Finite-element analysis is used to study the thermal deformation of a multilayer mirror due to the heat load from the undulator beam at a low-emittance synchrotron source, specifically the ESRF-EBS upgrade beamline EBSL-2. The energy bandwidth of the double-multilayer monochromator is larger than that of the relevant undulator harmonic, such that a considerable portion of the heat load is reflected. Consequently, the absorbed power is non-uniformly distributed on the surface. The geometry of the multilayer substrate is optimized to minimize thermally induced slope errors. We distinguish between thermal bending with constant curvature that leads to astigmatic focusing or defocusing and residual slope errors. For the EBSL-2 system with grazing angles θ between 0.2 and 0.4°, meridional and sagittal focal lengths down to 100 m and 2000 m, respectively, are found. Whereas the thermal bending can be tuned by varying the depth of the `smart cut', it is found that the geometry has little effect on the residual slope errors. In both planes they are 0.1-0.25 µrad. In the sagittal direction, however, the effect on the beam is drastically reduced by the `foregiveness factor', sin(θ). Optimization without considering the reflected heat load yields an incorrect depth of the `smart cut'. The resulting meridional curvature in turn leads to parasitic focal lengths of the order of 100 m.
Keywords: X-ray optics; finite-element analysis; high-heat-load optics; multilayer monochromator.