## 2 Results and Discussion
### Generation of Water Window IAPs from an MIR Waveform Synthesizer
#### 2.2.2 Simulation Results  
First, we employ a two-color waveform synthesizer, which contains a main driving pulse at 1.95 \(\upmu\)m and a control pulse at 1.35 \(\upmu\)m. Their intensities are 2.9 \(\times\) 10\({}^{14}\) W/cm\({}^{2}\) and 5 \(\times\) 10\({}^{13}\) W/cm\({}^{2}\), respectively, and both have a pulse duration of 30 fs (FWHM). Their CEPs are \(-\pi\) and 1.5\(\pi\), respectively, and the time delay between the two pulses is 1.13 fs. Then, the intensity profile of the synthesized electric field is shown by the orange shaded curve in Figure 5a. The intensity ratio between the main peak and the second strongest side peak is similar to that of a 1.4-cycle laser at 1.7 \(\upmu\)m (green solid). Because HHG is very sensitive to the electric field strength, only the central electric field peak contributes to the harmonic continuum in the cutoff region. Thus, even though we employ multicycle laser pulses, the synthesized pulse is sufficient to generate supercontinuum HH spectra, that support IAPs. The generated HH spectrum under a 32 torr pressure in a 60-mm-long Ne gas cell is shown in Figure 5b, which is phase-matched near 320 eV. By increasing the Ne pressure to 101 torr in a 70-mm-long cell, the supercontinuum is phase-matched to higher photon energies near 532 eV, as shown in Figure 5c. Thus, we were able to generate HH supercontinua in the entire water window region, which covers the k-edges of carbon (284 eV) to oxygen (543 eV). To extract these supercontinuafrom driving laser pulses from the viewpoint in an experiment, we employ a 1000-A-thick Sn filter in the simulation. Then, we calculate the IAP envelope of the HH supercontinua with a 1000 A Sn filter in Figure 5b. The result is shown in Figure 6a. An IAP with 63 as (FWHM) duration is obtained as shown by the red solid curve. The TL duration is 27 as (FWHM). In addition, we calculate the IAP of the supercontinuum in Figure 5c after a 1000-A-thick Sn filter. Its pulse envelope with a duration of 66 as is shown in Figure 6b. The TL pulse is shown by the black curve and has a duration of 28 as. Note that the filters are not helpful for compensating the chirp of IAPs owing to the low GDD values, as discussed above. Next, we add one more control pulse to the waveform synthesizer. The wavelengths of the main driving pulse and the first and second control pulses are 1.9 \(\upmu\)m, 1.38 \(\upmu\)m, and 0.8 \(\upmu\)m, respectively. The intensities are 2.5 \(\times\) 10\({}^{14}\) W/cm\({}^{2}\), 5 \(\times\) 10\({}^{13}\) W/cm\({}^{2}\), and 5 \(\times\) 10\({}^{12}\) W/cm\({}^{2}\), respectively, with a pulse duration of 30 fs (FWHM). The CEPs are \(-\pi\), 1.5\(\pi\) and \(\pi\), respectively. The delays of first and second pulses are 1.13 fs and 0 fs, respectively. The electric field of the synthesizer is shown in Figure 7a. It can be seen that, except for the central main peak, the electric field peaks are more suppressed than those for the two-color case. Thus, the effective waveform for HHG is further shortened, and is similar to a 0.99-cycle pulse at 1.73 \(\upmu\)m (green), in terms of the intensity ratio between the main and second-strongest side peaks. Employing the synthesized electric field to drive HHG, we simulate HH spectra in an Ne gas cell. When the pressure is 32 torr in the 60-mm-long cell, an HH continuum spectrum with a peak intensity at 350 eV is obtained, as shown in Figure 7b. By increasing the pressure to 101 torr in the 70-mm-long cell, the HH continuum spectrum is phase-matched to a higher photon energy near 480 eV, as shown in Figure 7c. The calculated IAP for the spectrum in Figure 7b after a 1000-A-thick Sn filter is shown by the red curve in Figure 8a and has a pulse  
Figure 5: (**a**) intensity (\(E^{2}\)) of electric fields of a two-color synthesizer (orange shaded) and a 1.4-cycle laser pulse at 1.7 \(\upmu\)m (green). HH spectra obtained using 60-mm-long (**b**) and 70-mm-long (**c**) gas cells. Transmission of Sn filters are plotted in (**b**,**c**).