## 4 Conclusions  
In this study, we explored the convergence behavior of the standard GP algorithm and the recently introduced RANA approach for PG/TG FROG traces averaged over trains of pulses with unstable shapes and hence with coherent artifacts. As expected, we found that instability yields PG/TG FROG traces that do not correspond to traces from single pulses, which implies that these techniques are very useful for identifying instability, in addition to measuring the pulse intensity and phase vs. time. However, we found that  
Figure 5: The first and second columns show the typical spectra for trains 1 (top) to 3 (bottom). The third column plots the directly retrieved spectra from the marginal of the PG/TG FROG trace (obtained in and required for the RANA approach) alongside the average spectra—i.e., the spectrometer measurement—for the three trains with average (**a**) \(\tau_{FWHM}=26\) fs, (**b**) \(\tau_{FWHM}=54\) fs, and (**c**) \(\tau_{FWHM}=108\) fs. The retrieved spectra from PG/TG FROG trace marginals match the average spectra of the unstable pulse train. the standard GP algorithm often fails to clearly distinguish between discrepancies caused by instability artifacts and those due to algorithm stagnation, making it difficult for it to identify instability accurately. This necessitates multiple retrieval attempts to find the lowest \(G\) error, with no definitive way to determine if the discrepancies are due to instability or algorithm stagnation. In contrast, our results show that, for traces of trains with unstable pulse shapes, the RANA-retrieved trace consistently achieves the minimum discrepancy with the measured trace, resulting in the lowest \(G\)' error, that is, _converges_ in all cases that we considered. Thus, the RANA approach, established to have 100% reliability with stable pulses, also proves highly effective in evaluating pulse-train stability. This reliable convergence indicates that any discrepancies between the measured and retrieved traces are due to pulse-shape fluctuations, and not algorithm stagnation. Moreover, the retrieved field accurately reflects the average pulse length, spectral width, and TBP of the pulses in the train, providing the best available depiction of a typical pulse, although with somewhat less structure. It also provides some of the structure present in a typical pulse in the train. In summary, the RANA approach offers the most reliable FROG pulse-retrieval approach for both stable and unstable pulse trains, making it a highly reliable gauge of the average pulse length, TBP, and pulse-train stability or instability and, of course, the precise pulse intensity and phase vs. time and frequency for stable pulse trains. As no other pulse-measurement technique has, to our knowledge, been shown to possess all of these characteristics, RANA (in particular, in conjunction with PG/TG FROG) yields the best general performance of all existing pulse-measurement techniques. As an aside, we note that there are many sources of pulse-shape instability, and each is likely to have its own shape probability distribution. But, at this point in time, the goal is simply to reliably identify the presence of instability so that future researchers may be able to sort out the actual cause(s) and pulse-shape probability distribution in a given case. The RANA approach is only an algorithmic approach for retrieving the best possible pulse from a given trace, even in the presence of instability, and to indicate the presence of instability in the pulse shapes, as we show here. It is not an attempt to provide a model for the instability itself, rather it is merely an indicator of its presence. It is, however, the first algorithm to reliably establish the presence or absence of instability and so is the first step toward identifying the instability, no matter what form the instability may take. Finally, it is worth considering possible future efforts on this subject. First, performing this analysis on larger sets of pulse trains would be helpful, as is always the case for iterative algorithms.