# Theoretical Study: High Harmonic Generation by Light Transients
## 3 Results and Discussion  
Figure 4a shows how the HHG cutoff, ionization fraction, and yield is affected when changing the relative amplitude between the three superimposed pulses. The energy ratio between the three pulses is varied from 10% to 100%. It can be observed that, while the ionization fraction for different energy ratios changes by less than 7%, the harmonic yield varies over more than eight orders of magnitude. This simulation loop identifies the amplitude ratio of (VIS:NIR:MIR) 0.1:0.1:1 as the optimum ratio to generate the highest cutoff energy. The inset in Figure 4 shows the waveform generating the highest cutoff. The dominant role of the MIR in extending the cutoff was expected, due to the higher electron excursion time. Figure 4b shows how the ionization fraction, cutoff, and yield varies when scanning the relative amplitude and delay of the superimposed pulses simultaneously. It can be seen that, unlike the results shown in Figure 4a, the ionization rate varies dramatically for different inter-pulse delays. In Figure 4c, the yield and energy of different waveforms for a small energy interval of 20 eV at 1.4 keV is shown. Our model enabled us to identify the optimum waveform in terms of the harmonic yield and ionization rate which is marked with a green circle. The waveform has an amplitude ratio of 0.1:0.1:1 and the relative delay of \(-\)6:3.6:\(-\)0.3 (fs). In addition, the other waveform marked with a blue circle is of particular interest, since it contains a relatively high amount of NIR pulses. This is an important factor as: (i) the quantum diffusion for such a waveform is smaller; and (ii) optical parametric amplifiers and optical components work more efficiently in this spectral region. Figure 5a,b illustrates the electric fields, ionization rates, and corresponding spectrograms of the optimized (green circle) and experimentally interesting (blue circle) waveforms. Both waveforms follow a similar trend: having a less than one-cycle MIR pulse, modulated on either the leading or trailing edge, or both edges by the fast oscillating fields in the NIR and VIS. This leads to a suppression of the electric field, except at the sub-cycle part of the electric field, which has a major role in generating the highest harmonics. In addition, the ionization rate prior to the arrival of the main sub-cycle of the electric field, and the launch as well as recollision of the high-energy trajectories, are controlled. As shown in Figure 4b, at a relative inter-pulse delay of \(-\)6:\(-\)1.5:5.5 (fs), the maximum photon energy of over 1.7 keV, with an ionization rate of 25% is reached. Compared to the light transient shown in Figure 2, this waveform leads to an order of magnitude higher harmonic yield and three time less ionization rate. Figure 4: Relation between cutoff energy, ionization rate and harmonic yield for different: (**a**) energy ratio of the superimposed few-cycle pulses at zero delay; and (**b**) energies and delays. The highest cutoff energy (marked in Panel (**a**) with a red circle) corresponds to a waveform with energy ratio of 0.1:0.1:1. Inset shows the temporal electric field of the best waveform at zero delay. (**c**) The optimized and experimentally interesting waveforms in this energy region are marked in green and blue, respectively. The logarithmic color scale represents the harmonic yield in all three panels. Table 1 compares the ionization rate, required peak intensity, and the harmonic yield for generating 1.7 keV isolated attosecond pulse, for six different cases. The driving field in case 1 is a 20 fs Gaussian pulse at 4 \(\upmu\)m. Such short pulses at 4 \(\upmu\)m are not yet available, however it can be seen how the harmonic yield in this case drops dramatically. case 2 shows the results of the sub-cycle waveform described in [6], with 90% ionization rate. Case 3 shows the results of the sub-cycle waveform described in [18], at a higher peak intensity and ionization rate, compared to the other cases.