## 2 Results and Discussion  
### Advantages of an MIR Waveform Synthesizer for Generating an Isolated Attosecond Pulse  
The mechanism of HHG can be understood through the behaviour of an electron inside a strong laser field using a three-step model [41], which involves tunnel ionization of the electron, acceleration by the laser field, and a return to the parent ion with the release of the gained energy in the form of a photon. Thus, following the three-step model, it is very helpful to analyze the attosecond pulse generation in HHG by calculating the return kinetic energy (RKE) of electrons after ionization in a classical manner. Then, the energy of the emitted photons in HHG can be calculated as the sum of RKE and the ionization potential of the atom or molecules. In this classical calculation, an Ne atom is selected, which has an ionization potential of 21.56 eV. As a simple example, we perform the calculation using a single-cycle (FWHM) laser pulse. The electric field with a CEP of 0 and a Gaussian pulse envelope is shown in Figure 2a. Figure 2b shows the calculation results for driving laser wavelengths of \(\lambda\) and 2\(\lambda\) with the same intensity. Peak A and Peak B represent different cutoff energies when electrons are ionized near each electric field peak in adjacent half cycles, which are indicated by \(\mathfrak{t}_{a}\) and \(\mathfrak{t}_{b}\) in Figure 2a, respectively. The short and long trajectories result from different ionization times in the same half cycle of the electric field peak. In an experiment, only one trajectory can be selected by placing the gas medium before (long trajectory) or after (short trajectory) the laser focus. Thus, a supercontinuum bandwidth, which determines the shortest attainable attosecond pulse duration under a TL condition, is determined by the difference between the cutoff energies of Peak A and Peak B. It is clear that the laser field with 2\(\lambda\) wavelength not only markedly increases the cutoff energy (\(E_{\text{cutoff}}=I_{\text{P}}+3.17~{}U_{\text{P}}\), where \(U_{\text{P}}\propto\lambda^{2}\) and \(I_{\text{P}}\) is the ionization potential) but also significantly extends supercontinuum bandwidth, which is approximately four times (\(\propto\lambda^{2}\)) that for the laser field at the wavelength of \(\lambda\). Hence, a longer driving laser wavelength is very helpful for obtaining a broader supercontinuum spectrum with higher photon energy, which is further proved by the calculated photon energy under different emission times (group delay (GD)) in Figure 2c under different driving wavelengths. To examine the temporal chirp (called the attochirp) of the photons in Figure 2c under the same laser intensity but with different wavelengths, the GDD is calculated and shown in Figure 2d. Short and long trajectories are positively and negatively chirped, respectively. The photon energies with minimum chirps (minimum GDD values for short or long trajectories) are the photons with approximately half of the cutoff energy, which are located in the plateau regions of high harmonics (HH) spectra. Comparing the GDD values at the same photon energies below the cutoff region (plateau region), short trajectories with less atto-chirp are obtained for shorter driving wavelengths. However, the most important feature is that a longer driving laser wavelength not only markedly increases the cutoff energy, but also significantly reduces attochirp at higher photon energies. It should be noted that, by increasing laser intensity for a shorter wavelength driving laser, the cutoff energy can also be extended with reduced attochip. However, the significantly increased ionization ruins phase matching between driving laser and HHs, which generally should be avoided in HHG. Moreover, mixing a weak control pulse, i.e., synthesizing a waveform, is very helpful for reducing attochirp [42; 43].