# Disentangling Long Trajectory Contributions in Two-Colour High Harmonic Generation
## 1 Introduction  
The fundamental process of High Harmonic Generation (HHG) was first discovered in the late 1980s [1; 2]. Starting from ultrashort few-femtosecond (fs = 10\({}^{-15}\) s) driving pulses (typically in the visible or near-infrared wavelength domain), HHG is an extremely nonlinear frequency up-conversion process, creating Extreme Ultra-Violet (XUV) pulses with a pulse duration shorter than a femtosecond (the current world record is 43 attoseconds (as = 10\({}^{-18}\) s) [3]). Since then, HHG-created XUV pulses have found diverse applications in ultrafast science [4; 5; 6; 7]. First, the intuitive three-step-model of HHG was developed within a single-atom response [8; 9; 10], which introduced the concept of short and long trajectories in HHG. Then, the macroscopic medium was also taken into account including propagation and phase matching [11; 12; 13], which typically favours the short trajectories. Finally, the temporal coherence was explored [14], which showed two distinct spatial regions with different coherence times in the far-field pattern of the high harmonics depending on the two different trajectories. As the general understanding became better, HHG spectroscopy was also proposed [15; 16], which combines sub-Angstrom spatial with attosecond temporal resolution. [PERSON] then proposed a new approach for HHG with a two-colour field in an orthogonal configuration that enables controlling and decoupling the angle of ionisation and the angle of recollision and therefore increasing the dimensionality of HHG spectroscopy [17; 18]. Here, we explore the contribution of the long trajectories and the parent ion interactions in two-colour HHG using Classical Trajectory Monte Carlo (CTMC) simulations. The work presented in this article focuses on HHG in helium gas. The intuitive three-step-model of HHG was developed by [8; 9; 10]. In the first, quantum mechanical step, a strong driving field tunnel ionises an electron, which is subsequently accelerated classically by the strong laser field. In the case of small ellipticity or linear polarisation, photoelectrons can be driven back into their parent ion by the laser field. These photoelectrons can then either rescatter, or recombine again, representing the third step. In the case of recombination, a high energy photon is emitted, where the frequency of the high harmonic is determined by energy conservation\[\omega_{n}=E_{\rm kin}+I_{\rm p}, \tag{1}\]  
with \(\omega_{n}=n\cdot\omega\), \(I_{\rm p}\) the ionisation potential and \(E_{\rm kin}\) the kinetic energy of the photoelectron when recombining. Atomic units are used throughout this manuscript, unless otherwise specified. The linearly polarised driving field is given by  
\[F_{x}(t)=f(t)F_{0}\cos\left(\omega t\right)\hat{x}, \tag{2}\]  
with envelope \(f(t)\), maximum field strength \(F_{0}\), optical frequency \(\omega\), and polarisation axis \(x\). The standard Strong Field Approximation (SFA) [19] neglecting the ion Coulomb field after ionisation, and dipole approximation [20] neglecting the spatial dependence of the laser field are used, as well as a single active electron assumption [21]. Depending on time \(t_{i}\) when a photoelectron is liberated, the trajectories will return at a different time \(t_{r}(t_{i})\) and carry a different kinetic energy \(E_{t_{i}}\), indicated by the colour code in Figure 1. For most energies, there are two trajectories, a long trajectory that starts soon after the peak of the field (greyed out in Figure 1), and a short trajectory that starts later in the cycle (full colours). The energy cutoff of the returning photoelectrons is provided by the single trajectory separating the domains of short and long trajectories. The cutoff energy is calculated to be  
\[E_{\rm cutoff}=n_{\rm max}\omega=3.17U_{\rm p}+I_{\rm p}, \tag{3}\]  
which corresponds to the maximum possible electron kinetic energy at the time of recombination [22].