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.

In this paper, the X-ray diffraction experiment with multiple crystals configuration and its application in characterizing the emission characteristics of double crystal monochromator (DCM) on BL09 beamline of Shanghai Synchrotron Radiation Facility (SSRF) are reported. It is a non-dispersion configuration when the second crystal of DCM and the crystal analyzer form a (+n, -n) type experimental configuration. The rocking curve of the analyzer crystal can only reflect the operation of the DCM. It is the dispersion configurations when the second crystal of DCM and the crystal analyzer form the (+n, +n), (+n, -m) and (+n, +m) type experimental configuration. The width of the analyzer crystal rocking curve includes not only the intrinsic bandwidth of the DCM and the analyzer crystal, but also the angular divergence of the beamline. In this paper, we use the method of DuMond diagram to explicitly illustrate the characteristics of the output beam of the DCM which can be measured under the above two kinds of experimental configurations, and distinguish the diffraction characteristics of different experimental configurations at the same time. Finally, the angular bandwidth and the relative energy bandwidth of the DCM are 5.40(4) arcsec and1.30(1) x 10(-4) @ 10 keV, respectively, which are obtained by deconvolution of the analyzer crystal in (+1, -1) nondispersive configuration. The angular distribution and the relative energy bandwidth of the synchrotron radiation beams are 26(1) arcsec and 6.3(2) x 10(-4) @ 10 keV, respectively, which are obtained by deconvolution of the analyzer crystal and removal of dispersion broadening in (+1, +1), (+1, -3) and (+1, +3) dispersion configuration. After removing the influence of the DCM, the obtained angular divergence of the light source by 25(1) arcsec @ 10 keV, is consistent with the theoretical value of the bending source. In addition, under a series of different white beam entrance slit widths, we characterize the divergence and bandwidth of the beam emitted from the monochromator by the method of multi-crystals dispersion configuration and the DuMond diagram.

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.

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), time-resolved resonant inelastic X-ray scattering (tr-RIXS) has attracted more attention. The preliminary designs of the tr-RIXS branchline and expected performance characteristics at the Shenzhen Superconducting Soft X-ray Free Electron Laser (${\mathrm{S}}^3\mathrm{FEL}$) are presented. A start-to-end simulation of the tr-RIXS branchline based on the 6-D phase space ray-tracing method of beamline simulation software package FURION was performed. The simulation design satisfies the key requirements of the tr-RIXS branchline, including spatial dispersion in the vertical dimension, temporal resolution, energy resolution, efficient utilization of SASE spectral photons, and spatial uniformity of the beam spot sizes across different wavelengths.