Modern X-ray free-electron lasers (XFELs) can generate pulses with durations ranging from femtoseconds to attoseconds. The numerical evaluation of ultra-short XFEL pulses through beamline systems is a critical process of beamline system design. However, the bandwidth of such ultra-short XFEL pulses is often non-negligible, and the propagation cannot be simply approximated using the central wavelength, especially in dispersive beamline systems. We developed a numerical model which is called Fourier optics based Ultrashort x-Ray pulse propagatION tool (FURION). This model can not only be used to simulate dispersive beamline systems but also to evaluate non-dispersive beamline systems. The FURION model utilizes Fresnel integral and angular spectrum integral to perform ultra-short XFEL pulse propagation in free space. We also present the method for XFEL pulse propagation through different types of dispersive gratings, which are commonly used in soft X-ray beamline systems. By using FURION, a start-to-end simulation of the FEL-1 beamline system at Shenzhen superconducting soft X-ray free electron laser (S³FEL) is carried out. This model can also be used to evaluate gratings-based spectrometers, beam splitters, pulse compressors, and pulse stretchers. This work provides valuable insights into the start-to-end simulation of X-ray beamline systems.

In soft X-ray free-electron laser (FEL) beamlines, variable-line-spacing (VLS) gratings are often used as dispersive components of monochromators and spectrometers due to their combined dispersion and focusing properties. X-ray FEL pulses passing through the VLS grating can result in not only transverse focusing but also spatiotemporal coupling effects, such as pulse front tilt, pulse front rotation, and pulse stretching. In this paper, we present a theoretical study of the spatiotemporal response of concave VLS gratings to ultra-short X-ray pulses. The theoretical analysis indicates that the tilt angle of the non-zero diffraction orders varies with the propagation distance, and disappears at the focus, where the focal lengths and pulse stretching differ for different diffraction orders. The model demonstrates the pulse duration after the concave VLS grating is the convolution of the initial pulse duration and the stretching term induced by dispersion, while the beam size at the focus in x dimension is the convolution of the geometric scaling beam size and the dispersion term. This work provides a mathematical explanation for the spatiotemporal response of concave VLS grating to ultra-short X-ray pulses and offers valuable insights into the design of FEL grating monochromators, spectrometers, pulse compressors, and pulse stretchers.

Free electron laser (FEL) is capable of producing ultra-short X-ray pulses. The estimation of X-ray pulse propagation is the key process of X-ray FEL beamline design. By using the Kostenbauder matrix approach, the evolution of an ultra-short pulse in a beamline system can be calculated. Therefore, it is of significant importance to investigate the Kostenbauder matrices of different kinds of X-ray optics. In this work, we derive a unified 6 × 6 optical matrix to describe various kinds of X-ray optical elements, including varied-line-spacing (VLS) toroidal grating, VLS spherical grating, VLS cylindrical grating, VLS plane grating, toroidal grating, spherical grating, cylindrical grating, plane grating, toroidal mirror, spherical mirror, cylindrical mirror, and plane mirror. These optics are usually adopted in soft X-ray regime. We apply this method to describe the transverse focusing, pulse front tilt, and pulse stretching after an X-ray pulse going through a VLS plane grating monochromator (VLS-PGM). We also use this approach to simulate a grating compressor which can be used to compress chirped soft X-ray pulse. This work is helpful in the design and optimization of X-ray beamline systems.

Free-electron laser (FEL) facilities operating at MHz repetition rates can emit lasers with average powers reaching hundreds of watts. Partial absorption of this power induces thermal deformation of a few micrometres on the mirror surface. Such deformation degrades the characteristics of the reflected photon beam, leading to focal spot aberrations and wavefront distortions that fail to meet experimental requirements. A robust method is necessary to correct the mirror surface shape to meet the Maréchal criterion. This paper proposes a thermal deformation compensation scheme for offset mirrors operating at MHz repetition rates using a piezoelectric deformable mirror. The mirror is side-mounted with slots filled with an indium–gallium alloy, which house copper tubes for water cooling. Eighteen groups of piezo actuators are symmetrically attached to the top and bottom surfaces. The scheme incorporates finite-element analysis for simulation and post-processing verification, utilizing a differential evolution (DE) algorithm for global optimization. The DE algorithm effectively addresses the voltage constraints that the traditional singular value decomposition algorithm cannot handle. Under an X-ray wavelength of 1 nm, the peak-to-valley (PV) height error of the mirror was reduced from 1340.8 nm to 1.1 nm, and the root-mean-square (RMS) height error decreased from 859.1 nm to 0.18 nm. The slope error was corrected to 154 nrad PV and 24 nrad RMS. Significant results were also achieved at an X-ray wavelength of 3 nm. Wave-optics simulations verified the reliability of this approach, and effects on key mirror parameters and conditions were systematically analysed.

Chirped X-ray pulse compression is a promising approach for generating ultra-short X-ray free electron laser (XFEL) pulses. The design of X-ray pulse compressors requires the careful control of group delay dispersion (GDD), which plays a critical role in achieving optimal compression. However, the penetration dispersion of crystals and multilayers can induce an extra GDD, which may result in over-compression or under-compression. In this study, we investigate the penetration dispersion of crystals and multilayers theoretically and numerically. Our results indicate that the extra GDD induced by the penetration effect increases as the bandwidth of the rocking curve decreases. Moreover, the extra GDD is nonlinear and can be mitigated by optimizing the configuration of X-ray pulse compressors. This work provides insights into the dispersion compensation and configuration optimization of X-ray pulse compressors, which are essential for generating ultra-short XFEL pulses.

With the rapid development of X-ray free-electron lasers (XFELs) that can generate ultrashort X-ray pulses with a duration range from attoseconds to femtoseconds, the study of ultrashort XFEL pulse propagation in beamline systems is increasingly important, especially in dispersive beamline systems. We developed a 6D phase space ray-tracing method to simulate pulse propagation in dispersive soft X-ray optical systems. We validated this method by simulating a typical dispersive optical system: a grating monochromator. The simulation indicated that the spatiotemporal properties such as pulse front tilt, pulse front rotation and angular dispersion can be described. Using this approach, we performed a start-to-end simulation of the Shenzhen Superconducting Soft X-ray Free Electron Laser (S³FEL) FEL-1 beamline. Compared with the 3D pulse propagation method based on Fourier optics, this significantly reduces the simulation time. Our work provides a useful tool for X-ray beamline systems design.