## 2 Background
### Double Pulses in FROG
Equivalently, the spectral structure of the FROG trace is constrained by the fact that the autoconvolution of the pulse spectrum must be equal to the frequency marginal (the sum of the trace over delay) [11]. Integrating the smooth averaged trace over delay--computing the frequency marginal--yields a spectral profile that is smooth. Thus, the lack of spectral structure in the trace requires the spectrum to be smooth, but the temporal structure requires the pulse spectrum to be modulated. There is no electric field that satisfies both conditions. Consequently, the averaged FROG trace of double pulses with varying relative phase cannot correspond to any single pulse. This is a very useful and important feature of FROG--this discrepancy indicates the presence of instability. Indeed, such discrepancies were also found to occur in the presence of partial mode-locking in our previous work and so meant that FROG is a good indicator of that type of instability. This works because the FROG trace is two-dimensional, whereas the pulse is only one-dimensional, and the resulting over-determination of the pulse by the trace yields this additional information. In the case of unstable double-pulsing, a variable pulse separation causes the position of the side lobes to vary. It also causes variations the periodicity of the fringes in the center lobe. Averaging over a range of pulse separations results in less intense, temporally wider side lobes in the averaged trace. Broadening the side lobes also causes the trace to become invalid for a single electric field. The spectrum of the side lobes is unchanged, but becoming longer in time with an equally broad spectrum requires consequently the addition of some phase variations, such as chirp. However, since the central lobe does not become wider in time, the field must also be nearly transform-limited. Variations in either pulse separation or relative phase lead to measured traces that are impossible to match with a single electric field. As a result, the FROG pulse-retrieval algorithm responds to these averaged spectrograms of unstable pulses cannot be predicted and is best studied using simulations. On the other hand, the averaged SHG FROG trace still contains a significant amount of information about the ensemble of double pulses. In particular, the SHG FROG trace of a double pulse is easily recognized, and the lack of fringes in the central lobe is a clear indicator that there are variations in the relative phase of the second pulse. This allows a simple post-processing technique to find the relative intensity of the satellite pulse. Specifically, the relative energy of the two pulses can always be determined without ambiguity (even if the order in which they arrive is ambiguous) based on the energy contained in each lobe in the trace. Because the side lobes are a cross-correlation between the first and second pulses, their energy is proportional to the product of the energy of each individual pulse. The center lobe contains the autocorrelation of each pulse, and therefore its energy is proportional to the sum of the squares of the energy in each individual pulse. If one pulse has an energy of 1 and the other pulse has relative energy \(\alpha\), we can write expressions for the energy in the center lobe \(M_{c}\) and a side lobe \(M_{s}\) as:  
\[1+\alpha^{2}\propto M_{c},\alpha\propto M_{s} \tag{12}\]  
The constants of proportionality are the same for each expression, and the exact energy of the pulses is often not of interest. Consequently, it makes sense to take the ratio of these two expressions. Multiplying through by \(\alpha\) and rearranging gives a quadratic expression that is easily solved:  
\[\alpha^{2}-\frac{M_{c}}{M_{s}}\alpha+1=0 \tag{13}\]  
The energy in each lobe can be determined by integrating the corresponding sections of the measured FROG trace over frequency and delay. One the ratio of the energies has been found, then this quadratic expression can be solved to yield the relative energy, \(\alpha\), of the second pulse.