X-ray beam quality after a mirror reflection: Experimental and simulated results for a toroidal mirror in a 4 th generation storage-ring beamline.

Background: The surface errors found in X-ray mirrors constitute a limiting factor for preserving beam quality. This is particularly important when the X-ray beam has low emittance and a significant coherence fraction, like in newly upgraded synchrotron storage rings. Methods: We studied the fringes observed in the image of an undulator-produced X-ray beam reflected by a high-quality toroidal mirror. The measurements and simulations were performed using different conditions: a photon beam either monochromatic or with large bandwidth, reflected by a mirror with variable curvature. Results: The experimental data are compared with up-to-date simulation including partial coherence. Conclusions: The observed fringes in the unfocused beam correlate with low spatial frequency structures in mirror profiles, irrespective of beam coherence. Both classical ray tracing and partially coherent simulations through coherent mode decomposition are confirmed as accurate methods for such simulations.

In this study, researchers focused on the surface errors found in X-ray mirrors and their impact on beam quality. These errors can be problematic, especially when dealing with X-ray beams coming from low emittance (a measure of beam size and divergence) electron beam sources and a significant coherence fraction (indicating the level of wavefront coherence). The researchers specifically investigated the fringes observed in the image of an X-ray beam produced by an undulator and reflected by a high-quality toroidal mirror. They conducted measurements and simulations under different conditions, such as using a monochromatic photon beam or one with a wide range of wavelengths, and varying the curvature of the mirror.

Tilting refractive x-ray lenses for fine-tuning of their focal length.

In this work, we measure and model tilted x-ray refractive lenses to investigate their effects on an x-ray beam. The modelling is benchmarked against at-wavelength metrology obtained with x-ray speckle vector tracking experiments (XSVT) at the BM05 beamline at the ESRF-EBS light source, showing very good agreement. This validation permits us to explore possible applications of tilted x-ray lenses in optical design. We conclude that while tilting 2D lenses does not seem interesting from the point of view of aberration-free focusing, tilting 1D lenses around their focusing direction can be used for smoothly fine-tuning their focal length. We demonstrate experimentally this continuous change in the apparent lens radius of curvature R: a reduction up to a factor of two and beyond is achieved and possible applications in beamline optical design are proposed.

A fast and lightweight tool for partially coherent beamline simulations in fourth-generation storage rings based on coherent mode decomposition.

A new algorithm to perform coherent mode decomposition of undulator radiation is proposed. It is based on separating the horizontal and vertical directions, reducing the problem by working with one-dimension wavefronts. The validity conditions of this approximation are discussed. Simulations require low computer resources and run interactively on a laptop. The focusing with lenses of the radiation emitted by an undulator in a fourth-generation storage ring (EBS-ESRF) is studied. Results are compared against multiple optics packages implementing a variety of methods for dealing with partial coherence: full two-dimension coherent mode decomposition, Monte Carlo combination of wavefronts from electrons entering the undulator with different initial conditions, and hybrid ray-tracing correcting geometrical optics with wave optics.

Polished diamond X-ray lenses.

High-quality bi-concave 2D focusing diamond X-ray lenses of apex-radius R = 100 µm produced via laser-ablation and improved via mechanical polishing are presented here. Both for polished and unpolished individual lenses and for stacks of ten lenses, the remaining figure errors determined using X-ray speckle tracking are shown and these results are compared with those of commercial R = 50 µm beryllium lenses that have similar focusing strength and physical aperture. For two stacks of ten diamond lenses (polished and unpolished) and a stack of eleven beryllium lenses, this paper presents measured 2D beam profiles out of focus and wire scans to obtain the beam size in the focal plane. These results are complemented with small-angle X-ray scattering (SAXS) measurements of a polished and an unpolished diamond lens. Again, this is compared with the SAXS of a beryllium lens. The polished X-ray lenses show similar figure errors to commercially available beryllium lenses. While the beam size in the focal plane is comparable to that of the beryllium lenses, the SAXS signal of the polished diamond lenses is considerably lower.

Wavelet-transform-based speckle vector tracking method for X-ray phase imaging.

We introduce a new X-ray speckle-vector tracking method for phase imaging, which is based on the wavelet transform. Theoretical and experimental results show that this method, which is called wavelet-transform-based speckle-vector tracking (WSVT), has stronger noise robustness and higher efficiency compared with the cross-correlation-based method. In addition, the WSVT method has the controllable noise reduction and can be applied with fewer scan steps. These unique features make the WSVT method suitable for measurements of large image sizes and phase shifts, possibly under low-flux conditions, and has the potential to broaden the applications of speckle tracking to new areas requiring faster phase imaging and real-time wavefront sensing, diagnostics, and characterization.

X-ray optics and beam characterization using random modulation: theory.

X-ray near-field speckle-based phase-sensing approaches provide efficient means of characterizing optical elements. Presented here is a theoretical review of several of these speckle methods within the framework of optical characterization, and a generalization of the concept is provided. As is also demonstrated experimentally in a parallel paper [Berujon, Cojocaru, Piault, Celestre, Roth, Barrett & Ziegler (2020), J. Synchrotron Rad. 27, (this issue)], the methods theoretically developed here can be applied to different beams and optics and within a variety of situations where at-wavelength metrology is desired. By understanding the differences between the various processing methods, it is possible to find and implement the most suitable approach for each metrology scenario.

X-ray optics and beam characterization using random modulation: experiments.

A parallel paper [Berujon, Cojocaru, Piault, Celestre, Roth, Barrett & Ziegler (2020), J. Synchrotron Rad. 27, 284-292] reviewed theoretically some of the available processing schemes for X-ray wavefront sensing based on random modulation. Shown here are experimental applications of the technique for characterizing both refractive and reflective optical components. These fast and accurate X-ray at-wavelength metrology methods can assist the manufacture of X-ray optics that transport X-ray beams with a minimum amount of wavefront distortion. It is also recalled how such methods can facilitate online optimization of active optics.

Modelling phase imperfections in compound refractive lenses.

A framework based on physical optics for simulating the effect of imperfect compound refractive lenses (CRLs) upon an X-ray beam is described, taking into account measured phase errors obtained from at-wavelength metrology. A CRL stack is modelled, with increasing complexity, as a single thin phase element, then as a more realistic compound element including absorption and thickness effects, and finally adding realistic optical imperfections to the CRL. Coherent and partially coherent simulations using Synchrotron Radiation Workshop (SRW) are used to evaluate the different models, the effects of the phase errors and to check the validity of the design equations and suitability of the figures of merit.

A hierarchical approach for modeling X-ray beamlines: application to a coherent beamline.

Different approaches to simulate a modern X-ray beamline are considered. Several methodologies with increasing complexity are applied to discuss the relevant parameters that quantify the beamline performance. Parameters such as flux, dimensions and intensity distribution of the focused beam, and coherence properties are obtained from simple analytical calculations to sophisticated computer simulations using ray-tracing and wave optics techniques. A latest-generation X-ray nanofocusing beamline for coherent applications (ID16A at the ESRF) has been chosen to study in detail the issues related to highly demagnifying synchrotron sources and exploiting the beam coherence. The performance of the beamline is studied for two storage rings: the old ESRF-1 (emittance 4000 pm) and the new ESRF-EBS (emittance 150 pm). In addition to traditional results in terms of flux and beam sizes, an innovative study on the partial coherence properties based on the propagation of coherent modes is presented. The different algorithms and methodologies are implemented in the software suite OASYS. These are discussed with emphasis placed upon the their benefits and limitations of each.

Memory and CPU efficient computation of the Fresnel free-space propagator in Fourier optics simulations.

We describe a version of the paraxial free-space Fourier optics propagator for numerical wave propagation simulations that eliminates the need for a dense sampling of an input electric field with phase dominated by quadratic terms developing at some distance from the source or from the radiation beam waist. This propagator requires considerably (two to three orders of magnitude as observed in routine simulations) less memory and CPU resources than the standard Fresnel free-space propagator while preserving its levels of accuracy and generality. This method has been successfully used in "Synchrotron Radiation Workshop" code for more than a decade. It has greatly contributed to the applicability of the code, and more generally the applicability of the Fourier optics methods, to wave-optics based simulations of radiation propagation through optical systems of beamlines at high-brightness and high-coherence synchrotron light sources.