The x-ray coherence length lc must be greater than
the maximum path difference between any pair of inter-
fering rays, i. e. lc > wθmax, where w is the width of
the sample and θmax is the maximum diffraction angle.
For our geometry and wavelength, θmax= 0.12 radian and
the resolution limit is 8.4 nm. For the 5 µm aperture (ef-
fectively the monochromator exit slit) shown in Fig. 1,
the resolving power is about 500, the coherence length
is then 1µm and the maximum sample illumination area
8×8 µm2. Similarly the (spatial) coherence patch pro-
vided by the 5 µm aperture is 10×10 µm2. Allowing for
an empty (but still coherently illuminated) band around
the sample, its allowed size is thus < 4×4 µm2.
We consider now the sampling of the diffraction pat-
tern.The Shannon interval for frequency-space sam-
pling of the intensity is 1/(2w) = ∆/λz where z is the
sample-to-detector distance and ∆ is the detector-plane
increment (a 25 µm CCD pixel in our case). For our
λ and z values this also leads to a maximum sample
width of 4 µm. This is correct (Shannon) sampling of
the diffraction-plane intensity and twofold oversampling
in each direction of the diffraction-plane wave amplitude.
Note that these limits on the sample size arising from
coherence and sampling considerations are not the only
ones in effect.
We have carried out three series of experiments, all
using test samples made from 50 nm gold balls. The
first [9] demonstrated the basic 2D technique with im-
age reconstruction using a support function determined
by scanning electron microscopy. The second [9] used
a sample intentionally prepared in two separated parts,
and reconstruction was achieved using information from
the 2D diffraction pattern alone. The third series used
a miniature sample-rotation device to collect several to-
mographic data sets. The picture of the device shown in
Fig. 1 (picture width = 7 cm) shows the sample and its
rotation spindle and driver. The small black square is
the frame of a Si3N4window on which the 3D sample is
deposited in a 2.5-µm-wide microfabricated tetrahedron
(Fig. 2). Not shown is an angular Vernier scale that
was used to measure the rotation angle. Using this ap-
paratus, a set of 150 views with at least a 100 second
exposure time per view required about 10 hours. The
3D data generated by the object shown in Fig. 2 are still
being analysed.

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FIG. 1: Experimental chamber layout (left) and rotation device for tomographic recordings (right).
FIG. 2:
image of a silicon nitride pyramid membrane, and the right picture shows a top view of a similar pyramid that has been
decorated with 50 nm diameter gold spheres. The silicon nitride is 100 nm thick, and the pyramid is hollow. These objects are
ideal for testing since they can be well characterized in the SEM, have extent in all three dimensions, and can be treated as an
analogue to a molecule
Three-dimensional test objects fabricated for x-ray diffraction imaging. The left picture shows a SEM perspective
IV.RECONSTRUCTIONS
Samples were made by placing a droplet of solution
containing ‘gold conjugate’ colloidal gold balls on a sil-
icon nitride window (thickness 100 nm) and allowing it
to dry. The gold balls formed several single layered (2D)
clusters on the SiN membranes, as imaged by a field-
emission scanning electron microscope (SEM).
In the first experiment the use of a Si3N4 window of
the order of 5 microns width ensured that the sample
was isolated and of the required size [9]. The structure
contained at least one isolated ball generating a ‘refer-
ence wave’ which interfered with the signals from other
clusters to form a hologram.
The autocorrelation function obtained by Fourier
transforming the intensity of the diffraction pattern in-
cluded an image of every cluster convolved with the single
ball, and these images formed a faithful representation of
the structure in real space, without iterative processing.
Not all the clusters could be imaged this way, since some
of the intra-cluster distances were overlapping. The com-
plex transmission function of the Si wedge at the corner
of the window generated a complex object that was dif-
ficult to reconstruct by phase retrieval. For real objects,
we have seen in simulations that using a support of tri-
FIG. 3: Comparison of reconstructed soft X-ray image (mid-
dle) and SEM images of gold ball clusters (left). Each ball
has a diameter of 50nm [H. He, et al. Phys. Rev. B, 67,
174114 (2003)]. Also shown (right) is a movie (1.1 MB) of
the reconstruction as it iterates. Each frame of the movie dis-
plays the current estimate of the image intensity on the left
and the image support on the right [11].
angular shape was sufficient to obtain the image. In the
experimental case, we had to use a support obtained from
a low resolution image of the object obtained by scanning

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electron microscopy (SEM). In the next experiment the
object was isolated by means of sweeping the sample par-
ticles with an Atomic Force Microscope (AFM). The re-
construction has been obtained using the standard HIO
algorithm [2] with missing data due to the beam stop
and its supports left unconstrained during the iterations.
Rather than rely on a low-resolution secondary imaging
method to obtain an estimate of the support, we obtained
an estimate of the object support from the support of its
autocorrelation. The object support was then adjusted
manually as the reconstructed image became clearer [10].
A recently developed algorithm allows both initial sup-
port selection and adjustment to be performed automati-
cally [11], extending the technique to objects other than a
few separated clusters. This algorithm has been success-
fully applied on simulated 3D diffraction patterns, and
it is being tested on the experimental 3D data recently
recorded. We are also investigating the use of the real
space algorithm [5] to reconstruct the clusters in Fig. 3
and the 3D data. While this algorithm does require prior
information in the form of a low-resolution image, it may
be advantageous for the reconstruction from sparse, ir-
regular, incomplete and noisy data.
V.DOSE AND FLUX LIMITATIONS TO
PERFORMANCE
In order to assess the promise of CXDI we have car-
ried out calculations intended to determine the fluence
(total photons per unit area) and dose (absorbed energy
per unit mass) required to make a 3D CXDI image at
given resolution. The basis of the calculation was the
so-called “dose fractionation” theorem [16, 17]. The the-
orem states that the radiation dose required to measure
the strength of a voxel is the same for a multi-view 3-D
experiment (with appropriate reconstruction algorithms)
as it is for measurement of the voxel alone, provided that
the voxel size and the statistical accuracy are the same
in both cases. The results are therefore based on a cal-
culation of diffraction by a single voxel. The conclusions
were as follows:
1. The dose (D) and fluence (N) needed to produce
P scattered x-rays per voxel into a detector collect-
ing the angle required for resolution d, are given by
N = P/σsand D = µhνP/εσswhere µ is the ab-
sorption coefficient, hν the photon energy and ε the
density.
2. The scattering cross section σsof the voxel of size
d × d × d is given by σs = r2
[18]) where reis the classical electron radius, λ the
wavelength and ρ the complex electron density.
eλ2|ρ|2d4(see also
3. For resolution d, the dose and fluence scale as d−4
4. For light elements ρ is fairly constant above about
1 keV so the energy dependence of the dose is ex-
pected to be quite flat in the range 1- 10 keV.
5. N is dominated by the cross section and scales with
the square of photon energy. Moreover, the coher-
ent power of a source of brightness B is B (λ/2)2.
This implies a fourth-power fluence penalty for in-
creasing the x-ray energy.
6. Therefore one should use the lowest possible x-ray
energy consistent with (roughly) λ < d/2
7. The dose for detecting a 10 nm protein feature
against a background of water according to the
Rose criterion [19] is shown in Fig.
quired imaging dose in the energy range 1-10 keV
is roughly 109Gy
4. The re-
Quantitative dose limit: to obtain an estimate of the
resolution limit we have plotted a variety of literature
values of the dose needed to destroy biological features
as a function of feature size (Fig. 1). Also plotted is the
required imaging dose. The tentative conclusion from the
graph is that the resolution of the crossover, i.e. about
10 nm, should be possible for unlabelled life-science sam-
ples although for material-sciences samples the radiation
tolerance (and thus the resolution) can be much higher.
Quantitative fluence limit: in our latest experiment
at ALS, we collected a full 3D data set at a resolution
that we believe to be around 10 nm (although this is not
yet supported any reconstructions) in about 10 hours.
We project that a beam line optimized for this experi-
ment operating on the ALS after its planned performance
upgrade would collect diffraction data about 104times
faster than now. From 2, above, this should allow the
step from 10 nm to around 1 nm resolution for sufficiently
radiation-resistant samples.
VI.FEMTOSECOND CXDI, BEYOND THE
RADIATION-DAMAGE LIMIT
A way of overcoming the radiation damage limit in x-
ray imaging is to use pulses of x-rays that are shorter
in duration than the damage process itself. This idea of
flash imaging, first suggested by Solem [20, 21], has been
proposed to be extended all the way to atomic resolution
using femtosecond pulses from an x-ray free-electron laser
(XFEL) [22]. The methodology of CXDI could be used
in this case to image single molecules [22, 23, 24]. The
general concept for imaging non-periodic samples is to
inject reproducible samples (macromolecules, complexes,
or virus particles) into the XFEL beam, to intersect one
particle (and record one diffraction pattern) per pulse.
In the general XFEL experiment the particle orienta-
tion in three dimensions will be random and unknown,
and the individual diffraction patterns will be noisy
(especially at the highest diffracted angles where the
highest-resolution orientation information resides). Gen-
eral ideas and methods, developed for “single-particle”
reconstructions in cryo-electron microscopy [25, 26], for

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FIG. 4: (left) Plot of dose against x-ray energy. (right) Plot of dose against resolution.
extracting tomographic information from a huge ensem-
ble of randomly oriented noisy images can be applied
here. Just as in single-particle EM, the limitations to im-
age resolution are the ability to sort patterns into classes
of like-oriented particles, so they can be averaged to im-
prove signal to noise, and the reproducibility of particles.
A statistical analysis [27] has shown that signal levels
much less than a single count per pixel are adequate to
be able to classify diffraction patterns. Even so, due to
the exceedingly small scattering cross section of a single
macromolecule, pulse fluences of >1012photons per (100
nm)2are required to achieve the required signal levels.
This corresponds to a dose per particle of > 1014Gray.
The dose could be reduced substantially if methods could
be employed to orient the particle in free space (even if
only along a single axis) or if symmetric nanoclusters of
particles could be formed.
We have performed calculations to determine, given
the required fluence, the longest pulse duration possi-
ble to acquire the diffraction pattern before disruption of
the electron density of the particle at the atomic scale.
The calculation uses a hydrodynamic method treating
the electrons and the ions as two separate fluids that
interact through the Coulomb force and ionization pro-
cesses, the assumption being that a large enough macro-
molecule can be treated as a continuum. Although it
does not treat the atomic motions as accurately as the
molecular dynamics model [22], the hydrodynamic model
is computationally faster and it also enables the inclu-
sion of additional physics effects, in particular the inclu-
sion of the redistribution of free electrons throughout the
molecule.
The dominant interactions between the matter and x-
ray pulse are K-shell photoionization, (producing dam-
age) and elastic scattering producing the diffraction pat-
tern. Following x-ray absorption in about 2 to 10 fs [28],
the K-shell holes decay by emitting Auger electrons with
energies in the range of 250 to 500 eV [29]. As the photo
and Auger electrons escape they leave behind a molecule
with increasing charge and the growing population of low
energy secondary electrons becomes trapped and grows
still further by collisional ionization (causing the bulk of
the damage).
The trapped electrons quickly relax in energy and po-
sition to form a neutralizing cloud around the positively
charged ions. The particle assumes roughly a two-zone
structure, consisting of a neutral core and a positively
charged outer shell similar to Debye shielding.
longer timescale of several 10’s fs, a macroscopic motion
of the whole molecule, called a Coulomb explosion, takes
place, leading to an outward radial motion of the ions.
The ion motion is greatest within the charged outer layer
of the molecule. In the core of the molecule the elec-
tron temperature is highest, with the greatest amount
of ionization and blurring of the electron density. Pre-
liminary results from the hydrodynamic model indicate
that collisional ionization is very rapid and may limit the
maximum pulse length to a value smaller than the 10-
20 fs inferred by considering only photoionization, Auger
ionization and atomic motion. The limits imposed by
collisional ionization might be overcome by developing
a method to reconstruct atomic positions from partially
ionized atoms.
To date it has not been possible to acquire experimen-
tal data on the rate of the Coulomb explosion and the
effect on electron density of ultrafast high-intensity x-
ray pulses. We plan to investigate this at the Tesla Test
Facility Phase 2 [30], a soft-x-ray FEL that will be oper-
ational in 2004 initially at a wavelength of 20 nm. The
first experiments that are planned are to determine the
onset of the Coulomb explosion of particles in an aerosol,
as a function of pulse fluence, as measured by diffraction
from the particles by the x-ray pulses themselves.
On a
VII. CONCLUSIONS
We have discussed advantages and limitations of the
CXDI technique, and our progress in implementing the
experiments and reconstruction techniques. Preparation
of special objects containing an isolated gold ball near
an unknown compact object are shown experimentally
to allow simple (although low resolution) image recon-
struction without iteration based on the autocorrelation
function (Fourier transform holography). Such an im-

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age may also provide a support function for the retrieval
of higher-resolution images using the HIO algorithm. We
have developed an algorithm that obtains the support au-
tomatically from the autocorrelation function by directly
Fourier transforming the recorded diffraction intensity.
This eliminates the need for an auxiliary imaging exper-
iment to obtain the support function.
Our last experiment shows that the exposure times for
10 nm resolution are reasonable (about 10 hours) and
this resolution is near the dose limit for life-science ex-
periments. Thus we conclude that 3D 10-nm resolution
life science experiments on frozen hydrated samples will
be possible, and even better resolutions might be achiev-
able with labeling techniques. Calculations indicate that
a factor of about 104improvement in imaging speed can
be expected using a dedicated CXDI beam line at the
upgraded ALS. This is important for applications in ma-
terial science where the dose limitation is much less se-
vere. Based on the inverse fourth power scaling of the
required flux with resolution, this should allow a factor
ten improvement in resolution to about 1 nm.
These results have prompted us to develop two par-
allel projects. One is the development of a program for
cryo-CXDI at ALS, and the other aims at femtosecond
imaging of single molecules at the LCLS and other 4th
generation sources.
The work in CXDI at ALS will be performed in collab-
oration with groups from NSLS and Stony Brook, who
have constructed an instrument designed for collecting
3D data sets [31], and which will be installed at ALS.
The instrument incorporates a set of in-vacuum step-
per motors for precision positioning the required aper-
tures, beam stop, and diagnostics. It incorporates an
air-lock and precision goniometer (developed for electron
tomography) for easy introduction and manipulation of
the specimen. The specimen holder is designed so as to
be able to rotate the specimen over the angular range
from -80 to +80 degrees before the specimen supporting
grid obscures the beam. A unique feature of the instru-
ment is that a zone plate can be positioned between the
specimen and the CCD, thereby a low resolution image
of the specimen can be recorded. This feature helps align
the specimen, and provides information on its “support”.
The key idea to achieving atomic resolution imaging of
radiation-sensitive single molecules and particles is that
the damage limit may be overcome with the use of very
short x-ray pulses to capture the data before damage
occurs. Three problems need to be assessed to be able
to perform single molecule imaging experiments: fem-
tosecond damage and experiment modeling; image ori-
entation, characterization and reconstruction; and single
molecule sample handling.
Fast and accurate hydrodynamic models to describe
the interaction of a molecule with femtosecond x-ray
pulses needs confirmation from experimental results of
short pulse, high-field x-ray-matter interaction experi-
ments planned in the near future. The recorded diffrac-
tion patterns must be classified, averaged to increase the
signal-to-noise ratio, and oriented for the final 3D recon-
struction. Methods developed for single-particle electron
microscopy can be applied here. We must also develop
methods to inject samples into the beam and, if possible,
orient the particles (or at least influence their orienta-
tion). Preliminary lower resolution experiments on sin-
gle particles will provide the answers to the experimental
needs to build an experiment at LCLS to perform single-
molecule imaging.
We have established the fundamental boundaries of ap-
plicability of CXDI to problems in materials and biolog-
ical sciences. With the rapidly growing importance of
nanostructures, and the potential that future develop-
ments in this area have for major breakthroughs in the
fundamental and applied sciences and in technology, the
addition of a new probe that can image thick objects at
very high 3d spatial resolution will have a decisive impact
on nanoscience and technology. Even more ambitiously,
atomic-resolution CXDI at an XFEL, has the potential
to provide atomic structure determination of biological
systems.
Acknowledgments
We would like to thank Rick Levesque of LLNL for de-
signing and building the rotation stage, and Cindy Lar-
son and Sherry Baker (both LLNL) for SEM imaging.
This work was performed under the auspices of the U.S.
Department of Energy by the Lawrence Livermore Na-
tional Laboratory under Contract No. W-7405-ENG-48
and the Director, Office of Energy Research, Office of Ba-
sics Energy Sciences, Materials Sciences Division of the
U. S. Department of Energy, under Contract No. DE-
AC03-76SF00098. SM acknowledges funding from the
National Science Foundation. The Center for Biophoton-
ics, an NSF Science and Technology Center, is managed
by the University of California, Davis, under Cooperative
Agreement No. PHY0120999.
[1] D. Sayre, “Some implications of a theory due to Shan-
non”, Acta Cryst. 5, 843 (1952).
[2] J. R. Fienup, “Phase retrieval algorithms: a compari-
son”, Appl. Opt. 21, 2758-2769 (1982).
[3] V. Elser, “Phase retrieval by iterative projections”, J.
Opt. Soc. Am. A 20, 40-55 (2003)
[4] A. Sz¨ oke, “Holographic microscopy with a complicated
reference”, J. Image. Sci. Technol. 41, 332-341 (1997).
[5] S. P. Hau-Riege, H. Sz¨ oke, H. N. Chapman, A. Sz¨ oke,
“SPEDEN: Reconstructing single particles from their
diffraction patterns”, Submitted, (2003).
[6] J. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extend-

Page 7

7
ing the methodology of x-ray crystallography to allow
imaging of micrometre-sized non-crystalline specimens”,
Nature 400, 342-344 (1999).
[7] Miao, J., T. Ishikawa, B. Johnson, E. H. Anderson, B.
Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction
microscopy”, Phys. Rev. Lett. 89, 088303 (2002).
[8] Williams, G. J., M. A. Pfeifer, I. A. Vartanyants, I. K.
Robinson, “Three-Dimensional Imaging of Microstruc-
ture in Au Nanocrystals”, Phys. Rev. Lett. 90, 175501
(2003).
[9] H. He, S. Marcesini, M. R. Howells, U. Weierstall,
G. Hembree, J. C. H. Spence, “Experimental lensless soft
x-ray imaging using iterative algorithms: phasing diffuse
scattering”, Acta. Cryst. A59, 143-152 (2003).
[10] H. He, S. Marchesini, M. R. Howells, U. Weierstall,
H. Chapman, S. Hau-Riege, A. Noy, J. C. H. Spence,
“Inversion of x-ray diffuse scattering to images using pre-
pared objects”, Phys. Rev. B 67, 174114 (2003).
[11] S. Marchesini, H. He, H. N. Chapman, A. Noy, S. P. Hau-
Riege, M. R. Howells, U. Weierstall, J. C. H. Spence,
“Imagingwithout lenses”,
(2003).
[12] Z. M. Zuo, I. Vartanyants, M. Gao, R. Zang, L. A. Na-
gahara, “Atomic resolution imaging of a carbon nan-
otube from diffraction intensities”, Science 300, 1419-
1421 (2003).
[13] U. Weierstall, Q. Chen, J. C. H. Spence, M. R. How-
ells, M. Isaacson, R. R. Panepucci, “Image reconstruc-
tion from electron and x-ray diffraction patterns using
iterative algorithms: theory and experiment”, Ultrami-
croscopy 90, 171-195 (2002).
[14] J. C. H. Spence, U. Weierstall, M. R. Howells, “Phase
recovery and lensless imaging by iterative methods in op-
tical, X-ray and electron diffraction”, Phil. Trans. Roy.
Soc. Lond. A 360, 875-895 (2002).
[15] M. R. Howells, P. Charalambous, H. He, S. Marchesini,
J. C. H. Spence, “An off-axis zone-plate monochromator
for high-power undulator radiation”, in Design and Mi-
crofabrication of Novel X-ray Optics, D. Mancini, (Ed),
Vol. 4783, SPIE, Bellingham, 2002.
[16] R. Hegerl, W. Hoppe, “Influence of electron noise on
three-dimensional image reconstruction”, Zeitschrift f¨ ur
Naturforschung 31a, 1717–1721 (1976).
[17] B. F. McEwen, K. H. Downing,
“The relevance of dose-fractionation in tomography of
arXiv:physics/0306174,
R. M. Glaeser,
radiation-sensitive specimens”, Ultramicroscopy 60, 357-
373 (1995).
[18] B. L. Henke, J. W. M. DuMond, “Submicroscopic struc-
ture determination by long wavelength x-ray diffraction”,
Journal of Applied Physics 26, 903–917 (1955).
[19] A. Rose, “Television pickup tubes and the problem of
vision”, in Advances in Electronics, Marton, L., (Ed),
Vol. 1, New York, 1948.
[20] J. C. Solem, G. C. Baldwin, “Microholography of Living
Organisms”, Science 218, 229-235 (1982).
[21] J. C. Solem, G. F. Chapline, “X-Ray Biomicrohologra-
phy”, Opt. Eng. 23,193-202 (1984).
[22] R. Neutze, R. Wouts, D. v. d. Spoel, E. Weckert, J. Ha-
jdu, “Potential for biomolecular imaging with femtosec-
ond x-ray pulses”, Nature 406, 752-757 (2000).
[23] A. Sz¨ oke, “Time resolved holographic diffraction with
atomic resolution”, Chem. Phys. Lett. 313, 777-788
(1999).
[24] J. Miao, K. O. Hodgson, D. Sayre, “An approach to
three-dimensional structures of biomolecules”, Proc. Nat.
Acad. Sci. 98, 6641-6645 (2001).
[25] Franck, J., Three-Dimensional Electron Microscopy of
Macromolecular Assemblies, Academic Press, San Diego,
1996.
[26] M. v. Heel, B. Gowen, R. Matadeen, E. V. Orlova, R.
Finn, T. Pape, D. Cohen, H. Stark, R. Schmidt, M.
Schatz, A. Patwardhan, “Single-particle electron cryo-
microscopy:towards atomic resolution”, Quart. Rev.
Biophys. 33, 269-307 (2000).
[27] G. Huldt, A. Sz¨ oke, J. Hajdu, “Single Particle Diffraction
Imaging: Image Classification”, Submitted, (2003).
[28] E. J. McGuire, “K-Shell Auger Transition Rates and Flu-
orescence Yields for Elements Be-Ar”, Phys. Rev. 185,
1-6 (1969).
[29] I. F. Furguson, Auger Microprobe Analysis, Adam Hilger,
New York, 1989.
[30] TESLATest Facility
http://www-hasylab.desy.de/facility/fel/vuv/main.htm
(2003).
[31] T. Beetz, C. Jacobsen, C. C. Cao, J. Kirz, O. Mentez,
C. Sanches-Hanke, D. Sayre, D. Shapiro, “Development
of a novel apparatus for experiments in soft x-ray diffrac-
tion imaging and diffraction tomography”, J. de Phys. IV
104, 351-359 (2003).
web page,