# The Role of Electron Trajectories in XUV-Initiated High-Harmonic Generation
## 1 Introduction  
Attosecond spectroscopy has enabled the direct observation of ultrafast electron dynamics in atoms, molecules, and solids on its natural time scale, the attosecond time domain (\(1\) as \(=10^{-18}\) s) [1; 2; 3]. It is based on the production of attosecond extreme ultraviolet (XUV) pulses through a process known as high-harmonic generation (HHG) [4]. In HHG, the interaction of a bound electron with a strong laser field leads to tunneling ionization. The liberated electron recollides with the parent matter, which is followed by the emission of bursts of XUV radiation with attosecond duration [5; 6; 7]. At the heart of HHG is the notion of electron trajectories, which lead to a direct mapping between time and energy and the emergence of a cutoff in the HHG spectrum. The cutoff energy is given by \(E_{\text{cutoff}}\approx I_{\text{p}}+3.17\,U_{\text{p}}\), where \(I_{\text{p}}\) is the ionization potential of the parent matter and \(U_{\text{p}}\) is the ponderomotive energy. Electron trajectories are classified into short and long trajectories according to the duration of their travel [8; 9; 10]. Macroscopic pulse propagation effects in the HHG medium typically lead to an enhancement of the short trajectories while the long trajectories are usually suppressed and possess a high degree of divergence [11; 12]. The quantity that governs the macroscopic response in HHG is the reciprocal intensity parameter \(\alpha\) for a specific harmonic and trajectory [13]. The parameter describes the scaling of the spectral phase \(\phi\) of a specific harmonic as a function of intensity \(I\), where to a good approximation, \(\phi(I)\approx-\alpha I\). Experimentally, \(\alpha\) has been resolved with the help of the reconstruction of attosecond beating by interference of two-photon transitions (RABBIT) [14], cross-correlation frequency-resolved optical gating (XFROG) [15; 16], two-source HHG interferometry [17], and quantum path interference inHHG [12]. The intensity dependence of \(\phi\) leads to an instantaneous frequency shift \(\delta\Omega(t)\) of the harmonics over the range of the driving laser pulse envelope, \(\delta\Omega(t_{1})\approx\alpha\,\partial I(t_{1})/\partial t_{1}\), where \(t_{1}\) is the time of recollision [15; 16]. This results in a pronounced frequency broadening of the long trajectory harmonics because their \(\alpha\) parameters are significantly larger than those of their short trajectory counterparts (cf. [10; 13]). A major limitation of HHG is imposed by the inherent coupling between the ionization and recollision steps of the interaction, both governed by the strong infrared (IR) field. XUV-initiated HHG replaces the tunnel ionization step in HHG by XUV-driven photoionization, overcoming this limitation [18; 19; 20; 21; 22; 23; 24]. Controlling the delay between an ionizing XUV field and a strong IR laser pulse determines the time frame when electrons are promoted to the continuum and undergo IR-driven recollision. Figure 1a shows the principle of XUV-initiated HHG. An XUV pulse promotes an electron to an excited or continuum state close to the ionization threshold. Subsequently, the IR field induces a recollision trajectory, leading to the emission of XUV light carrying higher energy. XUV-initiated HHG is related to XUV-initiated photoelectron rescattering [25; 26], laser-assisted photoemission [27; 28], and XUV-seeded cluster ionization avalanching [29]. In this work, we study the response of the electron trajectories to the instantaneous intensity of the IR field in XUV-initiated HHG. The ability to scan the delay between the ionizing XUV pulse train and the IR pulses allows the detection and isolation of various electron trajectories through their intensity-dependent spectral shifts. Such spectral shifts provide a direct insight into the reciprocal intensity parameter \(\alpha\) of XUV-initiated HHG, validating the semiclassical picture that describes the mechanism.