## 4 Results
### Small, Correlated Phase and Separation Variation  
While many reports of unstable multi-pulsing do not explicitly discuss relative phase, it is likely that the relative phase is usually random. Some simulations show that, under certain unusual conditions, the phase and the separation between pulses can vary together in interesting ways [19]. To explore this type of scenario, we include a third test case where the phase varies over a range of less than \(2\pi\) and follows a specific path as the pulse separation changes (see Figure 5c). The temporal separation ranges from 590 to 800 fs while the relative phase varies from \(\pi/4\) to \(3\pi/4\). This is case contains a larger range of separations than earlier cases, and the side lobes in its autocorrelation are broader and less intense as a result. The average spectrum is quite modulated due to the small range of relative phases.  
The average SPIDER trace of this pulse train also has an envelope that is strongly modulated, but the spectral phase retrieved from the trace is nearly flat. The retrieved temporal pulse shape has two very low-intensity satellite pulses with relative phases of \(-\pi/2\) and \(\pi/2\) separated from the main pulse by 700 fs, which is the average separation of the double pulses in the pulse train. The measured FROG trace does have fringes in the center lobe, but they are blurred by the variations in relative phase.  
Figure 5: SPIDER and FROG measurements of a train of unstable double pulses with correlated separation and relative phase (case C). (**a**) The average autocorrelation is shown in red, and the black dotted line shows the autocorrelation of a pulse with an average separation; (**b**) Average spectrum of the double pulse train; (**c**) Polar plot of the pulse separation as a function of relative phase; (**d**) Average SPIDER measurement of the whole pulse train; (**e**) Average FROG measurement of the whole pulse train; (**f**) Retrieved FROG trace with G error 0.0016; (**g**) Temporal intensity and phase of the pulse retrieved by SPIDER; (**h**) Spectral intensity and phase of the pulse retrieved by SPIDER; (**i**) Temporal intensity and phase of the pulse retrieved by FROG; (**j**) Spectral intensity and phase of the pulse retrieved by FROG.  
The side lobes in the measured trace are more diffuse than in previous cases because of the larger range of temporal pulse separations. The retrieved trace appears much more similar to the measured trace than in previous cases, but there are still clear discrepancies between the two. The retrieved time-domain pulse shape includes a small satellite separated by about 700 fs with a relative phase of \(-\pi/2\). Interestingly, this satellite pulse has a different temporal width from the main pulse, unlike retrieved satellites for other cases. This retrieved satellite is significantly wider, with a FWHM width of about 250 fs rather than 150 fs. The difference in temporal widths is also visible in the retrieved spectrum, which is more deeply modulated in the middle than at the edges because the spectrum of the satellite is narrower. Despite the longer retrieved satellite pulse, the side lobes in the retrieved trace are still narrower than the side lobes in the measured trace. Energy analysis of the trace gives a relative energy of 0.98 between the pulses. In contrast, the satellite in the retrieved pulse has a relative energy of 0.46. Even though the FROG algorithm does not give the correct relative energy, that information is still present in the measured FROG trace.