The Mutual Optical Intensity (MOI) model is further developed to simulate the propagation of partially coherent X-ray vortex beams through free space. The intensity, phase, and amplitude distributions of the degree of coherence in the far field zone are numerically calculated using the MOI model. The effects of the coherence property and topological charge on the vortex beam are analyzed. For Gaussian Schell-model vortex beams, a vortex structure appears around the coherence singularity at the amplitude distribution of the degree of coherence relative to the central point. The number and rotation direction of the vortex structure are consistent with the magnitude and sign of the topological charge of the partially coherent vortex beam. As the spatial coherence decreases, the size of the vortex structure for the phase and amplitude distributions of the degree of coherence gradually increase until they disappear. The intensity of the dark hollow region increases with increasing topological charge. In addition, for the Laguerre–Gaussian Schell-model vortex beam, there is no singularity or vortex structure at the amplitude distribution of the degree of coherence relative to the central point. Therefore, the vortex structure around the coherence singularity at the amplitude distributions of the degree of coherence relative to the central point can be used to measure the vortex beam types and topological charge.

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.

The vibrations of X-ray optics play a crucial role in determining the stability of synchrotron radiation light sources and modern X-ray free-electron (X-FELs). Therefore, it is of great importance to perform the numerical analysis of vibrations in X-ray beamline design. This paper presents a method based on matrix optics, which can be used to rapidly analyze the degradation of beam performance, such as stability, beam size, and coherence, caused by the vibrations of optical elements. The analytical results derived from matrix optics are in good agreement with the ray-tracing simulations performed using SHADOW3. The proposed method can analytically describe the impact of vibrations on the beam quality, facilitating rapid optical vibration evaluation in complex beamline configurations and establishing quantitative stability specifications for critical optical elements.