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.

A novel dual-frequency real-time feedback system has been developed to simultaneously optimize and stabilize beam position and energy at the hard X-ray nanoprobe beamline of the Shanghai Synchrotron Radiation Facility. A user-selected cut-off frequency is used to separate the beam position signal obtained from an X-ray beam position monitor into two parts, i.e. high-frequency and low-frequency components. They can be real-time corrected and optimized by two different optical components, one chromatic and the other achromatic, of very different inertial mass, such as Bragg monochromator dispersive elements and a pre-focusing total external reflection mirror. The experimental results shown in this article demonstrate a significant improvement in position and energy stabilities. The long-term beam angular stability clearly improved from 2.21 to 0.92 µrad RMS in the horizontal direction and from 0.72 to 0.10 µrad RMS in the vertical direction.