# Disentangling Long Trajectory Contributions in Two-Colour High Harmonic Generation
## 4 Macroscopic Propagation  
Both the semiclassical three-step-model [8; 9; 10] as well as the quantum theory [22] describe the single atom response of HHG. In order to have efficient production of high-order harmonic radiation, phase matching conditions for constructive interference at harmonic order \(n\),  
\[\mathbf{k}_{n}=n\cdot\mathbf{k}_{1}+\mathbf{K}, \tag{6}\]  
must be satisfied over the complete propagation distance of the driving laser field through the target interaction region. All new parameters and quantities are defined in Table 1. Equation (6) takes account of the Gouy-phase of a Gaussian beam, the phase of the electron wave packet (described by the classical action) and the phase due to the time passed until the recombination. On the other hand, it neglects dispersion effects induced by the neutral and ionised gas. Experimental setups are typically optimised for short trajectory phase matching on axis (see also Ref. [18] and their supplementary material). In that case, the phase matching condition for long trajectories is best fulfilled for emitted wave vectors diverging from the optical axis [27]. These contributions to the final HHG spectrum can easily be filtered out spatially, as was done in Ref. [18]. However, there still might be a small contribution of long trajectories to the low harmonics even when looking along the axis. The phase mismatch is then given by  
\[\delta k=k_{n}-|n\cdot\mathbf{k}_{1}+\mathbf{K}(z,r=0)|. \tag{7}\]  
For an interaction region on axis (\(r=0\)) in forward propagation along the axis, this mismatch can be estimated as  
\[\delta k(z,n,\tau)=n\frac{\omega}{c}-\left|n\left(\frac{\omega}{c}-\frac{2b}{b ^{2}+4z^{2}}\right)+\frac{F_{0}^{2}z_{0}^{2}z}{2\omega^{2}(z^{2}+z_{0}^{2})^{ 2}}\cdot\tau\right|, \tag{8}\]  
where we used approximations described in Ref. [27]. Figure 5A shows the phase mismatch \(\delta k\) of the wave vectors for two sample excursion times \(\tau_{\text{short}}=25\) au and \(\tau_{long}=80\) au at harmonic order \(n=22\), and the gas jet positioned after the focus of the beam (\(z>0\)). In an optimal position of the jet after the focus, long trajectories are very efficiently, but not completely, suppressed along the axis (corresponding to large \(|\delta k|\)). Starting from the phase mismatch \(\delta k(z,n,\tau)\) Equation (8), one can estimate the total amplitude efficiency with the help of a geometric series  
\[GS(z,n,\tau,N)=\sum_{t=0}^{N-1}\exp\left(i\delta k(z,n,\tau)\frac{j}{N}\cdot t \right)\frac{1}{N}. \tag{9}\] \begin{table}
\begin{tabular}{l l} \hline \(z\) & longitudinal position in direction of beam propagation \\ \(r\) & radial position away from the beam axis \\ \(t_{i}\) & exit time \\ \(t_{r}\) & return time \\ \(\tau=t_{r}-t_{i}\) & excursion time \\ \(\mathbf{k}_{n}=n\frac{\omega}{c}\xi_{1}\) & radiated harmonic \\ \(\mathbf{k}_{1}=\frac{\omega}{c}z+\
abla\arg\left(\frac{1}{b+2 iz}\exp\left\{-\frac {\omega}{b+2 iz}\right\}\right)\) & fundamental Gaussian beam \\ \(S_{p}(t_{r},t_{i})\) & classical action for electron wave packet propagation \\ \(\mathbf{K}(z,r)=\
abla\left(n\omega t_{r}-S_{p}(t_{r},t_{i})\right)\) & intrinsic atomic phase \\ \(N\) & total number of interacting targets along the axis \\ \(j\) & gas jet diameter \\ \hline \end{tabular}
\end{table} 
Table 1: Quantities in phase matching. This geometric series oscillates around a value for low \(N\), and converges for larger \(N\gtrsim 40\). The limit of \(N\to\infty\) is given by  
\[GS(z,n,\tau)=\frac{i(1-\exp(ij\delta k(z,n,\tau)))}{j\delta k(z,n,\tau)}. \tag{10}\]  
A typical XUV pulse contains several orders of magnitude more photons than \(N=40\) per harmonic order, which justifies taking the limit of \(N\to\infty\). Then, the absolute square  
\[PM(z,n,\tau)=GS(z,n,\tau)GS^{*}(z,n,\tau)=\frac{2}{j^{2}\delta k^{2}(z,n,\tau) }-2\frac{\cos(j\delta k(z,n,\tau))}{j^{2}\delta k^{2}(z,n,\tau)} \tag{11}\]  
yields the efficiency for the intensity of the high-order harmonic radiation.