## 2 Results and Discussion
### Generation of Water Window IAPs from an MIR Waveform Synthesizer
#### 2.2.1 Simulation Method  
The HHG in an atomic gases can be theoretically described by the response of a single atom and the collective response of the macroscopic gas to the laser and harmonic fields. In our calculation, we apply the strong-field approximation model (SFA) [45] to calculate the harmonic radiation. The time-dependent dipole momentum is [45]  
Figure 4: HHs spectra calculated for a single atom for each electric field in Figure 3a.  
Figure 3: Reducing attochirp using a waveform synthesizer. (**a**) Electric field of a single-color 1.95 μm pulse (single cycle, CEP \(=-\pi\)) with intensities of \(2.9\times 10^{14}\) W/cm\({}^{2}\) (black, \(E_{\text{f}1}\)) and \(4.44\times 10^{14}\) W/cm\({}^{2}\) (blue, \(E_{\text{f}2}\)) and a two-color (red, \(E_{\text{f}3}\)) synthesizer (Intensities are 2.9 \(\times\) 10\({}^{14}\) W/cm\({}^{2}\) at 1.95 μm and 5 \(\times\) 10\({}^{13}\) W/cm\({}^{2}\) at 1.356 μm); (**b**) photon energies under each group delay (emission time); (**c**) GDD values calculated from (**b**).  
\[d_{nl}(t) = i\int_{-\infty}^{t}dt^{\prime}\left[\frac{\pi}{\varepsilon+i(t-t^{ \prime})/2}\right]^{3/2} \tag{2}\] \[\times d_{\rm rec}\left[p_{\rm st}(t^{\prime},t)+A(t)\right]d_{\rm ion }\left[p_{\rm st}(t^{\prime},t)+A(t^{\prime})\right]\] \[\times\exp\left[-iS_{\rm st}(t^{\prime},t)\right]E(t^{\prime})g(t ^{\prime})+c.c.,\]  
where \(E(t)\) is the driving laser field, \(A(t)\) is the vector potential, and \(\varepsilon\) is a positive regularization constant. \(d_{\rm rec}\) and \(d_{\rm ion}\) are the recombination and ionization matrix elements, respectively. \(p_{\rm st}\) and \(S_{\rm st}\) are the stationary momentum and quasiclassical action, respectively. \(g(t)\) represents the ground-state amplitude, which is calculated by the Ammosov-Delone-Krainov (ADK) tunneling model [47]. Then, the harmonic spectrum is obtained as the Fourier transform of the time-dependent dipole acceleration \(a(t)\),  
\[a_{q}=\frac{1}{T}\int_{0}^{T}a(t)\exp(-iq\omega t), \tag{3}\]  
where \(a(t)=\hat{d_{nl}}(t)\), and \(T\) and \(\omega\) are the duration and frequency of the driving pulse, respectively. \(q\) is the harmonic order.  
To simulate the collective response of the macroscopic gas, we separately numerically solve the Maxwell wave equations for the fields of the laser pulse \(E_{l}\) and the harmonics \(E_{h}\) in a cylindrical coordinate,  
\[\
abla^{2}E_{l}(r,z,t)-\frac{1}{c^{2}}\frac{\partial^{2}E_{l}(r,z,t)}{ \partial t^{2}}=\frac{\omega_{p}^{2}(r,z,t)}{c^{2}}E_{l}(r,z,t), \tag{4}\]  
\[\
abla^{2}E_{h}(r,z,t)-\frac{1}{c^{2}}\frac{\partial^{2}E_{h}(r,z,t)}{ \partial t^{2}} = \frac{\omega_{p}^{2}(r,z,t)}{c^{2}}E_{h}(r,z,t)+\mu_{0}\frac{ \partial^{2}P_{nl}(r,z,t)}{\partial t^{2}}, \tag{5}\]  
where \(\omega_{p}=[4\pi n_{\rm e}(r,z,t)]^{1/2}\) is the plasma frequency, \(P_{nl}(r,z,t)=[n_{0}-n_{e}(r,z,t)]d_{nl}(r,z,t)\) is the nonlinear polarization of the medium, \(n_{0}\) is the gas density, and \(n_{e}(t)=n_{0}\left[1-\exp\left(-\int_{-\infty}^{t}w(t^{\prime})dt^{\prime} \right)\right]\) is the free-electron density in the gas, where \(w(t)\) is the electron ionization rate. The equations here take into account both temporal plasma-induced phase modulation and the spatial plasma-lensing effects on the driving field. Equations (4) and (5) can be numerically solved by the Crank-Nicholson method as described in Refs. [48; 49]. Finally, the high harmonics at the exit of the gas medium are multiplied by the transmissivity of the gas [31].