# The Role of Electron Trajectories in XUV-Initiated High-Harmonic Generation
## 4 Discussion
### Frequency Shifts of Short Trajectory Harmonics  
Here we investigate the origin of the weak delay-dependent spectral tilt of the odd harmonics in the range H19-H25 (Figure 3a). The TDSE calculation (Figure 3b) agrees well with the experiments and qualitatively reproduces all the experimental observations. Many features in the calculation, however, are not found in the experiments. For instance, we note the absence of strong inter-harmonic signal at negative delays between H21 and H23 at \(\Delta t\)\(\sim\)\(-\)7 IR cycles. We attribute this to the limitations of our single-atom response calculation, which does not suppress multiple-return trajectories [37]. In order to more clearly visualize the delay-dependent frequency shift effect on the odd harmonics, we show a normalized color plot for both the measured signal (Figure 4a) and the TDSE calculation (Figure 4b). The spectral tilt is clearly visible in both experimental and theoretical results and increases with ascending harmonic order. The spectral tilt in the TDSE calculation is comparatively steeper than the experimental one. This difference can be attributed to an additional chirp of the IR pulse in the experiment [15] and the choice of the IR envelope function as a squared-cosine pulse in the TDSE.  
The measured delay-dependent frequency shift enables us to determine the \(\alpha\) parameter for each spectral component directly from the experiments. Scanning \(\Delta t\) controls the overlap of the APT with the IR envelope, which determines the instantaneous IR intensity and the time window when photoelectrons are produced. Therefore, we can associate the delay with a mean ionization time \(\bar{t}_{0}\approx\Delta t\) and a recollision time \(\bar{t}_{1}\approx\Delta t+t_{\text{travel}}\). Here, we approximate the travel time \(t_{\text{travel}}\) with a cutoff trajectory, \(t_{\text{travel}}\approx 0.6\,T_{\text{opt}}\), where \(T_{\text{opt}}\) is the optical cycle duration of the IR field. We can establish a relationship between the frequency shift and the delay for a specific trajectory:  
\[\delta\Omega(\Delta t)\approx\alpha\left.\frac{\partial I(t)}{\partial t} \right|_{\Delta t+0.6T_{\text{opt}}}. \tag{1}\]  
We retrieve \(\alpha\) parameters corresponding to specific positions \((\delta\Omega,\Delta t)\) in the spectrum, assuming a cosine-square IR field envelope function as in the TDSE. From Equation (1), we estimate \(\alpha=(4\pm 2)\times 10^{-14}\) cm\({}^{2}\) W\({}^{-1}\) for the spectral tilts of H19, H21, and H23. The lower bound of this range is in agreement with short trajectory \(\alpha\) parameters in tunneling-driven HHG [13; 16]. In agreement with previous studies [22; 24], we conclude that the trajectory picture of HHG is valid in XUV-initiated HHG and that short trajectories dominate the odd harmonic signal.  
In addition, we compare our experimental findings with the CCTSM, which takes Coulomb effects and the XUV excitation into account. To this end, we calculate the \(\alpha\) parameter for the \"downhill\" and \"uphill\" short trajectories. Figure 4c,d show \(\alpha\) for H19 and H21, respectively, with incident harmonic H15. We find that \(\alpha\) is nearly constant with intensity. In order to compare the experimental and CCTSM results, we determine \(\delta\Omega\) for XUV-initiated HHG from incident harmonics H13 and H15, assuming again a 25 fs pulse. Figure 4e,f display the CCTSM results, overlaid with the experimental and TDSE results, respectively. Downhill (black curves) and uphill (blue curves) short trajectories yield almost indistinguishable frequency shifts. The calculated tilt is in good agreement with experiments and the TDSE for H23, while at lower harmonics the slope of the calculated tilt decreases and changes sign, unlike the experimental observations.