## 3 Results
Also, as such an artifact cannot occur in a stable-pulse or single-pulse FROG trace (unlike autocorrelation), all FROG algorithms necessarily ignore it, using it and other information in the trace to retrieve the best possible typical pulse, as we found in our earlier work for SHG FROG [59]. This is also illustrated in the worst-case retrieved traces for both algorithms we studied, shown in Figure 3b,c, where no such vertical line appears for either algorithm. In other words, even in the worst cases, no such artifact occurs in retrieved traces. Do note, however, that the standard GP algorithm is somewhat confused by this effect and retrieves a pulse with a larger area, that is, a larger TBP. This occurs, not only for the worst case but also for the average retrieved pulse for GP. The RANA approach does much better and returns fairly accurate TBPs in all three cases. Figure 3d-g shows every retrieved pulse field using both the GP algorithm (d,f) and the RANA approach (e,g) from a randomly chosen noisy trace of train #3, with 108 fs average temporal FWHM. The standard GP algorithm yields a variety of pulse shapes, based to some extent on the initial guess, with GP-retrieved pulses showing a large range of variations in the temporal and spectral FWHMs. Additionally, the worst-case (and many other unshown) GP-retrieved traces bear no resemblance to the measured trace. In contrast, pulses retrieved using the RANA approach, although somewhat less structured than the actual pulses, do show some of the typical pulse structure and have pulse lengths and spectral widths that closely match those of the measured pulses in the unstable train. Although we should have no expectation of convergence to identical pulses on all runs of any algorithm in the presence of instability, when the trace corresponds to no single pulse, impressively, the maximum and minimum pulse lengths retrieved by RANA are  
quite similar, indicating that, even in the presence of instability, it achieves a reasonable \"typical\" pulse length. Table 2 shows the average \(G^{\prime}\) errors and _TBPs_ for the 100 RANA and GP retrievals from the various FROG traces. For the least unstable case, the two approaches yield similar results, indicating that the standard GP algorithm performs reasonably well for somewhat complex and unstable pulse trains. But, for the two most unstable trains, RANA yields smaller _TBPs_ and standard deviations than GP, indicating both better convergence and also less variations in the retrieved pulses. The larger standard deviations of the _TBP_ values of the standard GP algorithm indicate that, when convergence is not achieved, the retrieved fields exhibit significant variations, resulting in inconsistent and arguably unreliable results. Figure 3: (**a**) The trace generated by the unstable train of pulses with an average temporal FWHM of 108 fs. Note the narrow coherent artifact centered at zero delay. (**b**) The reconstructed trace obtained using the GP algorithm for the “worst-case retrieval”, for which the \(G^{\prime}\) error is the largest. (**c**) The “worst-case” retrieval from the use of the RANA approach. Note the much smaller trace area and hence much smaller TBP for the RANA approach. (**d,e**) Two retrieved temporal intensities (both shown as dashed black and red dashed curves and corresponding to the maximum and minimum retrieved FWHMs), and all the retrieved phases (blue) for the GP algorithm (**d**) and RANA approach (**e**) are shown. Note the large differences between the two intensity curves for the GP algorithm in (**d**) and the much smaller, essentially indistinguishable intensity curves for the RANA approach in (**e**). The phase curves have arbitrary constants added to them in order to separate them, and those grouped together at the top of (**d**) correspond to the clearly stagnated results and are more complex on average than the converged results shown below them. (**f,g**) Analogous quantities for the spectra, except using green instead of red.