## 3 Results and Discussion
In the simulations shown in Figure 6, the first iteration has been performed by scanning the attenuation parameter from \(\alpha=10^{9}\) to \(\alpha=10^{10}\) in steps of \(\Delta\alpha=5\times\)108, and the angular position has been scanned from \(\theta=-720^{\circ}\) to \(\theta=720^{\circ}\) in steps of \(\Delta\theta=2^{\circ}\). The optimal values of the fitness parameters from the first iteration in this case are \(\alpha=10^{9}\) and \(\theta=466^{\circ}\), and \(\chi^{2}=0.6789\). Once the optimal values have been determined from the first iteration, further iterations are performed by scanning only near the first iteration optimal region, which substantially improves the performance of the iterative method. In the present simulation the value of the fitness function converges at approximately the 5\({}^{th}\) iteration. The optimal values of the fitness parameters from the 5\({}^{th}\) iteration are \(\alpha=10^{10}\) and \(\theta=496^{\circ}\), and the value of the fitness function is \(\chi^{2}=0.4927\). In Figure 6a the spectral amplitudes that result from the first and 5\({}^{th}\) iterations are shown. As it can be observed, the effect of the XUV pulses on the spectral amplitude is readily small, which is expected since the perturbation of the IR pulse is weak (\(\alpha=10^{9}-10^{10}\)). The intensity of the optimal XUV pulse after the 5\({}^{th}\) iteration is shown in Figure 6b, and the inset shows the evolution of the value of the normalized \(\chi^{2}\) function for the increasing iterations. Clearly, as the iteration procedure evolves, the value of the fitness function improves such that at the 5\({}^{th}\) iteration an optimal compressed pulse is achieved. As shown in the inset of Figure 6b, the value of the normalized \(\chi^{2}\) function becomes basically constant after the 5\({}^{th}\) iteration, and therefore no substantial improvements are achieved by considering further iterations. This simulation therefore shows the robustness of the control method, which is able to achieve optimal fitness targets both by using relatively weak perturbations of the IR driving pulse with a single iteration and by considering very weak perturbations in a multiple iteration procedure. **Figure 5. (color online) Time-frequency analysis of the original (**a**) and optimized (**b**) spectra, for the case shown in Figure 4.  
From the simulations that have been carried out, it can in general be concluded that the optimized configuration is not critically sensitive to the value of the intensity of the XUV\({}_{1}\) pulse, and also that  
Figure 6: (Color online) Optimization of the HHG spectral region 21-51 eV obtained by considering several iterations of the feedback loop for \(\alpha\) scanned from \(\alpha=10^{9}\) to \(\alpha=10^{10}\) (**a,b**) and from \(\alpha=10^{11}\) to \(\alpha=10^{12}\) (**c,d**). (**a,b**): The scan in the first iteration in \(\theta\) is from \(\theta=-720^{\circ}\) to \(\theta=720^{\circ}\). In the subsequent iterations, \(\theta\) is scanned from \(\theta=200^{\circ}\) to \(\theta=700^{\circ}\). In (**a**) the spectral intensity of the original and optimized spectra are shown for the first and 5\({}^{th}\) iterations, as indicated. The optimal values of the fitness parameters for the 5\({}^{th}\) iteration are in this case \(\alpha=10^{10}\) and \(\theta=496^{\circ}\). In (**b**) the temporal evolution of the intensity of the XUV pulse resulting from the second interaction region after the 5\({}^{th}\) iteration (loop) is shown. The inset in (**b**) shows the value of the normalized \(\chi^{2}\) parameter as a function of the iteration number. From approximately the 5\({}^{th}\) iteration the value of \(\chi^{2}\) and the intensity profile of the resulting XUV pulse do not change substantially. (**c**) and (**d**): The scan in the all the iterations in \(\theta\) is from \(\theta=-720^{\circ}\) to \(\theta=720^{\circ}\). In (**c**) the spectral intensity of the original and optimized spectra are shown for the first and 8\({}^{th}\) iterations, as indicated.