## 2 Background
### Double Pulses in FROG  
General analytical consideration of FROG traces is usually impractical because expressions for self-referenced spectrograms tend to be quite complicated. Considering double pulses is simpler, however, and, assuming that the two pulses are both flat-phase Gaussians simplifies the calculation further. While these assumptions are restrictive, the resulting calculation is still quite informative for more general situations. Writing the time-domain electric field for a Gaussian double pulse:  
\[E(t)=\exp\Bigl{(}-t^{2}/\sigma_{t}^{2}\Bigr{)}+\exp\left(-(t+t_{s})^{2}/\sigma _{t}^{2}+i\theta\right) \tag{7}\]  
the corresponding SHG FROG measurement is:  
\[I_{SHG~{}FROG}(\omega,T)=\left|\int_{-\infty}^{\infty}E(t)E(t-T)\exp(-i\omega t )dt\right|^{2} \tag{8}\]  
\[=\left|\int_{-\infty}^{\infty}\biggl{[}\exp\left(\frac{-t^{2}}{\sigma_{t}^{2 }}\right)+\exp\left(\frac{-(t+t_{s})^{2}}{\sigma_{t}^{2}}\right)\biggr{]}. \biggl{[}\exp\left(\frac{-(t-T)^{2}}{\sigma_{t}^{2}}\right)+\exp\left(\frac{-( t+t_{s}-T)^{2}}{\sigma_{t}^{2}}\right)\biggr{]}\exp(-i\omega t)dt\biggr{|}^{2} \tag{9}\]  
where \(T\) is again the delay between pulse replicas. After some calculations, this expression becomes:  
\[I_{SHG~{}FROG}(\omega,T)\propto\exp\left(\frac{-T^{2}}{\sigma_{t}^{2}}\right) \exp\left(\frac{-\omega^{2}\sigma_{t}^{2}}{4}\right)\left[ \begin{array}{c} \cos^{2}\left(\frac{\omega t_{s}}{2}+\theta\right)+\\ 2\cos\left(\frac{\omega t_{s}}{2}+\theta\right)\exp\left(\frac{-t_{s}^{2}}{2 \sigma_{t}^{2}}\right)\cosh\left(\frac{t_{s}T}{\sigma_{t}^{2}}\right)+\\ 2\exp\left(\frac{-t_{s}^{2}}{\sigma_{t}^{2}}\right)+\exp\left(\frac{-t_{s}^{2} }{\sigma_{t}^{2}}\right)\cosh\left(\frac{2t_{s}T}{\sigma_{t}^{2}}\right) \end{array} \right] \tag{10}\]  
The second and third terms inside the brackets are negligibly small because the pulse separation \(t_{s}\) is typically larger than \(\sigma_{t}\). The last (fourth) term can be written in a more intuitive form, resulting:  
\[ \begin{array}{c}I_{SHG~{}FROG}(\omega,T)\propto\exp\left(\frac{-T^{2}}{ \sigma_{t}^{2}}\right)\exp\left(\frac{-\omega^{2}\sigma_{t}^{2}}{4}\right) \cos^{2}\left(\frac{\omega t_{s}}{2}+\theta\right)+\\ \frac{1}{2}\exp\left(\frac{-\omega^{2}\sigma_{t}^{2}}{4}\right)\left[\exp \left(\frac{-(T-t_{s})^{2}}{\sigma_{t}^{2}}\right)+\exp\left(\frac{-(T+t_{s}) ^{2}}{\sigma_{t}^{2}}\right)\right]\end{array}  \tag{11}\]  
This expression describes the archetypal SHG FROG trace for double-pulses. The Gaussian frequency and delay amplitudes are equivalent to the FROG trace for a simple flat-phase Gaussian pulse. As is the case for all FROG traces of double-pulses, spectrogram has three intensity lobes, centered at \(T=0\) and \(T=\pm t_{s}\) respectively. The middle lobe (the first term) has fringes that depend on the pulse separation as well as relative phase. The side lobes are half as intense as the center lobe but have the same temporal and spectral widths as does the center lobe. Allowing the relative phase \(\theta\) to vary freely erases the fringes in the center lobe of the trace (see, for example, Figure 3e). Significantly, the resulting trace, with three smooth (fringeless) lobes, which is the sum of many different traces corresponding to the many different double-pulse relative phases, is not a valid FROG trace for any pulse, no matter how complex. This is because the trace constrains both the temporal and spectral structure of the integrated electric field over many different pulses. Lobes separated in delay in a FROG trace clearly require the field to have satellite pulses,as in intensity autocorrelation. However, important for our purposes, stable satellite pulses also cause spectral fringes, and therefore the FROG trace should have frequency fringes in the center lobe. In fact, double pulses are used for calibration of the frequency axis in FROG for exactly this reason.