The possibility of imaging a real-space object from the measurement of the coherent scattered intensity is based on oversampling in the reciprocal space. From precise measurements of the modulus of the object Fourier transform (FT), the phase and the amplitude of the real object have to be obtained. This needs oversampling by at least a factor of two as compared with the Nyquist sampling theorem, which states
is the sample size.
The basic algorithms are iterative Fourier transforms between estimates
of the scattering amplitude
in the reciprocal space and estimates
in the real space, where
refers to the k
th cycle. Constraints are added in order that the calculation converges towards the solution.
The main constraint in the reciprocal space is that the modulus of the estimation of the amplitude in the reciprocal lattice is
, which means that at each cycle
is replaced by
is the phase of
, wherever the intensity has been measured. In real space, the constraint is that of a finite support, which can be adjusted during cycling. This support must agree with the oversampling condition but, the higher the oversampling, the easier the convergence (Miao et al.
). This is the Gerchberg–Saxton algorithm, also called error reduction (ER) (Gerchberg & Saxton, 1972
). This algorithm is usually combined with the hybrid input–output (HIO) algorithm of Fienup (1982
), where the finite support constraint is relaxed. The constraint of real and positive
is also often used in astronomy and in imaging problems.
An excellent description of the sampling problems can be found in van der Veen & Pfeiffer (2004
). Many simulations based on these algorithms have been published in order to discuss the need for oversampling (Miao et al.
; Mielenz, 1999
) or the influence of experimental noise on the resulting object image (Marchesini, He et al.
This technique was first tested on very simple strongly scattering objects like a two-dimensional pattern of gold dots (Miao et al.
). The soft X-ray scattering was recorded with BI-CCDs and the resolution was of the order of tens of nanometres. Simple well prepared objects were studied (He, Marchesini, Howells, Weierstall, Chapman et al.
) and the reconstructed image could be compared with scanning-electron-microscopy images (He, Marchesini, Howells, Weierstall, Hembree & Spence, 2003
). These first experiments were carried out on two-dimensional samples prepared with 100 nm diameter gold balls on an SiN window. The samples were studied in transmission and various methods were used to compensate for lack of measurements in the beam stop.
In He, Marchesini, Howells, Weierstall, Hembree & Spence (2003
), the Patterson function (PF) was calculated by carrying out a FT of the measured spectrum. This function is the autocorrelation of the electron density of the sample and, from the properties of the convolution products, with well separated clusters, direct information on the distance between clusters was obtained from the observation of the PF. For a cluster well separated from a single gold ball, the shape of the cluster could be obtained. This property was systematically used by Eisebitt, Lörgen et al.
), where the scattering of a hole and a well separated sample was measured. As the two amplitudes coherently interfered, the PF directly gives the sample shape as the convolution of a point hole (a Dirac distribution) and the sample.
This holographic method using heterodyning between the scattering of a point source and the sample was applied to the study of the magnetic map of a Pt–Co multilayer. Magnetic scattering is observed in the vicinity of the L
edge of transition metals (Menteş et al.
; Chesnel et al.
), which can be controlled by tuning the energy and the polarization of the soft X-ray beam. In Eisebitt, Lüning et al.
), a hologram is measured between a hole and a distant sample and only a single Fourier transform is necessary to obtain an image of the magnetic configuration. The image resulting from the Fourier inversion was successfully compared with magnetic force microscopy measurements. Another method used was to illuminate the sample with the reference wave of a well defined 2.5 µm pinhole (Eisebitt et al.
) and to observe the changes in magnetic scattering with X-ray energy and polarization.
The magnetic configuration of multilayers was also studied in the symmetric Laue reflection configuration and speckles were used in order to image the configurational changes obtained by applying a magnetic field (Chesnel, Belakhovsky et al.
; Chesnel, van der Laan et al.
; Deutsch & Mai, 2005
Except for magnetic measurements, first experiments used soft X-rays because for wavelengths larger than 1 nm the number of photons per coherence area is larger, and also because the resolution requirements were lower (Sayre et al.
). First two-dimensional reconstructions were obtained with fixed samples and area detectors. The samples were studied in transmission and various methods were used to compensate for the lack of measurements in the beam stop. Obviously, samples were very thin and it was difficult to study three-dimensional systems, which are thicker. For buried structures (Miao et al.
), the three-dimensional
space was sampled by a reduced set of two-dimensional diffraction patterns and short wavelengths (
Å) from an Si111
monochromator were used. For the large number of three-dimensional
values that were not measured, no constraint was imposed on the modulus. For systems with a limited number of different layers, like a set of metallic patterns buried close to the surface of a silicon wafer, the three-dimensional image was well reconstructed. This technique was extended to the study of gold particles deposited at the surface of an SiN pyramidal membrane (Marchesini, Chapman et al.
). As these particles were deposited on a (non-planar) surface, the full sampling of the three-dimensional
space was not necessary. These measurements need the development of new set-ups of high reliability in order to position and to rotate the samples. One can observe in passing that recently developed set-ups use many tools (sample holders, cryo holder, …) from electron-microscope techniques (Beetz et al.
). Techniques are developed for solving the inversion problem with blank parts in the scattering planes. An excellent summary of the method is found in Chapman et al.