Wavefront sensing at X-ray free-electron lasers

X-ray microscopy and talbot imaging with the matter in extreme conditions X-ray imager at LCLS

The last decade has shown the great potential that X-ray Free Electron Lasers (FEL) have to study High Energy Density (HED) physics. Experiments at FELs have made significant breakthroughs in Shock Physics and Dynamic Diffraction, Dense Plasma Physics and Warm Dense Matter Science, using techniques such as isochoric heating, inelastic scattering, small angle scattering and X-ray diffraction. In addition, and complementary to these techniques, the coherent properties of the FEL beam can be used to image HED samples with high fidelity. We present new imaging diagnostics and techniques developed at the Matter in Extreme Conditions (MEC) instrument at Linac Coherent Light Source (LCLS) over the last few years. We show results in Phase Contrast Imaging geometry, where the X-ray beam propagates from the target to a camera revealing its phase, as well as in Direct Imaging geometry, where a real image of the sample plane is produced in the camera with a spatial resolution down to 200 nm. Last, we show an implementation of the Talbot Imaging method allowing both X-ray phase and intensity measurements change introduced by a target with sub-micron resolution.

Review and experimental comparison of speckle-tracking algorithms for X-ray phase contrast imaging

X-ray speckles have been used in a wide range of experiments, including imaging (and tomography), wavefront sensing, spatial coherence measurements, X-ray photon correlation spectroscopy and ptychography. In this review and experimental comparison, we focus on using X-ray near-field speckle grains as wavefront markers and numerical methods for retrieving the phase information they contain. We present the most common tracking methods, introducing the existing algorithms with their specifications and comparing their performances under various experimental conditions. This comparison includes applications to different types of samples: phantoms for quantitative analysis and complex samples for assessing image quality. Our goal is to unify concepts from several speckle tracking methods using consistent terminology and equation formalism, while keeping the discussion didactic and accessible to a broad audience.

Picometre-level surface control of a closed-loop, adaptive X-ray mirror with integrated real-time interferometric feedback

We provide a technical description and experimental results of the practical development and offline testing of an innovative, closed-loop, adaptive mirror system capable of making rapid, precise and ultra-stable changes in the size and shape of reflected X-ray beams generated at synchrotron light and free-electron laser facilities. The optical surface of a piezoelectric bimorph deformable mirror is continuously monitored at 20 kHz by an array of interferometric sensors. This matrix of height data is autonomously converted into voltage commands that are sent at 1 Hz to the piezo actuators to modify the shape of the mirror optical surface. Hence, users can rapidly switch in closed-loop between pre-calibrated X-ray wavefronts by selecting the corresponding freeform optical profile. This closed-loop monitoring is shown to repeatably bend and stabilize the low- and mid-spatial frequency components of the mirror surface to any given profile with an error <200 pm peak-to-valley, regardless of the recent history of bending and hysteresis. Without closed-loop stabilization after bending, the mirror height profile is shown to drift by hundreds of nanometres, which will slowly distort the X-ray wavefront. The metrology frame that holds the interferometric sensors is designed to be largely insensitive to temperature changes, providing an ultra-stable reference datum to enhance repeatability. We demonstrate an unprecedented level of fast and precise optical control in the X-ray domain: the profile of a macroscopic X-ray mirror of over 0.5 m in length was freely adjusted and stabilized to atomic level height resolution. Aside from demonstrating the extreme sensitivity of the interferometer sensors, this study also highlights the voltage repeatability and stability of the programmable high-voltage power supply, the accuracy of the correction-calculation algorithms and the almost instantaneous response of the bimorph mirror to command voltage pulses. Finally, we demonstrate the robustness of the system by showing that the bimorph mirror's optical surface was not damaged by more than 1 million voltage cycles, including no occurrence of the `junction effect' or weakening of piezoelectric actuator strength. Hence, this hardware combination provides a real time, hyper-precise, temperature-insensitive, closed-loop system which could benefit many optical communities, including EUV lithography, who require sub-nanometre bending control of the mirror form.

A phenomenological model of the X-ray pulse statistics of a high-repetition-rate X-ray free-electron laser

Many coherent imaging applications that utilize ultrafast X-ray free-electron laser (XFEL) radiation pulses are highly sensitive to fluctuations in the shot-to-shot statistical properties of the source. Understanding and modelling these fluctuations are key to successful experiment planning and necessary to maximize the potential of XFEL facilities. Current models of XFEL radiation and their shot-to-shot statistics are based on theoretical descriptions of the source and are limited in their ability to capture the shot-to-shot intensity fluctuations observed experimentally. The lack of accurate temporal statistics in simulations that utilize these models is a significant barrier to optimizing and interpreting data from XFEL coherent diffraction experiments. Presented here is a phenomenological model of XFEL radiation that is capable of capturing the shot-to-shot statistics observed experimentally using a simple time-dependent approximation of the pulse wavefront. The model is applied to reproduce non-stationary shot-to-shot intensity fluctuations observed at the European XFEL, whilst accurately representing the single-shot properties predicted by FEL theory. Compared with previous models, this approach provides a simple, robust and computationally inexpensive method of generating statistical representations of XFEL radiation.

Hard X-ray stereographic microscopy for single-shot differential phase imaging

The characterisation of fast phenomena at the microscopic scale is required for the understanding of catastrophic responses of materials to loads and shocks, the processing of materials by optical or mechanical means, the processes involved in many key technologies such as additive manufacturing and microfluidics, and the mixing of fuels in combustion. Such processes are usually stochastic in nature and occur within the opaque interior volumes of materials or samples, with complex dynamics that evolve in all three dimensions at speeds exceeding many meters per second. There is therefore a need for the ability to record three-dimensional X-ray movies of irreversible processes with resolutions of micrometers and frame rates of microseconds. Here we demonstrate a method to achieve this by recording a stereo phase-contrast image pair in a single exposure. The two images are combined computationally to reconstruct a 3D model of the object. The method is extendable to more than two simultaneous views. When combined with megahertz pulse trains of X-ray free-electron lasers (XFELs) it will be possible to create movies able to resolve 3D trajectories with velocities of kilometers per second.

Fast wavefront sensing for X-ray optics with an alternating speckle tracking technique

Advances in accelerator technologies have enabled the continuous development of synchrotron radiation and X-ray free electron laser (XFEL) sources. At the same time, it has been critical to perform in-situ wavefront sensing to aid delivery of high-quality X-ray beams to the end users of these facilities. The speckle-based scanning technique has obtained popularity due to its high spatial resolution and superior sensitivity compared to other wavefront sensing methods. However, these advantages often come at the expense of longer data acquisition times since multiple images have to be collected to derive the necessary wavefront information. Whereas initial speckle tracking techniques could obtain wavefront information relatively quickly, the installation of additional hardware was routinely required to do so. Here, we propose a novel speckle-based approach, termed Alternating Speckle Tracking (AST), to perform fast wavefront sensing within a conventional beamline setup. The wavefront information derived from the new technique has proven to be valuable for many applications that require temporal resolution. Importantly, both horizontal and vertical wavefront information can be simultaneously derived by moving the speckle generator along the diagonal direction. We expect this method will be widely used by the synchrotron radiation and XFEL community in the future.

Precise wavefront characterization of x-ray optical elements using a laboratory source

Improvements in x-ray optics critically depend on the measurement of their optical performance. The knowledge of wavefront aberrations, for example, can be used to improve the fabrication of optical elements or to design phase correctors to compensate for these errors. At present, the characterization of such optics is made using intense x-ray sources, such as synchrotrons. However, the limited access to these facilities can substantially slow down the development process. Improvements in the brightness of lab-based x-ray micro-sources in combination with the development of new metrology methods, particularly ptychographic x-ray speckle tracking, enable characterization of x-ray optics in the lab with a precision and sensitivity not possible before. Here, we present a laboratory setup that utilizes a commercially available x-ray source and can be used to characterize different types of x-ray optics. The setup is used in our laboratory on a routine basis to characterize multilayer Laue lenses of high numerical aperture and other optical elements. This typically includes measurements of the wavefront distortions, optimum operating photon energy, and focal length of the lens. To check the sensitivity and accuracy of this laboratory setup, we compared the results to those obtained at the synchrotron and saw no significant difference. To illustrate the feedback of measurements on performance, we demonstrated the correction of the phase errors of a particular multilayer Laue lens using a 3D printed compound refractive phase plate.

Single-exposure X-ray phase imaging microscopy with a grating interferometer

The advent of hard X-ray free-electron lasers enables nanoscopic X-ray imaging with sub-picosecond temporal resolution. X-ray grating interferometry offers a phase-sensitive full-field imaging technique where the phase retrieval can be carried out from a single exposure alone. Thus, the method is attractive for imaging applications at X-ray free-electron lasers where intrinsic pulse-to-pulse fluctuations pose a major challenge. In this work, the single-exposure phase imaging capabilities of grating interferometry are characterized by an implementation at the I13-1 beamline of Diamond Light Source (Oxfordshire, UK). For comparison purposes, propagation-based phase contrast imaging was also performed at the same instrument. The characterization is carried out in terms of the quantitativeness and the contrast-to-noise ratio of the phase reconstructions as well as via the achievable spatial resolution. By using a statistical image reconstruction scheme, previous limitations of grating interferometry regarding the spatial resolution can be mitigated as well as the experimental applicability of the technique.

Methods of Coherent X-Ray Diffraction Imaging

Abstract

Methods of coherent X-ray diffraction imaging of the spatial structure of noncrystalline objects and nanocrystals (nanostructures) are considered. Particular attention is paid to the methods of scanning-based coherent diffraction imaging (ptychography), visualization based on coherent surface scattering with application of correlation spectroscopy approaches, and specific features of visualization using X-ray free-electron laser radiation. The corresponding data in the literature are analyzed to demonstrate the state of the art of the methods of coherent diffraction imaging and fields of their application.

Binary Amplitude Reflection Gratings for X-ray Shearing and Hartmann Wavefront Sensors

New, high-coherent-flux X-ray beamlines at synchrotron and free-electron laser light sources rely on wavefront sensors to achieve and maintain optimal alignment under dynamic operating conditions. This includes feedback to adaptive X-ray optics. We describe the design and modeling of a new class of binary-amplitude reflective gratings for shearing interferometry and Hartmann wavefront sensing. Compact arrays of deeply etched gratings illuminated at glancing incidence can withstand higher power densities than transmission membranes and can be designed to operate across a broad range of photon energies with a fixed grating-to-detector distance. Coherent wave-propagation is used to study the energy bandwidth of individual elements in an array and to set the design parameters. We observe that shearing operates well over a ±10% bandwidth, while Hartmann can be extended to ±30% or more, in our configuration. We apply this methodology to the design of a wavefront sensor for a soft X-ray beamline operating from 230 eV to 1400 eV and model shearing and Hartmann tests in the presence of varying wavefront aberration types and magnitudes.

Quantitative X-ray Channel-Cut Crystal Diffraction Wavefront Metrology Using the Speckle Scanning Technique

A speckle-based method for the X-ray crystal diffraction wavefront measurement is implemented, and the slope errors of channel-cut crystals with different surface characteristics are measured. The method uses a speckle scanning technique generated by a scattering membrane translated using a piezo motor to infer the deflection of X-rays from the crystals. The method provides a high angular sensitivity of the channel-cut crystal slopes in both the tangential and sagittal directions. The experimental results show that the slope error of different cutting and etching processes ranges from 0.25 to 2.98 μrad. Furthermore, the results of wavefront deformation are brought into the beamline for simulation. This method opens up possibilities for new high-resolution applications for X-ray crystal diffraction wavefront measurement and provides feedback to crystal manufacturers to improve channel-cut fabrication.

Correction of the X-ray wavefront from compound refractive lenses using 3D printed refractive structures

A refractive phase corrector optics is proposed for the compensation of fabrication error of X-ray optical elements. Here, at-wavelength wavefront measurements of the focused X-ray beam by knife-edge imaging technique, the design of a three-dimensional corrector plate, its fabrication by 3D printing, and use of a corrector to compensate for X-ray lens figure errors are presented. A rotationally invariant corrector was manufactured in the polymer IP-S^TM using additive manufacturing based on the two-photon polymerization technique. The fabricated corrector was characterized at the B16 Test beamline, Diamond Light Source, UK, showing a reduction in r.m.s. wavefront error of a Be compound refractive Lens (CRL) by a factor of six. The r.m.s. wavefront error is a figure of merit for the wavefront quality but, for X-ray lenses, with significant X-ray absorption, a form of the r.m.s. error with weighting proportional to the transmitted X-ray intensity has been proposed. The knife-edge imaging wavefront-sensing technique was adapted to measure rotationally variant wavefront errors from two different sets of Be CRL consisting of 98 and 24 lenses. The optical aberrations were then quantified using a Zernike polynomial expansion of the 2D wavefront error. The compensation by a rotationally invariant corrector plate was partial as the Be CRL wavefront error distribution was found to vary with polar angle indicating the presence of non-spherical aberration terms. A wavefront correction plate with rotationally anisotropic thickness is proposed to compensate for anisotropy in order to achieve good focusing by CRLs at beamlines operating at diffraction-limited storage rings.

Ronchi shearing interferometry for wavefronts with circular symmetry

Ronchi testing of a focused electromagnetic wave has in the last few years been used extensively at X-ray free-electron laser (FEL) facilities to qualitatively evaluate the wavefront of the beam. It is a quick and straightforward test, is easy to interpret on the fly, and can be used to align phase plates that correct the focus of aberrated beams. In general, a single Ronchigram is not sufficient to gain complete quantitative knowledge of the wavefront. However the compound refractive lenses that are commonly used at X-ray FELs exhibit a strong circular symmetry in their aberration, and this can be exploited. Here, a simple algorithm that uses a single recorded Ronchigram to recover the full wavefront of a nano-focused beam, assuming circular symmetry, is presented, and applied to experimental measurements at the Matter in Extreme Conditions instrument at the Linac Coherent Light Source.

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.

Hard X-ray wavefront correction via refractive phase plates made by additive and subtractive fabrication techniques

Modern subtractive and additive manufacturing techniques present new avenues for X-ray optics with complex shapes and patterns. Refractive phase plates acting as glasses for X-ray optics have been fabricated, and spherical aberration in refractive X-ray lenses made from beryllium has been successfully corrected. A diamond phase plate made by femtosecond laser ablation was found to improve the Strehl ratio of a lens stack with a numerical aperture (NA) of 0.88 × 10-3 at 8.2 keV from 0.1 to 0.7. A polymer phase plate made by additive printing achieved an increase in the Strehl ratio of a lens stack at 35 keV with NA of 0.18 × 10-3 from 0.15 to 0.89, demonstrating diffraction-limited nanofocusing at high X-ray energies.

Focus characterization of an X-ray free-electron laser by intensity correlation measurement of X-ray fluorescence

This paper proposes and demonstrates a simple method using the intensity correlation of X-ray fluorescence to evaluate the focused beam size of an X-ray free-electron laser (XFEL). This method was applied to the sub-micrometre focused XFEL beam at the SPring-8 Angstrom Compact Free Electron Laser, and the beam size evaluated using the proposed method was consistent with that measured using the knife-edge scan method. The proposed method is readily applicable to extremely small X-ray spots and can be applied for the precise diagnostics of sub-10 nm focused X-ray beams which have recently emerged.

Reflective binary amplitude grating for soft x-ray shearing and Hartmann wavefront sensing

We demonstrate a reflective wavefront sensor grating suitable for the characterization of high-quality x-ray beamlines and optical systems with high power densities. Operating at glancing incidence angles, the optical element is deeply etched with a two-level pattern of shearing interferometry gratings and Hartmann wavefront sensor grids. Transverse features block unwanted light, enabling binary amplitude in reflection with high pattern contrast. We present surface characterization and soft x-ray reflectometry of a prototype grating array to demonstrate function prior to wavefront measurement applications. A simulation of device performance is shown.

Maximum likelihood reconstruction for grating-based X-ray microscopy

The combination of grating-based phase-contrast imaging with X-ray microscopy can result in a complicated image formation. Generally, transverse shifts of the interference fringes are nonlinearly dependent on phase differences of the measured wave front. We present an iterative reconstruction scheme based on a regularized maximum likelihood cost function that fully takes this dependency into account. The scheme is validated by numerical simulations. It is particularly advantageous at low photon numbers and when the premises for deconvolution-based reconstructions are not met. Our reconstruction scheme hence enables a broader applicability of X-ray grating interferometry in imaging and wave front sensing.

Conception of diffractive wavefront correction for XUV and soft x-ray spectroscopy

We present a simple and precise method to minimize aberrations of mirror-based, wavelength-dispersive spectrometers for the extreme ultraviolet (XUV) and soft x-ray domain. The concept enables an enhanced resolving power $ E/\Delta E $E/ΔE, in particular, close to the diffraction limit over a spectral band of a few percent around the design energy of the instrument. Our optical element, the "diffractive wavefront corrector" (DWC), is individually shaped to the form and figure error of the mirror profile and might be written directly with a laser on a plane and even strongly curved substrates. Theory and simulations of various configurations, like Hettrick-Underwood or compact, highly efficient all-in-one setups for $ {{\rm TiO}_2} $TiO₂ spectroscopy with $ E/\Delta E \mathbin{\lower.3ex\hbox{$\buildrel{\displaystyle{\lt}}\over{\smash{\displaystyle\sim}\vphantom{_x}}$}} 4.5 \times {10^4} $E/ΔE∼_x<4.5×10⁴, are addressed, as well as aspects of their experimental realization.