# Micro-Focusing of Broadband High-Order Harmonic Radiation by a Double Toroidal Mirror
## 3 Wavefront Measurements and Optimization of the Focal Spot Size
### Alignment of the Focusing Optics  
In order to evaluate the sensitivity of the wavefront aberrations to the alignment of the focusing optics, we scanned the rotation stage and goniometers described in Figure 1c while recording single-shot wavefronts. As there is no absolute characterization of the stage rotation angles, we express them in number of steps. We extracted the RMS of the wavefronts by zonal reconstruction, as well as the corresponding Strehl ratio [43], which is defined as the ratio of the peak intensities of the focal distribution in the aberrated and unaberrated cases. A good approximation of the Strehl ratio \(S\), used in the present work, is given by \(S=\exp(-\sigma^{2}/\lambda^{2})\), where \(\sigma\) is the wavefront RMS [44]. The results are presented in Figure 3, where Figures 3a, 3b and 3c correspond to scans of the rotation stage, and the horizontal and vertical goniometers, respectively, with the other axes around the optimum. Due to shot-to-shot fluctuations, for each point, the average value of several (\(\sim\) 5) single-shot measurements with corresponding RMS error bars is plotted. A sharp V-shaped trend is visible for every graph. It means that for each actuator, it is possible to find a position minimizing the aberrations. The RMS minima are 0.028 \(\pm\) 0.015 \(\lambda\) (\(\simeq\lambda/36\)) in Figure 3a, 0.088 \(\pm\) 0.012 \(\lambda\) in Figure 3b and 0.111 \(\pm\) 0.034 \(\lambda\) in Figure 3c. The values for Figure 3b and Figure 3c are slightly higher because of residual aberrations or misalignments. In order to better understand which aberrations are accounting for most of the wavefront distortions, we used the modal technique to decompose the wavefronts in a weighted sum of Zernike polynomials, as explained above. Our analysis shows that astigmatism is the dominant aberration, accounting for most of the deviation from a perfect wavefront. Figure 4 presents the Zernike coefficients for astigmatism (\(0^{\circ}\) and \(45^{\circ}\)) and coma (average value of \(0^{\circ}\) and \(90^{\circ}\)) as a function of the number of steps. For all three cases, the coma is found to be negligible, which means that if the first mirror introduces coma, it is completely compensated by the second one, thus validating the double toroidal mirror Wolter configuration. The horizontal goniometer introduces mostly \(0^{\circ}\) astigmatism while the rotation stage and the vertical goniometer introduce \(45^{\circ}\) astigmatism. These different behaviors allow us to determine which axis is misaligned, and is thus very useful for fast optimization. For example, during the vertical goniometer scan shown in Figure 4c, there was residual \(0^{\circ}\) astigmatism (green dots), meaning that the horizontal goniometer was not perfectly aligned in this case. Good alignment could be reached after a few iterations. In order to validate these results, we carried out simulations of the XUV beam propagation through the setup with the ray-tracing software FRED Optical Engineering (Version 14, Photon Engineering LLC, Tucson, AZ, USA), based upon Gaussian beamlet decomposition [45]. The wavelength was chosen to be 42 nm, corresponding to harmonic 19. The results are plotted in Figure 4 as solid lines, using the same colors as the experimental data for the different coefficients. Here, the relative angle is indicated in degrees. The ray-tracing simulations show the same behavior as the experimental data, with the sharp V-shape for the dependence of the main Zernike coefficient for a given axis. The simulations also confirm that each goniometer is affecting mainly a specific aberration. For the Zernike coefficients, simulations and respectively experimental data have been normalized to 1 at \(-0.8\) degrees (respectively at \(-160\) steps) of the rotation stage in Figure 4a. We also used the zonal wavefront reconstruction method for both experiment and simulation to extract the wavefront RMS. This allowed us to estimate the step size as a function of rotation angle for each axis.