Hard-X-Ray Lensless Imaging of Extended Objects
J. M. Rodenburg,
1,
*
A. C. Hurst,
1
A. G. Cullis,
1
B. R. Dobson,
2
F. Pfeiffer,
3
O. Bunk,
3
C. David,
3
K. Jefimovs,
3
and I. Johnson
3
1
Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, United Kingdom
2
CCLRC Daresbury Laboratory, Daresbury, Warrington WA4 4AD, United Kingdom
3
Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland
(Received 29 September 2006; published 18 January 2007)
We demonstrate a hard-x-ray microscope that does not use a lens and is not limited to a small field of
view or an object of finite size. The method does not suffer any of the physical constraints, convergence
problems, or defocus ambiguities that often arise in conventional phase-retrieval diffractive imaging
techniques. Calculation times are about a thousand times shorter than in current iterative algorithms. We
need no a priori knowledge about the object, which can be a transmission function with both modulus and
phase components. The technique has revolutionary implications for x-ray imaging of all classes of
specimen.
DOI: 10.1103/PhysRevLett.98.034801 PACS numbers: 41.50.+h, 42.30.Kq, 42.30.Rx
Introduction.—Lensless x-ray diffraction microscopy is
one of the most promising methods for life and material
science imaging on the nanometer scale. The high scatter-
ing vectors (q values) available in the Fraunhofer diffrac-
tion plane can, in principle, yield wavelength-limited
resolution (i.e., approaching the nanometer scale), without
any of the limitations which apply to x-ray lenses. The key
task is to solve computationally for the phase of the scat-
tered intensity. Crystallographic methods are not appli-
cable, given that any object of microscopic interest is, by
definition, aperiodic. In the absence of periodicity, it is
possible to use a size constraint, either in the object itself or
in the extent of a collimated beam illuminating the object.
Solution of the phase problem is then rendered tractable via
certain iterative phase-retrieval methods [1–9]. Large
aperiodic objects have highly structured diffraction pat-
terns. To solve for a given field of view from a single
diffraction pattern, the detector must have a pixel size (in
the Fraunhofer plane) which is inversely proportional to
the size of the object. In terms of practical microscopy, this
is a severe limitation: Microscopists require the ability to
scan a large field of view in order to place an object of
interest in spatial context, preferably in real time. Current
phase-retrieval imaging methods suffer from further limi-
tations. Even if the diffraction plane sampling criterion is
satisfied, a unique solution often depends upon the object
being single-valued (for example, being a pure phase ob-
ject or a purely absorptive object). In fact, many objects of
practical interest impress both phase and modulus changes
onto the exit wave field. Current methods are also compu-
tationally intensive, typically requiring tens of thousands
of iterations (each involving two Fourier transforms over
the entire—even if limited —field of view), meaning that
any possibility of real-time imaging is remote.
Ptychographical iterative engine. —We demonstrate
here an x-ray microscope which overcomes all of these
limitations. The theory of the technique [10–13] has been
demonstrated to work at low resolution at light optical
wavelengths [14], but the scientific impact at x-ray wave-
lengths is revolutionary. We demonstrate a proof of prin-
ciple that x-ray imaging can be accomplished without the
need for any sophisticated optics. Given the imminent
development of much higher brightness fourth-generation
coherent sources, we believe that the method will open up
the possibility of real-time, wavelength-limited hard (or
soft) x-ray imaging of objects of any size. Our experimen-
tal approach relies on collecting a number of Fraunhofer
diffraction patterns (Fig. 1), each of which comes from a
different, but overlapping, region of the specimen which is
moved laterally across the illuminating beam. The key
development is to meld all of these data into a wide field
of view. The method is related to a direct (noniterative)
solution of the crystallographic phase problem first pro-
posed by Hoppe [15]. An aperture the size of the unit cell is
placed in real space over a periodic object. In the
Fraunhofer diffraction plane, each crystalline reflection is
now convoluted with the Fourier transform of the aperture
function. In one dimension, such an aperture is a ‘‘top hat’’;
hence, the diffracted peaks are convoluted with sinc func-
tions. At their point of overlap, the sinc functions add
according to the complex values determined by the phase
of the underlying diffracted beams. The intensity of each
adjacent diffracted peak is easily measured: The intensity
at the midpoint (the ‘‘in-between diffraction spots,’’ nowa-
days referred to as the ‘‘oversampled’’ diffraction spots [3])
gives a measure of their relative phase. Hoppe recognized
that this measure could still lead to an ambiguity of a
complex conjugate in each diffracted beam, as is usual in
the classic phase problem [16]. However, by moving the
aperture function to a new position displaced by less than
one unit cell, a new measure can be obtained for this phase
difference, and hence all phases are obtained unambigu-
ously. This method, relying on a convolution (or ‘‘fold-
ing’’) in reciprocal space was later called ‘‘ptychography’’
PRL 98, 034801 (2007)
PHYSICAL REVIEW LETTERS
week ending
19 JANUARY 2007
0031-9007=07=98(3)=034801(4) 034801-1 © 2007 The American Physical Society
