# Refined Ptychographic Reconstruction of Attosecond Pulses
## 4 Comparison and Discussion
This is probably due to the finite number of points chosen for the time axis in VTGPA. Figure 4f shows that rePIE is again considerably faster. For the full VTGPA approach convergence is reached in almost 100 iterations leading to an error of \(1.65\times 10^{-5}\). rePIE needs only 10 additional iteration of the VTGPA to reach convergence with an error of \(3.11\times 10^{-5}\), thus saving a great amount of computational time. Figure 3: Attosecond pulses with spectrum centred around the 21 st harmonic. (**a**) Temporal amplitude profile of the XUV pulse. (**b,c**) Associated spectra amplitude and phase. (**d**) Reconstructed IR femtosecond pulse. (**e**) Residual error between the simulated and reconstructed IR pulse. (**f**) Reconstruction error of the last 100 iterations for VTGPA and rePIE algorithms as a function of iteration number. In all the figures, orange full dots represent the ePIE results, green squared marks show the VTGPA results, while the rePIE output is represented by the blue open circles. To conclude our analysis, we study what happens when a considerable portion of the XUV spectrum lies below the target ionization potential. The results are reported in Figure 5 and show the different reconstructions for an attosecond pulse centred at the 15 th harmonic. The Ar ionization potential produces a clear vertical cut in the photoelectron spectrum of Figure 5b (black solid curve). In this case the CMA is certainly not justified. Indeed, the error of the ePIE reconstruction increases when compared to the two previous cases (see Figure 2). The spectral phase in Figure 5c shows some oscillations in the region between 15 and 20 eV and it considerably deviates from the theoretical curve at high energies. The extracted chirp is \(-0.03\) fs\({}^{2}\) instead of the theoretical value of \(-0.02\) fs\({}^{2}\). The error on the IR reconstruction (Figure 5e) is also appreciably bigger than the previous cases. Nevertheless, also in this case the majority of the reconstruction error comes from a wrongly reconstructed IR pulse and the XUV pulse is surprisingly well reconstructed also in this extreme case. A full VTGPA approach does not perform well at these energies. The artefact observed before in Figure 4b becomes stronger in this case, deeply affecting the time profile of the reconstructed electron burst. Since in this case also the reconstructed spectral phase is affected (the shaded green area in Figure 5c indicates the deviation from the theoretical input), we notice that here it will not be possible to filter out the artefact after reconstruction. Therefore, despite giving the best IR reconstruction (compare the shaded areas in Figure 5e), VTGPA will not reconstruct accurately the electron burst, especially in the energy region close to \(I_{p}\). This can be a problem for short attosecond pulses, characterized by a broad bandwidth in the VUV-XUV region [13]. Once again, rePIE brings together the positive aspects of the two algorithms. Even if it also shows an artefact below \(I_{p}\) (blue area in Figure 5b), it is considerably smaller that the one obtained with full VTGPA and the spectral phase is no longer strongly affected. Compared to ePIE, rePIE reconstructed spectral phase follows more accurately the simulations also at high energies. The extracted value of the chirp is \(-0.0206\) fs\({}^{2}\), in excellent agreement with the theoretical values. The IR residual error is also strongly reduced (blue area in Figure 5e). Finally, as shown in Figure 5f, not only rePIE is faster than full VTGPA, but also it converges to a solution characterized by a smaller residual error of \(2.22\times 10^{-5}\) instead of \(3.25\times 10^{-5}\) for full VTGPA. We recently applied rePIE to actual experimental attosecond streaking traces centered at low electron kinetic energies. The results are reported in Ref. [13] and confirm what found by the theoretical investigation here presented. Figure 4: Attosecond pulses with spectrum centred around the 19 th harmonic. (**a**) Temporal amplitude profile of the XUV pulse.