Water cooled beryllium lenses have been successfully used in the white beam of an undulator; see Figure 1 below and Section 2.2 in Lengeler et al. (2005): Refractive x-ray lenses. A stack of 12 beryllium lenses, housed in an evacuated casing, was exposed to a white beam with a power density and a total power of about 100 W mm-2 and 40 W, respectively. The temperature was measured by thermocouples located near the beryllium platelets of the lenses.
Note that the temperature of beryllium lenses must nowhere exceed 120°C .