## 3 Results and Discussion
### Simulated Waveform Synthesis with cw Harmonic Comb  
The harmonic comb that is reported here has a fundamental wavelength of 3192 nm. The relative phase relationship of the harmonic comb is critical during waveforms synthesis. A fixed phase relationship in phase-matched three wave mixing process is \(\phi_{a}=\phi_{b}+\phi_{c}-\pi/2\), where subscripts a, b, and c represent the identities of the three optical waves [20]. Since all components of the cw harmonic comb in this work are generated from three wave mixing process; the deduced phase of \(\omega_{n}\) is \(\phi_{n}=n\phi_{i}-(n-1)\pi/2\), where \(\phi_{i}=(\phi_{3}+\pi)/3\). In this harmonic comb, the phase of the pump, \(\phi_{3}\) is the only unchangeable parameter in the system. The other components' phase will follow the phase of pump based on the deduced relation. By managing the phase and amplitude of each comb component periodic field waveforms of arbitrary shape could be synthesized. The calculated repetition rate of the synthesized pulse train with this comb is \(\sim\)94 THz, and has an equivalent period spacing of 10.6 fs. The shortest pulse synthesized within each period will be a transform-limited sub-cycle cosine pulse that has an electric field FWHM of 942 attoseconds. The FWHM of the intensity envelope of this cosine pulse is 1.59 fs. The simulated waveforms, synthesized with the comb produced in this experiment and when the spectral phases are adjusted to be equal, are plotted and shown as the red curve in Figure 4. By adjusting the amplitudes of the comb components to be equal, the narrowest pulses obtainable with this comb are plotted in blue in the figure. For a comb produced with a perfect divided-by-three cw SRO this pulse train will be continuous. In reality the SRO is not perfect and there is a frequency deviation \(\Delta\omega\) as described in Section 3.1 above. The time duration in which the pulses in the train will remain intact is dependent on this frequency deviation \(\Delta\omega\) of the SRO. With \(\Delta\omega\) at one standard deviation of the present case the period of repetition of the pattern of synthesized pulses is \(\sim\)2288 ns, corresponding to the time inverse of \(\Delta\omega\) of 437 kHz. Figure 5 shows the evolution of the synthesized waveform at this frequency deviation over a time span of 3000 ns, clearly indicating the pattern repeats itself every 2288 ns. Allowing for a drop of 10% in the maximum field strength as the criteria, then according to the numerical simulations, the sub-cycle cosine electric field maintains its strength for over 376 ns in every cycle. Since this is for a one standard deviation of the frequency deviation, 86% of the time this waveform has this shape for 376 ns or longer. We pointed out in Section 3.1 that the residual frequency deviation is due to the drift of the seed laser. Hence we believe that this duration can be extended to over 1000 ns by actively stabilizing the seed laser of the pump in this experiment. Figure 4: Synthesized waveforms calculated for the six component comb obtained in this experiment (red curve) and for the case where the components are reduced to equal amplitude (blue curve) for the zero detuning case. This will produce a continuous train of pulses of the calculated shape shown here. Figure 5: Numerical simulations of a subcycle cosine pulse electric field generated by this multi-harmonic optical comb with a frequency deviation of 437 kHz. The vertical scale is the time evolution of the pulse train from 0 to 3000 ns. The horizontal scale is the local time of the waveform displayed for about three cycles. The field pattern shifts with a phase shift of 2\(\pi\)/3 every 1/3 of a period by repeats and replicates itself in approximately 2288 ns which corresponds to the inverse of the frequency deviation used in the simulation.