An X-ray free-electron laser (XFEL) is an X-ray produced by a beam of free electrons that have been accelerated almost to the speed of light. XFELs produce laser beams with exceedingly high peak power intensity, which makes them attractive for applications in fundamental research, such as X-ray nonlinear optics and protein crystal structure determination, and also in medicine. It is important to precisely focus XFEL beams to achieve high performance. Lasers are typically focused using total reflection mirrors; however, conventional mirrors are unsuitable for formation of sub-10 nm X-ray beams because such mirrors cannot deliver the large numerical aperture required. To overcome this limitation, X-ray beams can be focused using multilayer mirrors. Unfortunately, it is difficult to manufacture such multilayer mirrors because very high fabrication accuracy is required.