## 1 Introduction
In the presence of instability, however, the convergence concept is more subtle. In this case, stagnation, as we recently defined it [59], occurs when the RMS difference between the measured and retrieved traces, usually referred to as \(G\), is higher than the lowest achievable \(G\) value for the given measured trace. But what is this latter value? Determining it requires running the relevant FROG algorithm numerous times (assuming that the algorithm converges at least once, which nearly always occurs in FROG in practice). The pulse with the lowest value of \(G\) is declared the converged case and hence the best estimate of the typical pulse. Then comparing its trace with the measured trace, it is evident that the more systematic error between the two traces the more instability is present. This is reasonable, but, unfortunately, running an algorithm many times is neither convenient nor always completely convincing. What is needed is an algorithm that always converges to the lowest possible \(G\) value, even in the presence of instability in the first place. Fortunately, in previous work, we showed that this can be done for the second-harmonic-generation (SHG) version of FROG. Specifically, we demonstrated that our recently introduced Retrieved-Amplitude N-grid Algorithmic (RANA) approach [60; 61; 62] not only achieves extremely reliable (100%) pulse-retrieval in SHG FROG for trains of stable pulse shapes, even in the presence of noise, but also does so for _unstable_ pulse trains and so reliably distinguishes between trains of stable and unstable pulse shapes. It also provides a reasonable estimate of the average pulse length, spectral width, and time-bandwidth product (\(TBP_{rms}\)) in the train. This is the case because the RANA approach is extremely reliable and, in studies involving tens of thousands of simulated pulses, even in the presence of significant noise, has _never_ stagnated. Specifically, we showed that it also did not stagnate even in the presence of pulse-shape instability [53]. It also provided many of the characteristics of a \"typical\" pulse in the unstable train [63] (although it tended to underestimate the amount of typical pulse structure). A quick review of the RANA approach [60; 61; 62] may be helpful at this point. It is not a pulse-retrieval algorithm in the usual sense. Rather, it is an approach that uses an existing algorithm (here the standard generalized projections algorithm) but takes advantage of the realization that one can actually retrieve an excellent approximate pulse spectrum directly from the measured trace. It then uses a \"multi-grid\" approach, which means using a small subset of the trace points in the first iterations of the algorithm, then more for later iterations, and the full dataset only for the final few iterations. It also continually discards poorer results along the way. It is illustrated in Figure 1.  
Anyway, an analogous never-stagnating algorithm--even in the presence of instability--is also highly desirable for the versions of FROG that are generally used to measure amplified pulses. These include the polarization-gating (PG) and transient-grating (TG) FROG variants (which are highly desirable because they eliminate the direction-of-time ambiguity of SHG FROG). Additionally, PG FROG is automatically phase-matched, and TG FROG is broadly phase-matched, so both are not usually limited by the pulse's bandwidth. PG FROG and the more common version of TG FROG are mathematically equivalent. The mathematical relation for their measured trace is:  
\[I_{FROG}^{PG/TG}(\omega,\tau)=\left|\int_{-\infty}^{\infty}E(t)\left|E(t-\tau )\right|^{2}\exp(-i\omega t)\,dt\right|^{2} \tag{1}\]  
Figure 1: Graphical illustration of the RANA approach’s multi-grid operation [61; 62]. The direct spectrum retrieval is not shown, but it is straightforward and described in detail in References [61; 62]. where \(E(t)\) is the pulse's complex electric field as a function of time, \(t\). Also, \(\omega\) is the angular frequency, and \(\tau\) is the delay between the pulses.