Near-perfect diffracting crystals have many uses in x-ray optics including as monochromators, energy analyzers, and phase retarders. The usefulness of a particular Bragg reflection is often related to its angular acceptance and efficiency, as is determined by the reflection’s structure factor. Silicon crystals, which belong to the same face-centered cubic space group 𝐹𝑑3̅𝑚 as germanium and diamond, are readily available in large and highly pure ingots. Combined with their high thermal conductivity and low thermal expansion, this makes them suitable for synchrotron x-ray beamlines. However, less symmetric trigonal crystals such as sapphire, lithium niobate, and α-quartz offer a better choice of high-energy-resolution Bragg reflections near backscattering with less likelihood of parasitic Bragg reflections. Because these crystals’ atoms vibrate anisotropically and shift relative to each other with temperature, the temperature dependence of their structure factors is not a given by a simple Debye-Waller factor. Also, many crystal structures may be described by several different conventions of origin and lattice vectors. A Python three software package, PyCSFex, is presented here for the rapid calculation of large numbers of structure factors of any crystal described in any convention. It can run on its own or as part of an already existing software package. Users can extend the package to new crystals by writing their own material files. α-Quartz is chosen as an example because it has already been successfully used in backscattering x-ray energy analyzers and presents the complexities previously mentioned.
Diffracting crystals are extensively used at synchrotron beamlines as x-ray monochromators and phase retarders. Imperfect growth processes, surface damage occurring during fabrication, and strain caused by poor clamping methods can all degrade the quality of these crystals and the x-ray beams diffracted by them. Because x-ray topography of these crystals can reveal both the location and the magnitude of these defects, it is now regularly used as an acceptance test for diffracting crystal optics at the Diamond Light Source synchrotron. Before installation on beamlines, crystal optics are inspected at the versatile bending-magnet B16 Test Beamline, where a variety of topographic techniques have been implemented with both white and monochromatic x-ray beams. A set of digital detectors permits rocking curve imaging with a choice of fields of view and spatial resolution down to 2 μm. Test crystals may be mounted in a variety of geometries according to need. For inspecting monochromator crystals fabricated for imaging applications, both on-the-fly scans and stitching techniques have been used to compose maps of surface defects. First crystals of multi-crystal monochromators have been tested under realistic cryocooled conditions, and their design has been improved to minimize strain. The Diamond Light Source’s x-ray topography program serves not only its own beamlines, but also industrial users and other x-ray synchrotron facilities.
A Bragg-case X-ray dynamical diffraction propagator has already been integrated into the “Synchrotron Radiation Workshop” (SRW) physical optics simulation software package. Previous benchmarking tests on crystal monochromators assumed thicknesses many times the extinction length, for which transmission is negligible. This paper reports tests of this propagator applied to thin crystals in transmission. The chosen example is a phase retarder, which allows users to alter the polarization of an X-ray beam. Phase retarders are often applied to studies of solid-state materials with hard X-rays, which current medium-energy storage ring synchrotron sources typically produce only with linear polarization. Correct designs of phase retarders require the accurate determination of both the intensity and the phase of the diffracted wave in all polarization states. First, to approximate an incident plane wave, SRW is used to simulate the passage of a Gaussian beam of very large radius of curvature through the phase retarder. Then, the phase retarder’s effects on a typical undulator beam are simulated and the results are compared. Because X-ray phase retarders are highly sensitive to angular alignment, tolerances in misalignment are also determined. SRW simulations are compared with experimental data from the Integrated In Situ and Resonant Hard X-ray Studies (ISR) beamline at NSLSII. The design of phase retarders can therefore be optimized for X-ray beamlines that must combine variable polarization with focusing or other properties.
The rapid development of new-generation synchrotron facilities with excellent coherence demands more accurate evaluation of beamline performance. A perturbation theory based on wave optics is proposed in this work to describe the effect of imperfections on the performance of x-ray optical elements. It shows that the perturbed performance of the non-ideal optical element could be derived from the perfect performance of the ideal optic through a convolution operation. The semi-analytical approach proposed here provides a new way to improve the simulation efficiency for imperfect optical elements. The finite aperture effect on diffraction-limited optics and focal shape distortion by surface height error are treated to show the application of the proposed method.
Brilliant beams of hard x-rays, with geometrical cross-sections below 50×50 nm2, are a standard research tool for
synchrotron users. With the advent of lower emittance sources, such as NSLSII, Petra III and Max IV, and planned
upgraded lattices, such as APS-2, SPING8-II, ESRF II and DLS II, nanofocusing optics operating in transmission mode
will become more competitive than they are currently. In general, they suffer from lower efficiency than reflective
optics, however they often have easier set-up and alignment, combined with a smaller footprint. Fabrication and
exploitation of ultra-short focal refractive lenses has not witnessed the same progress in the last decade as other optics,
such as multilayer mirrors and multilayer Laue lenses. This paper reports on current status of high-resolution lithography
for fabricating silicon lenses and on proposed designs for a new class of refractive lenses with zero aberrations and good
efficiency. The new designs are created with geometrical parameters matching the spatial resolution achieved by modern
lithography and silicon etch technology.
Deformable, piezo bimorph mirrors are often used to expand X-ray beams to a continuous range of sizes. However,
optical polishing errors present on all X-ray mirrors introduce striations into the reflected beam. To counteract them, reentrant
surface modifications with alternating concave and convex curvature have been proposed and applied to mirrors
of fixed shape or bimorph mirrors. For the latter, a new method of constructing re-entrant surface modifications on
segments of unequal length is described. This allows the re-entrant modification required for a desired beam size at the
focal point to be matched to the bimorph mirror’s polishing errors, thus reducing the voltage variations. Optical
profilometry using the Diamond-NOM showed that a 5-segment and a 7-segment modification could be suitably applied
to a deformable bimorph mirror. X-ray tests showed that striations caused by the 5-segment modification in the beam at
the focus are concentrated at the beam edges, while the beam center is left clear. This is in contrast to simple defocusing,
in which a strong side shoulder appears. The 7-segment modification produces a pattern of evenly spaced striations. The
intensity spikes seen with the re-entrant modifications are caused chiefly by the finite curvature of the mirror at the
turning points. The question of whether deformable bimorph mirrors with different piezo response functions could
sharpen the curvature changes will be investigated. Optimal modifications of continuous curvature, which could more
realistically be applied, will be sought.
Modern, third-generation synchrotron radiation sources provide coherent and extremely bright beams of X-ray radiation.
The successful exploitation of such beams depends to a significant extent on imperfections and misalignment of the
optics employed on the beamlines. This issue becomes even more critical with the increasing use of active optics, and the
desire to achieve diffraction-limited and coherence-preserving X-ray beams. In recent years, significant progress has
been made to improve optic testing and optimization techniques, especially those using X-rays for so-called atwavelength
metrology. These in-situ and at-wavelength metrology methods can be used not only to optimize the
performance of X-ray optics, but also to correct and minimize the collective distortions of upstream beamline optics,
including monochromators, and transmission windows. An overview of at-wavelength metrology techniques
implemented at Diamond Light Source is presented, including grating interferometry and X-ray near-field speckle based
techniques. Representative examples of the application of these techniques are also given, including in-situ and atwavelength
calibration and optimization of: active, piezo bimorph mirrors; Kirkpatrick-Baez (KB) mirrors; and
refractive optics such as compound refractive lenses.
A full wave propagation of X-rays from source to sample at a storage ring beamline requires simulation of the electron beam source and optical elements in the beamline. The finite emittance source causes the appearance of partial coherence in the wave field. Consequently, the wavefront cannot be treated exactly with fully coherent wave propagation or fully incoherent ray tracing. We have used the wavefront code Synchrotron Radiation Workshop (SRW) to perform partially coherent wavefront propagation using a parallel computing cluster at the Diamond Light Source. Measured mirror profiles have been used to correct the wavefront for surface errors.
Up to now simulation of perfect crystal optics in the “Synchrotron Radiation Workshop” (SRW) wave-optics computer
code was not available, thus hindering the accurate modelling of synchrotron radiation beamlines containing optical
components with multiple-crystal arrangements, such as double-crystal monochromators and high-energy-resolution
monochromators. A new module has been developed for SRW for calculating dynamical diffraction from a perfect
crystal in the Bragg case. We demonstrate its successful application to the modelling of partially-coherent undulator
radiation propagating through the Inelastic X-ray Scattering (IXS) beamline of the National Synchrotron Light Source II
(NSLS-II) at Brookhaven National Laboratory. The IXS beamline contains a double-crystal and a multiple-crystal highenergy-
resolution monochromator, as well as complex optics such as compound refractive lenses and Kirkpatrick-Baez
mirrors for the X-ray beam transport and shaping, which makes it an excellent case for benchmarking the new
functionalities of the updated SRW codes. As a photon-hungry experimental technique, this case study for the IXS
beamline is particularly valuable as it provides an accurate evaluation of the photon flux at the sample position, using the
most advanced simulation methods and taking into account parameters of the electron beam, details of undulator source,
and the crystal optics.
Grazing incidence mirrors are now a standard optic for focusing X-ray beams. Both bimorph and mechanically bendable mirrors are widely used at Diamond Light Source because they permit a wide choice of focal lengths. They can also be deliberately set out of focus to enlarge the X-ray beam, and indeed many beamline teams now wish to generate uniform beam spots of variable size. However, progress has been slowed by the appearance of fine structure in these defocused beams. Measurements showing the relationship between the medium-frequency polishing error and this structure over a variety of beam sizes will be presented. A theoretical model for the simulations of defocused beams from general mirrors will then be developed. Not only the figure error and its first derivative the slope error, but also the second derivative, the curvature error, must be considered. In conclusion, possible ways to reduce the defocused beam structure by varying the actuators' configuration and settings will be discussed.
Until now, a treatment of dynamical diffraction from perfect crystals has been missing in the "Synchrotron
Radiation Workshop" (SRW) wavefront propagation computer code despite the widespread use of crystals on X-ray synchrotron beamlines. Now a special propagator" module for calculating dynamical diffraction from a perfect crystal in the Bragg case has been written in C++, integrated into the SRW C/C++ library and made available for simulations using the Python interface of SRW. The propagator performs local processing of the frequency-domain electric field in the angular representation. A 2-D Fast Fourier Transform is used for changing the field representation from/to the coordinate representation before and after applying the crystal propagator. This ensures seamless integration of the new propagator with the existing functionalities of the SRW package, allows compatibility with existing propagators for other optical elements, and enables the simulation of complex beamlines transporting partially coherent X-rays. The code has been benchmarked by comparison with predictions made by plane-wave and spherical-wave dynamical diffraction theory. Test simulations for a selection of X-ray synchrotron beamlines are also shown.
The diffraction of an X-ray wavefront from a slightly distorted crystal can be modeled by the Takagi-Taupin
theory, an extension of the well-known dynamical diffraction theory for perfect crystals. Maxwell's equations
applied to a perturbed periodic medium yield two coupled differential equations in the incident and diffracted
amplitude. These equations are discretized for numerical calculation into the determination of the two amplitudes
on the points of an integration mesh, beginning with the incident amplitudes at the crystal's top surface. The
result is a set of diffracted amplitudes on the top surface (in the Bragg geometry) or the bottom surface (in
the Laue geometry), forming a wavefront that in turn can be propagated through free space using the Fresnel-
Huygens equations. The performance of the Diamond Light Source I20 dispersive spectrometer has here been
simulated using this method. Methods are shown for transforming displacements calculated by finite element
analysis into local lattice distortions, and for efficiently performing 3-D linear interpolations from these onto the
Takagi-Taupin integration mesh, allowing this method to be extended to crystals under thermal load or novel
mechanical bender designs.
Bimorph mirrors are used on many synchrotron beamlines to focus or collimate light. They are highly adaptable because
not only their overall figure but also their local slope errors can be corrected. However, the optimization procedure is
complex. At Diamond Light Source, highly repeatable and accurate pencil beam measurements are used to determine a
mirror's slope errors. These data are then used by automated scripts to calculate the necessary corrections. This procedure
may be applied to any type of active mirror, but for hard X-ray mirrors, diffraction from the slits must be considered.
Imaging of domains is a key step in understanding the microstructure and hence the properties of ferroelectric single
crystals. This understanding is essential for exploiting engineered domain configurations to achieve enhanced
performance. In this paper, single crystals of Barium Titanate are observed by reflection topography using unfocussed
monochromatic synchrotron X-ray light. A 10 x 10 mm polished surface of an unpoled crystal was mapped to form a
composite image, indicating a fine structure of a- and c-domains. By making use of the angular separation of the
diffracted reflections and specimen rocking, the relative tilts between adjacent domains about two orthogonal axes were
found. Angular resolution better than 0.1mrad in tilt measurements allowed the local elastic curvature of lattice planes to
be observed. The resulting composite images show well defined boundaries between regions of distinct microstructure,
and give an indication of the proportion of the domain types present. Over large regions of the crystal the domain
structure was finer than the X-ray camera resolution of 6.5μm; AFM and SEM imaging of domains was then used to
confirm the typical domain spacing. The results are interpreted in the context of models of compatible microstructure in
tetragonal crystals using microscopy of etched crystals to assist the interpretation. The technique shows promise for
mapping fine microstructure in single crystals, through the use of high resolution X-ray cameras, and is successful in
revealing lattice orientation information that is not normally available in optical or AFM measurements.
The Diamond Light Source beamline I12 (JEEP) is installing a monochromator for high-energy (50-150 keV) X-rays. It consists of two highly asymmetrically cut silicon crystals diffracting in the Laue case. These crystals will be bent to increase the bandpass to several hundred eV. It is necessary to estimate the tolerances for the
angular alignment and the bending radii, and to account for gravitational and thermal distortions. A simple ray-tracing model has been developed for use with finite-element analyses. For simple cases, this model is backed by more precise wave-optical calculations.
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