## 2 Background  
### Double Pulses in SPIDER  
Previous publications addressing instability in SPIDER have included some analytical considerations of averaged SPIDER measurements [7; 8], and we will use those calculations as a starting point for considering the specific case of double pulses. Given a spectral shear \(\Omega\) and an internal pulse separation \(T\) in the SPIDER apparatus, a general expression for the ideal SPIDER trace of a single pulse as a function of the pulse complex electric field \(E(\omega)\) is:  
\[S_{SPIDER}\propto|E(\omega)+E(\omega+\Omega)\exp(i\omega T)|^{2}. \tag{1}\]  
To understand how the pulse is retrieved from a SPIDER trace, this equation may be rewritten in terms of the spectral amplitude \(|E(\omega)|\) and the derivative of the spectral phase (the group delay) \(\tau(\omega)\) as:  
\[S_{SPIDER}\propto|E(\omega)|^{2}+|E(\omega+\Omega)|^{2}+2|E(\omega)||E(\omega+ \Omega)|\cos(\tau(\omega)\Omega+\omega T) \tag{2}\]  
The spectral fringes in the SPIDER trace depend on the group delay \(\tau(\omega)\), and the spectral phase can be retrieved by integrating the measured group delay. To understand the effect of multiple pulsing, we consider the SPIDER trace of a double pulse comprising two identical pulses with a temporal separation \(t_{s}\) and a relative phase \(\theta\). The SPIDER trace of this double pulse is then:  
\[S_{SPIDER}\propto\left| \begin{array}{c}E(\omega)+E(\omega+\Omega)\exp(i \omega T)+E(\omega)\exp(i\theta+i\omega t_{s})+\\ E(\omega+\Omega)\exp(i\omega T)\exp(i\theta+i(\omega+\Omega)t_{s})\end{array}  \right|^{2} \tag{3}\]  
Expanding Equation (3) and collecting terms results in a lengthy, but tractable expression:  
\[\left. \begin{array}{c}S_{SPIDER}\propto|E(\omega)|^{2}[1+\cos(\theta+\omega t _{s})]+|E(\omega+\Omega)|^{2}[1+\cos(\theta+(\omega+\Omega)t_{s})]+\\ \left.\begin{array}{c}\cos(\tau(\omega)\Omega+\omega T)+\cos(\tau(\omega) \Omega+\omega T+\Omega t_{s})\\ +\cos(\tau(\omega)\Omega+\omega(T+t_{s})+\theta+\Omega t_{s})\\ +\cos(\tau(\omega)\Omega+\omega(T-t_{s})-\theta)\end{array}\right\}\end{array}  \tag{4}\]  
The first two terms above describe the SPIDER trace's overall intensity envelope. This envelope contains spectral oscillations characteristic of a double pulse. The remaining four terms describe the SPIDER signal's interference fringes. The first two oscillatory terms are typical fringes that would appear in common SPIDER measurements of each single pulse. As in all SPIDER measurements, each individual pulse interferes with a spectrally sheared replica of itself, creating fringes that are a function of the group delay. The second pair of fringe terms depend on both pulses in the double pulse. Here the first of the two pulses interferes with the sheared replica of the second pulse, and the second pulse interferes with the sheared replica of the first pulse. These fringe terms have different periodicities due to the different relative delays, and they also depend on the relative phase \(\theta\) between the two pulses. If \(\theta\) varies randomly from 0 to \(2\pi\), then in an averaged measurement some of the terms in Equation (4) would average to zero. Specifically, the spectral modulations in the envelope even out and the last two fringe terms disappear. In this case, the SPIDER-trace expression the simplifies to:  
\[S_{SPIDER}\propto|E(\omega)|^{2}+|E(\omega+\Omega)|^{2}+|E(\omega)||E(\omega+ \Omega)|\cdot\{\cos(\tau(\omega)\Omega+\omega T)+\cos(\tau(\omega)\Omega+ \omega T+\Omega t_{s})\} \tag{5}\]  
Notice that this expression differs only slightly from the general expression for SPIDER in Equation (2). In this expression there are two sets of fringes instead of just one. The two fringe terms are nearly identical and only differ by a factor of \(\Omega t_{s}\) (the pulse separation times the spectral shear). To put the spectral shear in context, the temporal range in a SPIDER measurement is \(2\pi/\Omega\). \(\Omega t_{s}\) is therefore \(2\pi\) times the ratio of the pulse separation and the SPIDER temporal range.