# Refined Ptychographic Reconstruction of Attosecond Pulses
## 3 Reconstruction Results  
As discussed above, the most commonly used reconstruction approaches are based on the FROG-CRAB method in combination with several reconstruction algorithms [20; 29; 30]. The main limitation of this approach lies in the fact that the spectrogram in Equation (1) needs to be factorized as an internal product between a pulse \(P(t)\) and gate \(G(t)\) functions, which depend only on time. To do so, we need to remove the momentum dependence from the quantum phase \(\phi(p,t)\). This is is obtained by substituting \(p\) with the value of the electron central momentum \(p_{c}\) in Equation (2). This procedure is called central momentum approximation (CMA) and allows Equation (1) to be rewritten in the form:  
\[S(p,\tau)\simeq\left|\int dtE_{el}(t,\tau)e^{-i\phi(p_{c},t)}e^{i(\frac{p^{2}}{ \tau}+l_{p})t}\right|^{2}=\left|\int dtP(t-\tau)G(t)e^{i(\frac{p^{2}}{\tau}+l_{ p})t}\right|^{2}, \tag{6}\]  
Figure 1: Theoretical simulations. (**a**) Dashed lines represent the original attosecond spectral amplitude (black) and phase (blue) for a pulse with central energy corresponding to the 21 st harmonic of the infrared (IR) pulse (\(\omega_{XUV}=32.96\) eV). Solid lines display amplitude (violet) and phase (green) of the corresponding calculated electron burst resulting from Ar ionization [24]. The vertical black dash-dotted line marks the position of the Ar ionization potential. (**b**) Temporal behaviour of the extreme-ultraviolet (XUV) attosecond pulse (black dashed) and the attosecond electron burst (solid violet). (**c**) Associated attosecond streaking trace. In false color it is shown the amplitude of the spectrogram \(\sqrt{S(p,\tau)}\). (**d**–**i**) Same quantities for a single attosecond pulse with central energy corresponding to the 19 th harmonic at 29.82 eV and the 15 th harmonic at 23.54 eV, respectively. The other calculations parameters are reported in the text. As a result of the atomic cross-section spectral filtering, the final photoelectron spectral amplitude bandwidths are: 8.95, 9.75, and 9.44 eV for the results in the first, second, and last row, respectively. where the pulse function is \(P(t)=E_{el}(t)\) and the gate function is \(G(t)=e^{-i\phi(p_{c},t)}\), a pure phase gate. If one applies the CMA, the spectrogram in Equation (6) can be interpreted as a blind FROG trace and both pulse and gate can be retrieved with a FROG-like iterative algorithm. We underline that in a real attosecond streaking experiment FROG-like approaches are not sensitive to the actual pulse intensity as identical pulses with different peak intensities will produce the same spectrogram besides a constant multiplication factor. The attosecond pulse peak intensity can be estimated by measuring the actual XUV beam profile, the pulse energy and duration. Therefore, a precise time characterization of the attosecond pulse represents the first step towards a correct estimation of its peak intensity. Among all the proposed algorithms, the extended ptychographic iterative engine (ePIE) proved to be superior in terms of robustness, flexibility, and convergence speed [20; 31]. Figure 2 shows the residual reconstruction error obtained by subtracting the simulated streaking traces displayed in Figure 1c,f,i from their reconstructions for \(\omega_{XUV}=21\omega_{IR}\) (Figure 2a), \(\omega_{XUV}=19\omega_{IR}\) (Figure 2b), and \(\omega_{XUV}=15\omega_{IR}\) (Figure 2c). The first row of Figure 2 shows the residual reconstruction error obtained when ePIE is used for the reconstruction. In this case the residual error map shows a top-bottom asymmetry (negative values at high energies and positive values at low energies) which originates from the CMA at the heart of the ePIE algorithm [16]. Since the CMA is less accurate for low electron final momenta, the residual error increases upon decreasing \(\omega_{XUV}\), as clearly visible comparing Figure 2a,c.