## 3 Results  
As mentioned earlier, if an algorithm reliably converges, even in the presence of instability, we would expect it to achieve the minimum \(G\) or \(G^{\prime}\) error for all reasonable initial guesses. If not, then we would expect to see variations in these values depending on the initial guess. Figure 2 depicts all the \(G^{\prime}\) errors obtained from 100 runs using both the standard GP algorithm and RANA approach on noisy traces. Figure 2: \(G^{\prime}\) errors for retrieval of three noisy PG/TG FROG traces, which are contaminated with 3% additive and 5% multiplicative noise for three unstable pulse trains with an average temporal FWHM of (**a**) 26 fs, (**b**) 54 fs, and (**c**) 108 fs. The \(G^{\prime}\) errors are obtained from pulse recovery over 100 runs using the RANA approach (blue triangles) and the standard GP algorithm (red circles) on noise-filtered traces. The RANA approach consistently achieves the minimum \(G^{\prime}\) error, whereas the GP algorithm alone does not, necessitating additional algorithm runs in practice. In all six cases, the impact of random trace noise is small. On the other hand, in all cases, the effects of pulse-shape instability dominate, yielding the resulting large values of \(G^{\prime}\). However, stagnation contributes increasingly to the resulting values of \(G^{\prime}\) for the standard GP algorithm as instability increases. On the other hand, there is a negligible contribution to \(G^{\prime}\) from stagnation in all three RANA cases. This is the main result of this work. These results clearly illustrate that the RANA approach converges to the smallest value of \(G\)' for all runs and for all instability values. In other words, it converged in all cases. In contrast, the standard GP algorithm does fairly well for the least unstable pulse train, but, for increasing instability, it shows significant variability in the resulting \(G\)' errors, often requiring multiple attempts to achieve the minimum achievable \(G\)' error. For pulse trains with an average temporal FWHM of 26 fs, the standard GP algorithm successfully converged to acceptable pulses with traces closely matching the measured ones in 92% of the trials. This performance decreased to 86% and 44% for pulse trains with average FWHMs of 54 fs and 108 fs, respectively, that is, more instability. While this performance is not perfect and quite undesirable for the less stable trains, it is nevertheless considerably better than that of GP for SHG FROG that we found in our previous study [59]. It is important to note that, in our studies, the impact of trace noise was minor compared to that of pulse-shape instability. As is well known, random noise in FROG traces can be minimized using simple preprocessing, including Fourier filtering and background subtraction. In stable pulse trains, the converged \(G\)' error should, and does, align with the average random noise remaining in the measured trace after preprocessing; \(G\)' errors larger than this necessarily indicate pulse-train instability, at least for the RANA approach. For the standard GP algorithm, these discrepancies are a sum of stagnation and instability effects, and the best way to distinguish them using the standard GP algorithm at this time appears to be running the algorithm many times and using the result with the smallest \(G\) or \(G\)' error, as we have done here and as is often done in practice. Of course, this is far less desirable than running a more reliable algorithm, such as RANA, only once. Figure 3a shows the resulting multi-shot trace of train #3. Note the obvious thin vertical line in it, i.e., the coherent artifact, which is intimately related to that of autocorrelation (recall that FROG is a spectrally resolved _autocorrelation_). Of course, while FROG traces exhibit this artifact, this does not pose a problem in FROG, first because its presence alerts the algorithm and the user (as for autocorrelation) to the instability, and our results confirm this.