From analytical to data-driven: multi-method optimization of a bendable mirror system for dynamic focusing

Accurate and efficient tuning of small-radius-of-curvature (RoC) bendable mirrors remains challenging under variable beamline conditions, while conventional analytical tuning is often inadequate. To address this problem, this study presents the design, modeling, and multi-method optimization of a small-RoC bendable mirror system developed as a preparatory study for the Shenzhen Superconducting Soft X-ray Free-Electron Laser. The system adopts a four-point bending structure with an asymmetric support layout, and integrated force sensors are used for direct real-time force monitoring. A physically grounded analytical route based on classical bending theory is first used to assess the basic deformation capability of the system. A data-driven route is then developed by combining a transfer learning neural network (TLNN) with a differential evolution (DE) algorithm. The TLNN surrogate model is trained on simulated force-profile pairs generated by finite element analysis and refined using matrix-style measured force-profile pairs. The results show that, within the investigated center RoC range of 450-700 m, the differences in fitted image distance and center RoC are maintained within 5 mm and 0.55 m, respectively. The root-mean-square height prediction errors are at the nanometre level. Furthermore, the TLNN-DE framework is able to recover the end-force configuration corresponding to a prescribed target surface profile. The deviation between the recovered and measured end forces is kept within 0.5 N. These results demonstrate that the proposed framework provides an effective and transferable route for data-driven optimization of bendable mirrors and other active or adaptive optical systems in free-electron laser and synchrotron radiation facility beamlines.