## 2 Background
### Double Pulses in SPIDER
The second fringe term has a phase offset from the first fringe term that is proportional to how large the pulse separation is compared to the temporal range of the SPIDER measurement. This ratio depends on both the pulse's characteristics and the device parameters. When the pulse separation is stable, there is a consistent phase offset between the fringes. Because the fringe terms are otherwise identical, their sum gives the same group delay as the individual terms. Thus, if the relative phase between the two pulses varies randomly, SPIDER will interpret double pulses as a single pulse with the temporal width of one of the individual pulses. In other words, ones of the pulses will be ignored completely. The size of the phase offset, which is controlled by the ratio of the pulse separation to the temporal support, determines how much background is present in the trace. If the separation between pulses is small compared to the temporal range (that is, support), then the sum of the two fringe terms cannot be distinguished from a single cosine and has almost no background. As the pulse separation increases, the fringe terms are less in phase and the background increases (see Figure 2). Regardless of the amount of background, a SPIDER measurement of a double pulse with varying relative phase will always yield a single pulse. What is particularly interesting about this case is that it is possible to have very good fringe visibility and still have a very wrong measurement. In contrast with studies of other types of unstable pulses in SPIDER, having very low background is not a guarantee of measuring a pulse correctly. If the pulse separation is not stable, then the phase offset between the two fringe terms is variable. There will be partial cancellation of fringes from the unstable pulse and some background present. As found previously, the fringes from the second pulse will vanish from the measurement if its relative arrival time varies by the temporal support. In real SPIDER measurements, there is another issue that may affect how double pulses are measured. Because the spectral shear experienced by the pulse replicas is generated by interacting with different parts of an extremely chirped pulse, parts of the pulse occurring at different times will experience slightly different frequencies. While the frequency difference between the original pulse and the replica should remain constant in time, a double pulse will end up with slightly different frequencies for both pulses in the original pulse-shape and for both pulses in the replica. The modified expression for the SPIDER signal is:  
\[S_{SPIDER}\propto\left| \begin{array}{c}E(\omega)+E(\omega+\Omega)\exp(i \omega T)+E(\omega+\delta)\cdot\exp(i\theta+i\omega t_{\mathrm{s}})+\\ E(\omega+\Omega+\delta)\cdot\exp(i\omega T+i\theta+i(\omega+\Omega)t_{\mathrm{s }})\end{array} \right|^{2} \tag{6}\]  
Figure 2: Fringe terms and SPIDER signal evaluated for equal-energy double pulses with stable pulse separation, averaged over relative phases from 0 to 2\(\pi\). The last two fringe terms in (4) average to zero and are not plotted. In (**a**), the pulse separation and shear are chosen so that \(\Omega t_{\mathrm{s}}=0.75\pi\). Because the two surviving fringe terms are not in phase, there is some cancellation in their sum and the trace shows some background; In (**b**), the pulse separation is chosen such that \(\Omega t_{\mathrm{s}}=0.3\pi\). The fringe terms are much closer to being in phase, and there is a small amount of background in the SPIDER signal. where \(\delta=\Omega t_{s}T\) is the difference in shear experienced by the two pulses in the double pulse. Compared to Equation (3), both the second pulse and its sheared replica experience an additional shear of \(\delta\). This is not likely to have a large impact on the measurement, and this effect is not taken into account by the typical SPIDER retrieval procedure.