## 2 Results and Discussion
### Advantages of an MIR Waveform Synthesizer for Generating an Isolated Attosecond Pulse
The synthesized field in this paper is expressed as  
\[Ef_{s}=\sum_{i=1}^{N}E_{i}e^{-2\ln 2\left(\left(t-\Delta t_{i}\right)/\tau_{i} \right)^{2}}\cos(\omega_{i}(t-\Delta t_{i})+\phi_{i}), \tag{1}\]  
Figure 2: **(a)** electric field of a single cycle pulse; **(b)** classical calculation of photon energies with two cutoffs occurring in two subsequent half cycles of electric field, for driving laser wavelengths of \(\lambda\) and \(2\lambda\); **(c)** classical calculation of photon energies under different wavelengths as a function of recollision time (group delay); **(d)** GDD values calculated from **(c)**. where the subscript denotes the field with \(i=1\) indicating the main driving pulse, \(E\) denotes the amplitude of the electric field, \(\tau\) denotes the pulse duration (FWHM), \(\Delta t\) denotes the delay, \(\omega\) denotes the angular frequency, and \(\phi\) denotes the CEP. First, we employ a single-color pulse at 1.95 \(\upmu\)m with a single-cycle pulse duration, an intensity of 2.9 \(\times\) 10\({}^{14}\) W/cm\({}^{2}\), and a CEP of \(-\pi\), which we call \"one color 1\" (see Figure 3). The electric field (\(E\!f_{1}\)) is plotted as a black curve in Figure 3a. The GD and GDD of the generated photons are given by black curves in Figure 3b,c, respectively. The cutoff energy is 350 eV. The minimum absolute GDD (larger values indicate larger atto-chirp) in the plateau region is \(\sim\)3000 \(as^{2}\) for both short and long trajectories. By mixing a weak control pulse at 1.356 \(\upmu\)m with an intensity of 5 \(\times\) 10\({}^{13}\) W/cm\({}^{2}\) (17% of that of the main driving laser) and the same pulse duration, a CEP of 0.5\(\pi\), and a delay of \(-\)1.13 fs (quarter cycle of the control pulse), the electric field (\(E\!f_{s}\)) is modified as shown by the red curve in Figure 3a. Then, the cutoff energy is significantly increased by approximately 1.5-fold to 530 eV as shown by the red curve in Figure 3b. More importantly, the absolute GDD values are greatly reduced to \(\sim\)1500 \(as^{2}\), which is approximately half that in the one-color case. For comparison, we also employ the same single-color pulse as that of \(E\!f_{1}\) but with an increased laser intensity of 4.44 \(\times\) 10\({}^{14}\) W/cm\({}^{2}\) (\(E\!f_{2}\)), for which the electric field waveform is shown by the blue curve (labelled by \"one color 2\") in Figure 3a. The cutoff energy, as shown by the blue curve in Figure 3b, is 530 eV, which is the same as that of the two-color field (\(E\!f_{s}\)). However, the temporal window of the emission (see the group delay) is larger than that for the two-color field, indicating that the attochirp is also larger. The minimum absolute GDD is \(\sim\)1850 \(as^{2}\), which is much less than that for the single-color field (\(E\!f_{1}\)) with a lower intensity, but still larger than that for the two-color synthesized field (\(E\!f_{s}\)). At the cutoff region, the GDD is even larger. Thus, a weak field employed in a waveform synthesizer helps to reduce the attochirp compared with that for a single-color field, making it helpful for generating short IAPs without chirp compensation. For example, for a TL duration of 50 as (FWHM) at 280 eV (cutoff region) generated by a single-color field \(E\!f_{1}\), the actual duration of the IAP is \(\sim\)200 as owing to the attochirp. In the two-color synthesized field \(E\!f_{s}\), the actual IAP duration is 120 as at 450 eV (cutoff region), for which the pulse duration is significantly reduced. In the single-color field \(E\!f_{2}\), the actual IAP duration is 150 as at 450 eV (cutoff region), which is also longer than that for the two-color synthesized field. In fact, by constructing an optimum waveform using a synthesizer, it is possible to obtain IAPs without any attochirp [44]. Moreover, using the above electric fields, we calculate the HHG spectrum of a single Ne atom using the Lewenstein model [45] as shown in Figure 4. The Ne medium is kept neutral with the ionization ratio maintained at less than 1% for all the laser fields.