# Refined Ptychographic Reconstruction of Attosecond Pulses
## 4 Comparison and Discussion  
In this section we compare the results obtained with ePIE, VTGPA, and rePIE for the three attosecond pulses previously presented. We will start from the attosecond pulse centred around the 21 st harmonic of the IR field for which the CMA is supposed to hold and later move towards lower photon energies. Figure 3 shows in details the reconstructed XUV and IR pulses for ePIE (orange dots), VTGPA (green squares), and rePIE (blue open circles), in the case of attosecond pulses with spectrum centred around the 21 st harmonic. The vertical dash-dotted line in Figure 3b,c shows the position of the Ar ionization potential (15.76 eV). All methods reconstruct the correct XUV pulse in time (Figure 3a), spectral amplitude (Figure 3b), and phase (Figure 3c). Also the IR pulse is reconstructed with high accuracy (Figure 3d). The error on the IR retrieval, defined as reconstructed IR electric field minus the theoretical input field, is shown in Figure 3e. VTGPA gives the best reconstruction while ePIE is characterized by the largest error. Even if this could be a limiting factor for ePIE in certain kind of experiments, like attosecond IR transient polarization spectroscopy [32], in general it represents another strength of ePIE. Indeed, the error introduced by the CMA is projected on the IR pulse and the XUV reconstruction is unaffected. Being considerably faster than VTGPA, the use of ePIE is particularly useful for attosecond metrology experiments where the CMA is a good approximation. rePIE gives slightly better IR reconstruction than ePIE, but does not reach the degree of precision of VTGPA. Nevertheless, rePIE converges much faster than VTGPA. In Figure 3f we show the behaviour of the reconstruction error for the last 100 iteration of the VTGPA code in both cases: Pure VTGPA (green) and rePIE (blue). While pure VTGPA takes around 100 iterations (roughly 11 h of computing time) to converge to an error of \(1.45\times 10^{-5}\), after only 10 iterations (a bit more than 1 h) rePIE has already reached the convergence with an overall comparable error of \(3.2\times 10^{-5}\). Figure 4 shows the same quantities in the case of XUV attosecond pulses centred around the 19 th harmonic. In this spectral region the CMA starts to be less accurate and for ePIE we observe very small deviations of the reconstructed spectral phase of the XUV pulse, compared to the simulations (compare green and black curve in Figure 4c). Nevertheless, the phase is properly reconstructed and the deviations are observed only at the tails of the XUV spectrum where the spectral power is small. The error on the IR pulse reconstruction is comparable to what previously found for the pulse centred around the 21 st harmonic (compare orange coloured areas in Figures 3e and 4e). Therefore, ePIE can still reconstruct properly the XUV pulse, even if the best IR reconstruction is achieved with VTGPA also in this case. Nevertheless, we observe that full VTGPA reconstruction starts to face a problem due to the filtering action of the atomic cross-section. The reconstructed spectral amplitude (Figure 4b) displays a second shoulder located right below the Ar ionization potential, \(I_{p}\) (shown by the green shaded area in Figure 4b). This artefact probably originates from the abrupt cut of the photon spectrum at \(I_{p}\). We remark that, since the spectral phase is properly reconstructed (Figure 4c), one can remove the artefact after reconstruction, by filtering out the spectral components below \(I_{p}\) and thus retrieve the perfect time envelope of the electron burst. As in the previous case, rePIE gives an outcome that lies in between the results of the two previous algorithms. The attosecond pulse is perfectly reconstructed and there is no artefact below \(I_{p}\) in the spectral domain. The IR pulse residual error (Figure 4e) is smaller than the one of the previous case (Figure 3e). The stronger deviation is observed around the tails of the pulse where also the VTGPA result deviates considerably.