An improved algorithm for thermal compensation of synchrotron radiation optical mirrors based on Hessian matrix

The heaters-based thermal-compensated adaptive adjustment of a reflection mirror at Shanghai high repetition rate X-ray Free-Electron Laser and extreme light facility (SHINE) is presented here based on finite element analysis. The correction performance of different control algorithms [singular value decomposition and gradient descent (GD)] is analyzed and compared. This study has demonstrated that a significant control algorithm can further improve the surface shape accuracy of the mirror. After optimizing the mirror control algorithm, the calculated slope errors and height errors of the mirror are reduced to nearly less than 50 nrad rms and 0.5 nm rms, respectively. The optimization result indicates that the GD control algorithm based on the Hessian matrix exhibits superior performance and practicality compared to the control algorithm before optimization.