We present a method for quantifying ultrashort pulse-shape instability in a train of pulses using multi-shot second-harmonic-generation frequency-resolved optical gating (SHG FROG). All versions of multi-shot FROG have previously shown the ability to
qualitatively distinguish stable from unstable pulse trains, as systematic differences
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We present a method for quantifying ultrashort pulse-shape instability in a train of pulses using multi-shot second-harmonic-generation frequency-resolved optical gating (SHG FROG). All versions of multi-shot FROG have previously shown the ability to
qualitatively distinguish stable from unstable pulse trains, as systematic differences appear between measured and retrieved FROG traces when instability is present. This has proved possible because the recently introduced retrieved-amplitude N-grid algorithmic (RANA) approach provides highly reliable pulse retrieval, even for unstable pulse trains and in the presence of noise, thereby eliminating the possibility that algorithm stagnation, which also yields such systematic differences, could be confused for such instability. In other words, RANA’s excellent performance ensures that any non-random discrepancies between measured and retrieved FROG traces reflect physical pulse-shape instability rather than algorithmic stagnation. To
quantify such instability, we now introduce an instability parameter,
. It involves an extension of the well-known statistical “Runs” test, which has been used for decades to test for systematic error in fits to one-dimensional (1D) data. A runs test counts the “runs”—consecutive points in the plot of the difference between the data and fit with the same sign (+ or −), yielding an evaluation of the goodness of the fit, largely independent of random error. Specifically, the more runs, the better the fit. However, because FROG traces are functions of two variables, we must extend the usual 1D runs test to two dimensions, that is, to enumerate the 2D runs—“hills” and “valleys” in the difference between measured and retrieved 2D FROG traces. Many small 2D runs indicate only random noise-like differences, that is, a good fit, and, hence, a stable pulse train, whereas few large runs reflect systematic error, that is, a poor fit, and, hence, pulse-shape instability. Finally, because random noise could contribute numerous meaningless runs in the wings of a FROG trace, where the intensity is near zero, we must also weight each hill and valley by its average measured trace intensity in order to minimize its effects. We show that
R is intuitive and reasonable and, in addition, is independent of pulse complexity and trace size. As a result, it provides a clear metric of pulse-shape stability vs. instability.
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