## 2 Experimental Section  
The experiments were carried out with the fully diode-pumped POLARIS laser system operational at the Helmholtz-Institute and the Institute of Optics and Quantum Electronics in Jena, Germany. A schematic overview of the laser architecture is shown in Figure 1a). During the experiments the laser used a high-contrast XPW front end where the relative intensity of the amplified spontaneous emission (ASE) 200 ps before the peak of the main pulse was as low as \(2\times 10^{-13}\) with respect to the main pulse intensity. A detailed description of the XPW front end, which was used at an intensity of \(9\times 10^{11}\,\mathrm{W}/\mathrm{cm}^{2}\) and was operated reaching an XPW-efficiency of 13%, can be found in [22]. Six amplifiers and a nanosecond stretcher-compressor system are used to amplify and compress the laser pulses where a detailed description can be found in [10].  
Additionally to the results presented in [10] we installed a beam line from the last amplifier A5 to the existing pulse compressor [23]. Here, the pulses were compressed to a full width at half maximum (FWHM) pulse duration of \(t_{\mathrm{pulse}}=143\,\mathrm{fs}\) assuming a Gaussian pulse shape. In Figure 1b) the \(2^{\mathrm{nd}}\)-order autocorrelation measurement is shown which was performed with a home-made \(2^{\mathrm{nd}}\)-order autocorrelator, where the pulse is focused in one dimension into a BBO-crystal using a cylindrical lens. The inset in Figure 1b) displays the spatially resolved autocorrelation signal. During the experiments we achieved a maximum pulse energy of \(E_{\mathrm{pulse}}=20.3\,\mathrm{J}\) (measured before compression) and 14 J in the target chamber with a measured transmission efficiency of \(\eta=69\,\%\) of the grating compressor. The near-field profile is displayed in Figure 1c). The repetition rate \(f_{\mathrm{Rep}}\) was limited by the pump induced thermal load in the active media of the last two amplifiers and therefore set to \(f_{\mathrm{Rep}}=0.025\,Hz\) (\(f_{\mathrm{Rep}}\) = 0.1 Hz for A3, 0.5 Hz for A2.5 and 1 Hz for A1 and A2). Finally, the pulses were focussed in the target chamber with a f/2-focussing parabola (\(f\) = 300 mm). In Figure 1d) an image of the focal spot is shown. The adaptive mirror was set to correct the static wavefront aberrations of the beamline and laser optics. In this measurement the wavefront was flattened with an adaptive optics system and the pulses were strongly  
Figure 1: Setup and measurements of POLARIS. (**a**) Schematic setup. The system uses six amplifier and two stretcher-compressor stages to finally deliver 14 J pulse energy in 143 fs pulse duration on target; (**b**) Second-order autocorrelation measurement of the compressed A5-pulses. The upper inset displays the spatially resolved autocorrelation signal; (**c**) Measured near-field profile of the final amplifier with an output energy of 20.3 J; (**d**) Measured transverse far-field profile of the A5-amplified pulses. The area within which the intensity is larger than \(I_{\mathrm{max}}/2\) is 8.5 \(\mu\mathrm{m}^{2}\) and contains 27 % of the pulse energy (\(q\) = 0.27).  
attenuated in the laser chain. The area within which the intensity is larger than \(I_{\mathrm{max}}/2\) is 8.5 \(\mu\mathrm{m}^{2}\) and contains 27 % of the pulse energy (_q_ = 0.27). \(q\) gives the ratio of pulse energy which is contained within the area where the intensity is larger than \(I_{\mathrm{max}}/2\) (\(A_{\mathrm{FWHM}}\)) with respect to the total pulse energy. These pulses were used in the following experiments for the second harmonic generation.