# Disentangling Long Trajectory Contributions in Two-Colour High Harmonic Generation
## 5 CTMC Simulations
### Long Trajectories Contribution  
In Section 5.1, it was found that the maximizing phase \(\varphi_{\text{max}}\) in the CTMC agrees reasonably with the experimental results, except for low harmonics slightly above the HHG threshold \(n<25\). For these low harmonic orders, the \(\varphi_{\text{max}}\) extracted from CTMC shows an opposite slope from the experimental displacement gate plot. This is the range of energies where most long trajectories are initiated since the field is the strongest.  
In order to estimate the efficiency of the HHG build up, the phase matching efficiency Equation (11) is calculated for all numerically computed returning trajectories. The final phase matching corrected probability weight \(\Gamma_{\text{PM}}\) of a returning trajectory is given by  
\[\Gamma_{\text{PM}}=\Gamma(t_{i},p_{i})\cdot PM(n,\tau), \tag{15}\]  
where \(\Gamma(t_{i},p_{i})\) is the ionisation probability at time \(t_{i}\) with initial transverse momentum \(p_{i}\). For the following calculations, the jet centre is assumed to be located at \(z=5\,\text{mm}\), with a jet diameter of \(j=0.95\,\text{mm}\)[43].  
Figure 6 contrasts two displacement gate plots. In Figure 6A, only short trajectories are considered, with their probability weights given by \(\Gamma(t_{i},p_{i})\). In Figure 6B, the HHG spectra were calculated by summing up the corrected weights \(\Gamma_{\text{PM}}\) from Equation (15), including also long trajectories. The effect for higher harmonic orders is relatively small. However, including long trajectories does yield a change in the slope of the \(\varphi_{\text{max}}\) curve, similar to that found in the experiment (see Figure 9). Figure 6B is much noisier than the plot focusing on only short trajectories Figure 6A. While there are more trajectories included in Figure 6B, the increased number is not enough to completely compensate for the additional statistical fluctuation due to the introduced complex phase for all trajectories.  
The shape and the value of \(PM(n,\tau)\) strongly depends on the detailed parameters of the beam geometry, such as the spot size on the focusing mirror, the exact geometry of the gas jet in the interaction region with the laser beam, the minimal beam waist, the confocal parameter of the focused beam, and the jet position relative to the focus. Performing the CTMC analysis with just slightly different beam geometry leads to differences in the extracted \(\varphi_{\text{max}}\) larger than their respective 95% confidence intervals for the fits in the extraction process (shown as the band covering the data points in displacement gate plots). Considering this, it is difficult to predict accurately the ratio of short and long trajectories corresponding to the experimental conditions. However, including long trajectories based  
Figure 8: \(\varphi_{\text{max}}\) extracted from CTMC with Coulomb (white) or CTMC without Coulomb (cyan) compared to the experimental displacement gate (colour code) [18].  
on phase matching efficiency always shifted the \(\varphi_{\max}(n)\) curve to larger phases, with the strongest influence on the lower harmonics. Taking into account long trajectories therefore leads to qualitatively better agreement with the experimental displacement gate, compared to the version where all long trajectories are excluded a priori.  
Additionally, the oscillation contrast \(C(n)\) of the displacement gate plot especially for the lower harmonic orders is reduced by adding long trajectories. Compare again the two plots in Figure 6. Figure 10 highlights that the displacement gate contrast in the CTMC with only short trajectories is much stronger (blue) than the contrast in the experiment of [18] (red). In the case when long trajectories are considered as well (green), the contrast drops to comparable values.