## 4 Attosecond Pulse Trains with Four Harmonics
### Complex Waveform Synthesis Using the Phase Shifter PS\({}_{2}\)  
A phase scan was conducted by introducing the delay using the phase shifter PS\({}_{2}\). The evolution of the oscillating components \((P_{7,8},P_{8,9})\) and \((P_{8,9},P_{9,10})\) are presented in Figure 5. A delay of 31.45 as was introduced between one panel and the next. The delay 31.45 as corresponds to a phase increment of 2.02 rad for \(\Delta\varphi_{7,8,9}\) and a phase decrement of 1.80 rad for \(\Delta\varphi_{8,9,10}\). The periodicity of correlation distributions \((P_{7,8},P_{8,9})\), \((P_{8,9},P_{9,10})\) based on \(\Delta\varphi_{7,8,9}\), \(\Delta\varphi_{8,9,10}\) can be easily established from the evolution of the correlation plots. Considering the evolution of the correlation plot of \((P_{8,9},P_{9,10})\) from Panels a to d, the total phase variation corresponds to a value 5.40 rad, which is close to 2\(\pi\), and hence, the corresponding distributions are very similar. Figure 5: Experimental correlation plots for a phase scan using PS\({}_{2}\) in the four-harmonic case. (**a**–**g**) Evolution of the correlation plots of the oscillating components of the sidebands \((P_{8,9},P_{9,10})\) for increasing values of the delay \(\tau_{z2}\) introduced by the phase shifter PS\({}_{2}\). (**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 \(\tau_{z2}\) introduced by the phase shifter PS\({}_{2}\). The color scale indicates the density of single-shot experimental points normalized to unity for each panel. Similarly, for the evolution of the correlation plot of (\(P_{7,8},P_{8,9}\)) from Panels j (k) to m (n), the total phase added is equal to 6.06 rad, reproducing a correlation plot in close similarity to Panel j (k). The phase differences \(\Delta\varphi_{7,8,9}\) and \(\Delta\varphi_{8,9,10}\) were obtained following the method described in [23; 26]. From the amplitudes of the harmonics (\(F_{q}\)) obtained from the photoelectron spectra in the absence of the dressing NIR field, the intensity profile of the attosecond pulse trains is calculated using the equation:  
\[ \begin{split} I(t)&\propto\sum_{q=7}^{10}F_{q}^{2} f^{2}(t)+2F_{7}F_{8}f^{2}(t)\,\cos[\omega_{\text{UV}}t+\chi-\frac{2\Delta \varphi_{7,8,9}}{3}-\frac{\Delta\varphi_{8,9,10}}{3}]\\ &+2F_{8}F_{9}f^{2}(t)\,\cos[\omega_{\text{UV}}t+\chi+\frac{\Delta \varphi_{7,8,9}}{3}-\frac{\Delta\varphi_{8,9,10}}{3}]\\ &+2F_{9}F_{10}f^{2}(t)\,\cos[\omega_{\text{UV}}t+\chi+\frac{ \Delta\varphi_{7,8,9}}{3}+\frac{2\Delta\varphi_{8,9,10}}{3}]\\ &+2F_{7}F_{9}f^{2}(t)\,\cos[2\omega_{\text{UV}}t+2\chi+\frac{ \Delta\varphi_{7,8,9}}{3}-\frac{\Delta\varphi_{8,9,10}}{3}]\\ &+2F_{8}F_{10}f^{2}(t)\,\cos[2\omega_{\text{UV}}t+2\chi+\frac{2 \Delta\varphi_{7,8,9}}{3}+\frac{\Delta\varphi_{8,9,10}}{3}]\\ &+2F_{7}F_{10}f^{2}(t)\,\cos[3\omega_{\text{UV}}t+3\chi]\end{split}  \tag{6}\]  
where \(F_{q}\), \(f(t)\) are the amplitude and envelope of the harmonic H\({}_{q}\). Phase \(\chi\) (=\((\varphi_{10}-\varphi_{7})/3\)) introduces a temporal shift of the intensity profile of the attosecond waveforms with respect to the envelope of the single harmonic and does not affect the shape of the individual attosecond pulses within the pulse train. Reconstructed attosecond pulses corresponding to Figure 5 are displayed in Figure 6. The pulse duration (\(f(t)\)) for each harmonic is considered equal and taken to be 50 fs FWHM [28]. From the equation, the shortest attosecond pulses result when both the phase difference terms \(\Delta\varphi_{7,8,9}\) and \(\Delta\varphi_{8,9,10}\) are \(\approx\)0. Panel d displays the shortest attosecond pulses with a pulse duration of 216 \(\pm\) 7 as FWHM with the error in the pulse duration calculated from the error in the extracted phase differences and the obtained amplitudes of the harmonics.