Jean-Pierre Guigay

Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography.

A well-known problem in x-ray microcomputed tomography is low sensitivity. Phase contrast imaging offers an increase of sensitivity of up to a factor of 10(3) in the hard x-ray region, which makes it possible to image soft tissue and small density variations. If a sufficiently coherent x-ray beam, such as that obtained from a third generation synchrotron, is used, phase contrast can be obtained by simply moving the detector downstream of the imaged object. This setup is known as in-line or propagation based phase contrast imaging. A quantitative relationship exists between the phase shift induced by the object and the recorded intensity and inversion of this relationship is called phase retrieval. Since the phase shift is proportional to projections through the three-dimensional refractive index distribution in the object, once the phase is retrieved, the refractive index can be reconstructed by using the phase as input to a tomographic reconstruction algorithm. A comparison between four phase retrieval algorithms is presented. The algorithms are based on the transport of intensity equation (TIE), transport of intensity equation for weak absorption, the contrast transfer function (CTF), and a mixed approach between the CTF and TIE, respectively. The compared methods all rely on linearization of the relationship between phase shift and recorded intensity to yield fast phase retrieval algorithms. The phase retrieval algorithms are compared using both simulated and experimental data, acquired at the European Synchrotron Radiation Facility third generation synchrotron light source. The algorithms are evaluated in terms of two different reconstruction error metrics. While being slightly less computationally effective, the mixed approach shows the best performance in terms of the chosen criteria.

Optimization of phase contrast imaging using hard x rays

X ray radiography and tomography are important tools in medicine as well as in life science and materials science. Not long ago an approach called in-line holography based on simple propagation became possible using partially coherent synchrotron beams like the ones available at the European Synchrotron Radiation Facility (ESRF). Theoretical and experimental work by Cloetens et al. [Appl. Phys. Lett 75, 2912 (1999)] have shown that quantitative retrieval of the optical phase, from a set of radiographs taken at different sample-to-detector distances, is feasible. Mathematically speaking we are dealing with a direct method based on linearization in order to solve an inverse nonlinear problem. The phase retrieval can be combined with classical tomography in order to obtain a three-dimensional representation of the object’s electron density (holotomography). In order to optimize the image contrast for the numerical phase retrieval process, we have carried out calculations resulting in an optimized choice of v...

X-ray dynamical diffraction: the concept of a locally plane wave

The long distance between the source and the experiment and the small source size, now available at third-generation synchrotron sources, leads to new optical characteristics for X-ray diffraction. It is shown that, under certain conditions, the intensity received by a point located on the exit surface of a crystal is described by the diffraction of a locally plane wave. Each point along the surface is influenced by a plane wave with a varying departure from Bragg angle. In such a case, it is possible to visualize the rocking curve of the crystal as a function of the position along the exit surface. This represents a topographic method to obtain the reflectivity curve of the crystal instead of the usual goniometric method. Applications will be described in a forthcoming paper.

An analytical approach to estimating aberrations in curved multilayer optics for hard x-rays: 2. Interpretation and application to focusing experiments

Aberration effects are studied in parabolic and elliptic multilayer mirrors for hard x-rays, basing on a simple analytical approach. The interpretation of the underlying equations provides insight into fundamental limitations of the focusing properties of curved multilayers. Using realistic values for the multilayer parameters the potential impact on the broadening of the focal spot is evaluated. Within the limits of this model, systematic contributions to the spot size can be described. The work is complemented by a comparison with experimental results obtained with a W/B(4)C curved multilayer mirror.

An analytical approach to estimating aberrations in curved multilayer optics for hard x-rays: 1. Derivation of caustic shapes

An analytical approach has been developed to derive aberration effects in parabolic and elliptic multilayer optics with weak interaction between photons and matter. The method is based on geometrical ray tracing including refraction effects up to the first order of the refractive index decrement delta. In the parabolic case, the derivation leads to simple parametric equations for the caustic shape. In the elliptic case, the analytical results more involved, but can be well approximated by the parabolic solution. Both geometries are compared with regard to the fundamental impact on their focusing properties.

Bent crystals in Laue geometry: dynamical focusing of a polychromatic incident beam

The main limitation of Laue geometry for the achievement of a small focus size is the focus broadening caused by the intrinsic Darwin width and the spread of the beam in the Borrmann triangle, resulting from propagation inside the crystal. A method, based on dynamical focusing, is suggested that allows improvement of the quality of high-energy polychromatic focusing by bent crystals in Laue geometry.

Influence of the transverse and longitudinal coherence in the dynamical theory of x-ray diffraction

We investigate dynamical diffraction phenomena of a quasi-monochromatic x-ray beam emitted by a small source at a large distance from the sample. Under certain conditions of coherence the diffraction intensity profile in a recording plane reproduces the angular profile of the rocking curve of the crystal. The coherence requirements are stronger for thicker crystals. This has been verified in experiments at the ESRF. In these experiments the effective crystal thickness was varied by rotating the crystal around an axis perpendicular to the reflecting planes.

Optimized x-ray multilayer mirrors for single nanometer focusing.

We present numerical simulations optimizing the layer shapes of curved focusing x-ray multilayer mirrors deployed at synchrotron radiation facilities using a wave-optical model. The confocal elliptical shapes of the inner layers are corrected for refraction based on the modified Bragg law. Simulated wave amplitudes are further propagated to the focal region, promising nanometer focusing.

Quantitative phase tomography by holographic reconstruction

The coherence of third generation synchrotron beams makes a trivial form of phase-contrast radiography possible. It is based on simple propagation and corresponds to the defocusing technique of electron microscopy. Most of the work until now uses this technique to detect phase discontinuities associated with edges in the specimen. The tomographic reconstruction was initially performed using the algorithm for X-ray absorption tomography, a temporary and obviously unsatisfactory approach. This results in a decent solution for some cases, but gives rise to artifacts and does not reveal quantitatively the inner structure of the object. However the Fresnel diffraction fringes contain in an entangled form the phase modulation by the sample. We have successfully implemented a method for quantitative phase tomography based on the propagation technique. The reconstruction of the 3D refractive index distribution involves two steps. First the phase modulation is numerically retrieved from the combination of several images recorded at different distances. This holographic reconstruction process is repeated for a large number of angular positions of the specimen. Then the conventional filtered backprojection algorithm is used to determine the three dimensional distribution from the retrieved phase maps. The reconstructed 3D image of a complicated polystyrene foam sample has a straightforward interpretation with a spatial resolution limited by the detector to 1 - 2 microns.

Wave-optical theory of nanofocusing x-ray multilayer mirrors

We have derived a wave-optical model of curved nanofocusing x-ray multilayer mirrors used at synchrotron radiation sources, using a Takagi-Taupin-like approach. In a first approximation, the individual layers are assumed to be confocal elliptical. This assumption leads to a convenient spatial description in elliptical coordinates. As a first optimization, we study a rotation-like modification and compare numerical simulations to established results for planar multilayers.