Fig. 1. Layout of the optical design in Beamline FEL-I

Fig. 2. Diagram of compound refractive lens (CRL) structural design (a) Compound refractive lens structure includes a Kovar fixing structure and a CRL focusing mirror, (b) Design parameters of the CRL

Fig. 3. Mesh model of compound refractive lens (CRL) with Kovar fixation structure (a), magnified finite element mesh at the CRL center (b)

Fig. 4. Non-linear thermal expansion coefficient (a) and conductivity coefficient (b) of beryllium^[19]

Fig. 5. Heat flow density distribution at photon energy 5 keV (a), 7 keV (b), the Gaussian distribution of heat flow density (c) (color online)

Fig. 6. Variations of thermal stress and temperature of the CRL with spherical radius under the 1 MHz repetition rate

Fig. 7. Temperature (a) and thermal stress (b) distribution contour of the CRL within R=0.3 mm and photo energy 7 keV under the 1 MHz repetition rate (color online)

Fig. 8. Temperature (a) and thermal stress (b) distribution contour of the CRL within R=2 mm and photo energy 5 keV under the 1 MHz repetition rate (color online)

Fig. 9. Optimized maximum repetition rate of the CRL (color online)

Fig. 10. Thermal stress and temperature comparison before and after optimization