## 1 Introduction
From the birth of the field of ultrafast optics, shot-to-shot variations in the pulse shape, that is, the pulse intensity and phase vs. time, of such laser pulses have presented a particularly difficult challenge for ultrashort pulse laser measurement [49; 50]. When faced with a train of pulses with unstable pulse shapes, the temporal intensity autocorrelation (the first method for measuring ultrashort pulses) produces a broad background with a narrow spike atop it. The width of this spike, which has come to be known as the _coherence spike_ or _coherent artifact_, corresponds only to the potentially much shorter coherent temporal component of the unstable pulses in the train. The coherent artifact is therefore shorter than the typical pulse in the unstable train. While it is an obvious spikelike feature when the instability is significant, it can actually be quite misleading when the instability is not so significant; in this case, it may blend into the rest of the trace, resulting in a mistakenly short pulse length by a factor of two or more, as well as the mistaken conclusion of a stable pulse shape. While it is now possible to measure the complete intensity and phase vs. time for a stable pulse train [51], many such measurement techniques have not been thoroughly studied for the case of pulse-shape instability and, unfortunately, like autocorrelation, can be badly confused by it. Currently, no device dedicated to the independent verification of pulse-shape instability exists, and, as a result, the responsibility of identifying the existence, severity, and kind of instability falls to the pulse-measurement methods themselves, which are not, in general, designed for this challenge. As a result, insufficient progress has been made in addressing this problem, and recent work [52; 53; 54; 55; 56] has revealed, among other results, the startling fact that, in the presence of pulse-shape instability, some widely used (mostly interferometric) methods measure _only_ the coherent artifact. There are numerous complications in attempting to measure such an unstable pulse train by any technique, as all measurement techniques necessarily provide a single resulting pulse from a given measured trace, and no one pulse can properly represent the many possible different pulses over which a measurement is made. As a result, the task is fundamentally impossible, and one must settle for quantities that are in some sense averaged. Unfortunately, this can yield results that are of little to no value. Over the centuries, traditional spectrometers have provided a simple \"average\" spectrum. Of course, such measurements average out any spectral structure and so necessarily deliver a smoother spectrum than is in fact present. An extreme example of the highly misleading information provided by such a measurement is the multi-shot spectrometer measurement of the spectrum of a supercontinuum pulse from microstructure fiber, which yielded an extremely smooth spectrum, despite the fact that each individual pulse spectrum was wildly different and actually comprised over a _thousand_ sharp spikes [57; 58]. As a result, a _typical_ spectrum, which often differs significantly from the _average_ spectrum, could be, and often is, significantly more complex, but it would be much more informative. Remarkably, in the above-mentioned study, the Cross-correlation Frequency-Resolved Optical Gating (XFROG) technique [57; 58] used to measure the continuum averaged over _100 billion_ pulses nevertheless actually provided such a typical spectrum. As a result, XFROG has become the standard method for measuring such light pulses. The spectral _phase_ requires even more significant consideration. It is well known that, for a given spectrum, the shortest pulse corresponds to a spectral phase that is flat, while that of a longer, typically more complex, pulse is necessarily complex [57].