# Micro-Focusing of Broadband High-Order Harmonic Radiation by a Double Toroidal Mirror
## 3 Wavefront Measurements and Optimization of the Focal Spot Size
### Size of the Focal Spot  
To estimate the focal spot size, the measured wavefront and spatial intensity distribution are back-propagated to the focus by applying a Fourier transform. The wavefront \(W(x,y)\) is written as the sum of the first three terms in the expansion (tilts and defocus, the piston is ignored) plus a residual wavefront \(\tilde{W}(x,y)\). The contributions of defocus and tilts to the wavefront are strong and usually dominate the higher-order aberrations. Their back-propagation requires a very fine sampling of the \(xy\)-plane over the pupil such that the wavefront never changes by more than a fraction of the wavelength from one sample point to the next. We thus need to treat the defocus and tilt contributions separately in order to optimize the numerical calculation. The wavefront tilts represent the propagation direction of the beam and they can be added later to obtain the final result. The electric field \(E(x,y,z)\), at the position of the Hartmann mask (\(z=0\)), is described by  
\[E(x,y,0)=U(x,y,0)\ e^{-ik\frac{p}{2}x^{2}2(x,y)}\ \ \ \ \ \ \text{with}\ \ \ \ \ \ U(x,y,0)=\sqrt{I(x,y,0)}\ e^{-ik\tilde{W}(x,y)}. \tag{4}\]  
Figure 4: Relative Zernike coefficients for the main aberrations as a function of the angle of the (**a**) rotation stage; (**b**) horizontal goniometer; and (**c**) vertical goniometer. The experimental data are represented as circled colored dots, plotted as a function of the number of steps (top axis). The ray-tracing simulation data is represented as solid lines, plotted as a function of the angle in degrees (bottom axis). Red corresponds to \(45^{\circ}\) astigmatism, green to \(0^{\circ}\) astigmatism, and white to the averaged coma. Figure 5: Intensity and wavefront of the XUV beam before (**a,c**) and after (**b,d**) focusing. \(I(x,y,z=0)\) is the spatial intensity distribution measured at the position of the mask and \(k=2\pi/\lambda\) the wavevector. Since the defocus component \(c_{2}^{0}Z_{2}^{0}\) describes a quadratic phase, \(E(x,y,0)\) represents the field after an ideal lens with focal length \(f=r^{2}/(4c_{2}^{0})\), with \(r\) being the pupil radius, and \(U(x,y,0)\) being the field before the lens. The field in the focus is then given by the Fourier transform of \(U(x,y,0)\)  
\[I(x^{\prime},y^{\prime},z_{foc}=f)=|E(x^{\prime},y^{\prime},z_{foc})|^{2}\ \propto\Big{|}\iint U(x,y,0)\ e^{i2\pi(v_{x}x+vyy)}\ dx\ dy \Big{|}^{2}. \tag{5}\]  
Taking into account the tilts' contribution to the focal spot position, the rescaled coordinates in the focal plane are \(x^{\prime}=v_{x}\lambda f+c_{1}^{-1}f/r\) and \(y^{\prime}=v_{y}\lambda f+c_{1}^{1}f/r\). The defocus component of the wavefront determines the focus position along the beam path, whereas the tilts determine its position in the \(x^{\prime}y^{\prime}\)-plane. With this method, we back-propagated the least aberrated field (optimized alignment of the focusing setup), presented in Figure 5b,d, and obtained the focal spot plotted in Figure 6a. The focus size is estimated to be 3.6\(\times\)4.0 \(\upmu\)m\({}^{2}\) (FWHM). We also performed ray-tracing simulations for aligned focusing optics, assuming a perfect Gaussian beam, with the experimentally measured divergence. The focal spot is shown in Figure 6b with its profiles. The FWHM beam size is 3.0 \(\times\) 3.2 \(\upmu\)m\({}^{2}\). The focal spot size obtained by back-propagation of the wavefront is in good agreement with the one obtained by the ray-tracing simulations. The 20% difference in size can be explained by the difficulty of estimating precisely the divergence of the beam used for the ray-tracing simulations, and by distortions of the experimental intensity profile. The intensity of the focused attosecond pulse train is estimated to 7 \(\times\) 10\({}^{12}\) W/cm\({}^{2}\), using the back-propagated spot size of 3.6 \(\times\) 4.0 \(\upmu\)m\({}^{2}\) and typical energies of 5 nJ, measured in the focal region with a calibrated Andor XUV camera and an XUV photodiode.