## 4 Attosecond Pulse Trains with Four Harmonics
### Complex Waveform Synthesis Using the Phase Shifter PS\({}_{3}\)  
A phase scan conducted using the phase shifter PS\({}_{3}\) has an immediate effect, and as the phase altered does not affect the phase difference \(\Delta\varphi_{8,9,10}\), the correlations plots of the oscillating components (\(P_{8,9},P_{9,10}\)) should not show any change as a function of phase. However, the phase difference \(\Delta\varphi_{7,8,9}\) varies, and hence, the evolution of the correlation plots of the oscillating components (\(P_{7,8},P_{8,9}\)) is expected. In Figure 7, the correlation plots of the two pairs of oscillating components corresponding to different phase difference values are displayed. Panels a-g (i-n) show the evolution of the correlation distribution of the oscillating components \(P_{8,9}\) and \(P_{9,10}\) (\(P_{7,8}\) and \(P_{8,9}\)). The phase scan was carried out at a fixed PS\({}_{2}\) value corresponding to Figure 5d. The delay \(\tau_{53}\) introduced in each step was 31.45 as, corresponding to a reduction of 1.57 rad in each step. The correlation plots of the oscillating components (\(P_{8,9},P_{9,10}\)) do not show a variation in the correlation distribution as the delay \(\tau_{53}\) is varied. An addition of four delay steps corresponds to a reduction of 6.28 rad in \(\Delta\varphi_{7,8,9}\). From Panels h-n, a periodic repetition of the correlation distribution for every four panels can be observed. These results demonstrate the possibility to selectively manipulate the phase difference \(\Delta\varphi_{7,8,9}\) in waveform shaping. The power of attosecond waveform synthesis to vary the phase of the harmonics of the harmonic comb independently allows the investigation of different dynamics dependent on the phase of the input radiation. Figure 6: Reconstructed attosecond pulses for a phase scan using the phase shifter PS\({}_{2}\). (**a–g**) Evolution of the intensity profile of the attosecond pulses in the attosecond pulse train generated with the harmonic comb containing the harmonics H\({}_{7}\), H\({}_{8}\), H\({}_{9}\), and H\({}_{10}\) of the seed laser, for different values of the phase shifter (PS\({}_{2}\)), corresponding to Figure 5a,h (a), Figure 5b,j (b), Figure 5c,j (c), Figure 5d,k (d), Figure 5e,l (e), Figure 5f,m (f), and Figure 5g,n (g). The error in the reconstruction is indicated as a shaded region in each panel. Figure 7: Experimental correlation plots for a phase scan using PS\({}_{3}\) in the four-harmonic case. (**a–g**) Evolution of the correlation plots of the oscillating components of the sidebands \((R_{8},g,R_{9,10})\) for increasing values of the delay \(\tau_{5,3}\) introduced by the phase shifter PS\({}_{3}\). (**h–n**) Evolution of the correlation plots of the oscillating components of the sidebands \((P_{7,8},P_{8,9})\) for increasing values of the delay \(\tau_{5,3}\) introduced by the phase shifter PS\({}_{3}\). The delay \(\tau_{5,3}\) introduced by the phase shifter PS\({}_{3}\) affects only the phase of the harmonic H\({}_{7}\), resulting in fixed correlation plots of \((R_{8},g,R_{9,10})\) (**a–g**). The color scale indicates the density of single-shot experimental points normalized to unity for each panel. Figure 8 presents the reconstructed attosecond waveforms for the different values of \(\Delta\varphi_{7,8,9}\) as the delay \(\tau_{s3}\) is varied. The shortest attosecond waveform is produced when the phase difference \(\Delta\varphi_{7,8,9}\approx 0\), corresponding to a correlation distribution of (\(P_{7,8}\), \(P_{8,9}\)) close to a straight line. The reconstructed waveform displayed in Figure 8b presenting the shortest pulse duration of 210 \(\pm\) 5 as corresponds to the correlation plots presented in Panels b,i of Figure 7. An attosecond waveform with double or multiple peaks results when \(\Delta\varphi_{7,8,9}\approx\pi\), producing a distribution close to a straight line with a negative slope. The correlation plots in Figure 7d,k and the reconstructed waveform in Figure 8d represent such a scenario.