## 2 Design  
### Optics  
The optical parameters of the elliptical cylinder shaped focusing mirrors designed for the optical system are summarized in Table 1. The focusing system has a wide spatial acceptance of 970 \(\upmu\)m and 920 \(\upmu\)m in the vertical and horizontal directions, respectively, which is considerably larger than the incident beam size of 450 \(\upmu\)m (FWHM) at EH5 of BL3 at a photon energy of 10 keV. The designed K-B optics have an aperture receiving 80% of the incident pulse energy at a photon energy of 10 keV. We designed the working distance, (length between the downstream edge of the optics and the focus), to be 115 mm, which is long enough to conduct various experiments such as two-color double-pulse experiments and X-ray non-linear optics experiments. The focused beam sizes at a photon energy of 10 keV were geometrically calculated to be 182 and 87 nm (FWHM) using a demagnification ratio based on the light source size and the focal lengths. The focus size is broader than that of the two-stage focusing system because of the moderate demagnification factor. At a photon energy of 10 keV, the calculated focus sizes are three times larger than the diffraction-limited beam sizes of 56 and 28 nm (FWHM) determined using the numerical aperture of the optical system. These optics are advantageous for maintaining the optimized focusing condition because the alignment tolerances of both the incident angle and the focal length are approximately 10 times larger than those of the previous two-stage focusing optics system.  
The incident angles of the mirrors were designed for use under total reflection conditions of up to 12 keV. Figure 1 presents the calculated reflectivity of the mirrors plotted against the incident photon energy. The reflectivity curves for a rhodium surface coating with a density that is 100% of the bulk density are presented in Figure 1b,c. The reflectivity of the two mirrors is calculated to be 80% up to a photon energy of 12 keV and above 50% at a photon energy of 15 keV. The rhodium-coated mirrors are free from absorption edge within a photon energy range of 5-15 keV. The radiation dose to the rhodium in the optical system is estimated to be well below the threshold level for radiation damage from intense XFEL [17]. With these design parameters, the pulse energy, fluence and intensity of the  
\begin{table}
\begin{tabular}{c c c} \hline
**Optical Parameters** & **Vertical Focusing Mirror** & **Horizontal Focusing Mirror** \\ \hline Surface profile & Elliptical cylinder & Elliptical cylinder \\ Substrate material & Single crystal silicon & Single crystal silicon \\ Surface coating & Rhodium & Rhodium \\ Mirror substrate size & \(250\times 50\times 50\) mm\({}^{3}\) & \(250\times 50\times 50\) mm\({}^{3}\) \\ Glancing angle on optical axis & 4.0 mrad & 3.8 mrad \\ Focal length & 500 mm & 240 mm \\ Semimajor axis & 110.25 m & 110.25 m \\ Semiminor axis & 41.95224 mm & 27.62842 mm \\ Diffraction-limited focus size in FWHM at 10 keV & 56 nm & 28 nm \\ Effective mirror length & 242 mm & 242 mm \\ Spatial acceptance & 970 \(\upmu\)m & 920 \(\upmu\)m \\ \hline \end{tabular}
\end{table}
Table 1: Optical parameters of the focusing mirrors.  
focus were calculated to be \(\sim\)200 \(\upmu\)J, \(2\times 10^{6}\) J/cm\({}^{2}\) and \(2\times 10^{20}\) W/cm\({}^{2}\) at a photon energy of 10 keV, when the incident pulse energy to the optics and the pulse duration were assumed to be 300 \(\upmu\)J and 7 fs.