# Micro-Focusing of Broadband High-Order Harmonic Radiation by a Double Toroidal Mirror
## 1 Introduction  
High-order harmonic generation (HHG) [1,2] in gases is one of the main sources of attosecond pulses in the extreme ultraviolet (XUV) and soft X-ray regime [3,4]. These pulses, generated through highly nonlinear interaction between laser pulses and atoms, are coherent and of broad bandwidth, typically several tens of eV wide. If sufficiently intense, they can be used to investigate multiphoton phenomena in atoms and molecules to unravel ultrafast dynamics [5, 6, 7, 8]. These nonlinear experiments require high intensity, on the order of 10\({}^{12}\)-10\({}^{14}\) W/cm\({}^{2}\). Numerous studies report optimization of the pulse energy. The highest energies, up to the \(\upmu\) level, were demonstrated through loose focusing scaling techniques [9, 10, 11, 12, 13]. The generation of extremely short pulses of the order of 100 as or shorter was also reported [14, 15]. Short isolated attosecond pulses in the \(\upmu\) range have even been generated using a combination of two-color field synthesis and energy-scaling aforementioned techniques [16]. However, few studies tackle the problem of focusing the HHG radiation [17, 18]. The challenge is tofocus an XUV beam to a small focal spot, while maintaining the broad bandwidth of the radiation and the short pulse duration, as well as minimizing the loss in the pulse energy. Refractive optics cannot be used for XUV radiation, since it is strongly absorbed when propagating in materials. Spherical mirrors, used at near-normal incidence, allow high-aperture beams to be focused on small focal spots. However, they are effectively designed only for narrow bandwidth radiation. Although larger bandwidths can be handled by aperiodic multilayer mirrors, their reflectivity is often very low [18; 19]. Diffractive optics, such as zone plates [20], also have low transmission and are designed for monochromatic radiation. The best option to focus a broadband XUV beam with high reflectivity is therefore by using curved mirrors at grazing incidence [21]. Ellipsoidal mirrors are in principle able to perfectly focus the radiation in the second focus of the ellipsoid, if the point source is located in the first focus. An ellipsoidal mirror was used to focus harmonics between 24 and 38 nm (33 to 51 eV) with a full width at half maximum (FWHM) measured to be 2.5 um [22]. However, these mirrors are challenging to manufacture as well as to align, since strong aberrations appear if the source is extended or if it moves slightly out of the first focus of the ellipsoid [23]. Configurations with multiple toroidal mirrors have been used [21] to overcome the limitation of ellipsoidal mirrors. The concept is to mutually compensate the coma [24] with successive mirrors. A Z-shape configuration with two toroidal mirrors was used to focus high-order harmonics between 30 and 70 nm (17 to 38 eV) to an 8-um FWHM spot [17; 25]. In the 1950s, [PERSON] [26] demonstrated that some combinations of two confocal conic sections, such as ellipsoids, paraboloids, and hyperboloids, can minimize aberrations. This idea is widely used in astronomy for X-ray telescopes [27], for focusing neutron beams [28; 29], and for inertial confinement fusion imaging experiments [30; 31]. Focusing of XUV radiation requires not only good optics but also precise alignment and therefore accurate measurement techniques. The focus size can be determined by knife-edge [32], point diffraction [33], slit diffraction [34], lateral shearing interferometry [35], or direct imaging with a microscope [17]. This direct technique requires the insertion of a component in the focus, which hinders simultaneous application experiments. Wavefront sensors [36], on the other hand, can be located far from the focus and thus provide on-line diagnostics of beam spatial characteristics. Shack-Hartmann sensors, based on microlens arrays, are routinely used in the visible or infrared regime, but their use in the XUV regime is more challenging [37]. Therefore, the Hartmann technique, based upon diffraction through small apertures, is often used [36].