# Generation of Ultrafast Optical Pulses via Molecular Modulation in Ambient Air
## 3 Results and Discussions
In case of 5-sidebands contributing to generation at the second focus (FWM2), Equation (1) can be written as:  
\[I_{\text{AS3}}\ \sim\ |E_{\text{AS2}}E_{\text{AS1}}^{*}+E_{\text{AS1}}E_{ \text{p}}^{*}+E_{\text{P}}E_{\text{S1}}^{*}+E_{\text{S1}}E_{\text{S2}}^{*}|^{2} \tag{3}\]  
Figure 3: Intensity modulation of the AS3 sideband measured with the spectrometer after the second interaction region (FWM2 in Figure 1). The experimental data—dots with error bars, calculated profiles—solid line (for 5 sidebands) and broken line (for 3 sidebands). Best fit to the data shown in Figure 3 gives the initial relative phases among the five sidebands as: AS2--4.4 rad, AS1--0 rad, P--0 rad, S1--1.8 rad, S2--5.4 rad. Note that among the five sideband phases only three are independent, and two can be made zero, as long as the total waveform duration is much longer than the optical period and carrier-envelope phase effects can be neglected. In addition, we checked what is expected in the case of only three sidebands contributing to the molecular excitation in the second focal region (FWM2, Figure 1). In that case, Equation (1) becomes:  
\[I_{\text{AS3}}\ \sim\ |E_{\text{AS2}}E_{\text{AS1}}^{*}+E_{\text{AS1}}E_{\text{ P}}|^{2} \tag{4}\]  
Here, by the best-fit procedure, we extract the following initial phases: AS2--4.3 rad, AS1--0 rad, P--0 rad. The results of the two theoretical calculations, along with the experimental measurements, are shown in Figure 3 (solid and dashed curves). As we can see from Figure 3, both theoretical curves generally match the experimental results, however, there are discrepancies at several experimental points. One of the possible explanations for the mismatch between the experimental and theoretical data is the non-perfect phase-matching in the Raman generation process in the second focal region. In the above-described theory, we assume the ideal phase-matching case, i.e., the zero wave vector mismatch for the included sidebands. Even in air where the dispersion in the visible, near-ultraviolet and, infrared spectral regions is small, it cannot be completely neglected, particularly in the case of broadband multi-sideband generation. The experimentally measured modulation of the AS3 signal with the change of the optical path suggests that we can partially compensate for the dispersion of air by the rotating pair of glass plates. Moreover, the modulation confirms the coherent nature of the sidebands and suggests a way, by the manipulation of the relative phases among the sidebands, to control the waveform of the formed pulse train. Essentially, variable dispersion due to the tilted glass plates affords sufficient phase control, when the number of sidebands is not too large. In principle, more sideband phases can be more precisely controlled if several different types of glass are used [17; 25]. To determine the intensity envelope of the pulse-train waveform (which corresponds to the superposition of sideband fields) we used the following equation:  
\[I(t)\ \sim\ |\sum_{q}|E_{q}|exp\big{[}i(\omega_{q}t+\phi_{q}^{0}+\Delta\phi_{ q}(\alpha)\big{]}\big{]}^{2} \tag{5}\]  
Here \(\phi_{q}^{0}\) are the initial sideband phases obtained from the best fit procedure described above. The more explicit notation \(\Delta\phi_{q}(\alpha)\) emphasizes the fact that the individual sideband phases vary with the varying glass-plate tilt angle \(\alpha\). Each phase here is a different function of \(\alpha\), due to glass dispersion. Figure 4 shows the predicted waveforms obtained for various points on the AS3 modulation curve in Figure 3.  
We calculated the temporal waveforms for six points: (A), (B), (C), (D), (E) and (F), shown in Figure 3. Each of these points corresponds to a different incident angle on the rotating BK7 glass. As we can see, in the case of the 5-sidebands contribution (Figure 4A-C) the overall change of the pulse train lead to shortening of each individual pulse: from 7.5 fs to 3 fs FWHM duration at the optimum angle (30.3\({}^{\circ}\)).