# Micro-Focusing of Broadband High-Order Harmonic Radiation by a Double Toroidal Mirror
## 2 Experimental Setup
### Micro-Focusing  
With the intent of maintaining the broad spectrum and high throughput, the XUV radiation is focused in the experimental region by means of two toroidal mirrors in grazing incidence configuration. The XUV beam is not recollimated prior to refocusing, but instead the object, i.e., the generation volume in the gas cell, is imaged into the interaction region in the application chamber. The object arm measures approximately 6 m, while the image arm measures 170 mm, resulting in a demagnification of 35. The combination of the two toroidal mirrors is designed to achieve a total deflection of 60\({}^{\circ}\), arising from a 15\({}^{\circ}\) grazing incidence on each mirror.  
Figure 1: **(a)** Scheme of the intense extreme ultraviolet (XUV) beamline; (b) typical high-order harmonic generation (HHG) spectrum; and (c) double toroidal mirror setup in a Wolter configuration. The black lines indicate the translation axes (straight solid arrows) and the pivoting axes for the revolutions (curved solid arrows). AR: anti-reflection; FS: fused silica.  
Figure 1c shows a schematic representation of the Wolter assembly. It is composed of two toroidal mirrors which are 30 mm long and 10 mm high. The center-to-center distance between the two mirrors is 30 mm. The mirrors are coated with gold because of its broadband reflectivity in the XUV range. In fact, between 20 and 50 eV, the gold layer yields a 61% reflectivity for s-polarized light and for a 15\({}^{\circ}\) grazing angle. The theoretical throughput of the optical setup is then 37%. However, the roughness of the surface is not taken into account in this calculation and the real throughput is expected to be smaller. The two mirrors were installed and pre-aligned by the manufacturer Thales SESO (Aix-en-Provence, France). This assembly is mounted on a 5-axis stage, whose degrees of freedom are sketched in Figure 1c. These axes are actuated by piezo-driven stick-and-slip positioners in open-loop configuration.  
As mentioned in the introduction, ellipsoidal mirrors are in principle the ideal solution for imaging. However, the toroidal surface represents a good local approximation to the ellipsoidal surface with the advantage that the two focal lengths, \(f_{T}\) and \(f_{S}\), corresponding to the tangential (the plane which includes both the optical axis and the normal to the mirror surface) and the sagittal (normal to the tangential plane and containing the optical axis) planes, respectively, are not coupled and can be set independently according to the following equations:  
\[f_{T}=\frac{R_{T}\sin\theta}{2}\,, \tag{1}\] \[f_{S}=\frac{R_{S}}{2\sin\theta}\,, \tag{2}\]  
where \(R\) denotes the radius of curvature, and the subscripts \(T\) and \(S\) refer to the tangential and sagittal planes, respectively. \(\theta\) is the grazing angle. Table 1 shows the curvatures and focal lengths of the individual mirrors in the two planes, as well as the combined focal length.  
The curvatures of each of the two mirrors are adjusted so that the resulting focal lengths in both planes are almost the same, to prevent astigmatism. The equivalent focal length of the assembly is about 164 mm (from the center of the second mirror), calculated in the paraxial approximation. This implies that our source, located 6 m before the mirrors, is imaged at 170 mm from the center of the second mirror.