Error bars in frequency-resolved-optical-gating measurements of ultrashort laser pulses

Frequency resolved optical gating (FROG) has made it possible to unambiguously measure a pulse\´s intensity and phase vs. time (and frequency). But just how accurate is a given FROG reconstruction of a pulse? Some indication of the measured pulse accuracy is available from the "FROG error" - the rms difference between the measured and retrieved FROG traces. What is needed, however, is a method for determining the uncertainty in each of the retrieved intensity and phase points. In other words, we need to be able to place error bars on the intensity and phase at each time (and frequency). We present here a simple, robust, and general technique for placing error bars on the intensity and phase retrieved in a FROG measurement. It involves no additional measurements, instead operating with only a single measured trace. We apply this technique both to theoretical and experimental FROG measurements of pulses and show that it gives reasonable results. The technique we use is the "bootstrap" method. This implementation involves running the FROG algorithm several times for the measured FROG trace, but with a different set of points chosen at random, and tabulating the statistics of the retrieved intensity and phase values obtained during these runs. To choose the points, we took our original FROG trace and selected, at random (and with replacement), a number of points equal to the original. By allowing data points to be selected more than once, about 2/3 of the points are represented in the new trace. The FROG algorithm ignores points where no data was selected. This works because the FROG trace vastly over-determines the pulse and so running the algorithm with only a fraction of the points does not harm its accuracy.