# Micro-Focusing of Broadband High-Order Harmonic Radiation by a Double Toroidal Mirror
## 3 Wavefront Measurements and Optimization of the Focal Spot Size  
### Hartmann Wavefront Sensor  
The wavefront sensor, presented schematically in Figure 2a, is composed of a Hartmann mask and an XUV charge-coupled device (CCD) camera [36]. Flat, non-tilted wavefronts result in individual diffraction spots being equidistantly distributed on the camera chip (see black dots in Figure 2b), whereas in the case of distorted wavefronts, the positions deviate (red dots). These deviations are compared to a reference measurement with a known wavefront profile, here a spherical wavefront created from the diffraction of an HHG beam onto a pinhole. The local wavefront slopes in the mask plane, \((\partial W/\partial x,\partial W/\partial y)\) with \(W\) denoting the wavefront, are proportional to these deviations (\(\vec{h}\) in the figure), and are calculated for each hole. Subsequently, a wavefront reconstruction is performed to retrieve the wavefront over the whole aperture. To measure XUV wavefronts after the focusing setup, we used a wavefront sensor (collaboration between Imagine Optics and the Laboratoire d'Optique Applique, LOA) placed 50 cm after the focus. This distance was chosen to match the beam size to the sensor aperture. The Hartmann mask is a 100-\(\upmu\)m-thick nickel plate, which contains an array of 34 \(\times\) 34 holes. The holes are squares with a size of 110 \(\upmu\)m and a pitch of 387 \(\upmu\)m, and are tilted to prevent the diffraction patterns from consecutive holes to overlap. During previous calibration measurements performed at the LOA, the absolute accuracy of the sensor was estimated to be 0.6 nm root mean square (RMS), which corresponds to \(\lambda/70\) RMS for a wavelength of 42 nm. The sensor was mounted on a manual hexapod to ensure optimal alignment. The analysis was performed for the 19 th harmonic (42 nm/29.5 eV ), chosen as the weighted center of our spectrum. The validity of single-wavelength wavefront retrieval for broadband radiation in the visible regime was discussed with a Shack-Hartmann sensor [40] and was proven to be satisfying. Figure 2: (**a**) Positions of the sensor in the beamline; and (**b**) principle of the Hartmann wavefront sensor. WFS: Wavefront sensor. There are two main types of wavefront reconstruction from the measured slopes [41]. The first one is called zonal reconstruction, which corresponds to the direct numerical integration of the local slopes. The other type is modal reconstruction, a decomposition of the wavefront in the basis of orthogonal polynomials, which correspond to a set of different aberrations. It thus allows the aberrations to be extracted independently, with the drawback of fitting them over a circular pupil, which is not always matching real beam profiles and leads to slightly underestimated aberrations [42]. The basis of orthogonal polynomials is provided by Zernike polynomials \(Z_{n}^{m}(x,y)\), which are often the preferred choice to extract and study aberrations independently. The wavefront is then expanded as:  
\[W(x,y)=\sum_{m,n}c_{n}^{m}Z_{n}^{m}(x,y), \tag{3}\]  
where \(c_{n}^{m}\) are the weights of the individual polynomials. The first coefficients in Equation (3) are known as piston (\(c_{0}^{0}\)), x-tilt (\(c_{1}^{-1}\)), y-tilt (\(c_{1}^{1}\)), defocus (\(c_{2}^{0}\)), astigmatism at \(0^{\circ}\) (\(c_{2}^{2}\)), astigmatism at \(45^{\circ}\) (\(c_{2}^{-2}\)), and coma at \(0^{\circ}\) (\(c_{3}^{1}\)) and \(90^{\circ}\) (\(c_{3}^{-1}\)). The coefficients can be determined by fitting the polynomials' derivatives to the measured local slopes. For the analysis of the wavefronts presented in this work, when using the zonal method, the pupil was chosen to be the entire illuminated area, about 10 \(\times\) 10 mm\({}^{2}\). When using the modal method, a circular pupil with a radius of 2.3 mm was centered around the peak of the beam profile. The analysis was performed using the first 32 Zernike polynomials with a standard peak-to-valley (PV) normalization. All wavefronts presented here are single-shot measurements.