# Refined Ptychographic Reconstruction of Attosecond Pulses
## 3 Reconstruction Results
In all the ePIE reconstructions presented here, the original spectrograms of Figure 1 are down-sampled to 4096 points in energy (energy step \(\Delta E=49\) meV) and 199 in delay (delay step \(\Delta\tau=0.25\) fs) before reconstruction. The algorithm runs over 2000 iterations, which is more than enough to guarantee convergences after about 7 min of computational time on a common personal computer. CMA can be avoided in FROG-CRAB reconstruction by implementing a method recently proposed by [PERSON] and coworkers [18], based on a a different algorithm called VTGPA. VTGPA does not use fast Fourier transformation and can handle the momentum dependence in the gate function,  
Figure 2: (**a–c**) Residual error for the reconstructed spectrograms which have as input the traces in Figure 1c,f,i, respectively. The first raw, extended Ptychographic Iterative Engine (ePIE) outputs. Second and third rows, Volkov Transform Generalized Projection Algorithm (VTGPA) and refined extended Ptychographic Iterative Engine (rePIE) results. In all cases, VTGPA and rePIE show considerably low reconstruction error. hence giving more accurate results. The disadvantage of this method is a notable reduction of the computational speed. The second row in Figure 2 shows the residual error obtained with VTGPA after 100 iterations. VTGPA shows a superior convergence and the final residual error is considerably smaller and does not show any systematic up-down asymmetric structure (compare first and second rows in Figure 2). Since the code is considerably slower, we were forced to strongly reduce the energy and time resolution in the VTGPA reconstruction. The simulated spectrograms are re-sampled onto 179 energy points (\(\Delta E=196\) meV) and 100 delay points (\(\Delta\tau=0.2\) fs). In these conditions the same personal computer takes 11 h to finish the 100 iterations. Our new approach attempts to combine the positive aspects of ePIE and VTGPA in order to overcome the limitations introduced by CMA at acceptable total computational times. The idea is to perform a first fast ePIE reconstruction and then \"refine\" the obtained result with only a few iterations of the VTGPA to correct for the error introduced by the CMA. We dubbed this method rePIE (refined extended ptychographic iterative engine). The residual errors obtained with rePIE (2000 ePIE plus 100 VTGPA iterations) for the pulses centred at \(\omega_{XUV}=21\omega_{IR}\), \(19\omega_{IR}\), and \(15\omega_{IR}\) are shown in the third row of Figure 2. Details on the reconstructed pulses and speed convergence will be discussed in depth in the next section. Nevertheless, we can already notice that the residual spectrogram error is considerably better than the one obtained with ePIE and comparable to the VTGPA one. It is worth to notice that this is not a trivial outcome. As ePIE is based on CMA, the algorithm converges towards a solution which is the best approximation of the exact solution, but still satisfies the CMA. This point could represent a local attractor in the ensemble of the general solutions which are not based on the CMA. Therefore, there is no guarantee that the subsequent application of the VTGPA code will be able to steer the convergence away form this local minimum, towards a more general solution.