Minimization of thermal deformation in crystal optics for high-repetition-rate FEL

Minimizing thermal deformation in X-ray crystal optics is crucial for preserving coherence and wavefront in high-repetition-rate free-electron lasers (FELs). This study presents two approaches to reduce pulse-by-pulse transient thermal deformation in diamond crystals used in cavity-based X-ray FELs (CBXFELs): (i) cryogenic cooling with liquid nitrogen (LN₂), and (ii) second-order correction via focusing optics. We revisit the temperature-dependent thermal-mechanical properties of diamond and silicon, and implement a finite-element analysis method to accelerate convergence to a quasi-steady-state regime. Results show that LN₂-cooled diamond crystals meet the stringent deformation requirement of less than 15 pm RMS for the pulse at the mJ scale at 1 MHz repetition frequency, and up to 1.5 mJ for 100 kHz. Second-order correction by using focusing elements within the cavity can reduce the impact of thermal deformation for both LN₂ and water cooling.