A hybrid surface shape control method for optimizing thermal deformation of FEL reflection mirror

The surface shape errors induced by thermal deformation of the free electron lasers (FEL) beamline reflection mirror directly affect the quality of FEL beam. In order to reduce the thermal deformation and meet the stringent requirements of surface shape error of the FEL reflection mirror, this paper proposes a surface shape control method that combines mirror bending and active thermal control (ATC) technique, aiming to minimize surface shape errors and enhance the overall performance of the FEL beamline. A bending mechanism has been designed to achieve a bending sensitivity of 60 nrad RMS/N using only one bending actuator. It does not obstruct the FEL beam with extremely small incident angle. This mechanism can compensate for thermal deformation by eliminating the cylindrical surface component of the thermal deformation. Additionally, a novel ATC algorithm is proposed based on the Tikhonov regularization method. When combined with the bending mechanism, it proves effective in controlling the surface shape of the reflection mirror. The simulation results indicate that this surface control method can effectively reduce the surface shape errors of the reflection mirror. Taking the 3 nm wavelength FEL as an example, the RMS value of surface slope error of the reflection mirror is substantially reduced from 17.1974 grad (with heat load) to 0.0913 grad after bending. Based on the bending result, the ATC technique further reduces the slope error to 0.0303 grad RMS. For 1 nm and 2 nm wavelength FELs, the mirror slope errors ultimately reach 0.0147 grad RMS and 0.0295 grad RMS respectively. This surface shape control method ensures that the reflection mirror slope errors meet the beamline requirement, which is 0.05 grad RMS, thereby guaranteeing the beam quality.