# Generation of Ultrafast Optical Pulses via Molecular Modulation in Ambient Air
## 3 Results and Discussions  
Figure 2 shows the spectrum obtained after the 532 nm high reflection mirror. We observed several sidebands, from the second order anti-Stokes (AS2, 426 nm) to the second order Stokes (S2, 707 nm). Due to a high reflectivity at 532 nm for the mirror (HM), the intensity of the pump pulse was smaller than the first order anti-Stokes (AS1) and Stokes (S1) pulses. The upper axis shows the frequency shift from the pump beam. We see that the frequency spacing of the sidebands are in reasonable agreement with the nitrogen vibrational frequency (2331 cm\({}^{-1}\)), despite the small shift due to the spectrometer calibration. Hence, we conclude that the sidebands are generated via molecular modulation of nitrogen molecules. In what follows, we describe an experiment where the sidebands produced in the first focal region (FWM1, Figure 1) are refocused into the second region (FWM2, Figure 1) where we use their coherent interaction with nitrogen molecules for diagnostics of the resultant optical waveform. The refocused sidebands produced molecular coherence which is not only a function of their intensities but also their relative phases. Then, additional higher-order sidebands were produced in proportion to the magnitude of this molecular excitation. We controlled the relative phases of the primary sidebands by varying the amount of dispersion due to tilted BK7 glass plates in the beam path. Figure 3 shows the intensity modulation of the AS3 sideband at 386.87 nm after the FWM2 point, at various tilt angles of the BK7 plates. The relative phase among the sidebands changed because of the different optical path lengths and the dispersion. When the sidebands are mutually coherent they interfere with each other and produce short pulses. Figure 2: Spectral components of the coherent Raman generation in air: (**A**) as measured with the spectrometer. (**B**) Photo of the generated sidebands, taken with a digital camera, after a 532 nm high-reflection mirror. We assume that the driving force producing the molecular coherence is proportional to the absolute value of the electric field amplitude, squared. This assumption is justified when the resultant coherence \(\rho\) is relatively small (\(|\rho|<1/2\), which is the case under our experimental conditions), or when the molecular excitation is adiabatic [14]. The vibrational coherence, in turn, mixes with the highest-order anti-Stokes (AS2) sideband (which at this point is presumed to be of constant intensity) to produce the next anti-Stokes (AS3) sideband [21]. In this case, the AS3 signal can be evaluated by the equation:  
\[I_{\text{AS3}}\ \sim\ |\rho|^{2}\ \sim\ |\sum_{q}b_{q}E_{q}E_{q-1}^{*}|^{2} \tag{1}\]  
where \(b_{q}\) are constants from the interaction Hamiltonian for molecular nitrogen [22], and \(E_{q}\ =\ |E_{q}|exp\Big{[}i\psi_{q}\Big{]}\) is the slowly-varying complex electric field amplitude at frequency \(\omega_{q}\) corresponding to the \(q\)th-order (pump, Stokes, or anti-Stokes) sideband, such that the total electric field is:  
\[E(t)=Re\Bigg{\{}\sum_{q}E_{q}exp\Big{[}i\omega_{q}t\Big{]}\Bigg{\}} \tag{2}\]  
To properly evaluate phase \(\phi_{q}\) for each sideband we need to determine the optical path length, which depends on the wavelength of each sideband. Hence, we used the Sellmeier equation to estimate the refractive index and apply [PERSON]'s law to calculate refraction angles and the optical path length for each individual sideband \(E_{q}\)[23; 24]. The exact procedure is described in the Supplementary Materials. Then we used the obtained sideband phase in conjunction with Equation (1) to plot the theoretical curves of Figure 3. We took the power ratio between the sidebands AS2:AS1:P:S1:S2 as 0.05:1:0.3:1:0.8, corresponding to the experimentally measured spectra and assumed time-constant sideband intensities, essentially limiting our analysis to a time window of a few-picosecond centered around the peak of the applied pump/Stokes pulse envelope, and assuming that the total envelope duration was much longer than that.