The Shanghai High-repetition-rate XFEL and Extreme Light Facility (SHINE) is characterized by high brightness, full coherence, and high repetition rates. For focusing high-repetition-rate hard X-rays, compound refractive lenses (CRL) must withstand significant heat loads, which can potentially lead to optical performance failure due to high thermal stress.

This study aims to optimize the maximum allowable repetition rate of X-ray beams while maintaining thermal stress within permissible limits to ensure reliable optical performance of CRLs under high-repetition-rate operation.

Firstly, a finite element method (FEM) was employed to perform thermal-structural coupled analysis of beryllium CRLs with various internal spherical radii (0.3 mm, 1.0 mm, 2.0 mm, and 5.8 mm) under different photon energies (5 keV, 7 keV, and 15 keV). Then, the maximum thermal stress and temperature distributions were systematically evaluated under the initial 1 MHz repetition rate condition. Finally, the repetition rates were optimized to keep the maximum thermal stress below the permissible threshold of 120 MPa (50% of beryllium's yield strength), with special focus on the central region of the lens where stress concentration occurs.

Analysis results show that CRLs with internal radius R=0.3 mm experience a maximum thermal stress of 1 008 MPa at 7 keV photon energy at 1 MHz repetition rate, far exceeding the permissible limit. After optimization, the maximum thermal stress is reduced by 93% to 74 MPa by lowering the repetition rate to 100 kHz. The maximum temperature decreases from 201 °C to 36.6 °C, an 82% reduction. For larger radius CRLs (R=5.8 mm), the optimized repetition rate could be increased to 500 kHz for 7 keV X-rays while maintaining thermal stress below the threshold.

By optimizing the repetition rate according to CRL geometry and beam energy, thermal stress can be effectively controlled within safe limits. Increasing the internal spherical radius of CRLs from 0.3 mm to 5.8 mm allows for higher repetition rates (from 100 kHz to 500 kHz for 7 keV photons), thereby enhancing operational capabilities while ensuring reliable optical performance and structural integrity.