## 4 Results
### Phase and Separation Variation  
The second case allows variations in temporal pulse separation in addition to variations in relative phase. The pulse separation varies uniformly by slightly more than the FWHM pulse width--from 640 to 800 fs. Plotting the pulse separation as a function of phase results in a diffuse circle with an inner radius of 640 fs and an outer radius of 800 fs (see Figure 4c). The average autocorrelation changes very little (see Figure 4a). The side lobes are slightly broadened and their height is reduced by a small amount. Interestingly, despite varying the pulse separation by a little more than a pulse width,  
Figure 3: SPIDER and FROG measurements of a train of double pulses with stable separation and varying relative phase (case A). (**a**) The average autocorrelation; (**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 0.0140; (**g**) Temporal intensity (red) and phase (blue) of the pulse retrieved by SPIDER; (**h**) Spectral intensity (green) and phase (purple) of the pulse retrieved by SPIDER; (**i**) Temporal intensity (red) and phase (blue) of the pulse retrieved by FROG; (**j**) Spectral intensity (green) and phase (purple) of the pulse retrieved by FROG. the width of the side lobes in the autocorrelation has not increased by very much. As in the first case, the average spectrum is close to Gaussian because of the relative-phase variations. The average SPIDER trace for this case is extremely similar to the average trace for the previous case. The SPIDER measurement again does not see the satellite pulse at all in the averaged measurement. There is almost no background in the SPIDER trace. The averaged measured FROG trace has slightly wider side lobes, similar to the lobe broadening of the autocorrelation. The retrieved FROG trace has a different structure from the measured trace, with an extra pair of faint outer side lobes (see Figure 4f) that indicate the presence of a third satellite (see Figure 4i). The G error is 0.0124. The total energy in the retrieved satellite pulses is significantly less than the energy of the main pulse. Their relative energies are 0.44 and 0.25, for a combined relative energy in satellites of 0.69. Of course, the actual pulses are equal energy double pulses, and relative pulse energy derived from the measured trace is 0.97. Even though the side lobes in the measured trace are broadened by the variation in pulse separation, the relative energy in the two pulses is calculated with good accuracy. Each satellite is temporally separated from the main pulse by an amount equal to the average pulse separation in the actual pulse train (720 fs). The retrieved pulses have a small spectral chirp and have relative phases of approximately \(\pi/6\), 0, and \(-2\pi/3\) respectively. Running the FROG retrieval many times (using different initial guesses) generates retrieved pulses that all have very similar structure and nearly identical G errors. Some retrievals have positive chirp and some have negative chirp, and the ordering of the intensity and relative phase of satellites varies due to the SHG time direction ambiguity. Figure 4: SPIDER and FROG measurements of a train of unstable double pulses with varying separation and relative phase (case B).