Noise models for low counting rate coherent diffraction imaging

International audienceCoherent diffraction imaging (CDI) is a lens-less microscopy method that extracts the complex-valued exit field from intensity measurements alone. It is of particular importance for microscopy imaging with diffraction set-ups where high quality lenses are not available. The inversion scheme allowing the phase retrieval is based on the use of an iterative algorithm. In this work, we address the question of the choice of the iterative process in the case of data corrupted by photon or electron shot noise. Several noise models are presented and further used within two inversion strategies, the ordered subset and the scaled gradient. Based on analytical and numerical analysis together with Monte-Carlo studies, we show that any physical interpretations drawn from a CDI iterative technique require a detailed understanding of the relationship between the noise model and the used inversion method. We observe that iterative algorithms often assume implicitly a noise model. For low counting rates, each noise model behaves differently. Moreover, the used optimization strategy introduces its own artefacts. Based on this analysis, we develop a hybrid strategy which works efficiently in the absence of an informed initial guess. Our work emphasises issues which should be considered carefully when inverting experimental data

Inversion of the Diffraction Pattern from an Inhomogeneously Strained Crystal using an Iterative Algorithm

The displacement field in highly non uniformly strained crystals is obtained by addition of constraints to an iterative phase retrieval algorithm. These constraints include direct space density uniformity and also constraints to the sign and derivatives of the different components of the displacement field. This algorithm is applied to an experimental reciprocal space map measured using high resolution X-ray diffraction from an array of silicon lines and the obtained component of the displacement field is in very good agreement with the one calculated using a finite element model.Comment: 5 pages, 4 figure

X-ray lensless microscopy from undersampled diffraction intensities

International audienceX-ray coherent diffraction imaging including ptychography provides the nanoscale resolved three-dimensional description of matter. The combination of these approaches to the Bragg geometry case arouses a strong interest for its capability to provide information about strain state in crystals. Among the existing approaches, ptychography is particularly appealing because it allows the investigation of extended or weakly scattering samples. Coherent diffraction imaging approaches, based on redundancy in the collected diffraction intensity data set, are highly time consuming and rely on state-of-the-art mechanical setups, both being strong limitations for a general application. We show here that these can be overcome by regularization-based inversion algorithms introducing a priori structural knowledge. This method, which can be generalized to other wavelengths or beam sources, opens new possibilities for the imaging of radiation-sensitive specimens or very large samples

Coherent x-ray wavefront reconstruction of a partially illuminated Fresnel zone plate

International audienceA detailed characterization of the coherent x-ray wavefront produced by a partially illuminated Fresnel zone plate is presented. We show, by numerical and experimental approaches, how the beam size and the focal depth are strongly influenced by the illumination conditions, while the phase of the focal spot remains constant. These results confirm that the partial illumination can be used for coherent diffraction experiments. Finally, we demonstrate the possibility of reconstructing the complex-valued illumination function by simple measurement of the far field intensity in the specific case of partial illumination

Imaging the displacement field within epitaxial nanostructures by coherent diffraction: a feasibility study

International audienceWe investigate the feasibility of applying coherent diffraction imaging to highly strained epitaxial nanocrystals using finite-element simulations of SiGe islands as input in standard phase retrieval algorithms. We discuss the specific problems arising from both epitaxial and highly strained systems and we propose different methods to overcome these difficulties. Finally, we describe a coherent microdiffraction experimental setup using extremely focused x-ray beams to perform experiments on individual nanostructures

Three-dimensional high-resolution quantitative microscopy of extended crystals

International audienceHard X-ray lens-less microscopy raises hopes for a non-invasive quantitative imaging, capableof achieving the extreme resolving power demands of nanoscience. However, a limit imposedby the partial coherence of third generation synchrotron sources restricts the sample size tothe micrometer range. Recently, X-ray ptychography has been demonstrated as a solution forarbitrarily extending the fi eld of view without degrading the resolution. Here we show thatptychography, applied in the Bragg geometry, opens new perspectives for crystalline imaging.The spatial dependence of the three-dimensional Bragg peak intensity is mapped and the entiredata subsequently inverted with a Bragg-adapted phase retrieval ptychographical algorithm.We report on the image obtained from an extended crystalline sample, nanostructured froma silicon-on-insulator substrate. The possibility to retrieve, without transverse size restriction,the highly resolved three-dimensional density and displacement fi eld will allow for theunprecedented investigation of a wide variety of crystalline materials, ranging from life scienceto microelectronics

Oxygen Absorption in Free-Standing Porous Silicon: A Structural, Optical and Kinetic Analysis

Porous silicon (PSi) is a nanostructured material possessing a huge surface area per unit volume. In consequence, the adsorption and diffusion of oxygen in PSi are particularly important phenomena and frequently cause significant changes in its properties. In this paper, we study the thermal oxidation of p+-type free-standing PSi fabricated by anodic electrochemical etching. These free-standing samples were characterized by nitrogen adsorption, thermogravimetry, atomic force microscopy and powder X-ray diffraction. The results show a structural phase transition from crystalline silicon to a combination of cristobalite and quartz, passing through amorphous silicon and amorphous silicon-oxide structures, when the thermal oxidation temperature increases from 400 to 900 °C. Moreover, we observe some evidence of a sinterization at 400 °C and an optimal oxygen-absorption temperature about 700 °C. Finally, the UV/Visible spectrophotometry reveals a red and a blue shift of the optical transmittance spectra for samples with oxidation temperatures lower and higher than 700 °C, respectively