# Theoretical Study: High Harmonic Generation by Light Transients
## 4 Conclusions  
In conclusion, we demonstrated the prominent role of few-cycle pulse synthesis for optimizing the generation of isolated attosecond pulses at keV photon energies via HHG. We have shown that, in a sub-cycle pulse, the total electric field is confined to a sub-cycle period which results in an abrupt \begin{table}
\begin{tabular}{c c c c} \hline
**Case** & **Ionization Rate** & **Intensity (\(\times 10^{14}\))** & **Yield (arb. Units)** \\ \hline
1 & 0.01\% & 3.5 & 0.0006 \\
2 & 99\% & 52.8 & 1.05 \\
3 & 70\% & 52.8 & 1.05 \\
4 & 16\% & 13.9 & 41 \\
5 & 17.4\% & 13.5 & 46 \\
6 & 24.5\% & 15 & 78 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of the single dipole response of different waveforms interacting with He, for generation of isolated attosecond pulses at 1.7 keV photon energies. The driver pulses in each case are as follows: Case 1: 20 fs, Gaussian pulses at 4 \(\upmu\)m. Case 2: 1 fs sub-cycle pulses centered at 540 nm discussed in [6]. Case 3 Sub-cycle pulse from the laser architecture discussed in [18] and shown in Figure 1. Case 4: The optimized synthesized pulses considering only the VIS and MIR channels of the synthesizer shown in Figure 1. This case shows the influence of the NIR channel. Case 5: Gaussian 11.5 fs pulses at 2 \(\upmu\)m. Case 6: The optimized waveform (see Figure 5c) from the three channels of synthesizer shown in Figure 1. It can be seen that the optimized synthesized waveform has the highest yield with a seemingly low ionization rate. Figure 5: Electric field, ionization rate (**left**) and the spectrogram (**right**) of optimized waveforms with: (**a**) 0.1:0.1:1 energy ratio at \(-\)6:3.6:\(-\)0.3 (fs) delay generating 1.4 keV; (**b**) 0.1:0.3:1 energy ratio at \(-\)2.7:\(-\)6:4.8 (fs) delay generating 1.4 keV; and (**c**) 0.1:0.1:1 energy ratio at \(-\)6:\(-\)1.5:5.4 (fs) delay generating 1.7 keV. The logarithmic color scale in both panels represents the harmonic yield. onset of ionization within a single half-cycle [16]. However, they are not the optimum shape of the waveform for HHG at keV photon energies. The increase in harmonic yield in our optimized waveform for generating isolated attosecond pulses at 1.7 keV is due to the increased ionization within the main sub-cycle field and modification of the electron trajectories to have an earlier return time [32]. Our study shows that by moving the center of mass of the waveform's spectrum to longer wavelengths, the cutoff is increased. Using the optimized waveform for HHG suppresses the ionization probability. This is a major advantage for efficiently generating isolated attosecond pulses, as it allows for the most appropriate phase matching and energy-scaling schemes. Here, the generation of attosecond pulses benefits from the minimum ionization prior the arrival of the main electric field's sub-cycle, which minimizes the density of free electrons and, hence, the distortion of the driving wave and its dephasing with the generated harmonic wave. As a result, the coherent build up of the harmonic emission over an extended propagation is maximized. In addition, the probability of ionization outside the central cycle is more than two orders of magnitude lower than that at the field maximum and hence is negligible. Our study shows how the synthesis of multi-color broadband spectra enables fine tuning of the electron trajectories leading to the enhancement of the high harmonic yield and extension of the cutoff beyond what is achievable from the synthesis of semi-monochromatic fields at discrete frequencies [33; 34; 35]. Compared to the waveform in [6], the optimized waveform has a longer propagation length in the macroscopic scale due to the lower dispersion at a higher wavelength. This holds promise for compensation of the rapidly decreasing microscopic single-atom yield, by optimizing the dispersion of the free-electron plasma [28].