## 1 Introduction
Of course, simply averaging the spectral phase over many pulses with random complex spectral phases will, like a spectrometer-measured spectrum, also yield a much simpler and smoother curve, indeed, often a _flat_ spectral phase. As a result, measuring the average spectral phase of an unstable pulse train invariably erroneously yields a shorter pulse than is actually present and often yields the shortest possible pulse for a given measured spectrum (which, by the way, is also usually averaged over many pulses in these methods and so is also anomalously smooth). Indeed, measuring the average spectral phase will _always_ yield a pulse that is shorter than any of the individual pulses in the train, and often by a large factor. In other words, the average spectral phase is _the frequency-domain description of the coherent artifact_[51]. As a result, it is crucial that a pulse-measurement technique _not_ provide an average spectral phase, which is, in fact, an essentially useless quantity. If it does, it will invariably yield a shorter pulse than is, in fact, present--unless the pulse train is perfectly stable (and also spatially uniform). Indeed, such a method will be unable to differentiate a stable train of short, simple pulses from an unstable train of long, complicated pulses--the best- and worst-case scenarios, respectively, for most applications. Although critically important,this issue is often overlooked and/or poorly understood, and several popular techniques currently in use (and in use for decades) suffer from precisely this problem [52; 53; 54; 55; 56]. As with the spectrum, but much more importantly here, a pulse-measurement technique should ideally provide a _typical_ spectral phase, or at least a phase as close as possible to it, which, coupled with a typical spectrum, would then more accurately reflect the average pulse length in time. If this is not possible (and it usually is not), the measurement should at least provide a typical pulse length. Although single-shot measurement is the obvious solution to this problem, it is not possible for many laser systems, especially high-repetition-rate systems, for which even the shortest camera exposure times still capture multiple pulses in the train. And spatially averaging these quantities over a spatially complex beam would likely have precisely the same smoothing effect (although this effect has not yet been studied). Fortunately, some progress has been made: we earlier demonstrated that discrepancies between measured and retrieved Frequency Resolved Optical Gating (FROG) traces turn out to be a good indicator of instability [57; 58]. They are a beneficial result of FROG's overdetermination of the pulse: FROG's \(N\times N\) data array is used to measure only \(2N\) pulse parameters. So, a trace that averages over many different pulses cannot correspond to a single pulse. This has turned out to be a very helpful feature, allowing FROG to indicate instability by the presence of systematic discrepancies between measured and retrieved FROG traces. Unfortunately, possible pulse-retrieval algorithm stagnation can also yield similar discrepancies. Even in the absence of instability, iterative algorithms and, in particular, FROG's standard Generalized Projections (GP) algorithm can stagnate for complex pulses, yielding a retrieved pulse that bears little resemblance to a typical pulse and also depends on the initial guess. So, distinguishing between pulse-shape instability and algorithm stagnation--two very different issues--as the cause of such trace discrepancies is critical. This has required us to redefine what we mean by algorithm \"convergence\" when dealing with pulse-shape instability [59]. In the absence of instability, convergence is easy to identify; the only differences between the measured and retrieved traces are due to random noise in the measured trace. Stagnation can be easily visually identified by the presence of systematic errors in the difference between the two traces (assuming that the measurement was made correctly).